C323-RiskManagementPrinciplesandApplications-AssignmentOne

Mark Heath
C323 Risk Management: Principles & Applications
Assignment 1
_____________________________________________________________________________
Question 1a)
Mean Variance Approach - In order to build a portfolio of the two stocks, R1 BAG and R2
ABF, we can use Harry Markowitz’s Modern Portfolio theory, developed in the 1950’s which
won him the Nobel peace prize. Using mean returns and variances we can construct a portfolio
of the two stocks and know the standard deviations, risk, involved in that portfolio and also
establish the expected returns from that portfolio. In this case we will assume we are a rational
investor, we are risk averse and therefore our target is minimise the risk, or standard deviation, of
our portfolio.
The results for period 1, 31st January 2011 to 27th January 2014 are as follows: -
Result Stock R1 (BAG) A G Barr Stock R2 (ABF) Ass British
Foods
Expected Average Weekly Return
E (Ri)
0.44% 0.66%
Average Variance σ² 0.001330 0.000567
Standard Deviation σ 3.65% 2.38%
Covariance σR1R2 0.000111
Correlation Coefficient ⍴R1R2 0.128005
*All results are based on weekly returns calculated as: - Rt = (St - St-1) / St-1
Lets interpret the results.

Average Weekly Return –
𝑅̅𝑖 = ∑
𝑅𝑖𝑗
𝑀
𝑀
𝑗=1
We can see that the expected average weekly (average) return during period 1 from R2 ABF is
almost 50% higher than R1 BAG.
Variance & Standard Deviation –
𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = 𝜎𝑖
2
= ∑(𝑅𝑖𝑗 − 𝑅̅𝑖)
2
𝑀
𝑗=1
𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 = 𝜎𝑖 = √𝜎𝑖
2
Interestingly we would normally associate higher returns with higher variance, or risk, but in this
case the variance of the lower performing stock, R1 BAG, is higher than the better performing
stock, R2 ABF. Note standard deviation is square root of the squared variances, and is expressed
in the same unit as original figures, i.e. percentage in this case.
Covariance & Correlation –
𝐶𝑜𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = 𝜎12 = ∑(𝑅1𝑗 − 𝑅̅1)
𝑀
𝑗=1
(𝑅2𝑗 − 𝑅̅2)
𝐶𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 = 𝜌12 =
𝜎12
𝜎1 𝜎2
Measuring the covariance for period 1 between R1 BAG and R2 ABF the result is a low positive
number which indicates to us that the two stocks do not move together, however, the relationship
is not inverse, in that own when stock goes up the other goes down. In order to standardise the
covariance we can use the correlation coefficient (max +1, min -1) which gives us a result of
0.128, underlining our result from the covariance that the stocks are not highly correlated. With

this result we can be relatively satisfied that given our portfolio of 2 stocks, we are reasonably
well diversified.
Variance of a Portfolio –
As the number of assets get larger in a portfolio the effect of variances in individual assets is
diversified away, with the variance eventually converging on the covariance between the assets.
Individual risk is completely diversified away with variances in individual stocks having no
effect in a large portfolio.
The Opportunity Set
The expected returns from a portfolio can calculated from the weighted returns of the underlying
assets.
𝑅̅ 𝑝 = 𝑋𝐴 𝑅̅ 𝐴 + (1 − 𝑋𝐴) 𝑅̅ 𝐵
Where XA is the percentage of portfolio held in asset A. Note, we assume the investor is 100%
invested, so weighting for asset B is 1 – XA.
The variances, risks, associated with the portfolio, measured as standard deviation, are not
simply a weighting of the variances of the underlying assets, we must also take into account to
correlation coefficient between the two assets. We solve for the standard deviation of the
portfolio as follows.
𝜎 𝑝 = [ 𝑋𝐴
2
𝜎𝐴
2
+ (1 − 𝑋𝐴)2
𝜎 𝐵
2
+ 2𝑋𝐴(1 − 𝑋𝐴) 𝜌 𝐴𝐵 𝜎𝐴 𝜎 𝐵]
1
2
Using the above formulas for weighted expected returns and standard deviation of the portfolio
we are able to plot our portfolio possibilities at various weightings as below.

Fig.1
Calculating the expected return and standard deviation of the portfolio of the two different stocks
at different weighting (see appendix 1), starting from 100% invested in R1 BAG and varying in
steps of 5%, until we reach a scenario in which we are 100% invested in R2 ABF, we are able to
plot a chart as shown in Fig.1. The curve produced is known at the portfolio possibilities curve
and demonstrates the risk/return combinations of the portfolio. As a risk averse rational investor
we are looking for the lowest possible risk, i.e. the lowest standard deviation, which is the X axis
on the chart, and what the expected return is from the portfolio at that standard deviation. From
our chart we can see that the lowest possible standard deviation looks to be around 0.021 and the
expected return around 0.59%, i.e. indicated by point B on our chart. The curve from point B to
A is known as the efficient frontier, this is where increased risk is rewarded with increased
expected returns. Portfolios on this section of the curve (B to A) dominate section B to C where
increased risk with these portfolios are met with lower expected returns.
A
B
C

Because the weightings are in steps of 5% the technique is rather a blunt measure to find the
minimum variance. Using the following formula we are able to more accurately estimate the
weighting of the portfolio which gives the lowest variance: -
𝑋 𝑅2 =
𝜎 𝑅1
2
− 𝜎 𝑅2 𝜎 𝑅1 𝜌 𝑅1𝑅2
𝜎 𝑅2
2
+ 𝜎 𝑅1
2
− 2𝜎 𝑅2 𝜎 𝑅1 𝜌 𝑅1𝑅2
Solving the formula for XR2 we find that a weighting 0.708, or 70.8%, invested in R2 ABF gives
us the minimum variance of 0.021 and an expected return of 0.595%.
Given that we are rational investors, seeking minimum risk, we should construct our portfolio as
70.8% invested in R2 ABF, and 29.2% in R1 BAG.
Question 1b)
Period 2 - 3rd February 2014 to 26th January 2015
Result Stock R1 (BAG) A G Barr Stock R2 (ABF) Ass British
Foods
Expected Average Weekly
Return Ri
0.15% 0.35%
Average Variance σ² 0.017608 0.013423
Standard Deviation σ 13.3% 11.6%
Correlation Coefficient ⍴R1R2 0.027624
Looking at the performance of our constructed portfolio for the period 2 - 3rd Febraury 2014 to
26th January 2015, we see the weekly return is 0.29%. Compared with period 1 the expected
return has reduced by around 50%.
The performance of the individual stocks has slumped dramatically from period 1 to period 2,
with R1 BAG reducing from expected weekly return of 0.44% to 0.15%, and R2 ABF, reducing
from 0.66% to 0.35%. However, Markowitz’s Portfolio Theory has probably helped us achieve
higher than normal returns, for example, if we had arbitrarily chosen to invest 50% of our

portfolio in each stock our expected return would have been a lower 0.25%. Portfolio theory
assisted in managing the risk and weighting our portfolio accordingly. During period 2 the
correlation coefficient between the two stocks has fallen from 0.128 in period 1 to 0.028 in
period 2, indicating a divergence in performance of the stocks. It is also interesting to note the
standard deviation of both stocks does increase considerably from period 1 to period 2,
indicating an increase variance, risk, in the returns.

Question 2)
The Single Index Model - The theory behind the single index model is that stock returns are
directly related to market returns, i.e. when the market returns rise, stock returns rise, and vice
versa. Studies show that most stocks are positively correlated with the market. The single index
model reduces the number of calculations and workload required in calculating expected returns
for portfolios. Rather than calculating covariances for each pair of stocks, as with Markowitz’s
Modern Portfolio Theory, we calculate the covariance between each stock and the market index.
The benefits of reduced number of pairings and calculations increases as the number of stocks in
a portfolio increases. For example, using Markowitiz’s mean variance theory, for a portfolio of
60 stocks, there are 60 estimates of means, 60 estimates of variance, and 1770 estimates of
covariances, a total of 1890 estimates. Using the single index model there are 60 estimates of
Beta, 60 estimate of Alpha, 60 estimate of firm specific variance, 1 estimate of market mean and
estimate of market variance. A total of 182 estimates. The benefits of the single index model are
clear to see.
There are certain assumptions in place that need to hold in order for the single index model to be
effective: -
 stocks tend to move in a similar direction to the average return on the market,
 deviations (random) of returns from stock from expected values are uncorrelated with the
returns of the market, i.e. 𝑐𝑜𝑣 ( 𝑒𝑖, 𝑅 𝑚) = 0
 deviations (random) of returns from stocks from expected values are uncorrelated with
each other. i.e. 𝑐𝑜𝑣 (𝑒𝑖, 𝑒𝑗) = 0
The single index model has two major elements to calculate expected returns, alpha and beta.
First of all let’s look at beta. Beta is the market related part of the single index model, they use
past performance of the stock (returns) and the covariance of the stock with the with the market
index to estimate a constant. The constant is the measure of the expected change in returns (Ri)
for a given change in the market index (Rm). To calculate the beta (βi) value for a stock we
calculate the covariance of the stock with the market divided by the variance of the market.
𝛽̂
𝑖 =
𝜎 𝑖𝑚
𝜎 𝑚
2

The alpha element of the single index model is a random variable specifically related to the
stock, which is independent of the market return.
𝛼̂ 𝑖 = 𝑅̅𝑖 − 𝛽̂𝑖 𝑅̅ 𝑚
Variance of a stock is computed with the following formula.
𝜎𝑖
2
= 𝛽𝑖
2
𝜎 𝑚
2
+ 𝜎𝑒𝑖
2
And covariance between securities is computed as follows.
𝜎𝑖𝑗 = 𝛽𝑖 𝛽𝑗 𝜎 𝑚
2
The parameters for our single index model are as follows (based on weekly returns) : -
Result Stock R1 (BAG) A G Barr Stock R2 (ABF) Ass British Foods
Beta βi 0.440579 0.495408
Alpha αi 0.003970 0.006101
Variance σ2
i 0.001330 0.000567
Expected Return Rm 0.001077
Variance σ2
m 0.000431
The estimates of historical beta for our stocks R1 BAG, and R2 ABF, tell us that, whilst the beta
are positive, i.e. we can expect the stock returns to be positive as long as the market returns are
positive, the stock in actual fact are less volatile than the market as whole. In respect of R1 BAG,
for every 1% movement in the market we can estimate the stock to return 0.440579%, and for R2
ABF, we can estimate the stock to return 0.495408%.
These regression techniques help us estimate expected returns from stocks and portfolios against
our opinions on the performance of market indexes. Beta explains the return of stocks as a

function of the market index and is a measure of risk, volatility, of a stock. For example, we may
expect riskier technology stocks to have a beta of 1.5, indicate estimated returns 50% higher than
the index movements.
Weakness of beta are their tendency to overestimate high beta and underestimate low beta.
Blume and Vasicek put forward theories on how the reduce the bias in beta. Blume’s theory is by
far the simpler of the two theories. His research led him to find the following relationship: -
𝛽𝑖2 = 0.343 + 0.677 𝛽𝑖1
Βi1 is the beta from historical period and solving for formula gives us a new beta which will be
closer the mean than the historical beta. For example, our beta for R1 (BAG) becomes 0.44939,
original we solved beta for 0.440579
Vasicek’s model is more complex in that it uses the weighted averages of historical beta for the
stock and average beta of the market.
𝛽𝑖2 = 𝜔 𝑜 𝛽𝑖1 + 𝜔1 𝛽̅1 𝜔 𝑜 + 𝜔1 = 11
As with Blume’s model, βi1 is historical beta of the stock, β1 is the historical average beta of the
market. The additional terms, ω, are weightings on historical beta of the stock and the average
market beta.
Now, lets estimate expected returns (Ri) for our stocks using the single index model
𝑅𝑖 = 𝛼𝑖 + 𝛽𝑖 𝑅 𝑚
For R1 (BAG) = 0.003970 + 0.440579 * 0.001077
For R2 (ABF) = 0.006101 + 0.495408 * 0.001077
Result Stock R1 (BAG) A G Barr Stock R2 (ABF) Ass British Foods
Ri 0.44% 0.66%

Note, Ri, and σi are the same under Markowitz’s mean variance theory as they are under the
single index model.

Question 3)
10yr Govt Bond – Coupon Paying - 23rd
January - Yield
9%
10yr Govt Bond – Coupon Paying - 30th
January - Yield
11%
Payment on Redemption €1090
Modified Duration - the duration of a bond, otherwise known as Macaulay’s Duration, is the
average time for receipt of cash flows, similar to the concept of Internal Rate of Return studied in
corporate finance. The duration is measured in years, Bonds with higher durations are considered
to be riskier, with them being more sensitive to changes in interest rate movements. To calculate
Modified Duration first of all we must calculate the Duration, which is the weighted time to
receipt of the cash flows, divided by the value of the bond (equivalent to the present value of the
cash flows from the bond).
𝐷 =
∑ 𝑡𝑇
𝑡=1 .
𝐶𝑡
(1 + 𝑖)
𝑃
In our example of the Duration is 7 years. Whilst Duration is a measure of elasticity of the
reaction of the price of the bond when interest rates change, Modified Duration, provides us with
a percentage change in the price of a Bond as a function of the change in yield.
𝐷 𝑚 =
𝐷
(1 + 𝑖)
The result for Modfied Duration, Dm , is 6.42 years. Dm is used in the next formula to calculate
the percentage change in the bond price.
∆𝑃𝑜
𝑃𝑜
= −𝐷 𝑚. ∆𝑖

The result being, -13%, meaning a 2% increase in interest yield would decrease the bond price
by 13%. So from €1000 the bond would be worth an estimated €871.65. Note the inverse
relationship between interest rates and bond prices. However, Dm, is only an approximate
measure and is useful only for small interest rate changes. The relationship between Bond prices
and interest yield in not linear, rather the relationship is curved, or more accurately, Convex,
hence our next task, building in correction to the percentage price change to allow for convexity.
𝐶 =
1
2
.
∑ 𝑡𝑇
𝑡=1 .
𝐶𝑡
(1 + 𝑖)
𝑃
Solving for C in our example we arrive at a convexity of 44.14. Applying the factor for
convexity into the next stage to work out the percentage change we must establish the
proportional change,
∆(1 + 𝑖)
(1 + 𝑖)
Finally we can solve for percentage change in bond price by applying our calculations into the
following fomula,
∆𝑃𝑜
𝑃𝑜
= −𝐷 𝑚∆(1 + 𝑖) + 𝐶 [
∆(1 + 𝑖)
(1 + 𝑖)
]
In this case we can see that allowing for convexity has altered the percentage change from a
previous estimate of -13%, to -11%, if there was a two percentage increase in interest yields.
Hence, it is estimated the bond price would have fallen less than under the Macualay method,
from €1000 to €886.51.

Appendix 1 – table of results of standard deviation and expected returns as weightings
for asset R2.
XR2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.7080 0.75 0.8 0.85 0.9 0.95 1
SD(Rp) 0.036 0.034 0.033 0.031 0.030 0.028 0.027 0.026 0.025 0.024 0.023 0.022 0.022 0.021 0.021 0.021 0.021 0.021 0.022 0.022 0.023 0.024
E(Rp) 0.44 0.451 0.462 0.473 0.484 0.495 0.506 0.517 0.528 0.539 0.55 0.561 0.572 0.583 0.594 0.59576 0.605 0.616 0.627 0.638 0.649 0.66
References
Elton et al (2011), Modern Portfolio Theory and Investment Analysis: John Wiley &
Sons, Inc.

C323-RiskManagementPrinciplesandApplications-AssignmentOne

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