QTRRTC FULL SLIDE.pdf

Introduction to Financial Risk
Management
Presented by: Dung Tran

Black Monday (1987)
• The global, sudden, severe, and largely unexpected stock market crash on
October 19, 1987

Black Monday (1987)
• Causes: computer program-driven trading models that followed a portfolio
insurance strategy as well as investor panic.
• Investors hedge a portfolio of stocks against market risk by short-selling stock index
futures  limit the losses a portfolio might experience as stock price declines
without that portfolio's manager having to sell off those stocks
• Computer programs automatically began to sell stocks as certain loss targets were
hit, pushing prices lower  a domino effect as the falling markets triggered
more stop-loss orders.
• Before the crash:
• overvalued stock market – a strong bull that was overdue for a major correction
• a series of monetary and foreign trade agreements that depreciated the U.S. dollar in
order to adjust trade deficits and then attempted to stabilize the dollar at its new
lower value.

Financial crisis 2007-2008
• Cause and Effects
https://www.youtube.com/watch?v=N9YLta5Tr2A
• The collapse of the housing market — fueled by low interest rates,
easy credit, insufficient regulation, and toxic subprime mortgages —
led to the economic crisis.

Do stockholders care about volatile cash
flows?
• If volatility in cash flows is not caused by systematic risk, then
stockholders can eliminate the risk of volatile cash flows by
diversifying their portfolios.
• Stockholders might be able to reduce impact of volatile cash flows by
using risk management techniques in their own portfolios.

Questions
• Why do firms need to manage risks?
• How can risk management increase the value of a
corporation?

Intrinsic Value: Risk Management
Required investments
in operating capital
−
Free cash flow
(FCF) =
Weighted average
cost of capital
(WACC)
Market risk aversion
Firm’s debt/equity mix
1 2
1 2
FCF FCF FCF
Value
(1 WACC) (1 WACC) (1 WACC)


   
  
Input costs
Net operating
profit after taxes
Product prices
and demand
Firm’s business risk
Market interest rates
Foreign exchange rates

How can risk management increase the value of a
corporation?
Risk management allows firms to:
• Have greater debt capacity, which has a larger tax shield of interest
payments.
• Implement the optimal capital budget without having to raise
external equity in years that would have had low cash flow due to
volatility.
• (More . .)

Risk management allows firms to: (1)
• Avoid costs of financial distress.
• Weakened relationships with suppliers.
• Loss of potential customers.
• Distractions to managers.
• Utilize comparative advantage in hedging relative to hedging ability
of investors.
• Firms can hedge more efficiently than most investors due to lower transaction
costs and asymmetric information
• (More . .)

Risk management allows firms to: (2)
• Minimize negative tax effects due to convexity in tax code.
• Present value of taxes paid by companies with volatile earnings is higher than the PV of taxes
paid by stable companies
• Example: A: EBT of $50K in Years 1 and 2, total EBT of $100K,
• Tax = $7.5K each year, total tax of $15.
• Less volatile income
B: EBT of $0K in Year 1 and $100K in Year 2,
• Tax = $0K in Year 1 and $22.5K in Year 2. total 22.5
• Reduce borrowing costs by using interest rate swaps.
• Maximize bonuses if managerial compensation system has floor or ceiling—Bad
Reason!
• Managers’ bonus is higher if earnings are stable

LO 1.a: Explain the concept of risk and compare risk
management with risk taking.
• In an investing context, risk is the uncertainty surrounding outcomes. Investors
are generally more concerned about negative outcomes (unexpected investment
losses) than they are about positive surprises (unexpected investment gains).
• Natural trade-off between risk and return; opportunities with high risk have the
potential for high returns and those with lower risk also have lower return
potential.
• Risk is not necessarily related to the size of the potential loss. The more
important concern is the variability of the loss, especially an unexpected loss
that could rise to unexpectedly high levels.
• Many potential losses are large but are quite predictable and can be accounted for using
risk management techniques.

LO 1.a: Explain the concept of risk and compare risk
management with risk taking.
• Risk management: the sequence of activities aimed to reduce or
eliminate an entity’s potential to incur expected losses. On top of
that, there is the need to manage the unexpected variability of some
costs.
• In managing both expected and unexpected losses, risk management can be
thought of as a defensive technique.
• However, risk management is actually broader in the sense that it considers
how an entity can consciously determine how much risk it is willing to take to
earn future uncertain returns.
• Risk taking: the active acceptance of incremental risk in the pursuit of
incremental gains.
• opportunistic action.

LO 1.b: Describe elements of the risk management
process and identify problems and challenges that can
arise in the risk management process.
• The risk management process is a formal series of actions designed to determine
if the perceived reward justifies the expected risks. A related query is whether
the risks could be reduced and still provide an approximately similar reward.
• There are several core building blocks in the risk management process.
• Identify risks.
• Measure and manage risks.
• Distinguish between expected and unexpected risks.
• Address the relationships among risks.
• Develop a risk mitigation strategy.
• Monitor the risk mitigation strategy and adjust as needed.

• Figure 1.1 illustrates that risks can
move along a spectrum from being
expected (i.e., known) to being fully
unknown. The unknown category can
be subdivided into the known
unknowns (i.e., Knightian uncertainty)
and the unknown unknowns.
• The former are items that may impact
a firm, while the latter are truly
unknown (i.e., tail risk events). Where
possible, risk managers should move a
risk into the known category, but this
does not work for risks that cannot be
quantified

LO 1.b: Identify problems and challenges that can arise
in the risk management process.
• One of the challenges in ensuring that risk management will be beneficial to the
economy is that risk must be sufficiently dispersed among willing and able participants in
the economy.
• It has failed to consistently assist in preventing market disruptions or preventing financial
accounting fraud (due to corporate governance failures). For example, the existence of
derivative financial instruments greatly facilitates the ability to assume high levels of risk
and the tendency of risk managers to follow each other’s actions.
• The use of derivatives as complex trading strategies assisted in overstating the financial
position (i.e., net assets on balance sheet) of many entities and complicating the level of
risk assumed by many entities.
• Finally, risk management may not be effective on an overall economic basis because it
only involves risk transferring by one party and risk assumption by another party.

The Evolution of Risk Management
• Commodity futures contracts
• 2000 B.C.E in India
• 1800s: grain traders in Midwest
• Insurance
• Maritime: 1300s, Genoa
• Fire:
• 1680, London
• 1752, Benjamin Franklin and the Union Fire Company

1970s Bring Changes
• Risk increases:
• End of gold standard: increased exchange rate volatility
• OPEC: increased oil volatility
• Expansion of global trade and competition
• Risk management tools improve:
• Black-Scholes option pricing model leads to other derivative pricing models
• Technology
• Information collection and processing
• Computers that can easily conduct Monte Carlo simulation

1970s-1980s: Bribery and Fraud
• Foreign Corrupt Practices Act (FCPA), 1977
• To prevent corporate bribery
• Required accounting systems to be able to identify
funds used for bribery
• Savings & Loan Crisis, 1980s
• Bad business models, but also fraud
• Congress and SEC threaten to intervene in self-
regulatory activities and standards that previously had
been determined by the accounting profession

COSO: Committee of Sponsoring Organizations (1)
• In response to Congressional and SEC criticism in mid-1980s, a
group of five private accounting firms created a commission to
study accounting fraud and write a report.
• James Treadway (former SEC Commissioner) was chairman.
• The Treadway commission recommended that its sponsoring
organizations create guidelines for an accounting system that would be
able to detect fraud.
• Continued…

COSO: Committee of Sponsoring Organizations (2)
• The Committee of Sponsoring Organizations extended their original
framework to include enterprise risk management (ERM).
• The COSO ERM framework:
• Satisfies the regulatory requirements related to financial reporting required
by FCPA and SOX.
• Is widely used.

Seven Major Categories of Risk (1)
1. Strategy and reputation:
• Include competitors’ actions, corporate social responsibilities, the public’s
perception of its activities, and reputation among suppliers, peers, and
customers.
2. Control and compliance:
• Include regulatory requirements, litigation risks, intellectual property rights,
reporting accuracy, and internal control systems.
• Continued…

3. Hazards:
• Fires, floods, riots, acts of terrorism, and other natural or man-made
disasters.
• All downside, no upside.
4. Human resources:
• Risk related to recruiting, succession planning, employee health, and
employee safety.
• Continued…

5. Operations:
• Risk events include supply chain disruptions, equipment failures, product
recalls, and changes in customer demand.
6. Technology:
• Risk events related to innovations, technological failures, and IT reliability
and security.
• Continued…

7. Financial management:
• Foreign exchange risk
• Commodity price risk.
• Interest rate risk.
• Project selection risk.
• Liquidity risk.
• Customer credit risk.
• Portfolio risk.

What are some actions that companies can take to
minimize or reduce risk exposures? (1)
• Transfer risk to an insurance company by paying periodic premiums.
• Transfer functions which produce risk to third parties.
• Share risk with third party by using derivatives contracts to reduce
input and financial risks.
• (More...)

1990s-early 2000s: More Fraudulent Accounting
• Enron, Tyco, and more
• 2002, Congress passes the Sarbanes-Oxley (SOX) act
• Section 404: Annual report must include section that addresses the
accounting system’s internal control.
• Framework of system
• Assessment of system’s ability to detect fraud

Late 2000s-Now: Cumulative Impact of Regulatory
Environment
• Companies must demonstrate compliance with FCPA and SOX
• Need to have an enterprise risk management system that meets the
compliance requirement
• COSO provides ERM system that meets requirement– See next slide

Quantitative Risk Measures
• Economic capital is the amount of liquid capital necessary to cover known
losses.
• For example, if one-day VaR is $2.5 million and the entity holds $2.5 million in liquid
reserves, then they have sufficient economic capital (i.e., they are unlikely to go
bankrupt in a one-day expected tail risk event).
• Drawbacks of VaR:
• There are a few different versions of VaR used in practice.
• VaR uses several simplifying assumptions, and risk managers can alter the computed
value by adjusting the number of days or the confidence level used in the calculation.
• VaR is intended to determine a loss threshold level. It measures the largest loss at a
specified cutoff point, not the magnitude of tail risk.

What are some actions that companies can take to
minimize or reduce risk exposures? (2)
• Take actions to reduce the probability of occurrence of adverse
events.
• Take actions to reduce the magnitude of the loss associated with
adverse events.
• Avoid the activities that give rise to risk.

Qualitative Risk Assessment
• Scenario analysis is a process that considers potential future risk factors and the
associated alternative outcomes.
• The typical method is to compare a best-case scenario to a worst-case scenario, which shocks
variables to their extreme known values.
• This process factors the potential impact of several categories of risk and influences risk
manager decision making by attempting to put a value on an otherwise qualitative concept
(i.e., what-if analysis).
• This exercise is an attempt to understand the assumed full magnitude of potential losses
even if the probability of the loss is very small.
• Stress testing is a form of scenario analysis that examines a financial outcome
based on a given “stress” on the entity. This technique adjusts one parameter at a
time to estimate the impact on the firm.
• For example, examining the impact of a dramatic increase in interest rates on the value of a
bond investment portfolio.

Enterprise Risk Management
• In practice, the term enterprise risk management (ERM) refers to a general
process by which risk is managed within an organization.
• An ERM system is highly integrative in that it is deployed at the enterprise level and not
siloed at the department level.
• A top-down approach, risk is not considered independently, but rather in relation to its
potential impact on multiple divisions of a company.
• One challenge with the ERM approach is a tendency to reduce risk management
to a single value (e.g., either VaR or economic capital).
• This attempt is too simplistic in a dynamic-risk environment. Risk managers learned from the
financial crisis of 2007–2009 that risk is multi-dimensional, and it requires consideration from
various vantage points.
• Risk also develops across different risk types. The reality is that proper application of an ERM
framework requires both statistical analysis and informed judgment on the part of risk
managers.
• The ultimate goal of an ERM is to understand company-wide risks and to
integrate risk planning into strategic business planning.

Expected and Unexpected Loss
LO 1.d: Distinguish between expected loss and unexpected loss and
provide examples of each.
• Expected loss (EL) considers how much an entity expects to lose in
the normal course of business.
• These losses can be calculated through statistical analysis with relative
reliability over short time horizons.
• The EL of a portfolio can generally be calculated as a function of: (1) the
probability of a risk occurring; (2) the dollar exposure to the risk event; and
(3) the expected severity of the loss if the risk event does occur.
• Example: a business can use its operating history to reasonably estimate the
percentage of annual credit sales that will never be collected  bad debt
expense. A bank can calculate its expected loss on loans.

Expected and Unexpected Loss
LO 1.d: Distinguish between expected loss and unexpected loss and
provide examples of each.
• Unexpected loss considers how much an entity could lose in excess of
their average (expected) loss scenarios.
• There is considerable challenge involved with predicting unexpected losses
because they are, by definition, unexpected.
• Correlation risk: when unfavorable events happen together, the correlation
risk drives potential losses to unexpected levels.
• Example: During an economic recession, many more loan defaults are likely to
occur from borrowers than during an economic expansion. It is also likely that
many of these losses will be clustered at the same time  Unexpected loss to
commercial lenders.

The Relationship Between Risk and Reward
LO 1.e: Interpret the relationship between risk and reward and explain
how conflicts of interest can impact risk management.
• There is a natural trade-off between risk and reward. In general, the
greater the risk taken, the greater the potential reward. However, one
must consider the variability of the potential reward.
• The portion of the variability that is measurable as a probability
function could be thought of as risk (EL) whereas the portion that is
not measurable could be thought of as uncertainty (unexpected loss).

Market Risk (L.O. 1.f)
• Market risk: refers to the fact that market prices and rates are continually
in a state of change.
• Interest rate risk: uncertainty flowing from changes in interest rate levels. If market
interest rates rise, the value of bonds will decrease. Another form of interest rate risk
is the potential for change in the shape of (or a parallel shift in) the yield curve.
• Equity price risk: the volatility of stock prices. It can be broken up into two parts: (1)
general market risk, which is the sensitivity of the price of a stock to changes in
broad market indices, and (2) specific risk, which is the sensitivity of the price of a
stock due to company-specific factors (e.g., rising cost of inputs, strategic
weaknesses, etc.).
• Foreign exchange risk: monetary losses that arise from either fully or partially
unhedged foreign currency positions, resulted from imperfect correlations in
currency price movements as well as changes in international interest rates
• Commodity price risk: the price volatility of commodities (e.g., precious metals, base
metals, agricultural products, energy) due to the concentration of specific
commodities in the hands of relatively few market participants.

Credit Risk
• Credit risk refers to a loss suffered by a party whereby the counterparty
fails to meet its contractual obligations. Credit risk may arise if there is an
increasing risk of default by the counterparty throughout the duration of
the contract
• Default risk refers to potential nonpayment of interest and/or principal on a loan by
the borrower. The PD is central to risk management.
• Bankruptcy risk is the chance that a counterparty will stop operating completely. The
risk management concern is that the liquidation value of any collateral might be
insufficient to recover a loss flowing from a default.
• Downgrade risk considers the decreased creditworthiness of a counterparty,
resulting in a higher lending rate charged by creditors to compensate for the
increased risk.
• Settlement risk could be illustrated using a derivatives transaction between two
counterparties. At the settlement date, one of them is in a net gain (“winning”)
position and the other is in a net loss (“losing”) position. The position that is losing
may simply refuse to pay and fulfill its obligations. This risk is also known as
counterparty risk (or Herstatt risk1).

Liquidity Risk
• Funding liquidity risk occurs when an entity is unable to pay down (or
refinance) its debt, satisfy cash obligations to counterparties, or fund
capital withdrawals.
• Example: Mismatch between assets and liabilities in banks (e.g., short-term
deposits mismatched with longer-term loans). Improper risk management of
this fundamental mismatch led to bank defaults during the financial crisis of
2007–2009.
• Market liquidity risk (also known as trading liquidity risk) refers to
losses flowing from a temporary inability to find a needed
counterparty. This risk can cripple an entity’s ability to turn assets into
cash at any reasonable price..

Risk and Return (Part I)
Reading 5:
Modern Portfolio Theory and Capital Asset Pricing Model

Outline
• Modern portfolio theory
• The efficient frontier
• The capital market line
• The security market line (SML), beta, and the capital asset pricing
model (CAPM).
• Risk-adjusted measures of return

Modern Portfolio Theory (L.O. 5.a)
Harry Markowitz laid the foundation for modern portfolio theory in the early
1950s. Markowitz’s portfolio theory makes the following assumptions:
• Returns are normally distributed. This means that, when evaluating utility,
investors only consider the mean and the variance of return distributions. They
ignore deviations from normality, such as skewness or kurtosis.
• Investors are rational and risk-averse. Markowitz defines a rational investor as
someone who seeks to maximize utility from investments. Furthermore, when
presented with two investment opportunities at the same level of expected risk,
rational investors always pick the investment opportunity which offers the
highest expected return.
• Capital markets are perfect. This implies that investors do not pay taxes or
commissions. They have unrestricted access to all available information and
perfect competition exists among the various market participants.

• Because investors are risk-averse, they strive to minimize the risk of their
portfolios for a given level of target return. This could be achieved by investing in
multiple assets which are not perfectly correlated with each other (i.e., where
their correlation coefficients, ρ, are less than 1).
• When correlation is less
than 1, diversification
occurs and portfolio
variance declines below
the weighted average of
individual variances. The
lower the correlation,
the greater the benefit
becomes.

• By holding a sufficiently large, diversified portfolio, investors are able to reduce,
or even eliminate, the amount of company-specific (i.e., idiosyncratic) risk
inherent in each individual security
• By holding a well-diversified portfolio, the importance of events affecting
individual stocks in the portfolio is diminished, and the portfolio becomes mostly
exposed to general market risk.
Modern Portfolio Theory (L.O. 5.a)
well-diversified

The Efficient Frontier
• Rational investors maximize portfolio return per unit of risk. Plotting all those
maximum returns for various risk levels produces the efficient frontier.

The Efficient Frontier
• Point C is known as the global minimum variance portfolio because it is the
efficient portfolio offering the smallest amount of total risk.
• Points A and B are considered inefficient because there is always a portfolio
directly above them on the efficient frontier offering a higher return for the same
amount of total risk.
• Any portfolio below the efficient frontier is, by definition, inefficient, whereas any
portfolio above the efficient frontier is unattainable.
• In the absence of a risk-free asset, the only efficient portfolios are the portfolios
on the efficient frontier.
• Investors choose their position on the efficient frontier depending on their
relative risk aversion. A risk seeker may choose to hold Portfolio G whereas
another investor seeking lower risk may choose to hold Portfolio D.

The Capital Market Line (CML) (L.O. 5.d)
• Investors will combine the risk-free asset with a specific efficient portfolio that
will maximize their risk-adjusted rate of return.
• A common proxy used for the risk-free asset is the U.S. Treasury bill (T-bill).
• Thus, investors obtain a line tangent to the efficient frontier whose y-intercept is
the risk-free rate of return.
• Assuming investors have identical expectations regarding expected returns,
variances/standard deviations, and covariances/correlations (i.e., homogenous
expectations), there will only be one tangency line, which is referred to as the
capital market line (CML)

The Capital Market Line (CML)
• Market portfolio is the portfolio containing
all risky asset classes in the world (can be
proxied by a stock market index (S&P 500))
• All investors hold some combination of the
risk-free asset and the market (tangency)
portfolio, depending on their desired
amount of total risk and return.
• A more risk-averse investor (A) may invest
some of his money in the risk-free asset
with the remainder invested in the market
• At any point to the left of M, investors are
lending at the risk-free rate (some of their
money is invested in Treasuries), whereas
at points to the right of M, they are
borrowing at the risk-free rate (using
leverage).

The Capital Asset Pricing Model (CAPM)
• Developed by William Sharpe and John Lintner in the 1960s.
• CAPM builds on the ideas of modern portfolio theory and the CML in that
investors are assumed to hold some combination of the risk-free asset and the
market portfolio.
• Key assumptions:
• Information is freely available.
• Frictionless markets. There are no taxes and commissions or transaction costs.
• Fractional investments are possible. Assets are infinitely divisible, meaning investors can take
a large position as well as very small positions.
• Perfect competition. Individual investors cannot affect market prices through their buying and
selling activity and are, therefore, viewed as price takers.
• Investors make their decisions solely based on expected returns and variances. This implies
that deviations from normality, such as skewness and kurtosis, are ignored from the decision-
making process.
• Market participants can borrow and lend unlimited amounts at the risk-free rate.
• Homogenous expectations. Investors have the same forecasts of expected returns, variances,
and covariances over a single period.

Estimating and Interpreting Systematic Risk
• The expected returns of risky assets in the market portfolio are assumed to only depend on their
relative contributions to the market risk of the portfolio.
• The systematic risk of each asset represents the sensitivity of asset returns to the market return
and is referred to as the asset’s beta.
• Market beta is, by definition, equal to 1. Any security with a beta of 1 moves in a one-to-one
relationship with the market.
• Beta > 1: any security with a beta greater than 1 moves by a greater amount (has more market
risk) and is referred to as cyclical (e.g., luxury goods stock).
• Beta <1: Any security with a beta below 1 is referred to as defensive (e.g., a utility stock).
• Cyclical stocks perform better during expansions whereas defensive stocks fare better in
recessions.

EXAMPLE: Calculating an asset’s beta
• The standard deviation of the market return is estimated as 20%.
• If Asset A’s standard deviation is 30% and its correlation of returns with the
market index is 0.8, what is Asset A’s beta?
• If the covariance of Asset A’s returns with the returns on the market index is
0.048, what is the beta of Asset A?

Deriving the CAPM
• The intercept occurs when beta is equal to 0
(i.e., when there is no systematic risk). The only
asset with zero market risk is the risk-free asset,
which is completely uncorrelated with market
movements and offers a guaranteed return.
→The intercept of the SML is equal to the risk-
free rate of return, RF

This implies that the expected return of an investment depends on the risk-free rate
RF, the MRP, [RM − RF], and the systematic risk of the investment, β. The expected
return, E(Ri), can be viewed as the minimum required return, or the hurdle rate, that
investors demand from an investment, given its level of systematic risk.

Investment decision
• If an analyst determines that the expected return is different from the
required rate of return implied by CAPM, then the security may be
mispriced according to rational expectations. A mispriced security
would not lie on the SML
• Required rate of return (CAPM) > Expected return (analyst valuation)
→ Overvalued, plotted below SML
• Required rate of return (CAPM) < Expected return (analyst valuation)
→ Undervalued, plotted above SML

• EXAMPLE: Expected return on a stock
Assume you are assigned the task of evaluating the stock of Sky-Air, Inc. To evaluate
the stock, you calculate its required return using the CAPM. The following
information is available:
• Expected market risk premium 5%
• Risk-free rate 4%
• Sky-Air beta 1.5
Using CAPM, calculate and interpret
the expected return for Sky-Air.

Performance Evaluation Measures
Sharpe Performance Index
• SPI measures excess return (portfolio return in excess of the risk-free
rate) per unit of total risk (as measured by standard deviation).

Treynor Performance Index
• TPI measures excess return per unit of systematic risk.
• While the Sharpe measure uses total risk as measured by standard
deviation, the Treynor measure uses systematic risk as measured by beta.
• Beta and TPI should be more relevant metrics for well-diversified
portfolios.

An alternative approach is to calculate excess return relative to a target return or a
benchmark portfolio return.
• Tracking Error: Standard deviation of the difference between the portfolio return
and the benchmark return.
• Information Ratio: calculated by dividing the portfolio expected return in excess
of the benchmark expected return by the tracking error:

Risk and Return (Part II)
Reading 7
The Arbitrage Pricing Theory and Multifactor Models of Risk and
Return

Outline
• Arbitrage Pricing Theory
• Multifactor Model Inputs
• Applying Multifactor Models
• The Fama-French Three-factor Model

Arbitrage Pricing Theory
• Arbitrage is the simultaneous buying and selling of two securities to capture a
perceived abnormal price difference between the two assets.
• Example: The stock of Company X is trading at $20 on the New York Stock
Exchange (NYSE) while, at the same moment, it is trading for $20.05 on the
London Stock Exchange (LSE). A trader can buy the stock on the NYSE and
immediately sell the same shares on the LSE, earning a profit of 5 cents per share.
The trader can continue to exploit this arbitrage until the specialists on the NYSE
run out of inventory of Company X's stock, or until the specialists on the NYSE or
LSE adjust their prices to wipe out the opportunity.

• In 1976, Steven Ross proposed an alternative risk modeling tool to CAPM called
arbitrage pricing theory (APT)
• APT refers to a model that measures expected return relative to multiple risk
factors (a number of macroeconomic variables that capture systematic risk).
• Arbitrage pricing theory has very simplistic assumptions, including the following:
• Market participants are seeking to maximize their profits.
• Markets are frictionless (i.e., no barriers due to transaction costs, taxes, or lack of
access to short selling).
• There are no arbitrage opportunities, and if any are uncovered, then they will be very
quickly exploited by profit-maximizing investors.

• According to arbitrage pricing theory, the expected return for security i can be
modeled as:

• Chen, Roll, and Ross propose the following four factors as one way to structure an
APT model:
• The spread between short-term and long-term interest rates (i.e., the yield
curve)
• Expected versus unexpected inflation
• Industrial production
• The spread between low-risk and high-risk corporate bond yields
• APT model could include any number of variables that an analyst desires to
consider: macroeconomic variables or firm attributes (e.g., P/E multiples, revenue
trends, historical returns).

LO 6.c: Calculate the expected return of an asset using a single-
factor and a multifactor model.
Example:
• RHCI = E(RHCI) + βGDP*FGDP* + βCS*FCS* + eHCI
• The factor beta for CS surprises is 1.5.
• The expected CS growth rate is 1.0%.
• Given that CS presents a growth rate of 0.75%, calculate the RHCI
Answer:
• The CS surprise factor is −0.25% (= 0.75% − 1.0%)
• RHCI = 0.10 + 2.0(−0.006) + 1.5(−0.0025) + eHCI = 0.0843 = 8.43%
• This model predicts a value of 8.43%, which is much closer to the actual result of 8.25%.
This multifactor model is capturing more of the systematic influences.
• An analyst would likely keep exploring to find a third or fourth factor that would get them
even closer to the actual result. Once the proper risk factors have been included, the
analyst will be left with company-specific risk (ei) that cannot be diversified away.

Accounting for Correlation
• Arbitrage pricing theory relies on the use of a well-diversified portfolio.
• Diversification is enhanced when correlations between portfolio assets is low.
Assets have lower correlations when drawn from different asset classes (e.g.,
commodities, real estate, industrial firms, utilities).
• The presence of multiple asset classes will result in a divergent list of factors that
might impact the expected returns for a stock.
• Multifactor models are ideal for this form of analysis.
• The main conclusion of APT is that expected returns on well-diversified portfolios
are proportional to their factor betas. However, we cannot conclude that the APT
relationship will hold for all securities. We can conclude that the APT relationship
must hold for nearly all securities.

• One drawback of APT is that it does not specify the systematic factors, but
analysts can find these by regressing historical portfolio returns against factors
such as real GDP growth rates, inflation changes, term structure changes, risk
premium changes and so on.
• The idea behind a no-arbitrage condition is that if there is a mispriced security in
the market, investors can always construct a portfolio with factor sensitivities
similar to those of mispriced securities and exploit the arbitrage opportunity.
• As all investors would sell an overvalued and buy an undervalued portfolio, this
would drive away any arbitrage profit. This is why the theory is called arbitrage
pricing theory.

LO 6.e: Three options
1) Long Portfolio 1 and short Portfolio 2:
• Result in zero beta for GDP surprise
• Retain a 0.30 beta for consumer sentiment surprise and add a −0.25 beta (because the position is held
short) to unemployment surprise.
• It is possible to find a financial asset that only has an equal factor exposure to the single variable of GDP
surprise. In such a circumstance, the investor could neutralize the GDP surprise exposure and not add
any other new exposures
2) Long Portfolio 1 and short Portfolio 3:
• neutralize the consumer sentiment exposure while retaining GDP surprise and adding manufacturing
surprise.
3) Form a hedged portfolio (Portfolio H):
• Find derivatives that could hedge the 0.50 beta exposure to GDP surprise and the 0.30 beta exposure to
consumer sentiment surprise
• Form a hedged portfolio (Portfolio H) which has a 50% position in a derivative with exposure to only
GDP surprise, a 30% position in a derivative with exposure to only consumer sentiment surprise, and
the remaining 20% in the risk-free asset.
• Take a long position in Portfolio 1 and a short position in Portfolio H to effectively mitigate all exposure
to both GDP surprise and consumer sentiment surprise.

The Fama-French Three-Factor Model
• CAPM is a single-factor model:
• Because well-diversified portfolios include assets from multiple asset classes,
multiple risk factors will influence the systematic risk exposure of the portfolio.
Therefore, multifactor APT can be rewritten as follows:

The Fama-French Three-Factor Model
• Eugene Fama and Kenneth French (1996) specified a multifactor model with three factors:
1) a risk premium for the market
2) a factor exposure for “small minus big”
• Small minus big (SMB) is the difference in returns between small firms and large firms.
• This factor adjusts for the size of the firm because smaller firms often have higher returns than
larger firms (small firms are inherently riskier than big firms)
3) a factor exposure for “high minus low”.
• High minus low (HML) is the difference between the return on stocks with high book-to-market
values and ones with low book-to-market values.
• A high book-to-market value means that the firm has a low price-to-book metric (book-to-
market and price-to-book are inverses). Firms with lower starting valuations are expected to
potentially outperform those with higher starting valuations.
Data: https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html

Extension
• Mark Carhart (1997) added a momentum factor to the Fama and French model to
yield a four-factor model.
• Fama and French (2015) themselves proposed adding factors for:
• “robust minus weak” (RMW) that accounts for the strength of operating
profitability
• “conservative minus aggressive” (CMA) to adjust for the degree of
conservatism in the way a firm invests

Example
A company has a beta relative to the market (βM) of 0.85, an SMB factor sensitivity (βSMB)
of 1.65, and an HML factor sensitivity (βHML) of −0.25. The equity risk premium is 8.5%, the
SMB factor is 2.5%, the HML factor is 1.75%, and the risk-free rate is 2.75%. Given this
series of inputs, compute the expected return for this stock?
Answer:
• E(Ri) = RF + βi,MRPM + βi,SMBFSMB + βi,HMLFHML + ei
• E(Ri) = 0.0275 + 0.85(0.085) + 1.65(0.025) + −0.25(0.0175) + ei = 0.1366 = 13.66%
• Any return that is different from 13.66% is considered to be alpha (α). The source of this
alpha could be company-specific risk (ei), or it could be that other factors need to be
added to this multifactor model to better predict this stock’s future returns.

Random Variables and Probability Functions
• Discrete random variable (Bernoulli random variable): one that can take on only
a countable number of possible outcomes
• Example: the number of outcomes of a coin flip, the number of days in June that will have a
temperature greater than 35 °C
• Continuous random variable: uncountable number of possible outcomes.
• Example: The amount of rainfall that will fall in June
• For continuous random variables, we measure probabilities only over some positive interval,
(e.g., the probability that rainfall in June will be between 500 and 520 mm).
• A probability mass function (PMF), f (x) = P(X = x), gives us the probability that the
outcome of a discrete random variable, X, will be equal to a given number, x.
• A cumulative distribution function (CDF) gives us the probability that a random
variable will take on a value less than or equal to x [i.e., F(x) = P(X ≤ x)].

Expected value
• Expected value: weighted average of the possible outcomes of a random variable,
where the weights are the probabilities that the outcomes will occur.
E(X) = ΣPiXi= P1X1 + P2X2 + … + PnXn
In which Pi is the probability of outcome Xi to occur
• The following are two useful properties of expected values:
1. If c is any constant, then:
E(cX) = cE(X)
2. If X and Y are any random variables, then:
E(X + Y) = E(X) + E(Y)

MEAN, VARIANCE, SKEWNESS, AND KURTOSIS
• Skewness: a measure of a distribution’s symmetry, is the standardized third
moment.
• E{[X − E(X)]3} = E[(X − μ)3]
• Skew = 0 → perfectly symmetric distribution

The Normal Distribution
• Many of the random variables that are relevant to finance and other professional
disciplines follow a normal distribution.
• It is completely described by its mean, μ, and variance, σ2, stated as X ~ N(μ, σ2).
In words, this says, “X is normally distributed with mean μ and variance σ2.”
• Skewness = 0, meaning the normal distribution is symmetric about its mean, so
that P(X ≤ μ) = P(μ ≤ X) = 0.5, and mean = median = mode.
• Kurtosis = 3.
• A linear combination of normally distributed independent random variables is
also normally distributed.
• The probabilities of outcomes further above and below the mean get smaller and
smaller but do not go to zero (the tails get very thin but extend infinitely).

Confidence interval
• A confidence interval is a range of values around the expected outcome within
which we expect the actual outcome to be some specified percentage of the
time.
• A 95% confidence interval is a range that we expect the random variable to be in
95% of the time.
• For a normal distribution, this interval is based on the expected value (sometimes
called a point estimate) of the random variable and on its variability, which we
measure with standard deviation.
do tin cay

The standard normal distribution
• A standard normal distribution (i.e., z-distribution) is a normal distribution that
has been standardized so it has a mean of zero and a standard deviation of 1
• N~(0,1)

The standard normal distribution
• EXAMPLE: Standardizing a random variable (calculating z-values)
Assume the annual earnings per share (EPS) for a population of firms are normally
distributed with a mean of $6 and a standard deviation of $2. What are the z-values for EPS
of $2 and $8?
• Answer:
If EPS = x = $8, then z = (x − μ) / σ = ($8 − $6) / $2 = +1
If EPS = x = $2, then z = (x − μ) / σ = ($2 − $6) / $2 = –2
Here, z = +1 indicates that an EPS of $8 is one standard deviation above the mean, and z =
−2 means that an EPS of $2 is two standard deviations below the mean.

• EXAMPLE: Using the z-table (1)
Considering again EPS distributed with μ = $6 and σ = $2, what is the probability
that EPS will be $9.70 or more?
Answer:
The z-value for EPS = $9.70 is:
That is, $9.70 is 1.85 standard deviations above the mean EPS value of $6. From the
z-table, we have F(1.85) = 0.9678, but this is P(EPS ≤ 9.70).
P(EPS > 9.70) = 1 − 0.9678 = 0.0322, or 3.2%

LO 6.e: Explain how to construct a portfolio to hedge exposure
to multiple factors.
• Using calculated factor sensitivities, an investor can build factor portfolios, which
retain some exposures and intentionally mitigate others through targeted
portfolio allocations
• Example: take a long position in Portfolio 1 and a short position in Portfolio 2 to
mitigate all exposure to GDP surprise risk.

Student’s t-Distribution
• Student’s t-distribution is similar to a normal distribution, but has fatter tails (i.e.,
a greater proportion of the outcomes are in the tails of the distribution).
• When small samples (n < 30) from a population with unknown variance and a
normal, or approximately normal, distribution.
• When population variance is unknown and the sample size is large enough that
the central limit theorem will assure that the sampling distribution is
approximately normal

Student’s t-Distribution
• It is symmetrical.
• It is defined by a single parameter, the
degrees of freedom (df) (the number of
sample observations minus 1, n − 1, for
sample means.
• It has a greater probability in the tails
(fatter tails) than the normal distribution.
• As the degrees of freedom (the sample
size) gets larger, the shape of the t-
distribution more closely approaches a
standard normal distribution.

• The Chi-Squared Distribution
• The F-Distribution
• The Exponential Distribution
• The Beta Distribution
• Mixture distributions

Covariance
• Covariance is the expected value of the product of the deviations of
the two random variables from their respective expected values.
• Covariance measures how two variables move with each other or the
dependency between the two variables.
• Cov(X,Y) and σXY.
• Cov(X,Y) = E{[X − E(X)][Y − E(Y)]}
• Cov(X,Y) = E(X,Y) − E(X) × E(Y)

• EXAMPLE: Covariance
Assume that the economy can be in three possible states (S) next year: boom, normal, or
slow economic growth. An expert source has calculated that P(boom) = 0.30, P(normal) =
0.50, and P(slow) = 0.20. The returns for Stock A, RA, and Stock B, RB, under each of the
economic states are provided in the following table. What is the covariance of the returns
for Stock A and Stock B?
Answer:
E(RA) = (0.3)(0.20) + (0.5)(0.12) + (0.2)(0.05) = 0.13
E(RB) = (0.3)(0.30) + (0.5)(0.10) + (0.2)(0.00) = 0.14

Correlation
EXAMPLE: Correlation
Using our previous example, compute and interpret the correlation of the returns for Stocks A
and B, given that σ2(RA) = 0.0028 and σ2(RB) = 0.0124 and recalling that Cov(RA,RB) = 0.0058.
Answer:
σ(RA) = (0.0028)1/2 = 0.0529
σ(RB) = (0.0124)1/2 = 0.1114

Expected value
• EXAMPLE: Expected earnings per share (EPS)
The probability distribution of EPS for Ron’s Stores is given in the following figure.
Calculate the expected earnings per share.
Answer:
The expected EPS is simply a weighted average of each
possible EPS, where the weights are the probabilities of
each possible outcome.
E(EPS) = 0.10(1.80) + 0.20(1.60) + 0.40(1.20) + 0.30(1.00)
= £1.28

Sample moments
• Biased sample variance
• Unbiased sample variance
• Population variance
𝜎2 =
1
𝑁
෍
𝑖=1
𝑁
(𝑋𝑖 − 𝜇)2

MEAN, VARIANCE, SKEWNESS, AND KURTOSIS
• Kurtosis: is the standardized fourth moment.
• Kurtosis is a measure of the shape of a distribution, in particular the total probability
in the tails of the distribution relative to the probability in the rest of the distribution.
• The higher the kurtosis, the greater the probability in the tails of the distribution.
Positive Kurtosis
Negative Kurtosis

Time series
• Time series is data collected over regular time periods
• Example: monthly S&P 500 returns, quarterly dividends paid by a
company, etc.).
• Time series data have trends (the component that changes over
time), seasonality (systematic change that occur at specific times of
the year), and cyclicality (changes occurring over time cycles).

Covariance Stationary
• To be covariance stationary, a time series must exhibit the following
three properties:
1. Its mean must be stable over time.
2. Its variance must be finite and stable over time.
3. Its covariance structure must be stable over time.
• Covariance structure refers to the covariances among the values of a
time series at its various lags, which are a given number of periods
apart at which we can observe its values.

Autocovariance and Autocorrelation Functions
• The covariance between the current value of a time series and its
value τ periods in the past is referred to as its autocovariance at lag τ.
• Its autocovariances for all τ make up its autocovariance function. If a
time series is covariance stationary, its autocovariance function is
stable over time.
• To convert an autocovariance function to an autocorrelation function
(ACF), we divide the autocovariance at each τ by the variance of the
time series. This gives us an autocorrelation for each τ that will be
scaled between −1 and +1.

White noises
• A time series might exhibit zero correlation among any of its lagged values. Such a
time series is said to be serially uncorrelated.
• A special type of serially uncorrelated series is one that has a mean of zero and a
constant variance. This condition is referred to as white noise, or zero-mean
white noise, and the time series is said to follow a white noise process.
• One important purpose of the white noise concept is to analyze a forecasting
model. A model’s forecast errors should follow a white noise process

Autoregressive Processes
• The first-order autoregressive [AR(1)] process is specified in the form of a variable
regressed against itself in lagged form. This relationship can be shown in the following
formula:
yt = d + Φyt–1 + εt
where:
• d = intercept term
• yt = the time series variable being estimated
• yt–1 = one-period lagged observation of the variable being estimated
• εt = current random white noise shock (mean 0)
• Φ = coefficient for the lagged observation of the variable being estimated
• In order for an AR(1) process to be covariance stationary, the absolute value of the
coefficient on the lagged operator must be less than one (i.e., |Φ| < 1). Similarly, for an
AR(p) process, the absolute values of all coefficients should be less than 1.

Autoregressive Processes
• Autoregressive model predicts future values based on past values.
• For example, an autoregressive model might seek to predict a stock's future prices
based on its past performance.
• Based on the assumption that past values have an effect on current values.
• For example, an investor using an autoregressive model to forecast stock prices
would need to assume that new buyers and sellers of that stock are influenced by
recent market transactions when deciding how much to offer or accept for the
security.
• This assumption is not always the case.
• For example, in the years prior to the 2008 Financial Crisis, most investors were not
aware of the risks posed by the large portfolios of mortgage-backed securities held
by many financial firms. During those times, an investor using an autoregressive
model to predict the performance of U.S. financial stocks would have had good
reason to predict an ongoing trend of stable or rising stock prices in that sector.

Moving average process
• An MA process is a linear regression of the current values of a time series against both
the current and previous unobserved white noise error terms, which are random shocks.
MAs are always covariance stationary.
• The first-order moving average [MA(1)] process can be defined as:
yt = μ + θεt−1 + εt
where:
• μ= mean of the time series
• εt = current random white noise shock (mean 0)
• εt−1 = one-period lagged random white noise shock
• θ = coefficient for the lagged random shock
• The MA(1) process is considered to be first-order because it only has one lagged error
term (εt−1). This yields a very short-term memory because it only incorporates what
happens one period ago

Moving average process
• Example of daily demand for ice cream (yt):
yt = 5,000 + 0.3εt−1 + εt
• The error term is the daily change in demand.
• Using only the current period’s error term (εt), if the daily change is positive, then
we would estimate that daily demand for ice cream would also be positive.
• But, if the daily change yesterday (εt−1) was also positive, then we would expect
an amplified impact on our daily demand by a factor of 0.3.
• If the coefficient θ is negative, the series aggressively mean reverts because the
effect of the previous shock reverts in the current period

Quantitative Analysis
Reading 22
Non-Stationary Time Series

Time Trends
• Non-stationary time series may exhibit deterministic trends,
stochastic trends, or both.
• Deterministic trends include both time trends and deterministic
seasonality.
• Stochastic trends include unit root processes such as random walks

Time Trends
• Time trends may be linear or nonlinear.
• Linear
• Log-linear model
• Non-linear
• log-quadratic model

Seasonality
• Seasonality in a time series is a pattern that tends to repeat from year to year.
• Example: monthly sales data for a retailer. Because sales data normally varies according to the calendar,
we might expect this month’s sales (xt) to be related to sales for the same month last year (xt−12).
• Specific examples of seasonality relate to increases that occur at only certain
times of the year.
• Example: purchases of retail goods typically increase dramatically every year in the weeks leading up to
Christmas. Similarly, sales of gasoline generally increase during the summer months when people take
more vacations.
• Weather is another common example of a seasonal factor as production of agricultural commodities is
heavily influenced by changing seasons and temperatures.
• Seasonality in a time series can also refer to cycles shorter than a year.
• Example: Calendar effects (January effects)
• An effective technique for modeling seasonality is to include seasonal dummy
variables in a regression.

Unit roots
• We describe a time series as a random walk if its value in any given period is its
previous value plus-or-minus a random “shock.” Symbolically, we state this as
yt = yt−1 + εt.
• If it follows logically that the same was true in earlier periods,
yt−1 = yt−2 + εt−1
yt−2 = yt−3 + εt−2 and so forth
y1 = y0 + ε1.
• If we substitute these (recursively) back into yt = yt−1 + εt, we eventually get:
yt = y0 + ε1 + ε2 + … + εt−2 + εt−1 + εt.
That is, any observation in the series is a function of the beginning value and all the
past shocks, as well as the shock in the observation’s own period.

Random walk theory
• Random walk theory suggests that changes in stock prices have the same
distribution and are independent of each other.
• Therefore, it assumes the past movement or trend of a stock price or market
cannot be used to predict its future movement.
• In short, random walk theory proclaims that stocks take a random and
unpredictable path that makes all methods of predicting stock prices futile in the
long run.

Unit roots
• A key property of a random walk is that its variance increases with time. This
implies a random walk is not covariance stationary, so we cannot model one
directly with AR, MA, or ARMA techniques
• A random walk is a special case of a wider class of time series known as unit root
processes.
• The most common way to test a series for a unit root is with an augmented
Dickey-Fuller test

Derivatives
Reading 28-FRM
Introduction to Derivatives
(Includes content from Chapter 01 - J.Hull - Options,Futures
and Other Derivatives 8th edition)

What is a Derivative?
• A derivative security is a financial security whose value depends on,
or is derived from, the value of another asset.
• Examples: futures, forwards, swaps, options…
• This other security is referred to as the underlying asset.
• The underlying assets include stocks, currencies, interest rates,
commodities, debt instruments, electricity, insurance payouts, the
weather, etc.

Why are derivatives important?
• Derivatives play a key role in transferring risks in the economy
• Many financial transactions have embedded derivatives
• The real options approach to assessing capital investment decisions has become widely
accepted
• Derivatives can be used:
• For financial risk management (i.e., hedging)
• For speculation
• To lock in an arbitrage profit
• For diversification of exposures
• As added features to a bond (e.g., convertible, callable)
• As employee compensation in the case of stock options
• Within a capital project as an embedded option (e.g., real or abandonment options).
short term
long term

The Lognormal Distribution
• The lognormal distribution is generated by the function ex, where x is normally distributed.
• Because the natural logarithm, ln, of ex is x, the logarithms of lognormally distributed random
variables are normally distributed.
• The lognormal distribution is skewed to the right.
• „
. The lognormal distribution is bounded from below by zero so that it is useful for modeling asset
prices that never take negative values.

Size of OTC and Exchange-Traded Markets
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull
2012
5
Source: Bank for International Settlements. Chart shows total principal amounts for
OTC market and value of underlying assets for exchange market
otc > exchange

OTC trading
Advantages of OTC trading:
• Terms are not set by any exchange (i.e., not standardized so customization is
possible).
• Some new regulations since the credit crisis (e.g., standardized OTC derivatives
now traded on swap execution facilities, a central counterparty is now required
for standardized trades, and trades are now required to be reported to a central
registry)
• Greater anonymity (e.g., an interdealer broker only identifies the client at the
conclusion of the trade).
Disadvantages of OTC trading:
• OTC trading has more credit risk than exchange trading when it comes to
nonstandardized transactions.

The Lehman Bankruptcy (Business Snapshot 1.10)
• Lehman’s filed for bankruptcy on September 15, 2008. This was the biggest
bankruptcy in US history
• Lehman was an active participant in the OTC derivatives markets and got into
financial difficulties because it took high risks and found it was unable to roll
over its short term funding
• It had hundreds of thousands of transactions outstanding with about 8,000
counterparties
• Unwinding these transactions has been challenging for both the Lehman
liquidators and their counterparties
Options, Futures, and Other Derivatives, 8th Edition, Copyright
© John C. Hull 2012
7

Forward contracts
• An agreement to buy or sell an asset at a certain future time for a certain price.
• There is no standardization for forward contracts, and these contracts are traded
in the OTC market.
• Long position: agreeing to purchase the underlying asset at a future date for a
specified price.
• Short position: agreeing to sell the asset on that same date for that same price.
• Forward contracts are often used in foreign exchange situations as these
contracts can be used to hedge foreign currency risk.

Forward Price
• The forward price for a contract is the delivery price that would be applicable
to the contract if were negotiated today (i.e., it is the delivery price that would
make the contract worth exactly zero)
• The forward price may be different for contracts of different maturities
9

Forwards
• EXAMPLE: Calculating Forward Contract Payoffs
Compute the payoff to the long and short positions in a forward contract given that
the forward price is $25 and the spot price at maturity is $30.
• Answer:
Payoff to long position:
payoff = ST − K = $30 − $25 = $5
Payoff to short position:
payoff = K − ST = $25 − $30 = −$5

Example
• On May 24, 2010 the treasurer of a corporation enters into a long forward contract to
buy £1 million in six months at an exchange rate of 1.4422
• This obligates the corporation to pay $1,442,200 for £1 million on November 24, 2010
• What are the possible outcomes?
Answer:
• If the spot exchange rate rose to 1.5000, at the end of the 6 months, the forward
contract would be worth $57,800 (= $1,500,000 - $1,442,200) to the corporation. It
would enable £1 million to be purchased at an exchange rate of 1.4422 rather than
1.5000.
• If the spot exchange rate fell to 1.3500 at the end of the 6 months, the forward contract
would have a negative value of $92,200 (= $1,350,000 - $1,442,200) to the corporation
because it would lead to the corporation paying $92,200 more than the market price for
the GBP.
14
tính payoff

Futures Contracts
• Agreement to buy or sell an asset for a certain price at a certain time in the
future.
• Similar to forward contract, but futures contracts are highly standardized
regarding quality, quantity, delivery time, and location for each specific asset.
• Whereas a forward contract is traded OTC, a futures contract is traded on an
exchange.
• The commodities include pork bellies, live cattle, sugar, wool, lumber, copper,
aluminum, gold, and tin.
• The financial assets include stock indices, currencies, and Treasury bonds.
• Futures prices are regularly reported in the financial press.
15

Exchanges Trading Futures
• CME Group (formerly Chicago Mercantile Exchange and Chicago Board of Trade)
• NYSE Euronext
• BM&F (Sao Paulo, Brazil)
• TIFFE (Tokyo)
• and many more (see list at end of book)
16

Examples of Futures Contracts
Agreement to:
• Buy 100 oz. of gold @ US$1400/oz. in December
• Sell £62,500 @ 1.4500 US$/£ in March
• Sell 1,000 bbl. of oil @ US$90/bbl. in April
17

Options
• A contract that, in exchange for paying an option premium, gives the option buyer
the right, but not the obligation, to buy (sell) an asset at the prespecified
exercise (strike) price from (to) the option seller within a specified time period, or
depending on the type of option, a precise date (i.e., expiration date).
• A call option is an option to buy a certain asset by a certain date for a certain
price (the strike price)
• A put option is an option to sell a certain asset by a certain date for a certain price
(the strike price)
• CBOE (Chicago board options exchange)
18

American vs European Options
• An American-style option can be exercised at any time during its life (between
the issue date and the expiration date).
• A European-style option can be exercised only at maturity (at the actual
expiration date)
• American options will be worth more than European options when the right to
early exercise is valuable, and they will have equal value when it is not.
19

How do options differ from futures and forwards?
Options Forwards or Futures
Give the holder the right to buy or sell the
underlying asset, but the holder does not
have to exercise this right
The holder is obligated to buy or sell
the underlying asset
There is a cost to acquiring an option.
Option seller charges buyers a premium.
It costs nothing to enter into a forward
or futures contract

Google Call Option Prices (June 15, 2010; Stock Price is bid 497.07, offer 497.25)
Source: CBOE
21
Strike
Price
Jul 2010
Bid
Jul 2010
Offer
Sep 2010
Bid
Sep 2010
Offer
Dec 2010
Bid
Dec 2010
Offer
460 43.30 44.00 51.90 53.90 63.40 64.80
480 28.60 29.00 39.70 40.40 50.80 52.30
500 17.00 17.40 28.30 29.30 40.60 41.30
520 9.00 9.30 19.10 19.90 31.40 32.00
540 4.20 4.40 12.70 13.00 23.10 24.00
560 1.75 2.10 7.40 8.40 16.80 17.70
• The price of a call option
decreases as the strike price
increases, while the price of
a put option increases as the
strike price increases.
• Both types of option tend to
become more valuable as
their time to maturity
increases.
long maturity, higher volality, more profit

Google Put Option Prices (June 15, 2010; Stock Price is bid 497.07, offer 497.25)
Source: CBOE
22
Strike
Price
Jul 2010
Bid
Jul 2010
Offer
Sep 2010
Bid
Sep 2010
Offer
Dec 2010
Bid
Dec 2010
Offer
460 6.30 6.60 15.70 16.20 26.00 27.30
480 11.30 11.70 22.20 22.70 33.30 35.00
500 19.50 20.00 30.90 32.60 42.20 43.00
520 31.60 33.90 41.80 43.60 52.80 54.50
540 46.30 47.20 54.90 56.10 64.90 66.20
560 64.30 66.70 70.00 71.30 78.60 80.00

Types of option positions
• There are four types of option positions:
1. A long position in a call option
2. A long position in a put option
3. A short position in a call option
4. A short position in a put option.

Call Option Payoff
• The payoff on a call option to the option buyer is calculated as follows:
CT = max(0, ST − X)
where:
• CT = payoff on call option
• ST = stock price at maturity
• X = strike price of option
The payoff to the option seller is −CT [= −max(0, ST − X)].
We should note that max(0, St − X), where time, t, is between 0 and T, is the payoff
if the owner decides to exercise the call option early.

Call Option Profit
• The price paid for the call option, C0, is referred to as the call premium. Thus, the
profit to the option buyer is calculated as follows:
profit = CT − C0
where:
• C0 = call premium
• Conversely, the profit to the option seller is:
profit = C0 − CT

Put Option Payoff
• The payoff on a put option is calculated as follows:
PT = max(0, X − ST)
where:
• PT = payoff on put option
• ST = stock price at maturity
• X = strike price of option
The payoff to the option seller is −PT [=−max(0, X − ST)].
We should note that max(0, X − St), where 0 < t < T, is also the payoff if the owner
decides to exercise the put option early.

Put Option Payoff
• The price paid for the put option, P0, is referred to as the put premium. Thus, the
profit = PT − P0
where:
• P0 = put premium
• The profit to the option seller is:
profit = P0 − PT

For buyer:
• ST < X: buyer will
exercise the put
option
→Payoff = X - ST
→ Profit = X – ST – Po
• ST >X : buyer will not
exersise the put
option
→payoff = 0
→ Profit = - Po

• Po = 7; X= 70; If ST = 50
• Buyer:
Payoff = 70-50 = 20
Profit = 20 – 7 = 13
• Seller:
Payoff = 50-70 = -20
Profit = 50-70+7 = -13
x-st
x-st-po
st-x
st-x+po

• EXAMPLE: Calculating Payoffs and Profits From Options
Compute the payoff and profit to a call buyer, a call writer, put buyer, and put writer if the strike
price for both the put and the call is $45, the stock price is $50, the call premium is $3.50, and the
put premium is $2.50.
Answer:
Call buyer:
• payoff = CT = max(0, ST − X) = max(0, $50 − $45) = $5
• profit = CT − C0 = $5 − $3.50 = $1.50
Call writer:
• payoff = −CT = −max(0, ST − X) = −max(0, $50 − $45) = −$5
• profit = C0 − CT = $3.50 − $5 = −$1.50
Put buyer:
• payoff = PT = max(0, X − ST) = max(0, $45 − $50) = $0
• profit = PT − P0 = $0 − $2.50 = −$2.50
Put writer:
• payoff = −PT = −max(0, X − ST) = −max(0, $45 − $50) = $0
• profit = P0 − PT = $2.50 − $0 = $2.50

Swap
• A derivative contract through which two parties exchange the cash flows or
liabilities from two different financial instruments.
• Swaps can be used to efficiently alter the interest rate risk of existing assets and
liabilities.
• Interest rate swap: an agreement between two parties to exchange interest
payments based on a specified principal over a period of time. In a plain vanilla
interest rate swap, one of the interest rates is floating, and the other is fixed.
• A currency swap exchanges interest rate payments in two different currencies

Derivatives Traders
Types of traders:
• Hedgers
• Speculators
• Arbitrageurs
35

Hedgers
• Hedgers typically reduce their risks with forward contracts or options.
• By using forward contracts (at no cost), the trader is attempting to neutralize risk by fixing the
price the hedger will pay or receive for the underlying asset.
• Option contracts, in contrast, are more of an insurance policy that require the payment of a
premium, but will protect against downside risk while keeping some of the upside.
• An investor or business with a long exposure to an asset can hedge exposure by either entering
into a short futures contract or by buying a put option.
• An investor or business with a short exposure to an asset can hedge exposure by either entering
into a long futures contract or by buying a call option.

Value of Microsoft Shares with and without Hedging
38
20,000
25,000
30,000
35,000
40,000
20 25 30 35 40
Value of Holding
($)
Stock Price ($)
No Hedging
Hedging

• EXAMPLE: Hedging With a Forward Contract
Suppose that a company based in the United States will receive a payment of €10M in three
months. The company is worried that the euro will depreciate and is contemplating using a forward
contract to hedge this risk.
Compute the following:
1. The value of the €10M in U.S. dollars at maturity given that the company hedges the exchange
rate risk with a forward contract at 1.25 $/€.
2. The value of the €10M in U.S. dollars at maturity given that the company did not hedge the
exchange rate risk and the spot rate at maturity is 1.2 $/€.
Answer:
1. The value at maturity for the hedged position is:
€10,000,000 × 1.25 $/€ = $12,500,000
2. The value at maturity for the unhedged position is:
€10,000,000 × 1.2 $/€ = $12,000,000

• EXAMPLE: Hedging With a Put Option
Suppose that an investor owns one share of ABC stock currently priced at $30. The investor is worried
about the possibility of a drop in share price over the next three months and is contemplating purchasing
put options to hedge this risk. Compute the following:
1. The profit on the unhedged position if the stock price in three months is $25.
2. The profit on the unhedged position if the stock price in three months is $35.
3. The profit for a hedged stock position if the stock price in three months is $25, the strike price on the
put is $30, and the put premium is $1.50.
4. The profit for a hedged stock position if the stock price in three months is $35, the strike price on the
put is $30, and the put premium is $1.50.
Answer:
1. Profit = ST − S0 = $25 − $30 = –$5
2. Profit = ST − S0 = $35 − $30 = $5
3. Profit = ST − S0 + max(0, X − ST) − P0
= $25 − $30 + max(0, $30 − $25) − $1.50 = −$1.50
4. Profit = ST − S0 + max(0, X − ST) − P0
= $35 − $30 + max(0, $30 − $35) − $1.50 = $3.50

Speculators
• Speculators are effectively betting on future price movement.
• When a speculator uses the underlying asset, any potential gain or loss arises
only on the differential between the share purchase price and the future share
price.
• When a speculator uses options, the potential gain is magnified (assuming the
same initial dollar investment in shares as options) and the maximum loss is the
dollar investment in options.

• EXAMPLE: Speculating With Futures
An investor believes that the euro will strengthen against the dollar over the next three months and
would like to take a position with a value of €250,000. He could purchase euros in the spot market
at 0.80 $/€ or purchase two futures contracts at 0.83 $/€ with an initial margin of $10,000. Compute
the profit from the following:
1. Purchasing euros in the spot market if the spot rate in three months is 0.85 $/€.
2. Purchasing euros in the spot market if the spot rate in three months is 0.75 $/€.
3. Purchasing the futures contract if the spot rate in three months is 0.85 $/€.
4. Purchasing the futures contract if the spot rate in three months is 0.75 $/€.
Answer:
1. Profit = €250,000 × (0.85 $/€ − 0.80 $/€) = $12,500
2. Profit = €250,000 × (0.75 $/€ − 0.80 $/€) = −$12,500
3. Profit = €250,000 × (0.85 $/€ − 0.83 $/€) = $5,000
4. Profit = €250,000 × (0.75 $/€ − 0.83 $/€) = −$20,000

• EXAMPLE: Speculating With Options
An investor who has $30,000 to invest believes that the price of stock XYZ will increase over the
next three months. The current price of the stock is $30. The investor could directly invest in the
stock, or she could purchase 3-month call options with a strike price of $35 for $3. Compute the
profit from the following:
1. Investing directly in the stock if the price of the stock is $45 in three months.
2. Investing directly in the stock if the price of the stock is $25 in three months.
3. Purchasing call options if the price of the stock is $45 in three months.
4. Purchasing call options if the price of the stock is $25 in three months.
Answer:
1. Number of stocks to purchase = $30,000 / $30 = 1,000
Profit = 1,000 × ($45 − $30) = $15,000
2. Profit = 1,000 × ($25 − $30) = –$5,000
3. Number of call options to purchase = $30,000 / $3 = 10,000
Profit = 10,000 × [max(0, $45 − $35) − $3] = $70,000
4. Profit = 10,000 × [max(0, $25 − $35) − $3] = −$30,000

Arbitragers
• Arbitrageurs seek to earn a risk-free profit in excess of the risk-free rate through
the discovery and manipulation of mispriced securities.
• They earn a riskless profit by entering into equivalent offsetting positions in one
or more markets.
• Arbitrage opportunities typically do not last long as supply and demand forces
will adjust prices to quickly eliminate the arbitrage situation.

EXAMPLE: Arbitrage of Stock Trading on Two Exchanges
Assume stock DEF trades on the New York Stock Exchange (NYSE) and the Tokyo Stock Exchange
(TSE). The stock currently trades on the NYSE for $32 and on the TSE for ¥2,880. Given the current
exchange rate is 0.0105 $/¥, determine if an arbitrage profit is possible.
Answer:
• Value in dollars of DEF on TSE = ¥2,880 × 0.0105 $/¥ = $30.24
• Arbitrageur could purchase DEF on TSE for $30.24 and sell on NYSE for $32.
• Profit per share = $32 − $30.24 = $1.76
Arbitrage Example

Arbitrage Example
• A stock price is quoted as £100 in London and $140 in New York
• The current exchange rate is 1.4300
• What is the arbitrage opportunity?
46

Risks From Using Derivatives
• If the bet one makes starts going in the wrong direction, the results can be
catastrophic (e.g., Barings Bank).
• Traders with instructions to hedge a position may use derivatives to speculate
due to the massive potential payoffs if speculation succeeds. This risk is known as
an operational risk when it is done in an unauthorized manner.
• It is important to set up controls to ensure that trades are using derivatives in for
their intended purpose. Risk limits should be set, and adherence to risk limits
should be monitored.

Hedge Funds
• Hedge funds are not subject to the same rules as mutual funds
and cannot offer their securities publicly.
• Mutual funds must
• disclose investment policies,
• makes shares redeemable at any time,
• limit use of leverage
• take no short positions.
• Hedge funds are not subject to these constraints.
• Hedge funds use complex trading strategies are big users of
derivatives for hedging, speculation and arbitrage
48

Types of Hedge Funds
• Long/Short Equities
• Convertible Arbitrage
• Distressed Securities
• Emerging Markets
• Global macro
• Merger Arbitrage
49

Futures and Forwards
Reading 31 – Future Markets

Profit from a Short Forward Position
(K= delivery price=forward price at time contract is entered
into)
11
Profit
Price of Underlying
at Maturity, ST
K
lost

Some Terminology
• Open interest: the total number of contracts outstanding
• equal to number of long positions or number of short positions
• Settlement price: the price just before the final bell each day
• used for the daily settlement process
• Volume of trading: the number of trades in one day
Options, Futures, and Other Derivatives, 8th Edition,
Copyright © John C. Hull 2012
3

Convergence of Futures to Spot
4
• The spot (cash) price of a commodity or financial asset is the price for immediate delivery.
• The futures price is the price today for delivery at some future point in time (i.e., the
maturity date).
• The basis is the difference between the spot price and the futures price.
basis = spot price − futures price
• As the maturity date nears, the basis converges toward zero.
• Arbitrage will force the prices to be the same at contract expiration.
Time Time
Futures
Price
Futures
Price
Spot Price
Spot Price

Foreign Exchange Quotes for GBP, May 24, 2010
13
Bid Offer
Spot 1.4407 1.4411
1-month forward 1.4408 1.4413

Margin requirements
• Margin is cash or highly liquid collateral (i.e. marketable securities) placed in an
account to ensure that any trading losses will be met.
• The balance in the margin account is adjusted to reflect daily settlement
• Margins minimize the possibility of a loss through a default on a contract
• The maintenance margin is the minimum margin account balance required.
• An investor will receive a margin call if the margin account balance falls below the
maintenance margin. → The investor must bring the margin account back to the
initial margin amount.
6

Example of a Futures Trade
• An investor takes a long position in 2 December gold futures contracts on June 5
• contract size is 100 oz.
• futures price is US$1250
• initial margin requirement is US$6,000/contract (US$12,000 in total)
• maintenance margin is US$4,500/contract (US$9,000 in total)
7

A Possible Outcome
8
Day Trade
Price ($)
Settle
Price ($)
Daily
Gain ($)
Cumul.
Gain ($)
Margin
Balance ($)
Margin
Call ($)
1 1,250.00 12,000
1 1,241.00 −1,800 − 1,800 10,200
2 1,238.30 −540 −2,340 9,660
….. ….. ….. ….. ……
6 1,236.20 −780 −2,760 9,240
7 1,229.90 −1,260 −4,020 7,980 4,020
8 1,230.80 180 −3,840 12,180
….. ….. ….. ….. ……
16 1,226.90 780 −4,620 15,180
• By end of day 1, the futures
price has dropped by $9 from
$1,250 to $1,241.
Loss = $1,800 (= 200x$9), the
200 ounces of December gold,
which the investor contracted to
buy at $1,250, can now be
sold for only $1,241.
→ The balance in the margin
account would therefore be
reduced by $1,800 to $10,200.
• On Day 7, the balance in the
margin account falls $1,020 below
the maintenance margin level
→ margin call

Margin Cash Flows When Futures Price Decreases
10
Long Trader
Broker
Clearing House
Member
Clearing House
Clearing House
Member
Broker
Short Trader

Future markets
• The exchange guarantees that traders in the futures and over-the-counter (OTC)
markets will honor their obligations
• splitting each trade once it is made and acting as the opposite side of each position.
• The exchange acts as the buyer to every seller and the seller to every buyer.
• By doing this, the exchange allows either side of the trade to reverse positions at a future
date without having to contact the other side of the initial trade.
• This allows traders to enter the market knowing that they will be able to reverse their
position.
• Traders are also freed from having to worry about the counterparty defaulting
since the counterparty is now the exchange.
co day phan nay k ky

Future market quotes
• Each gold futures contract represents 100 ounces and is priced in U.S. dollars per ounce.
• The CME Group website (www.cmegroup.com)

Key Points About Futures
• They are settled daily
• Closing out a futures position involves entering into an offsetting trade
• Most contracts are closed out before maturity
Example: Closing a Futures Position
You have entered a long position in 30 December S&P 250 contracts, in
August. Come September, you decide that you want to close your position
before the contract expires. To accomplish this, you must short, or sell the
30 December S&P 250 contract. The clearing house sees your position as
flat because you are now long and short the same amount and type of
contract.
13

Types of trading orders
• Market orders: orders to buy or sell at the best price available.
• The key problem is that the transaction price may be significantly higher or lower than
planned.
• Discretionary order: a market order where the broker has the option to delay
transaction in search of a better price.
• Limit order: orders to buy or sell away from the current market price.
• A limit buy order is placed below the current price.
• A limit sell order is placed above the current price.
• Stop-loss order: used to prevent losses or to protect profits
• Stop-loss sell order: if the price falls to a certain price, the broker will sell the asset.
• Stop-loss buy order: usually combined with a short sale to limit losses.

Forward Contracts vs Futures Contracts
15
Contract usually closed out
Private contract between 2 parties Exchange traded
Non-standard contract Standard contract
Usually 1 specified delivery date Range of delivery dates
Settled at end of contract Settled daily
Delivery or final cash
settlement usually occurs prior to maturity
FORWARDS FUTURES
Some credit risk Virtually no credit risk

Example of short hedge
• Assume that it is May 15 today and that an oil producer has just negotiated a contract to
sell 1 million barrels of crude oil. It has been agreed that the price that will apply in the
contract is the market price on August 15. The oil producer is therefore in the position
where it will gain $10,000 for each 1 cent increase in the price of oil over the next 3
months and lose $10,000 for each 1 cent decrease in the price during this period.
• Suppose that on May 15 the spot price is $80 per barrel and the crude oil futures price
for August delivery is $79 per barrel. Because each futures contract is for the delivery of
1,000 barrels, the company can hedge its exposure by shorting (i.e., selling) 1,000 futures
contracts. If the oil producer closes out its position on August 15, the effect of the
strategy should be to lock in a price close to $79 per barrel.

Example of short hedge
• Suppose that the spot price on August 15 proves to be $75 per barrel. The company
realizes $75 million for the oil under its sales contract. Because August is the delivery
month for the futures contract, the futures price on August 15 should be very close to
the spot price of $75 on that date. The company therefore gains approximately $79 - $75
= $4 per barrel, or $4 million in total from the short futures position. The total amount
realized from both the futures position and the sales contract is therefore approximately
$79 per barrel, or $79 million in total.
• For an alternative outcome, suppose that the price of oil on August 15 proves to be $85
per barrel. The company realizes $85 per barrel for the oil and loses approximately $85 -
$79 = $6 per barrel on the short futures position. Again, the total amount realized is
approximately $79 million. It is easy to see that in all cases the company ends up with
approximately $79 million.

Long Hedge
• A long hedge occurs when the hedger buys a futures contract to hedge against
an increase in the value of the asset that underlies a short position.
• An increase in the value of the shorted asset will result in a loss to the short seller → The
long hedge offsets the loss in the short position with a gain from the long futures position
• Appropriate when you own an asset and expect to sell it in the future, or when you does not
currently own an asset but expect to purchase it in the future, and expect prices to rise.
20

Example of long hedge
• Suppose that it is now January 15. A copper fabricator knows it will require
100,000 pounds of copper on May 15 to meet a certain contract. The spot price
of copper is 340 cents per pound, and the futures price for May delivery is 320
cents per pound. The fabricator can hedge its position by taking a long position in
four futures contracts offered by the COMEX division of the CME Group and
closing its position on May 15.
• Each contract is for the delivery of 25,000 pounds of copper. The strategy has the
effect of locking in the price of the required copper at close to 320 cents per
pound.

Example of long hedge
• Suppose that the spot price of copper on May 15 proves to be 325 cents per pound.
Because May is the delivery month for the futures contract, this should be very close to
the futures price. The fabricator therefore gains approximately
100,000 x ($3.25 – $3.20) = $5,000
on the futures contracts. It pays 100,000 x $3:25 = $325,000 for the copper, making the
net cost approximately $325,000 – $5,000 = $320,000.
• For an alternative outcome, suppose that the spot price is 305 cents per pound on May
15. The fabricator then loses approximately
100,000 x ($3.20 – $3.05) = $15,000
on the futures contract and pays 100,000 x $3.05 = $305,000 for the copper. Again, the net
cost is approximately $320,000, or 320 cents per pound.

Advantages and disadvantages of hedging
(+) The objective of hedging with futures contracts is to reduce or eliminate the
price risk of an asset or a portfolio
(-) Despite the outcome being more certain with hedging, basis risk still exists
(-) Hedging can lead to less profitability if the asset being hedged ends up
increasing in value
(-) Shareholders can more easily hedge risk on their own by diversifying their
investments in terms of industry and/or geography
(-) If industry prices adjust to the changes, hedging could lose money and the
hedge is unnecessary.

Long Hedge for Purchase of an Asset
• Define
F1 : Futures price at time hedge is set up
F2 : Futures price at time asset is purchased
S2 : Asset price at time of purchase
b2 : Basis at time of purchase
25
Cost of asset S2
Gain on Futures F2 −F1
Net amount paid S2 − (F2 −F1) =F1 + b2

Short Hedge for Sale of an Asset
26
• Define
F1 : Futures price at time hedge is set up
F2 : Futures price at time asset is sold
S2 : Asset price at time of sale
b2 : Basis at time of sale
Price of asset S2
Gain on Futures F1 −F2
Net amount received S2 + (F1 −F2) =F1 + b2

Optimal Hedge Ratio
• We can account for an imperfect relationship between the spot and futures positions by calculating
an optimal hedge ratio that incorporates the degree of correlation between the rates.
• A hedge ratio is the ratio of the size of the futures position relative to the spot position. The optimal
hedge ratio, which minimizes the variance of the combined hedge position, is defined as follows:
where
sS is the standard deviation of DS, the change in the spot price during the hedging period,
sF is the standard deviation of DF, the change in the futures price during the hedging period
rSF is the coefficient of correlation between DS and DF.
• This is also the beta of spot prices with respect to futures contract prices:
27
F
S
F
S
HR


 ,
=

• EXAMPLE: Minimum Variance Hedge Ratio
Suppose a currency trader computed the correlation between the spot and futures
to be 0.925, the annual standard deviation of the spot price to be $0.10, and the
annual standard deviation of the futures price to be $0.125.
Compute the hedge ratio.
• Answer:
The ratio of the size of the futures to the spot should be 0.74.

Example
• Airline will purchase 2 million gallons of jet fuel in one month and
hedges using heating oil futures
• From historical data F =0.0313, S =0.0263, and = 0.928
29
7777
.
0
0313
.
0
0263
.
0
928
.
0 =

=
HR

• EXAMPLE: Hedging With Stock Index Futures
You are a portfolio manager with a $20 million growth portfolio that has a beta of
1.4, relative to the S&P 500. The S&P 500 futures are trading at 1,150, and the
multiplier is 250. You would like to hedge your exposure to market risk over the
next few months. Identify whether a long or short hedge is appropriate, and
determine the number of S&P 500 contracts you need to implement the hedge.
Answer:
You are long the S&P 500, so you should construct a short hedge and sell the
futures contract. The number of contracts to sell is equal to:
31

Adjusting the portfolio beta
• Hedging an existing equity portfolio with index futures is an attempt to reduce
the systematic risk of the portfolio → reduction of portfolio beta
• The appropriate number of contracts:
β be our portfolio beta, β* be our target beta after we implement the strategy with
index futures, P be our portfolio value, and A be the value of the underlying asset
(i.e., the stock index futures contract).
• Negative values indicate selling futures (decreasing systematic risk), and positive
values indicate buying futures contracts (increasing systematic risk).

EXAMPLE: Adjusting Portfolio Beta
Suppose we have a well-diversified $100 million equity portfolio. The portfolio beta relative
to the S&P 500 is 1.2. The current value of the 3-month S&P 500 Index is 1,080. The
portfolio manager wants to completely hedge the systematic risk of the portfolio over the
next three months using S&P 500 Index futures. Demonstrate how to adjust the portfolio’s
beta.
Answer:
In this instance, our target beta, β*, is 0, because a complete hedge is desired.
The negative sign tells us we need to sell 444 contracts.

Liquidity Issues
• In any hedging situation there is a danger that losses will be realized on
the hedge while the gains on the underlying exposure are unrealized
• This can create liquidity problems
• One example is Metallgesellschaft which sold long term fixed-price
contracts on heating oil and gasoline and hedged using stack and roll
• The price of oil fell.....
35

Pricing Financial Forwards and Futures
Reading 34

Consumption vs Investment Assets
• Investment assets are assets held by significant numbers of people
purely for investment purposes (Examples: gold, silver)
• Consumption assets are assets held primarily for consumption
(Examples: copper, oil)
37

Short Selling
• Short selling involves selling securities you do not own, and it is possible with
some investment assets.
• Your broker borrows the securities from another client and sells them in the
market in the usual way
• At some stage you must buy the securities so they can be replaced in the account
of the client
• You must pay dividends and other benefits the owner of the securities receives
• There may be a small fee for borrowing the securities
38

Short Selling
• In terms of motivations to sell securities short, the seller thinks the current price
is too high and that it will fall in the future, so the short seller hopes to sell high
and then buy low.
• Example: Ignoring all fees, if a short sale is made at $30 per share and the price
falls to $20 per share, the short seller can buy shares at $20 to replace the shares
borrowed and keep $10 per share as profit.

• EXAMPLE: Net Profit of a Short Sale of a Dividend-Paying Stock
Assume that a trader sold short XYZ stock in March by borrowing 200 shares and selling them for
$50/share. In April, XYZ stock paid a dividend of $2/share. Calculate the net profit from the short
sale assuming the trader bought back the shares in June for $40/share to replace the borrowed
shares and close out his short position.
Answer:
• The cash flows from the short sale on XYZ stock are as follows:
March: borrow 200 shares and sell them for $50/share +$10,000
April: short seller dividend payment to lender of $2/share −$400
June: buy back shares for $40/share to close short position −$8,000
Total net profit = +$1,600

Notation for Valuing Futures and Forward
Contracts
41
S: Spot price today
F: Futures or forward price today
T: Time until delivery date (in years)
r: annually compounded risk-free interest rate

Forward Price With No Income or Yield
• The right-hand side of Equation 1 is the cost of borrowing funds to buy the
underlying asset and carrying it forward to time T
• If F > S × (1 + r)T, arbitrageurs will profit by selling the forward and buying the
asset with borrowed funds.
• If F < S × (1 + r)T, arbitrageurs will profit by selling the asset, lending out the
proceeds, and buying the forward. Hence, the equality in Equation 1 must hold.

Forward Price With Income or Yield
• If the underlying pays a known amount of cash over the life of the forward
contract. Because the owner of the forward contract does not receive any of the
cash flows from the underlying asset between contract origination and delivery,
the present value of those cash flows must be deducted from the spot price when
calculating the forward price.

Example
• Forward Price When Underlying Asset Has a Cash Flow
Compute the price of a six-month forward on a coupon bond worth $1,000 that
pays a 5% coupon semiannually. A coupon is to be paid in three months. Assume
the annual risk-free rate is 4%.
Answer:
The income in this case is computed as:
I = 25 / 1.0250.25 = $24.84615
Using Equation 2:
F = ($1,000 − $24.84615) × (1 + 0.04)0.5 = $994.47

The Effect of a Known Dividend
• When the underlying asset for a forward contract pays a dividend, we assume that the
dividend is paid annually.
• Letting q represent the annually compounded dividend yield paid by the underlying
asset, Equation 1 becomes:
• EXAMPLE: Forward Price When the Underlying Asset Pays a Dividend
Compute the price of a six-month forward contract for which the underlying asset is a
stock index with a value of 1,000 and a continuous dividend yield of 1%. Assume the risk-
free rate is 4%.
Answer: F = 1,000 × (1.04 / 1.01)0.5 = 1,014.74

Value of a Forward Contract
• Because the forward price at every moment in time is computed to prevent
arbitrage, the value at inception of the contract must be zero. The forward
contract can take on a non-zero value only after the contract is entered into and
the obligation to buy or sell has been made
• If we denote the obligated delivery price after inception as K, then the value of
the long contract on an asset with no cash flows is computed as:
S − [K/(1 + r)T]
with cash flows (with present value I):
S − I − [K/(1 + r)T];
and with an annual dividend yield of q:
[S/(1 + q)T] − [K/(1 + r)T]

Example
• Value of a Stock Index Forward Contract
Using the stock index forward in the previous example, compute the value of a long
position if the index increases to 1,050 immediately after the contract is purchased.
Answer:
In this case, K = 1,014.74 and S = 1,050, so the value is:
(1,050 / 1.010.5) − (1,014.74 / 1.040.5) = 1,044.79 − 995.03 = 49.76

Forward Prices vs. Futures Prices
• The daily marking to market requirement on futures contracts and the unpredictable
changes in interest rates lead to price differences between futures and forwards.
• Assume asset prices are positively correlated to interest rates. A gain from an asset price
increase will be recognized immediately (due to daily settlement) and can be reinvested
at a high rate of interest. That makes a long futures contract a bit more desirable than a
long forward contract, so the former will be priced slightly higher.
• The opposite would hold true if asset prices are negatively correlated to interest rates—
the forward would be priced slightly higher in that case.
• Overall, the price differences are usually very small and can often be ignored → Use
Equation 1-3 for valuing both
• A futures contract may recognize an immediate profit but the forward contract would
only be able to recognize the present value of that profit.

Currency Futures
• Interest rate parity (IRP) states that the forward exchange rate, F [using the quote
format of XXXYYY (e.g., EURUSD)], must be related to the spot exchange rate, S,
and to the interest rate differential between the domestic (currency YYY) and the
foreign (currency XXX) country, rYYY − rXXX.

Example
• Calculate Forward Foreign Exchange Rate
Suppose we wish to compute the forward foreign exchange rate of a 10-month
futures contract on the Mexican peso. Each contract controls 500,000 pesos and is
quoted in terms of MXNUSD. Assume that the annually compounded risk-free rate
in Mexico is 14%, the annually compounded annual risk-free rate in the United
States is 2%, and the current exchange rate is MXNUSD 0.12.
Answer:
Applying Equation 4:
F = 0.12 × (1.02 / 1.14)10/12 = $0.10938 /peso

Stock Index Futures
• Stock index futures are valued similarly to forward contracts that pay dividends.
• If the average dividend yield for the contract term, q, is annually compounded, the
futures price of the stock index will be computed using Equation 3.
• With stock index futures, arbitrage opportunities will be present if:
F > S × [(1 + r) / (1 + q)]T
or F < S × [(1 + r) / (1 + q)]T
• If the futures price is greater than the theoretical value, an arbitrage profit is generated
by shorting the futures contracts and going long stocks underlying the index at the spot
price. (typically performed by pension funds that hold a portfolio of index stocks)
• If the futures price is lower than the theoretical value, an arbitrage profit is generated by
shorting stocks underlying the index and going long the futures contracts. (typically
performed by corporations or banks that hold shorter-term investments)

Options
Reading 36
Options Markets
1

Option Types
• A call is an option to buy
• A put is an option to sell
• A European option can be exercised only at the end of its life
• An American option can be exercised at any time
2

Option Positions
• Long call
• Long put
• Short call
• Short put
3

Standard symbols
X = strike price or exercise price specified in the option contract (a fixed
value)
St = price of the underlying asset at time t
Ct = the market value of a call option at time t
Pt = the market value of a put option at time t
t = the time subscript, which can take any value between 0 and T,
where T is the maturity or expiration date of the option

• Option contracts have asymmetric payoffs.
• The buyer of an option has the right to exercise the option but is not obligated to
exercise.
• The maximum loss for the buyer of an option contract is the loss of the price
(premium) paid to acquire the position, while the potential gains in some cases
are theoretically infinite.
• Because option contracts are a zero-sum game, the seller of the option contract
could incur substantial losses, but the maximum potential gain is the amount of
the premium received for writing the option.

Long Call
Profit from buying one European call option: option
price = $5, strike price = $100, option life = 2 months
9
30
20
10
0
-5
70 80 90 100
110 120 130
Profit ($)
Terminal
stock price ($)
A-X =C0 => A=105

Short Call
Profit from writing one European call option: option
price = $5, strike price = $100
10
-30
-20
-10
0
5
70 80 90 100
110 120 130
Profit ($)
Terminal
stock price ($)

Put Option Profit
• The price paid for the put option, P0, is referred to as the put premium. Thus, the
profit = PT − P0
where:
• P0 = put premium
• The profit to the option seller is:
profit = P0 − PT

Short Put
Profit from writing a European put option: option price = $7,
strike price = $70
15
-30
-20
-10
7
0
70
60
50
40
80 90 100
Profit ($)
Terminal
stock price ($)

Payoffs from Options
What is the Option Position in Each Case?
K = Strike price, ST = Price of asset at maturity
16
Payoff Payoff
ST ST
K
K
Payoff
Payoff
ST ST
K
K

Underlying Assets
Exchange-traded options trade on assets, including individual stocks, stock indices, and
exchange-traded funds (ETFs).
Stock options:
• Stock options are typically exchange-traded, American-style options.
• Each option contract is normally for 100 shares of stock. For example, if the last
trade on a call option occurred at $3.60, the option contract would cost $360.
• After issuance, stock option contracts are adjusted for stock splits but not cash
dividends.
• Primary U.S. exchanges: Chicago Board Options Exchange (CBOE), Boston Options
Exchange, NYSE Euronext, and the International Securities Exchange.
17

Convergence of Futures to Spot
5
EXAMPLE: Why the Futures Price Must Equal the Spot Price at Expiration
Suppose the current spot price of silver is $4.65. Demonstrate by arbitrage that the futures price of a
futures silver contract that expires in one minute must equal the spot price.
Answer:
• Suppose the futures price was $4.70. We could buy the silver at the spot price of $4.65, sell the
futures contract, and deliver the silver under the contract at $4.70. Our profit would be $4.70 −
$4.65 = $0.05. Because the contract matures in one minute, there is virtually no risk to this
arbitrage trade.
• Suppose instead the futures price was $4.61. Now we would buy the silver contract, take delivery
of the silver by paying $4.61, and then sell the silver at the spot price of $4.65. Our profit is $4.65
− $4.61 = $0.04. Once again, this is a riskless arbitrage trade.
• Therefore, to prevent arbitrage, the futures price at the maturity of the contract must be equal to
the spot price of $4.65.

Example
EXAMPLE: Index Options
Assume you own a call option on an index with an exercise price equal to 950. The
multiplier for this contract is 100. Compute the payoff on this option assuming that
the index is 956 at expiration.
Answer:
Payoff on an index call (long) = (Index at expiration – Exercise price) x Contract
multiplier.
= (956 − 950) × 100 = $600.
ST-X

Option Specification and Trading
Option Expiration
• On the CBOE, an option will be included in one of three maturity cycles:
• January cycle: January, April, July, October
• February cycle: February, May, August, November
• March cycle: March, June, September, December
• The actual day of expiration is the 3rd Friday of the expiration month.
• Short-term options (weeklys) are available. Long-term equity anticipation securities (LEAPS®)
are simply long-dated options with expirations greater than one year and up to three years.
Strike Prices
• Strike prices are dictated by the value of the stock. Low-value stocks have smaller strike
increments than higher-value stocks.
• All options of the same type (e.g., puts, calls) are called a class, and all options in a class with a
given expiration and strike price are called an option series (e.g., put options on Intel maturing
in September 2019).
20

The Effect of Dividends and Stock Splits
• In general, options are not adjusted for cash dividends. This will have option
pricing consequences that will need to be incorporated into a valuation model.
• Options are adjusted for stock splits.
• if a stock experiences a b-for-a split, the strike price becomes (a/b) of its
previous value and the number of shares underlying the option is increased
by multiples of (b/a).
• Stock dividends are dealt with in the same manner.
• Example: Consider a call option to buy 100 shares for $20/share. How
should terms be adjusted:
• for a 2-for-1 stock split?
• for a 25% stock dividend?
21
1,25:1 => 100x1,25 and 20/1,25

Trading
• Options are quoted relative to one underlying stock.
• To compute the actual option cost, the quote needs to be multiplied by 100. This is because an
options contract represents an option on 100 shares of the underlying stock.
• The quotes will also include the strike, expiration month, volume, and the option class.
• Market makers will quote bid and offer (or ask) prices whenever necessary. They profit on the
bid-offer spread and add liquidity to the market.
• Floor brokers represent a particular firm and execute trades for the general public. The order
book official enters limit orders relayed from the floor broker.
• An offsetting trade takes place when a long (short) option position is offset with a sale
(purchase) of the same option, which is often done when a trader is trying to exit a position. If a
trade is not an offsetting trade, then open interest increases by one contract.
22

Trading
• The number of options a trader can have on one stock is limited by the exchange → position
limit.
• Short calls and long puts are considered to be part of the same position.
• The exercise limit equals the position limit and specifies the maximum number of option
contracts that can be exercised by an individual over any five consecutive business days.
• Traders are subject to position limits and exercise limits to discourage them from potentially
manipulating the market.
Commissions
• Option investors must consider the commission costs associated with their trading activity.
• Commission costs often vary based on trade size and broker type (discount vs. full service).
• Brokers typically structure commission rates as a fixed amount plus a percentage of the trade
amount.
23

Other Option-like Securities
Employee Stock Options
• Issued as an incentive to company employees and provide a benefit if the stock
price rises above the exercise price.
• A vesting period often applies before the options may be exercised, so the
employee generally must still be employed by the company to receive the options
or else the options are forfeited.
• Employee stock options are not transferrable to a third party.
26

Herging strategies using
futures
Reading 32

Short Hedge
• A short hedge occurs when the hedger shorts (sells) a futures contract to
hedge against a price decrease in the existing long position.
• When the price of the hedged asset decreases, the short futures position realizes a
positive return, offsetting the decline in asset value
• A short hedge is appropriate when you own an asset and expect to sell it in the future, or
you do not currently own the asset but will purchase it in the future, and expect prices to
decline
• Example:
• A short hedge could be used by a farmer who owns some hogs and knows that they will
be ready for sale at the local market in two months.
• A US exporter who knows that he or she will receive euros in 3 months. The exporter will
realize a gain if the euro increases in value relative to the US dollar and will sustain a loss
if the euro decreases in value relative to the US dollar. A short futures position leads to a
loss if the euro increases in value and a gain if it decreases in value. It has the effect of
offsetting the exporter’s risk
17

Options Pricing Factors
The following six factors will impact the value of an option:
1. S0 = current stock price
2. X = strike price of the option
3. T = time to expiration of the option
4. r = short-term risk-free interest rate over T
5. D = present value of the dividend of the underlying stock
6. σ = expected volatility of stock prices over T
When evaluating a change in any one of the factors, hold the other factors
constant.
S rises => call rises
X rises => call falls (out of money cang co kha
nang xay rs
T rises => EU: unchaged => US: c rises
r rises => ST rises/X falls => call rises
st falls
high payoff => rises value

Effect of Variables on Option Pricing
31

American vs European Options
33
An American option is worth at least as much as the corresponding
European option
C  c
P  p
C/P: Value of American Call/Put
c/p: Value of European Call/Put

Upper Pricing Bounds
• A call option gives the right to purchase one share of stock at a certain price. No matter
what happens, the option can never be worth more than the stock. If it were, everyone
would sell the option and buy the stock and realize an arbitrage profit.
c ≤ S0 and C ≤ S0
• Similarly, a put option gives the right to sell one share of stock at a certain price. No
matter what happens, the put can never be worth more than the sale or strike price. If it
were, everyone would sell the option and invest the proceeds at the risk-free rate over
the life of the option.
p ≤ X and P ≤ X
• For a European put option, we can further reduce the upper bound. Because it cannot be
exercised early, it can never be worth more than the present value of the strike price:
p ≤ PV(X)

Lower Pricing Bounds for European Calls on
Non-Dividend-Paying Stocks
Consider the following two portfolios:
• Portfolio P1: one European call, c, with exercise price X plus a zero-coupon risk-free bond that
pays X at T.
• Portfolio P2: one share of the underlying stock, S.
At expiration, T:
• Portfolio P1 value = max(X,ST) = max (0,S-X) + X
0 + X = X
S-X + X = S
• Portfolio P2 value = ST
→ P1 ≥ P2 at expiration T. It always has to be true because if it were not, arbitrage would be
possible. Therefore,
c + PV(X) ≥ S0
• Because the value of a call option cannot be negative (if the option expires out-of-the-money, its
value will be zero), the lower bound for a European call on a non-dividend-paying stock is:
c ≥ max(S0 − PV(X), 0)
P2=st
X>st => value =X => p1>p2
st>X => value = st => p1=p2
=> p1>=p2 (always)
c>= S0- pv(X)

Lower Pricing Bounds for European Puts on
Non-Dividend-Paying Stocks
Consider the following two portfolios:
• Portfolio P3: one European put, p, plus one share of the underlying stock, S.
• Portfolio P4: zero-coupon risk-free bond that pays X at T.
At expiration, T:
• Portfolio P3 value = max(X, ST)
• Portfolio P4 value = X
→ P3 ≥ P4 at expiration T. It always has to be true because if it were not, arbitrage would be
possible. Therefore,
p + S0 ≥ PV(X)
• Because the value of a put option cannot be negative (if the option expires out-of-the-money, its
value will be zero), the lower bound for a European put on a non-dividend-paying stock is:
p ≥ max(PV(X) − S0, 0)
PV(X) = X/(1+r)^T
P3: put: max (x-st,0)
stock: St
X>=st => P3 = x-st+st
X<st => p3=0+st (ko exercise)
=> p3 = max(x,st)
p4=x
x>st=> p3=x=p4
x<st=>p3=st>p4

Computing Options Values Using Put-Call Parity
• Equivalencies for each of the individual securities in the put-call parity
relationship can be expressed as:
S = c − p + PV(X)
p = c − S + PV(X)
c = S + p − PV(X)
PV(X) = S + p − c
• The single securities on the left-hand side of the equations all have the same
payoffs as the portfolios on the right-hand side.
• For example, to synthetically produce the payoff for a long position in a share of
stock, you use the relationship: S = c − p + PV(X), meaning that the payoff on a
long stock can be synthetically created with a long call, a short put, and a long
position in a risk-free discount bond
c+pv(x) = s+p
s=c-p+pv(x)
c= s+p-pv(x)
p=c+pv(x)-s
pv(x)=s+p-c

Example
Call Option Valuation Using Put-Call Parity
Suppose that the current stock price is $52 and the risk-free rate is 5%. You have found a quote for a
three-month put option with an exercise price of $50. The put price is $1.50, but due to light trading
in the call options, there was not a listed quote for the three-month, $50 call. Estimate the price of
the three-month call option.
Answer:
Rearranging put-call parity, we find that the call price is:
call = put + stock − PV(X)
call = $1.50 + $52 − ($50 / 1.0125) = $4.12
This means that if a three-month, $50 call is available, it should be priced at $4.12 per share.
s=52
r=5%
x=50
p=1.5
c=s+p-pv(x) = 52+1.5-50/(1+5%/12)^3 = 4.12

Lower Pricing Bounds for an American Call Option
on a Non-Dividend-Paying Stock
• Recall that the lower pricing bounds for a European call option:
c ≥ max(S0 − PV(X), 0)
• Because the only difference between an American option and a European option is that the
American option can be exercised early, American options can always be used to replicate their
corresponding European options simply by choosing not to exercise them until expiration.
Therefore, it follows that:
C ≥ c ≥ max(S0 − PV(X), 0)
• Note that when an American call is exercised, it is only worth S0 − X. Because this value is never
larger than S0 − PV(X) for any r and T > 0, it is never optimal to exercise early. In other words, the
investor can keep the cash equal to X, which would be used to exercise the option early, and
invest that cash to earn interest until expiration. Because exercising the American call early means
that the investor would have to forgo this interest, it is never optimal to exercise an American call
on a non-dividend-paying stock before the expiration date (i.e., c = C).

Lower Pricing Bounds for an American Put Option
on a Non-Dividend-Paying Stock
• While it is never optimal to exercise an American call on a non-dividend-paying stock,
American puts are optimally exercised early if they are sufficiently in-the-money.
• If an option is sufficiently in-the-money, it can be exercised, and the payoff (X − S0) can be
invested to earn interest. In the extreme case when S0 is close to zero, the future value of
the exercised cash value, PV(X), is always worth more than a later exercise, X. We know
that:
P ≥ p ≥ max(PV(X) − S0, 0)
• However, we can place an even stronger bound on an American put because it can
always be exercised early:
P ≥ max(X − S0, 0)

Example
1. Minimum Prices for American vs. European Puts
Compute the lowest possible price for four-month American and European 65 puts on a stock that is
trading at 63 when the risk-free rate is 5%.
Answer:
• P ≥ max(0, X − S0) = max(0, 2) = $2
• p ≥ max(0, PV(X) − S0) = max(0, (65 / 1.0167) − 63) = $0.93
2. Minimum Prices for American vs. European Calls
Compute the lowest possible price for three-month American and European 65 calls on a stock that
is trading at 68 when the risk-free rate is 5%.
Answer:
• C ≥ max(0, S0 − PV(X)) = max(0, 68 − (65 / 1.0125)) = $3.80
• c ≥ max(0, S0 − PV(X)) = max(0, 68 − (65 / 1.0125)) = $3.80
PV = 65/(1+5%/12)^4

Relationship Between American Call Options and
Put Options
• Put-call parity only holds for European options. For American options,
we have an inequality. This inequality places upper and lower bounds
on the difference between the American call and put options.
S0 − X ≤ C − P ≤ S0 − PV(X)

EXAMPLE: American Put Option Bounds
Consider an American call and put option on stock XYZ. Both options have the same one-year
expiration and a strike price of $20. The stock is currently priced at $22, and the annual interest rate
is 6%. What are the upper and lower bounds on the American put option if the American call option
is priced at $4?
Answer:
The upper and lower bounds on the difference between the American call and American put
options are:
S0 − X ≤ C − P ≤ S0 − PV(X)
S0 − X = 22 − 20 = $2
S0 − PV(X) = 22 − (20 / 1.06) = 22 − 18.87 = $3.13
$2 ≤ C − P ≤ $3.13
or −$2 ≥ P − C ≥ −$3.13
Therefore, when the American call is valued at $4, the upper and lower bounds on the American put
option will be:
$2 ≥ P ≥ $0.87

Protective Puts
• A protective put (also called portfolio insurance or a hedged portfolio)
is constructed by holding a long position in the underlying security
and buying a put option
Protective put = Long the stock + long put
• You can use a protective put to limit the downside risk at the cost of
the put premium, P0.

Protective Puts
• X > S
Put profit:
X-S-Po
Stock profit:
X-S
= (X-S –Po)
+ (S-X) = - Po

Principal protected notes (PPNs)
• Principal protected notes (PPNs) are securities that are generated
from one option.
• Investors may participate in gains on a portfolio but do not suffer
from any losses.

Example
• EXAMPLE: Computing a Forward Price With No Interim Cash Flows
Suppose we have an asset currently priced at $1,000. The current
annually compounded risk-free rate is 4%. Compute the price of a six-
month forward contract on the asset.
Answer:
• F = $1,000 × 1.040.5 = $1,019.80

Bull and Bear Spreads
• profit = max(0, ST − XL) − max(0, ST − XH) − CLO + CHO

EXAMPLE: Bull Call Spread
An investor purchases a call for CL0 = $3.00 with a strike of XL = $40 and sells a call
for CH0 = $1.00 with a strike price XH of $50. Compute the profit of a bull call spread
strategy when the price of the stock is at $45.
Answer:
• profit = max(0, ST − XL) − max(0, ST − XH) − CLO + CHO
• profit = [max(0, ST − XL) − CLO ] + [CHO - max(0, ST − XH)]
• profit = max(0, 45 − 40) − max(0, 45 − 50) − 3 + 1 = $3.00

Bear Call Spreads
• A bear call spread is the sale of a bull spread. That is, the bear spread trader will
buy the call with the higher exercise price and sell the call with the lower exercise
price.
• This strategy is designed to profit from falling stock prices (i.e., a “bear” strategy).
• As stock prices fall, the investor keeps the premium from the written call, net of
the long call’s cost.
• The purpose of the long call is to protect from sharp increases in stock prices. The
payoff is the opposite (mirror image) of the bull call spread.

Bear Call Spreads
Profit = max(0, ST − XL) – CL0
+ CH0 – max(0, ST − XH)

Bear Put Spreads
• Puts can also be used to replicate the payoffs for both a bull call spread and
a bear call spread. In a bear put spread, the investor buys a put with a
higher exercise price and sells a put with a lower exercise price.
EXAMPLE: Bear Put Spread
An investor sells a put for PL0 = $3.00 with a strike of X = $20 and purchases a
put for PH0 = $4.50 with a strike price of $40. Compute the profit of a bear
put spread strategy when the price of the stock is at $35.
Answer:
• profit = max(0, XH − ST) − max(0, XL − ST) − PH0 + PL0
• profit = max(0, 40 − 35) − max(0, 20 − 35) − 4.50 + 3 = $3.50

Butterfly Spreads
• A butterfly spread involves the purchase or sale of three different call options.
• Here, the investor buys one European call with a low exercise price, buys another
European call with a high exercise price, and sells two European calls with an
exercise price in between (usually near the current stock price).
• The net cost of the butterfly spread is always positive because the payoff is
always zero or more; it will be zero for large moves in either direction.
• The buyer of a butterfly spread is essentially betting that the stock price will stay
near the exercise price of the written calls. However, the loss that the butterfly
spread buyer sustains if the stock price strays from this level is limited.

EXAMPLE: Butterfly Spread With Calls
An investor makes the following transactions in calls on a stock:
• Buys one call defined by CL0 = $7.00 and XL = $55.
• Buys one call defined by CH0 = $2.00 and XH = $65.
• Sells two calls defined by CM0 = $4.00 and XM = $60.
Compute the profit of a butterfly spread strategy with calls when the stock is at
$60.
• Answer:
profit = max(0, ST − XL) − 2max(0, ST − XM) + max(0, ST − XH) − CL0 + 2CM0 − CH0
profit = max(0, 60 − 55) − 2max(0, 60 − 60) + max(0, 60 − 65) − 7 + 2(4) − 2 = $4.00

Calendar Spreads
• A calendar spread is created by transacting in two options that have the same
strike price but different expirations.
• The strategy sells the short-dated option and buys the long-dated option.
• Notice that the payoff here is similar to the butterfly spread. The investor profits
slightly only if the stock remains in a narrow range (e.g., close to strike price), but
losses are limited to about the net option premium cost. In this case, the losses
are not symmetrical as they are in the butterfly spread.

Option Combination Strategies
Straddle
• A (long) straddle is created by purchasing a call and a put with the same strike
price (often near current stock price) and expiration.
• Given the need to pay for two option premiums, this strategy is only profitable
when the stock price moves significantly in either direction; it is a bet on volatility
but without certainty on the direction.
• Straddle payoffs are symmetric around the strike price.

Strips and Straps
• A strip involves purchasing two puts and one call with the same strike price and expiration, so it is
similar to a straddle.
• A strip is betting on volatility but is more bearish because it pays off more on the downside.
• A strap involves purchasing two calls and one put with the same strike price and expiration, so
again, it is similar to a straddle. A strap is betting on volatility but is more bullish since it pays off
more on the upside.

Plain vanilla interest rate swap
• In this swap arrangement, Company X agrees to pay Company Y a
periodic fixed rate on a notional principal over the tenor of the swap.
In return, Company Y agrees to pay Company X a periodic floating rate
on the same notional principal. Therefore, only the net payment is
exchanged.
• Use the London Interbank Offered Rate (LIBOR) as the reference rate
for the floating leg of the swap.
• Because the payments are based in the same currency, there is no
need for the exchange of principal at the inception of the swap. The
notional principal is used only to determine the respective interest
rates.

Example
• Companies X and Y enter into a two-year plain vanilla interest rate swap. The
swap cash flows are exchanged semiannually, and the reference rate is six-month
LIBOR. The fixed rate of the swap is 3.784%, and the notional principal is $100
million. We will compute the cash flows for Company X, the fixed payer of this
swap.

Example
• The gross cash flows for the end of the first period for both parties are
calculated in the following manner:
floating = $100 million × 0.03 × 0.5 = $1.5 million
fixed = $100 million × 0.03784 × 0.5 = $1.892 million

Plain vanilla interest rate swap
• Suppose that X has a two-year floating-rate liability, and Y has a two-
year fixed-rate liability. After they enter into the swap, interest rate
risk exposure from their liabilities has completely changed for each
party. X has transformed the floating-rate liability into a fixed-rate
liability, and Y has transformed the fixed-rate liability to a floating-rate
liability.

Another example
• Example: Two firms with different credit ratings, Hi and Lo:
• Hi can borrow fixed at 11% and floating at LIBOR + 1%.
• Lo can borrow fixed at 11.4% and floating at LIBOR + 1.5%.

Hi wants fixed rate, but it will issue floating and “swap” with Lo. Lo wants floating rate,
but it will issue fixed and swap with Hi. Lo also makes “side payment” of 0.45% to Hi.
Hi Lo
CF to lender −(LIBOR + 1%) −11.40%
CF Hi to Lo −11.40% +11.40%
CF Lo to Hi +(LIBOR + 1%) −(LIBOR + 1%)
CF Lo to Hi +0.45% −0.45%
Net CF −10.95% −(LIBOR + 1.45%)

Currency swap
• Currency swap exchanges both principal and interest rate payments with
payments in different currencies. The exchange rate used in currency swaps is the
spot exchange rate.
• Example: Two companies, A and B, enter into a fixed-for-fixed currency swap with
periodic payments annually.
Company A pays 6% in Great Britain pounds (GBP) to Company B and receives 5% in
U.S. dollars (USD) from Company B.
Company A pays a principal amount to B of USD 175 million, and B pays GBP 100
million to A at the outset of the swap.
Notice that A has effectively borrowed GBP from B, and so it must pay interest on
that loan. Similarly, B has borrowed USD from A.
• Every period (12 months), A will pay GBP 6 million to B, and B will pay USD 8.75
million to A. At the end of the swap, the principal amounts are re-exchanged.

Measuring Financial Risks
Reading 45

Mean-variance framework
• The traditional mean-variance model estimates the amount of
financial risk for portfolios in terms of the portfolio’s expected return
(i.e., mean) and risk (i.e., standard deviation or variance).
• Under the mean-variance framework, it is necessary to assume that
return distributions for portfolios are elliptical distributions, such as
normal distribution.

Value at Risk
• Value at risk (VaR)
• The worst possible loss under normal conditions over a specified
period.
• An estimate of the maximum loss that can occur with a given
confidence level.
• “For a given month, the VaR is $1 million at a 95% level of
confidence” → Under normal conditions, in 95% of the months (19
out of 20 months), we expect the fund to either earn a profit or lose
no more than $1 million.
• Delta-normal VaR = [μ – zσ] × asset value.

What is the monthly VaR for this security at a confidence level
of 95% ?
The value associated with a 95%
confidence level is a return of
−15.5%.
If you have $1,000,000 invested
in this security, the one-month
VaR is $155,000 (−15.5% ×
$1,000,000).

Example
EXAMPLE: Calculating value at risk
For a $100,000,000 portfolio, the expected 1-week portfolio return and
standard deviation are 0.00188 and 0.0125, respectively. Calculate the 1-
week VaR with a 95% confidence level.
Answer:
VaR = (μ − zσ)  × portfolio value
= [ 0.00188 − 1.65( 0.0125) ]  × $100,000,000
= − 0.018745 × $100,000,000
=   − $1,874,500
The manager can be 95% confident that the maximum 1-week loss will not
exceed $1,874,500.

Value at Risk - Limitations
• A major limitation of the VaR measure for risk is that two arbitrary parameters
are used in the calculation:
• Confidence level: the likelihood or probability that we will obtain a value
greater than or equal to VaR. VaR increases (at an increasing rate) when the
confidence level increases.
• Holding period: can be any predetermined time period measured in days,
weeks, months, or years. VaR will increase with increases in the holding
period. The rate at which VaR increases is determined in part by the mean of
the distribution

Value at Risk - Limitations
• VaR estimates are subjected to both model risk (risk of errors resulting from
incorrect assumptions used in the model) and implementation risk (risk of errors
resulting from the implementation of the model).
• VaR does not tell the investor the amount or magnitude of the actual loss. VaR
only provides the maximum value we can lose for a given confidence level. Two
different return distributions may have the same VaR, but very different risk
exposures.
• VaR measurements work well with elliptical return distributions, such as the
normal distribution. VaR is also able to calculate the risk for nonnormal
distributions; however, VaR estimates may be unreliable for nonnormal
distributions.
• VaR also violates the coherent risk measure property of subadditivity when the
return distribution is not elliptical

Call Option Profit
• The price paid for the call option, C0, is referred to as the call premium. Thus, the
profit = CT − C0
where:
• C0 = call premium
• Conversely, the profit to the option seller is:
profit = C0 − CT
if ST < X + C0 then: call buyer profit < 0 < call seller profit
if ST = X + C0 then: call buyer profit = 0 = call seller profit
if ST > X + C0 then: call buyer profit > 0 > call seller profit

Expected Shortfall
• Value at risk is the minimum percent loss, equal to a pre-specified worst case
quantile return (typically the 5th percentile return).
• Expected shortfall (ES) is the expected loss given that the portfolio return
already lies below the pre-specified worst case quantile return (i.e., below the
5th percentile return).
• Example: The 5th percentile return for the fund equals –20%. Therefore, 5% of the
time, the fund earns a return less than –20%. The value at risk is –20%. However,
VaR does not provide good information regarding the expected size of the loss if
the fund performs in the lower 5% of the possible outcomes. That question is
answered by the expected shortfall amount, which is the expected value of all
returns falling below the fifth percentile return (i.e., below –20%). Therefore,
expected shortfall will equal a larger loss than the VaR.

Expected Shortfall
• For a normal distribution with a mean equal to μ and a standard
deviation equal to σ, the expected shortfall will be:
In which, x is the confidence level and z equals the point in the
distribution that has a probability of being exceeded of x%.

Example
EXAMPLE: Calculating expected shortfall
For a $100,000,000 portfolio, the expected 1-week portfolio return and standard deviation
are 0.00188 and 0.0125, respectively. Calculate the 1-week expected shortfall with a 95%
confidence level.
• Note that this amount is larger than the VaR level calculated earlier of $1,874,500.

Expected shortfall vs. VaR
• ES satisfies all of the properties of coherent risk measurements including
subadditivity. VaR only satisfies these properties for normal distributions. For VaR,
the combined VaR may exceed the summation of the individual assets’ VaRs, thus
not always satisfying the subadditivity.
• ES provides an estimate of how large of a loss is expected if an unfavorable event
occurs. VaR does not provide any estimate of the magnitude of losses, only the
probability that they might occur.
• When adjusting both the holding period and confidence level at the same time,
an ES surface curve showing the interactions of both adjustments is convex. Thus,
the ES method is more appropriate than the VaR method in solving portfolio
optimization problems.
• ES has less restrictive assumptions regarding risk/return decision rules.

Calculating and Applying VaR
Reading 46

Example
EXAMPLE: Futures contract VaR
Determine how a risk manager could estimate the VaR of an equity index futures
contract. Assume a one-point increase in the index increases the value of a long
position in the contract by $500.
Answer:
This relationship is shown mathematically as: Ft = $500St , where Ft is the futures
contract and St is the underlying index. The VaR of the futures contract is calculated
as the amount of the index point movement in the underlying index, St , times the
multiple, $500 as follows:
VaR(Ft) = $500VaR(St).

Long Put
Profit from buying a European put option: option price
= $7, strike price = $70
14
30
20
10
0
-7
70
60
50
40 80 90 100
Profit ($)
Terminal
stock price ($)

Historical simulation approach
• The advantage of this approach is that it may identify a crisis event
that was previously overlooked for a specific asset class. The focus is
on identifying extreme changes in valuation.
• The disadvantage of the historical simulation approach is that it is
limited to actual historical data.

Delta-Normal Approach
• The delta-normal method explicitly assumes a distribution for the
underlying observations.
• The delta-normal method can be used for portfolios that are linearly
dependent on the underlying market variables.
• Assuming the returns on these variables is multivariate normal, then
portfolio value changes will be normally distributed. This makes VaR
and ES calculations more intuitive.

Underlying Assets
Index options
• Options on stock indices are typically European-style options and are cash settled.
Index options can be found on both the over-the-counter (OTC) markets and the
exchange-traded markets.
• The payoff on an index call is the amount (if any) by which the index level at
expiration exceeds the index level specified in the option (the strike price),
multiplied by the contract multiplier (typically 100).
ETF options
• While similar to index options, ETF options are typically American-style options and
utilize delivery of shares rather than cash at settlement.
18

Delta-Normal Approach
• The portfolio risk (i.e., variance) can be rewritten as a variation of the well-known
portfolio risk formula:
• Delta-Normal Approach

Delta-Normal Approach - Limitations
• This method works well for linear portfolios, but only an approximation for
nonlinear products (e.g., call option) or portfolios.
• Delta works well for small price changes but does not work well for large price
changes. This is because delta is a linear measure.
• The gamma parameter helps adjust for the delta’s linearity through curvature
(i.e., nonlinearity). Therefore, delta works much better with in-the-money
options than at-the-money options, because the gamma of in-the-money options
is much higher.

Delta-Normal Approach - Limitations
• In addition to call options, nonlinear products include Asian options, barrier
options, mortgage-backed securities (MBS), butterfly spreads
• Contrary to linear products, a normal distribution of the underlying asset price
translates into a skewed (nonnormal) distribution for nonlinear products.
• Therefore, the delta-normal method (which translates a normal distribution for
the asset price into a normal distribution of the derivative) would understate the
probability of high option values and would overstate the probability of low
option values.

Full Revaluation Approach
• The full revaluation approach calculates the VaR of the derivative by valuing the
derivative based on the underlying value of the index after the decline
corresponding to an x% VaR of the index.
• This approach is accurate, but can be highly computational. The revaluation of
portfolios that include more complex products (i.e., mortgage-backed securities,
or options with embedded features) are not easily calculated due to the large
number of possible scenarios.

Monte Carlo, Stress Testing, and Scenario Analysis
Useful methods in extending VaR techniques to more appropriately
measure risk for complex derivatives and scenarios
• Structured Monte Carlo
• Stress testing
• Worst-case scenario (WSC) analysis

The Monte Carlo Approach
• The Monte Carlo approach generates scenarios using random
samples and simulates thousands of valuation outcomes for the
underlying assets based on the assumption of normality.
• The VaR and ES for the portfolio of derivatives is then calculated from
the simulated outcomes.

• The Monte Carlo approach involves six steps:
1. Using current values of risk factors, value the portfolio today.
2. Apply sampling techniques from the multivariate normal probability
distribution for the change in x (Δxi).
3. Using the sampled values of Δxi, determine the values of the risk factors at
the end of the period.
4. Revalue the portfolio using the updated risk factor values.
5. Subtract the revalued portfolio value from the current value. This will
determine the amount of loss.
6. Repeat steps two–five to create a loss distribution.

• Once this process is complete, we can calculate daily VaR and expected loss, using
a similar approach as historical simulation.
• For example, if Monte Carlo produces 500 trials, the daily VaR with a 99%
confidence level will be the fifth worst loss (= 1% × 500), and the expected loss
will be the average of the four worst losses.
• Remember, to calculate longer time period VaR and expected losses, the daily
values will be multiplied by the square root of time:

Advantage:
• Able to address multiple risk factors by assuming an underlying distribution and
modeling the correlations among the risk factors.
For example, a risk manager can simulate 10,000 outcomes and then determine the
probability of a specific event occurring. In order to run the simulations, the risk
manager just needs to provide parameters for the mean and standard deviation
and assume all possible outcomes are normally distributed.
• Monte Carlo simulation can assume any distribution type as long as correlations
between the risk factors can be determined.
Disadvantage:
• The process is slow and computationally intensive. The Monte Carlo approach is
typically used for large portfolios which is time consuming.

Worst-Case Scenario Analysis
• The worst-case scenario (WCS) focuses on the distribution of worst possible
outcomes given an unfavorable event.
• An expected loss is then determined from this worst-case distribution analysis.
• Thus, the WCS information extends the VaR analysis by estimating the extent of
the loss given an unfavorable event occurs.
• For example, an investor may be concerned about the worst possible daily
outcome over a six-month period, and would look at the distribution of returns
during the six-month period.
• While useful, WCS should not be viewed as an alternative to VaR and ES
calculations.

Market risk – Measuring and
monitoring volatility
Reading 47
Lecturer: Dung Tran

Warrants
• Warrants are often issued by a company to make a bond issue more attractive
(e.g., equity upside) and will typically trade separately from the bond at some
point.
• Warrants are like call options except that, upon exercise, the company may issue
new shares and the warrant holders can purchase the shares at the exercise
price.
25

Implied Volatility
• Estimating future volatility using historical data requires time to adjust to current
changes in the market
• An alternative method for estimating future volatility is implied volatility.
Whereas volatilities calculated from historical data (including EWMA or GARCH)
are backward looking, implied volatility calculated from options prices is forward
looking.
• Option prices are dependent on volatilities, and as volatilities increase so do
options prices. As a result, volatilities are implied from options prices.
• Usually expressed as annual volatilities. If annual volatility is 18%, the daily
volatility would be 18% / 252 = 1.13% (assuming 252 trading days in a year).

The Chicago Board Options Exchange (CBOE)
Volatility Index (VIX)
• VIX is the most widely used index for publishing implied volatility
• The VIX demonstrates implied volatility on a wide variety of 30-day calls and puts
on the S&P 500 Index.
• Note that trading in futures and options on the VIX is a bet on volatility only.
Since its inception, the VIX has mainly traded between 10 and 20 (which
corresponds to volatility of 10%–20% on the S&P 500 Index options), but it
reached a peak of close to 80 in October 2008, after the collapse of Lehman
Brothers.
• The VIX is often referred to as the fear index by market participants because it
reflects current market uncertainties.
https://www.cboe.com/tradable_products/vix/

Convertible Bonds
• Contain a provision that gives the bondholder the option of exchanging the bond
for a prespecified number of shares of the company’s common stock.
• At exercise, the newly issued shares increase the number of shares outstanding
and debt is retired based on the amount of bonds exchanged for the shares.
27

Moneyness
• "In the money" (ITM) is an expression that refers to an option that possesses
intrinsic value. ITM thus indicates that an option has value in a strike price that is
favorable in comparison to the prevailing market price of the underlying asset:
• An in-the-money call option means the option holder has the opportunity to buy the security
below its current market price.
• An in-the-money put option means the option holder can sell the security above its current
market price.
• At the money (ATM) is a situation where an option's strike price is identical to the
current market price of the underlying security.
• “Out of the money" (OTM) is an expression that refers to an option that
possesses extrinsic value.
call: St>X; put: X>ST
call: ST<X; put: X<ST
call: ST=X
put: ST=X

Exponentially weighted moving average (EWMA)
• If the goal is to estimate the current level of volatility, we may want to weight
recent data more heavily.

• The estimated volatility on day n is derived by applying the weights to past squared
returns, where more recent days receive a heavier weighting than older days.
• The EWMA model employs decay factor λ, where 0 < λ < 1 that is used to weight
each day’ s percentage price change
• Conditional variance is estimated as:
w0 = 1- λ

• This model is simplified to just two periods of data (n–1 and n–2). In addition, we
can substitute w0 = (1 – λ) because the two weights must sum to one.
• EWMA example: see file EWMA.xls

• EXAMPLE: EWMA model
The decay factor in an exponentially weighted moving average model is estimated
to be 0.94 for daily data. Daily volatility is estimated to be 1%, and today’s stock
market return is 2%. Calculate the new estimate of volatility using the EWMA
model.

• The EWMA model was used by RiskMetrics, formerly a division of JPMorgan. The
RiskMetrics approach is just an EWMA model that uses λ = 0.94 for daily data and
λ = 0.94 for monthly data.
• Simplest interpretation of the EWMA model: the day-n volatility estimate is
calculated as a function of the volatility calculated as of day n–1 and the most
recent squared return.
• High values of λ will minimize the effect of daily percentage returns, whereas low
values of λ will tend to increase the effect of daily percentage returns on the
current volatility estimate.
• EWMA requires few data points

(G)ARCH Motivation
• Autoregressive Conditional Heteroscedasticity (ARCH) models specifically model
the volatility of time series.
• Originally suggested by Engle (1982).
• ARCH-type models (including GARCH) seek to capture the following typical
features of financial time series:
• Volatility clustering.
• Higher proportion of outliers than the Normal distribution (e.g. heavy tails).
• Asymmetry in extreme random shocks (negative shocks are more sudden and
frequent than positive shocks).

The GARCH (1,1) Model
• In GARCH (1,1) (Generalized autoregressive conditional heteroscedasticity)
model, we assign some weight to the long-run average variance rate.
• (1,1) refers to the weight given to one squared return (the most recently
observed) and one variance rate (most recent estimate).

Summary of effects
• S: For call options, as the current price of the stock (S) increases (decreases), the value of the call
increases (decreases). For put options, as S increases (decreases), the value of the put decreases
(increases). As an option becomes closer to or more in-the-money, its value increases.
• X: The effect of strike prices (X) on option values will be exactly the opposite of the effect of S. For
call options, as X increases (decreases), the value of the call decreases (increases). For put
options, as X increases (decreases), the value of the put increases (decreases).
• T: For American-style options, increasing time to expiration will increase the option value. With
more time, the likelihood of being in-the-money increases. Can’t conclude with the European-
style options
• r: As the risk-free rate increases, the value of the call (put) will increase (decrease).
• D: The option owner does not have access to the cash flows of the underlying stock, and the stock
price decreases when a dividend is paid. Thus, as the dividend increases, the value of the call
(put) will decrease (increase).
• As volatility increases, option values increase, due to the asymmetric payoff of options. Because
long option positions have a maximum loss equal to the premium paid, increased volatility only
increases the chances that the option will expire in-the-money.

• EXAMPLE: GARCH (1,1) model
The parameters of a generalized autoregressive conditional heteroskedastic
[GARCH (1,1)] model are ω = 0.000003, α = 0.04, and β = 0.92. If daily volatility is
estimated to be 1%, and today’s stock market return is 2%, calculate the new
estimate of volatility using the GARCH (1,1) model, and the implied long-run
volatility level.

GARCH model
• GARCH models do a very good job at modeling volatility clustering when periods
of high volatility tend to be followed by other periods of high volatility, and
periods of low volatility tend to be followed by subsequent periods of low
volatility.
• If GARCH models do a good job of explaining volatility changes, there should be
very little autocorrelation in ri
2 / σi
2 . GARCH models appear to do a very good job
of explaining volatility.

Mean Reversion and Long Time Horizons
• However, empirical data indicates that volatility exhibits a mean-
reverting characteristic. This means that if current volatility is high,
we expect it to decline; if it is low, we expect it to increase.
• If we expect volatility to decline, we will overestimate volatility if we
multiply the standard deviation of the return by the square root time.
• The term VL in GARCH (1,1) provides a pull back toward the long-term
average mean. EWMA does not provide this pull.

Correlation
• We can establish a general covariance formula using the EWMA
model between return X and return Y:

EXAMPLE: Calculating correlation
Suppose an analyst is looking to estimate the updated correlation between two asset
returns. The analyst observes on day n–1 that return X is 2% and Y is 4%, and the
correlation between X and Y is 0.3. The volatility of return X and Y is 1% and 2%,
respectively. The analyst estimates a value for λ of 0.92. Calculate the new coefficient
of correlation.
Answer:
• The covariance on day n–1 can be calculated as:
covn–1 = 0.3 × 0.01 × 0.02 = 0.00006
• For day n, the covariance is updated as follows:
covn = λcovn–1 + (1 – λ)Xn–1Yn–1 = 0.92 × 0.00006 + 0.08 × 0.02 × 0.04 = 0.0001192
• Assuming the same λ of 0.92, the volatilities of X and Y are now updated to 2.2%
and 3%, respectively. We can now calculate the new coefficient of correlation:
corr x,y =
0.0001192
0.022 × 0.03
= 0.018

Credit risk
• The risk arises from any non-payment or
rescheduling of any promised payment or
from credit migrations
• Credit risks are not normally distributed and
tend to be highly skewed, because
maximum gains are limited to receiving
promised payments while extreme losses
are very rare events – beta distribution
• The tail of the credit loss distribution,
however, is more difficult to model. In
practice, fitting the tail often involves
combining the beta distribution with a
Monte Carlo simulation

Bank’s economic capital
• The best estimate of the devaluation of a risky asset is expected loss. However, the
unexpected loss can exceed the expected loss by a wide margin.
• Banks set aside credit reserves in preparation for expected losses. A bank’s own estimate
of capital is called economic capital.
• How much capital a bank needs to hold depends on a bank’s estimate of possible losses,
but it also depends on its capital structure, including its level of debt relative to equity.
• Regulatory capital is the capital that regulators require banks to keep.
• Basel I in 1996
• Basel II in 2004
• Basel III in 2009
• Basel II features two approaches for calculating credit risk capital: (1) the standardized
approach, and the (2) internal ratings-based (IRB) approach. The standardized approach
involves the use of credit ratings.

Credit risk factors
• Probability of default (PD) is the likelihood that a borrower will default
• Exposure, also referred to as exposure at default (EAD), is the loss exposure
stated as a dollar amount (e.g., the loan balance outstanding).
• EAD can also be stated as a percentage of the nominal amount of the loan or
the maximum amount available on a credit line.
• Loss rate, also referred to as loss given default (LGD), represents the likely
percentage loss if the borrower defaults - the severity of a default.
• Both PD and LGD are expressed as percentages.
• LGD = 1 – recovery rate (RR) → factors that affect the loss rate will also impact
the recovery rate.

Unexpected Loss
• The actual loss in the event of default on its assets may be higher or lower
than the expected loss.
• The difference between the actual loss and expected loss is called the
unexpected loss (UL).
• UL is the average total loss above the expected loss. It represents the
variation in expected loss.
• The amount of economic capital needed to absorb credit losses is the
distance between the unexpected (negative) outcome and the expected
outcome for a given confidence level

Example
• Suppose XYZ bank has booked a loan
with the following characteristics: total
commitment of $2,000,000 of which
$1,800,000 is currently outstanding. The
bank has assessed an internal credit
rating equivalent to a 1% default
probability over the next year. The bank
has additionally estimated a 40% loss
rate if the borrower defaults. The
standard deviation of PD and LR is 5%
and 30%, respectively. Calculate the
expected and unexpected loss for XYZ
bank.

Measuring Credit Losses and Modeling Credit Risk
• For a portfolio consisting of n loans
• Loss on loan default (on ith loan with a face value of L) = Li(1 – RRi)
• RRi is the recovery rate in the event of default on the ith loan.
• Binomial distribution of loan losses in the portfolio:
• probability of losses = PDi
• probability of no losses = 1 – PDi
• Mean loss:
• Standard deviation of loss from the ith loan:

Measuring Credit Losses and Modeling Credit Risk
• Portfolio’s standard deviation
• Assume all loans have the same principal L and the standard deviation
of the loss from loan i is the same for all i
• Standard deviation of the loss from the loan portfolio as a percentage
of its size:

Example
Computing standard deviation of loss
Suppose that a bank has a portfolio with 10,000 loans, and each loan is EUR 1
million and has a 0.5% PD in a year. Also assume that the recovery rate is 30% and
correlation between losses is 0.2. Calculate the standard deviation of the loss from
the loan portfolio and the standard deviation of the loss as a percentage of its size.

Portfolio expected and unexpected loss
• ELp, is the sum of the expected losses of each asset:
• Portfolio unexpected loss (ULp) is the variance formula for an N-asset portfolio
• Risk contribution (RC), also known as the unexpected loss contribution (ULC)

Example
• Step 1: Compute EL for both assets.
ELa = EA x PD x LR = $8,250,000 x 0.005 x 0.50 = $20,625
ELb = = $1,800,000 x 0.01 x 0.40 = $7,200
• Step 2: Compute UL for both assets.
ULa = $8,250,000 x sqrt(0.005 x 0.252 + 0.5^2 x 0.022) = $167,558
ULb = $ 1,800,000 x sqrt(0.01 x0.32 + 0.4^2 x0.052) = $64,900
• Step 3: Compute ELp: ELp = $20,625 + $7,200 = $27,825
• Step 4: Compute ULp:
ULp = sqrt[167,5582 + 64,9002 + 2x0.3x167,558x64,900] = $197,009
• Step 5: Compute RC for both assets
RCa = 167,558 x (167,558 + 0.3 x 64,900)/197,009 = 159,070
RCb = 64,900 x (64,900 + 0.3 x 167,558)/197,009 = 37,939
RCa + RCb = 197,009 = ULp

External rating
• Three primary rating agencies in the U.S [Moody’s, Standard and Poor’s (S&P),
and Fitch] serve as external sources of credit risk data, providing independent
opinions on credit risk.
• External credit ratings convey information about either a specific instrument,
called an issue-specific credit rating, or information about the entity that
issued the instrument, which is called an issuer credit rating, or both.
• The rating is established for the purpose of assessing how likely it is that an
entity will default on its obligations.
• Why rating
• Borrowers: assure access to capital with reasonable cost
• Investors: use credit rating to estimate potential risk and return
• Regulatory agencies: establish capital and margin requirements

External rating
• Process
• Qualitative and quantitative analysis
• Meeting with the firm’s management
• Meeting of the committee in the rating agency assigned to rating the firm
• Notification of the firm being rated of the assigned rating,
• Opportunity for the firm to appeal the rating
• An announcement of the rating to the public.
• Relationship with price: a ratings downgrade is likely to make the price
decrease, and an upgrade is likely to make the price increase

External rating
• The higher the credit rating, the lower the default frequency.
• A key dividing line for both Moody’s and S&P are the Baa3 and BBB– ratings,
respectively. Any instruments with ratings at or above this line are considered
investment grade, whereas any instruments with ratings below this line are
considered noninvestment grade (speculative grade or junk bonds)
• Investment grade ratings: firms have the capacity to meet their obligations and
adequate protection in place to withstand adverse economic conditions or other
changes in circumstances.
• Noninvestment grade ratings: considerable capacity and uncertainty issues.
• The probability of default increases as a function of time for investment grade
bonds, but not for lowest rated bonds.

Covered Call
• A covered call position is to sell a call option on a stock that is owned by the
option writer.
Covered call = Long the stock + Short call
• By writing an out-of-the-money call option, the combined position caps the
upside potential at the strike price.
• In return for giving up any potential gain beyond the strike price, the writer
receives the option premium.
• This strategy is used to generate cash on a stock that is not expected to increase
above the exercise price over the life of the option.

Covered Call
• ST < X
Long stock:
ST - X
Short call
Co
Sum = ST – X +
Co

Option Spread strategies
• These strategies combine options positions to create a desired payoff profile.
• The differences between the options are either the strike prices and/or the time
to expiration.
• Bull and bear spreads, butterfly spreads, calendar spreads, and diagonal spreads.

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