Intro to Value at Risk (VaR)

Value at Risk (VaR) – Part 1 (LOs 7.1 – 8.7 , 12.1 – 13.9, 15.1-15.5) Intro to VaR (Allen Chapter 1) 1 VaR Mapping 4 VaR Methods 5 Cash flow at Risk (CFaR) 2 Putting VaR to Work (Allen Chapter 3) 3 Stress Testing 6

Value at Risk (VaR) in the Readings We are reviewing here (Sec II) Was reviewed in Quant (Sec I) To be reviewed in Investments (Sec V) Learning Outcome Location in Study Guide Reading LO 7.1 to 7.6 II. Market 1.A. Intro to VaR Allen Ch. 1 LO 7.7 to 7.15 II. Market 1.B. Putting VaR to work Allen Ch. 3 LO 8.1 to 8.7 II. Market 6.A. Firm-wide Approach to Risk Stulz Ch. 4 LO 9.1 to 9.11 V. Investment 6.A. Portfolio Risk Jorion Ch. 7 LO 10.1 10.7 I. Quant 3.A. Forecasting Risk and Correlation Jorion Ch. 9 LO 11.1 to 11.10 I. Quant 1. Quantifying Volatility Allen Ch. 2

We are reviewing here (Sec II) Was reviewed in Quant (Sec I) To be reviewed in Investments (Sec V) Value at Risk (VaR) in the Readings Learning Outcome Location in Study Guide Reading LO 12.1 to 12.6 II. Market 3.A. VaR Methods Jorion Ch. 10 LO 13.1 to 13.9 II. Market 3.B. VaR Mapping Jorion Ch. 11 LO 14.1 to 14.7 I. Quant 3.B. MCS Jorion Ch. 12 LO 15.1 to 15.5 I. Market 3.C Stress Testing Jorion Ch. 14 LO 16.1 to 16.3 I. Quant 4. EVT Kalyvas Ch. 4 LO 17.1 17.13 V. Investment 6.B. Budgeting in I/M Jorion Ch. 17

Value at Risk (VaR) Traditional metric was capital asset pricing model (CAPM) where beta is (traditional) risk metric But beta has a “tenuous connection” to actual returns CAPM is a one-factor model : too simplistic 1994, JP Morgan created an “open architecture metric” (i.e., not proprietary) called RiskMetrics TM In 1998 Bank for International Settlements (BIS) allowed banks to use internal models (e.g., VaR) to calculate their capital requirements VaR is relatively easy to calculate LO 7.1 Discuss reasons for the widespread adoption of VaR as a measure of risk

Value at Risk (VaR) LO 7.2 Define value at risk and calculate VaR for a single asset on both a dollar and percentage basis. Single-Period VaR (n=1)

Value at Risk (VaR) LO 7.2 Define value at risk and calculate VaR for a single asset on both a dollar and percentage basis.

Value at Risk (VaR) - % Basis -1.645  One-Period VaR (n=1) and 95% confidence (5% significance)

Value at Risk (VaR) – Dollar Basis -1.645  One-Period VaR (n=1) and 95% confidence (5% significance)

10-period VaR @ 5% significance -1.645  10 10-Period VaR (n=10) and 95% confidence (5% significance)

10-period (10-day) VaR 5% significance, annual  = +12% $100(1+  ) +0.48 ($95.30)

Absolute versus Relative VaR $100(1+  ) $100  $5.20  $4.73

Absolute VaR $100(1+  ) $100

Value at Risk (VaR) Calculate the VaR for a single asset, in both dollar terms and percentage terms. What is one-day VaR with 95% confidence, dollars and percentage terms?

Value at Risk (VaR) Calculate the VaR for a single asset, in both dollar terms and percentage terms. What is one-day VaR with 95% confidence?

Value at Risk (VaR) LO 7.3 Convert a daily VaR measure into a weekly, monthly, or annual VaR measure. Square-root rule J-day VaR = 1-day VaR  Square Root of Delta Time Assumes i.i.d. Independent  not (auto/serial) correlated Identically distr.  constant variance (homoskedastic)

Value at Risk (VaR) LO 7.3 Convert a daily VaR measure into a weekly, monthly, or annual VaR measure. Daily VaR is (-)$10,000. What is 5-day VaR?

Value at Risk (VaR) LO 7.3 Convert a daily VaR measure into a weekly, monthly, or annual VaR measure. 10-day VaR is (-)$1 million. What is annual VaR (assume 250 trading days)?

Value at Risk (VaR) Stationarity : the (shape of the) probability distribution is constant over time Random walk : tomorrow’s outcome is independent of today’s outcome Non-negative : requirement: assets cannot have negative value Time consistent : what is true for a single period is true for multiple periods; e.g., assumptions about a single week can be extended to a year Normal : expected returns follow a normal distribution LO 7.4 Discuss assumptions underlying VaR calculations

Value at Risk (VaR) LO 7.5 Explain why it is best to use continuously compounded rates of return when calculating VaR

Value at Risk (VaR) Absolute change (today’s price – yesterday’s price): violates stationarity requirement. Simple change ([today price – yesterday’s price]  [yesterday’s price]): satisfies stationarity requirement, but violates time consistency requirement. Continuous compounded return is best because it satisfies the time consistency requirement: the 2-period return = sum of 1-period returns.  The sum of two random variables that are jointly distributed is itself (i.e., the sum) normally distributed. LO 7.5 Explain why it is best to use continuously compounded rates of return when calculating VaR Except for interest rate variables: absolute 

Value at Risk (VaR) LO 7.6 Calculate portfolio VaR and describe the primary factors that affect portfolio risk

Value at Risk (VaR) LO 7.6 Calculate portfolio VaR and describe the primary factors that affect portfolio risk Directional Impacts Factor Impact on Portfolio volatility Higher variance  Greater asset concentration  More equally weighted assets  Lower correlation  Higher systematic risk  Higher idiosyncratic risk Not relevant

Value at Risk (VaR) Linear derivative . Price of derivative = linear function of underlying asset. For example, a futures contract on S&P 500 index is approximately linear Non-linear derivative . Price of derivative = non-linear function of underlying asset. For example, a stock option is non-linear LO 7.7 Differentiate between linear and non-linear derivatives. All assets are locally linear . For example, an option: the option is convex in the value of the underlying. The delta is the slope of the tangent line. For small changes, the delta is approximately constant.

Value at Risk (VaR) LO 7.8 Describe the calculation of VAR for a linear derivative.

Value at Risk (VaR) LO 7.9 Explain how the addition of second-order terms through the Taylor approximation improves the estimate of VAR for non-linear derivatives.

Value at Risk (VaR) LO 7.9 Explain how the addition of second-order terms through the Taylor approximation improves the estimate of VAR for non-linear derivatives. The Taylor approximation is not helpful where the derivative exhibits extreme non-linearities. Mortgage-backed securities (MBS) Fixed income securities with embedded options

Value at Risk (VaR) Full Revaluation Every security in the portfolio is re-priced. Full revaluation is accurate but computationally burdensome. Delta-Normal A linear approximation is created. This linear approximation is an imperfect proxy for the portfolio. This approach is computationally easy but may be less accurate. The delta-normal approach (generally) does not work for portfolios of nonlinear securities. LO 7.11 Explain the differences between the delta-normal and full-revaluation methods for measuring the risk of non-linear derivatives.

Value at Risk (VaR) LO 7.12 Describe the structured Monte Carlo approach to measuring VAR, and identify the advantages and disadvantages of the SMC approach. Advantage Disadvantage Structured Monte Carlo Able to generate correlated scenarios based on a statistical distribution By design, models multiple risk factors Generated scenarios may not be relevant going forward

Value at Risk (VaR) The problem with the SMC approach is that the covariance matrix is meant to be “typical” But severe stress events wreak havoc on the correlation matrix. That’s correlation breakdown. Scenarios can attempt to incorporate correlation breakdowns. One approach is to stress test (simulate) the correlation matrix. This is easier said than done; e.g., the variance-covariance matrix needs to be invertible. LO 7.13 Discuss the implications of correlation breakdown for scenario analysis.

Value at Risk (VaR) LO 7.14 Describe the primary approaches to stress testing and the advantages and disadvantages of each approach. Advantage Disadvantage Stress Testing Can illuminate riskiness of portfolio to risk factors Can specifically focus on the tails (extreme losses) Complements VaR May generate unwarranted red flags Highly subjective (can be hard to imagine catastrophes)

Value at Risk (VaR) The worst case scenario measure asks, what is the worst loss that can happen over a period of time? Compare this to VAR, which asks, what is the worst expected loss with 95% or 99% confidence? The probability of a “worst loss” is certain (100%); the issue is its location LO 7.15 Describe the worst case scenario measure as an extension to VAR.

Value at Risk (VaR) As an extension to VAR, there are three points regarding the WCS: The WCS assumes the firm increases its level of investment when gains are realized; i.e., that the firm is “capital efficient.” The effects of time-varying volatility are ignored There is still the extreme tail issue: it is still possible to underestimate the likelihood of extreme left-tail losses LO 7.15 Describe the worst case scenario measure as an extension to VAR.

Value at Risk (VaR) A cash flow at risk (CAR) of $X million at the Y% percent level says: The probability is Y percent that the firm's cash flow will be lower than its expected value by at least $X million. LO 8.1 Calculate cash flow at risk for a firm with normally distributed cash flows for any period, given the expected return and volatility of firm value, and interpret the CFAR measure.

Value at Risk (VaR) CFAR at p percent ( p %) is the cash flow shortfall (defined as expected cash flow minus [-] realized cash flow) such that there is a probability (%, percent) that the firm will have a larger cash flow shortfall. If realized, cash flow is C and expected cash flow is E ( C ), we have: LO 8.1 Calculate cash flow at risk for a firm with normally distributed cash flows for any period, given the expected return and volatility of firm value, and interpret the CFAR measure.

Value at Risk (VaR) A firm that depends entirely on cash flows as key to its growth opportunities should measure CFAR. A firm that depends on its market value (or the market value of its assets) should measure VAR. VAR is associated with the balance sheet : it focuses on market values. CFAR is associated with the income statement or cash flow statement : it focuses on risk to flows. LO 8.2 Describe the characteristics of firms for which either VAR or CFAR is the more appropriate measure of risk.

Value at Risk (VaR) LO 8.2 Describe the characteristics of firms for which either VAR or CFAR is the more appropriate measure of risk. VAR CFAR Balance sheet (asset values) Statement of cash flows Banks, financial services firm, funds Non-financial corporations  External markets for capital  Internal growth provides capital

Value at Risk (VaR) LO 8.3: Given the cost per dollar of VAR and the relevant betas, expected returns, and correlations, calculate the VAR impact and expected net gain of a project/trade that is not large relative to the firm’s portfolio of projects.

Value at Risk (VaR) LO 8.3: Given the cost per dollar of VAR and the relevant betas, expected returns, and correlations, calculate the VAR impact and expected net gain of a project/trade that is not large relative to the firm’s portfolio of projects. Impact of “small” project: Buy #1 and Sell #3 1% of portfolio

Value at Risk (VaR) LO 8.4: Evaluate the impact of a project that is large relative to the firm’s portfolio of projects on CFAR, and explain how the cost of additional CFAR impacts the capital budgeting decision.

Value at Risk (VaR) LO 8.5: Explain how to allocate CFAR and VAR to the existing activities of the firm and how to use these allocations to improve the evaluation of the economic profitability of these activities (projects, divisions, and trading).

Value at Risk (VaR) Reduce the cost of risk for a given level of VAR or CAR. reduce risk through project choice Use derivatives LO 8.6 Discuss how a firm can reduce the cost of VAR/CFAR.

Value at Risk (VaR) Increase equity capital Invest is less risky projects Use derivatives However, at least two factors prevent the elimination of firm-specific risk: Information asymmetries : outsiders cannot know the same as insiders. Moral hazard : Corporate managers can take unobserved actions that adversely affect the value of the contract. LO 8.7 Explain the limitations on project selection and the use of derivative instruments as ways to decrease VAR/CFAR.

Value at Risk (VaR) The local-valuation method is the use of partial derivatives as approximations. In local-valuation, value at risk (VaR) is determined by a linear relationship between the asset and the exposure. Full-valuation is more comprehensive. Full-valuation is used when local-valuation is inadequate. Under full-valuation, the entire portfolio is re-priced at various levels. The advantage is that exposure can be modeled as non-linear. LO 12.1: Explain the difference between local- and full-valuation methods.

Value at Risk (VaR) Delta-normal methods rely on first-order partial derivatives. They are locally accurate due to their linearity. Second-order terms improve the accuracy by accounting for curvature. For a bond portfolio, duration is the linear sensitivity of price to yield change. The second-order term, given by C below, accounts for the curvature and is called convexity: LO 12.2: Describe how the addition of second-order terms improves the accuracy of estimates for nonlinear relationships.

Value at Risk (VaR) The best methods depend on speed required and whether the portfolio contains much optionality: For large portfolios where optionality is not a major factor, the delta normal method is fast and efficient For fast approximations of option values, mixed methods are efficient; e.g., delta-gamma-Monte Carlo or grid Monte Carlo For portfolios with substantial optionality or long horizons, full valuation may be necessary LO 12.3: Compare the delta normal, historical simulation, and Monte Carlo simulation methods, and explain their appropriate uses.

Value at Risk (VaR) LO 12.3: Compare the delta normal, historical simulation, and Monte Carlo simulation methods, and explain their appropriate uses. Valuation Method Risk factor Local Full Analytical Delta-normal Not used Delta-gamma-delta Simulated Delta-gamma-Monte-Carlo Monte Carlo Grid Monte Carlo Historical

Value at Risk (VaR) LO 12.4: List the advantages and disadvantages of the delta-normal model. Advantage Disadvantage Delta-normal Easy to implement Computationally fast Can be run in real-time Amenable to analysis (can run marginal and incremental VaR) Normality assumption violated by fat-tails (compensate by increasing the confidence interval) Inadequate for nonlinear assets

Value at Risk (VaR) LO 12.5: List the advantages and disadvantages of the historical simulation method. Advantage Disadvantage Historical Simulation Simple to implement Does not require covariance matrix Can account for fat-tails Robust because it does not require distributional assumption (e.g., normal) Can do full valuation Allows for horizon choice Intuitive Uses only one sample path (if history does not represent future, important tail events not captured) High sampling variation (data in tail may be small) Assumes stationary distribution (can be addressed with filtered simulation)

Value at Risk (VaR) LO 12.6: List the advantages and disadvantages of the Monte Carlo simulation method. Advantage Disadvantage Monte Carlo Most powerful Handles fat tails Handles nonlinearities Incorporates passage of time; e.g., including time decay of options Computationally intensive (need lots of computer and/or time) Can be expensive Model risk Sampling variation

Value at Risk (VaR) LO 13.1 Describe the two components of the typical VAR model.

Value at Risk (VaR) First component is the data feed system: Historical data used to compute… Historical volatility and correlations, in turn used to produce… Estimated covariance matrix LO 13.1 Describe the two components of the typical VAR model. The second component is the “mapping system” which transforms (or maps) the portfolio positions into weights on each of the securities for which risk is measured.

Value at Risk (VaR) Large underlying cash markets — demonstrates demand for the underlying asset and implies the contract can be fairly priced High volatility of the underlying asset — leads to hedging needs and creates the possibility of profits for speculators Lack of close substitutes—contracts are best when they hedge price risks that cannot be met with existing contracts LO 13.2 Discuss the three qualities of successful futures contracts and why these are desired in the chosen risk factors.

Value at Risk (VaR) Specific risk is the risk inherent to (or unique to) the company or security. It is issuer-specific as opposed to market-related. A greater number of general risk factors should create less residual risk LO 13.3 Discuss how the residual specific risk is related to the number of risk factors.

Value at Risk (VaR) Principal mapping : bond risk is associated with the maturity of the principal payment only Duration mapping : the risk is associated with that of a zero-coupon bond with maturity equal to the bond duration Cash flow mapping : the risk of fixed-income instruments is decomposed into the risk of each of the bond cash flows LO 13.4 Explain the three approaches for mapping a portfolio onto the risk factors.

Value at Risk (VaR) LO 13.4 Explain the three approaches for mapping a portfolio onto the risk factors.

Value at Risk (VaR) LO 13.5 Decompose a fixed-income portfolio into positions in the standard instruments.

Value at Risk (VaR) LO 13.6 Calculate the VAR of a fixed-income portfolio using the delta-normal method, given the expected change in portfolio value and the standard deviation.

Value at Risk (VaR) The delta-normal method requires that a portfolio (or instrument) be expressed as a linear combination of risk factors. If a financial instrument can be “reduced” to such a linear expression—as many can—then it can be utilized by the delta -normal method. For almost the same reason as the previous AIM: because delta -normal method relies on a linear combination of risk factors. Options and derivatives exhibit non-linear relationships. LO 13.7: Explain why the delta-normal method can provide accurate estimates of VAR for many types of financial instruments.

Value at Risk (VaR) LO 13.8: Discuss why caution must be used in applying delta-normal VAR methods to derivatives and options, and describe when this is appropriate.

Value at Risk (VaR) LO 13.9: Explain what is meant by benchmarking a portfolio, and define tracking error.

Value at Risk (VaR) Generically, simple stress testing consists of three steps: Create a set of extreme market scenarios (i.e., stressed scenarios)—often based on actual past events; For each scenario, determine the price changes to individual instruments in the portfolio; sum the changes in order to determine change in portfolio value Summarize the results : show estimated level of mark-to-market gains/losses for each stressed scenario; show where losses would be concentrated. LO 15.1 Discuss the role of stress testing as a complement to the VAR measure, and describe the benefits and drawbacks of stress testing.

Value at Risk (VaR) LO 15.1 Discuss the role of stress testing as a complement to the VAR measure, and describe the benefits and drawbacks of stress testing. VAR vs. Stress Testing VAR Stress Testing No information on magnitude of losses in excess of VaR Captures the “magnitude effect” of large market moves. Little/no information on direction of exposure; e.g., is exposure due to price increase or market decline Simulates changes in market rates and prices, in both directions Says nothing about the risk due to omitted factors; e.g., due to lack of data or to maintain simplicity Incorporates multiple factors and captures the effect of nonlinear instruments.

Value at Risk (VaR) Unidimensional scenarios focus “stressing” on key one variable at time; e.g., shift in the yield curve, change in swap spread. Scenarios consist of shocking one variable at a time . The key weakness of a unidimensional analysis is that scenarios cannot, by definition, account for correlations. The multidimensional is more realistic and attempts to “stress” multiple variables and their relationships (correlations). Multidimensional scenario analysis consists of: First, posit a state of the world (high severity event) Then, infer movements in market variables Multidimensional analysis includes: Factor push method: first, shock risk factors individually. Then, evaluate a worst-case scenario. Conditional scenario method: systematic approach LO 15.2 Compare and contrast the use of unidimensional and multidimensional scenario analysis.

Value at Risk (VaR) Prospective scenarios try to analyze the implications of hypothetical one-off surprises; e.g., a major bank failure, a geopolitical crisis. Historical scenarios looks to actual past events to identify scenarios that would fall outside of the VaR window. Events that are often used include: The one-month period in October 1987 (S&P 500 index fell by > 21%) Exchange rate crisis (1992) and U.S. dollar interest rates changes (spring of 1994) The 1995 Mexican crisis The East Asian crisis (summer of 1997) The Russian devaluation of August 1998 and the Brazilian devaluation of 1999 LO 15.3 Compare and contrast the use of prospective scenarios and historical scenarios in multidimensional scenario analysis, and describe the advantages and disadvantages of each.

Value at Risk (VaR) LO 15.3 Compare and contrast the use of prospective scenarios and historical scenarios in multidimensional scenario analysis, and describe the advantages and disadvantages of each. Advantage Disadvantage Prospective Scenarios in MDA Relies on input of managers to frame scenario and therefore may be most realistic vis-à-vis actual extreme exposures May not be well-suited to “large, complex” portfolios FACTOR PUSH METHOD: ignores correlations Historical scenarios in MDA Useful for measuring joint movements in financial variables Typically, limited number of events to draw upon

Value at Risk (VaR) LO 15.4 Discuss an advantage and disadvantage of using the conditional scenario method as a means to generate a prospective scenario Advantage Disadvantage Conditional Scenario Method More realistically incorporates correlations across variables: allows us to predict certain variables conditional on movements in key variables Relies on correlations derived from entire sample period. Highly subjective

Value at Risk (VaR) Although not every scenario requires a response, an institution should address relevant scenarios. The institution can: Set aside economic capital to absorb worst-case losses Purchase protection or insurance Modify the portfolio Restructure the business or product mix to enhance diversification Develop a corrective or contingency plan should a scenario occur Prepare alternative funding sources in anticipation of liquidity crunches LO 15.5 Discuss possible responses when scenario analysis reveals unacceptably large stress losses.

VaR: Question • A portfolio has an initial value (V 0 ) of $200 and an annual standard deviation (  252 days) of 15%. What is the 1-day dollar VaR relative with 95% confidence? What is the 10-day dollar VaR relative with 95% confidence?

VaR: Answer 1 • A portfolio has an initial value (V 0 ) of $200 and an annual standard deviation (  252 days) of 15%. What is the 1-day VaR relative with 95% confidence?

VaR: Answer 2 • A portfolio has an initial value (V 0 ) of $200 and an annual standard deviation (  252 days) of 15%. What is the 10-day VaR relative with 95% confidence?

VaR: Question • A portfolio has an initial value (V 0 ) of $100 and an annual standard deviation (  252 days) of 15%. Now assume the annual expected return (  252 ) of the portfolio is +8% . What is the 10-day “absolute VaR” with 95% confidence?

VaR: Answer • A portfolio has an initial value (V 0 ) of $100 and an annual standard deviation (  252 days) of 15%. Now assume the annual expected return (  252 ) of the portfolio is +8% . What is the 10-day “absolute VaR” with 95% confidence?

VaR of nonlinear derivative • In the case of a linear derivative, the relationship between the derivative and the underlying asset is linear (delta, the “transmission parameter,” is constant) - Example: Futures contract on the S&P 500 Given that a futures contract is [$250 x Index], the VaR of the futures contract is: F t =250  VaR(S S&P 500 Index )

Taylor approximation AIM: Explain how the addition of second-order terms through the Taylor approximation improves the estimate of VAR for non-linear derivatives 1. Constant approximation 2. First-order (linear) approximation 3. Second-order (quadratic) approximation

Taylor approximation AIM: Explain how the addition of second-order terms through the Taylor approximation improves the estimate of VAR for non-linear derivatives

Taylor approximation AIM: Discuss why the Taylor approximation is ineffective for certain types of securities Does not perform well when the derivative shows extreme nonlinearities . For example: Mortgage-backed securities (MBS) Fixed income securities with embedded options When beta/duration can change rapidly, Taylor approximation (delta-gamma approximation) is ineffective – need more complex models.

Versus Full Re-value AIM: Explain the differences between the delta-normal and full-revaluation methods for measuring the risk of non-linear derivatives Delta-normal: linear approximation that assumes normality Option = F [Delta] Bond = F [Duration] Full-revaluation: linear approximation that assumes normality Computationally fast but… approximate Accurate but… computationally burdensome Re-price (simulate) the portfolio at several price levels

Structured Monte Carlo • Make an assumption about the behavior of the underlying (i.e., “create artificial variables with properties similar to portfolio risk factors”) Conduct several random trials (1000s, millions) Calculate VaR based on the worst %-ile outcome— just like identifying the worst 1% or 5% in a historical simulation Simulate with one variable (e.g., GBM) or several (Cholesky)

Structured Monte Carlo Advantages Can generate correlated scenarios based on statistical distribution Flexibility makes it “most powerful approach to VaR” (Jorian) Disadvantages Computational time requirements Simulations are not necessarily predictive (same problem as extrapolation)

Scenario Analysis Evaluate Correlation Matrix Under Scenarios ERM Crisis (92) Mexican Crisis (94) Crash Of 1987 Gulf War (90) Asian Crisis (’97/8)

Scenario Analysis AIM: Discuss the implications of correlation breakdown for scenario analysis Severe stress events wreak havoc on the covariance matrix

Scenario Analysis AIM: Describe the primary approaches to stress testing and the advantages and disadvantages of each approach Provide two independent sections of the risk report: VAR-based risk report, and Stress testing-based risk report. Either: (a) Plugs-in historical events , or (b) Analyzes predetermined scenarios

Scenario Analysis Historical events + Can inform on portfolio weaknesses - But could miss weaknesses unique to the portfolio Stress Scenarios + Gives exposure to standard risk factors - But may generate unwarranted red flags - May not perform well in regard to asset-class-specific risk AIM: Describe the primary approaches to stress testing and the advantages and disadvantages of each approach

Summary • VaR is the worst expected loss, with specified confidence of a given interval Taylor (Series) approximation corrects for the curvature (nonlinearity) but can only help so much – not for extreme non-linearities Monte Carlo simulation – given a model for underlying (stock) behavior, conducts many random trials Scenario analysis may improve by evaluating the portfolio against “actual” historical events

Random Variables + + + Short-term Asset Returns Probability distributions are models of random behavior + + + - - - - - - - ? ? ?

Random Variables Random variable (or stochastic variable): function that defines a point in the sample space of outcomes. LO 1.1 Define random variable, outcome, an event, mutually exclusive events, and exhaustive events.

Random Variables Random variable (or stochastic variable): function that defines a point in the sample space of outcomes. Outcome : the result of a single trial. LO 1.1 Define random variable, outcome, an event, mutually exclusive events, and exhaustive events.

Random Variables Random variable (or stochastic variable): function that defines a point in the sample space of outcomes. Outcome : the result of a single trial. Event: the result that reflects none, one, or more outcomes in the sample space. Events can be simple or compound . LO 1.1 Define random variable, outcome, an event, mutually exclusive events, and exhaustive events.

Random Variables Random variable (or stochastic variable): function that defines a point in the sample space of outcomes. Outcome : the result of a single trial. Event: the result that reflects none, one, or more outcomes in the sample space. Events can be simple or compound . Mutually exclusive events: cannot simultaneously occur. Probability of (A and B) = 0. Intersection is null set. LO 1.1 Define random variable, outcome, an event, mutually exclusive events, and exhaustive events.

Random Variables Random variable (or stochastic variable): function that defines a point in the sample space of outcomes. Outcome : the result of a single trial. Event: the result that reflects none, one, or more outcomes in the sample space. Events can be simple or compound . Mutually exclusive events: cannot simultaneously occur. Probability of (A and B) = 0. Intersection is null set. Exhaustive events: all outcomes described LO 1.1 Define random variable, outcome, an event, mutually exclusive events, and exhaustive events.

Random Variables Rolling a “total of seven” (craps) with two dice is one event consisting of six outcomes One Event: Roll a seven Six outcomes

Probability LO 1.2 Discuss the two defining properties of probability.

Conditional LO 1.3 Compare and contrast unconditional and conditional probabilities. Unconditional

Conditional LO 1.3 Compare and contrast unconditional and conditional probabilities. Conditional

Joint probability LO 1.4 Define joint probability and interpret the joint probability of any number of independent events. S= $10 S= $15 S=$20 Total T=$15 0 2 2 4 T=$20 3 4 3 10 T=$30 3 6 3 12 Total 6 12 8 26

Theorems LO 1.5 Apply the three theorems on expectations for two independent variables.

Theorems LO 1.5 Apply the three theorems on expectations for two independent variables. Independent = not correlated

Theorems LO 1.6 Apply the four theorems on variance for two independent variables.

Theorems LO 1.6 Apply the four theorems on variance for two independent variables. What is the variance of a single six-sided die?

Covariance & correlation LO 1.7 Define and calculate covariance and correlation.

Covariance & correlation LO 1.7 Define and calculate covariance and correlation. X Y 3 5 2 4 4 6

Covariance & correlation LO 1.7 Define and calculate covariance and correlation. X Y (X-X avg )(Y-Y avg ) 3 5 0.0 2 4 1.0 4 6 1.0 Avg = 3 Avg = 5 Avg = .67

Covariance & correlation X Y (X-X avg )(Y-Y avg ) 3 5 0.0 2 4 1.0 4 6 1.0 Avg = 3 Avg = 5 Avg = .67 s.d. = SQRT(.67) s.d. = SQRT(.67) Correl. = 1.0

Covariance & correlation LO 1.8 Use Bayes’ formula to determine the probability of causes for a given event.

Bayes’ formula LO 1.8 Use Bayes’ formula to determine the probability of causes for a given event.

Permutations & combinations LO 1.9 Determine the number of possible permutations of n objects taken r at a time and the number of possible combinations of n objects taken r at a time.

Permutations & combinations LO 1.9 Determine the number of possible permutations of n objects taken r at a time and the number of possible combinations of n objects taken r at a time. Given a set of seven letters: {a, b, c, d, e, f, g} How many permutations of three letters? How many combinations of three letters?

Distributions LO 2.1 Distinguish between discrete random variables and continuous random variables and contrast their probability distributions LO 2.2 Discuss a probability function, a probability density function, and a cumulative distribution function

Comparison Probability Density Function (pdf) Cumulative Distribution Discrete variable Continuous variable

Distributions Discrete uniform distribution Outcomes can be counted All equally likely Example: six-sided die LO 2.3 Describe a discrete uniform random variable and a binomial random variable

Distributions Discrete uniform distribution Outcomes can be counted All equally likely Example: six-sided die Binomial Two outcomes (success/fail, true/false, heads/tails) Binomial experiment: binomial variables, fixed trials, independent and identical outcomes (i.i.d.) LO 2.3 Describe a discrete uniform random variable and a binomial random variable

Distributions LO 2.3 Describe the continuous uniform distribution

Normal distribution LO 2.5 Identify the key properties of the normal distribution

Normal distribution LO 2.5 Identify the key properties of the normal distribution % of all (two-tailed) % “to the left” (one-tailed) Critical values Interval –math (two-tailed) VaR ~ 68% ~ 34% 1 ~ 90% ~ 5.0 % 1.645 (~1.65) ~ 95% ~ 2.5% 1.96 ~ 98% ~ 1.0 % 2.327 (~2.33) ~ 99% ~ 0.5% 2.58

Normal distribution Only two parameters required : mean (  ) and variance/standard deviation (  ) Symmetrical : coefficient of skewness = 0 No excess kurtosis (“normal not fat” tails): coefficient of kurtosis = 3 LO 2.5 Identify the key properties of the normal distribution

For Parametric VAR The normal distribution Advantage : Large data set not required Only need two parameters, mean and volatility Disadvantage : Does not reflect actual distributions. Actual returns exhibit tend to exhibit “fat tails.”

Poisson: Question We observe that a key operational process produces, on average, fifteen (15) errors every twenty-four hours What is probability that exactly three (3) errors will be produced during the next eight-hour work shift?

Poisson: Answer The model is Poisson (5) and we are solving for the P(X=3):

Distributions - Poisson Poisson distribution Discrete Occurrences over time Errors per 1,000 transaction Calls per hour Breakdowns per week Lambda (  ) is the mean and the variance LO 2.7 Calculate the expected value and variance of the Poisson distribution

Compared LO 2.8 Compare and contrast the binomial, normal, and Poisson distributions In Poisson, the expected value (the mean) = variance Variance is standard deviation 2

Lognormal LO 2.9 Contrast the lognormal and normal distributions

Lognormal Transform x-axis To logarithmic scale

Summary Random variables Discrete or Continuous We can ask a PDF question: P(X=) or a cumulative distribution function question P(X<=) Distributions are attempts to model random events. For example: Normal: daily stock price change Binomial: success/failure, heads/tails, 0/1 Poisson: errors/time, exceptions/time Key problem with normal: likely to underestimate extreme losses (“left-tail events”)

Sampling The population: entire group, often unknowable. Denoted by a capital “N.” The sample: subset of the population. Denoted with small “n” A parameter is a quantity in the f(x) distribution that helps describe the distribution For example, mean (  ) and the standard deviation (  ) LO 3.1: Define a population, a parameter, and a sample. We take a sample (from the population) in order to draw an inference about the population.

Frequencies LO 3.2: Construct a frequency distribution, calculate relative frequencies from a frequency distribution, and illustrate the use of a histogram and a frequency polygon to the present data.

Sampling Population mean (arithmetic) Sample mean (arithmetic) LO 3.3: Calculate and interpret the following measures: population mean, sample mean, arithmetic mean, geometric mean, mode, and median.

Sampling Geometric mean Mode: most frequent Median = 50 th percentile (%ile) Odd n: (n+1)/2 Even n: mean of n/2 and (n+2)/2 LO 3.3: Calculate and interpret the following measures: population mean, sample mean, arithmetic mean, geometric mean, mode, and median.

Geo & Arithmetic mean 2003 5.0% 2004 8.0% 2005 (3.0%) 2006 9.0% Geo. Arith.

Geo & Arithmetic mean 2003 5.0% 1.05 2004 8.0% 1.08 2005 (3.0%) 0.97 2006 9.0% 1.09  1.199 Geo. 4.641% Arith. 4.75%

Sampling LO 3.4: Discuss the properties of the sampling distribution of means, proportions, and differences and sums. The sampling distribution is the probability of the sample statistic

Sampling LO 3.4: Discuss properties of the sampling distribution of means, proportions, and differences and sums. Variance of sampling distribution of means: Infinite population or with replacement

Sampling LO 3.4: Discuss properties of the sampling distribution of means, proportions, and differences and sums. Variance of sampling distribution of means: Finite population (size N) and without replacement Variance of sampling distribution of means: Infinite population or with replacement

Sampling LO 3.4: Discuss the properties of the sampling distribution of means, proportions, and differences and sums. Standardized Variable: “ Asymptotically normal” even when population is not normally distributed !!

Sampling LO 3.4: Discuss the properties of the sampling distribution of means, proportions, and differences and sums. Standardized Variable: “ Asymptotically normal” even when population is not normally distributed !! Central limit theorem: Random variables are not normally distributed, But as sample size increases -> Average (and summation) tend toward normal

Sampling LO 3.4: Discuss the properties of the sampling distribution of means, proportions, and differences and sums. Sampling distribution of proportions Where p = probability of success

Sampling LO 3.4: Discuss the properties of the sampling distribution of means, proportions, and differences and sums. Sampling distribution of differences Two populations, Two samples, difference of the means

Sampling LO 3.4: Discuss the properties of the sampling distribution of means, proportions, and differences and sums. Sampling distribution of differences Two populations, Two samples, sum of the means

Variance LO 3.5: Calculate a sample variance, population variance, and standard deviation. Population Variance

Variance LO 3.5: Calculate a sample variance, population variance, and standard deviation. Sample Variance

Variance LO 3.5: Calculate a sample variance, population variance, and standard deviation. Sample Standard Deviation

Chebyshev’s LO 3.6: Determine the percentage of a distribution that lies a stated number of deviations from the mean using Chebyshev’s inequality. What is the probability that random variable X (with finite mean and variance) will differ by more than three (3) standard deviations from its mean?

Chebyshev’s LO 3.6: Determine the percentage of a distribution that lies a stated number of deviations from the mean using Chebyshev’s inequality. If k = 3, then P() = 1/(3 2 ) = 1/9 = 0.1111

Skewness & Kurtosis LO 3.7: Describe and interpret measures of skewness and kurtosis.

Intro to Value at Risk (VaR)

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Intro to Value at Risk (VaR)