# Class 1 Portfolio theory

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Class 1: Portfolio theory

PART ONE

1. (a) Why must it be the case that an investor with diminishing marginal utility of
wealth is risk-averse?
A risk averse investor is defined as someone who will refuse a fair game (where the
expected return is zero). They will presumably do this because the negative utility of a
loss of a marginal £1 is greater than the positive utility of a gain of a marginal £1.
This is the same as saying the investor has a diminishing marginal utility of wealth.
This leads to an upward sloping, concave trace in utility-wealth space.
Task: Draw the appropriate investor utility of wealth function

(b) Why are indifference curves of typical investors assumed to slope upward to the
right?
In modern portfolio theory agents have 2 conflicting objectives:
1. Maximise expected return;
2. Minimise risk.
These 2 must be balanced against each other. Risk averse investors are willing to
forgo some expected return in exchange for less risk. This trade off leads to upward
sloping, convex lines in expected return-standard deviation space. All portfolios along
a single line give the same utility.
Task: Draw the appropriate investor indifference curves, where portfolio A and B are
equally desirable.

(c) Why are the indifference curves of more risk-averse investors more steeply sloped
than those of investors with less risk aversion?
With greater risk aversion, an investor will require more return for each marginal
increase in risk. This leads to more steeply sloped indifference curves for more risk-
averse investors.
Task: Draw indifference curves showing different degrees of risk aversion.

(d) On an indifference curve map indicate 3 portfolios of different utility. Which
portfolio has the highest utility?
A risk-averse investor will find any portfolio that is “farther north-west” provides
greater utility than any portfolio lying on an indifference curve below it.
Task: Draw an indifference curve map indicating 3 portfolios of different utility
(e) Consider 3 stocks with the following expected returns and standard deviations:

Stock                         r                             
A                             16%                           13%
B                             14%                           9%
C                             15%                           8%

Are any of the stocks preferred over another by a risk-averse investor?
Our investor wants maximum return for minimum risk. Thus, s/he will prefer the
stock that offers the highest expected return to standard deviation ratio (this is
sometimes called the 'reward-to-volatility' ratio).

r     16
Stock A:            1.231
     13
r     14
Stock B:            1.556
      9
r     15
Stock C:            1.875
      8

Stock C offers the highest expected return per unit of risk and so this would be
preferred.

(f) Do you agree with the assumptions of nonsatiation and risk aversion. Make a case
for or against these assumptions.
Empirical evidence comes from (1) experimental evidence from simple choice
situations or (2) survey data on investor’s asset choices. This evidence indicates:
a. More to less & risk aversion is consistent with most evidence;
b. However, cognitive psychology/behavioural finance suggests situations where risk
aversion may not hold. For example, so-called framing effects. This economic
equivalent choices are treated differently when presented in different contexts. For
example, in a situation involving large expected losses, people do not seem to exhibit
risk-averse behaviour.

Additionally consider the risk preferences of participants in the gambling industries.
What type of risk behaviour are they exhibiting?

See text book for more details.
2. Given the following information about three stocks comprising a portfolio,
calculate each stock’s expected return. Then, using these individual securities’
expected returns, calculate the portfolio’s expected return.

Stock                   Initial    investment Expected end-of-     Proportion        of
value                 period investment    portfolio    initial
value                market value
A                       £500                  £700                 20%
B                       £200                  £300                 50%
C                       £1000                 £1050                30%

Answer: rp  0.2  0.4  0.5  0.5  (0.05  0.3)  0.345 or 35 % End of answer.

3. (a) Both the covariance and the correlation coefficient measure the extent to which
the returns on securities move together. What is the relationship between the two
statistical measures? Why is the correlation coefficient a more convenient measure?
T
Answer:  ij   pl ( xl  x )( y l  y )
l 1

 ij  0  i and j tend to move in the same direction
 ij  0  i and j tend to offset one another
 ij  0  i and j have little or no association
 ij   ij i j where  1   ij  1
Covariance measures the association of two assets, but it is silent on the strength on
this association. The covariance between two assets depends, in part, on how volatile
each asset is independently. Therefore, they are not comparable across pairs of assets.
Correlation is the standardized version of covariance. It is obtained by dividing the
covariance by the product of the individual standard deviations, and therefore is
comparable across different pairs of assets.

For example, if the correlation of rates of return between stocks A and B is 0.8 and
the correlation between C and D is 0.6, then we can state that stocks A and B have a
stronger positive dependency.

(b) Give an example of two common stocks that you would expect to exhibit a
relatively low correlation. Then give an example of two that would have a relatively
high correlation.
Answer: e.g. (i) HSBC & BT; (ii) VODAFONE & BT. End of answer.

(c) Given the following variance-covariance matrix for three securities, as well as the
percentage of the portfolio for each security, calculate the portfolio’s standard
deviation.
Security A                         Security B    Security C
Security A                459                                -211          112
Security B                -211                               312           215
Security C                112                                215           179
X A  0.55                         X B  0.25   X c  0.20

1/ 2
 3 3            
Answer:  p   X i X j  ij 
 i 1 j 1      
1/ 2
 3                3                3             
 p   X 1 X j 1 j   X 2 X j 2 j   X 3 X j 3 j 
 j 1            j 1             j 1           
 X 1 X 1 11  X 1 X 2 12  X 1 X 3 13  
1/ 2

  X 2 X 1 21  X 2 X 2 22  X 2 X 3 23  
                                           
 X 3 X 1 31  X 3 X 2 32  X 3 X 3 33 
                                           
(0.55  0.55  459)  (0.55  0.25  211)  (0.55  0.2  112)  
1/ 2

 (0.25  0.55  211)  (0.25  0.25  312)  (0.25  0.2  215)  
                                                                  
(0.2  0.55  112)  (0.2  0.25  215)  (0.2  0.2  179)
                                                                  

138.8475  29.0125  12.32  
1/ 2

  29.0125  19.5  10.75  
                             
12.32  10.75  7.16
                             

 153.62  12.39
1/ 2

4. Explain why a portfolio may have smaller standard deviation of return than the
individual securities that comprise it.
Answer: The risk equation suggests that, ceteris paribus, the smaller the covariance
between pairs of assets, the smaller the risk of the portfolio.

Portfolio return variances are generally smaller than asset return variances, while not
having systematically different expected returns.

Occurs because if you combine assets, which are to a degree, uncorrelated,
fluctuations in the return of one asset will be damped by fluctuations in the others.

Generally, if you combine 20 or more randomly selected securities in a portfolio, all
asset specific risk (e.g. unsystematic risk) is washed away and portfolio risk
comprises solely of that risk which is common to all securities, known as market or
systematic risk.

Task: Draw a diagram indicating the elimination of unsystematic risk; leaving
systematic risk.
The idea of combining low covariance pairs in a portfolio to eliminate unsystematic
risk is known as Markowitz diversification.

PART TWO

1. You are given the following information:

i    ri     iA    iB
Security A               15%    10%    1
Security B               40%    20%   0.6     1

Notation:

 i  standard deviation of the rate of return on security i = A, B, C.
 ij  correlation coefficient between the rate of return on assets i and j.
ri  expected rate of return on asset i.

(a) Construct and calculate the expected return and risk of the following two
portfolios. Compare the two portfolios in terms of their 'reward-to-volatility' ratio.

Portfolio I: equal weight to the two assets.
Portfolio II: 0.1*A + 0.9*B

n
rp   wi ri
i 1

 rp1  (0.5  0.1)  (0.5  0.20)  0.15or15%
 rp 2  (0.1  0.1)  (0.9  0.20)  0.19or19%

n   n
p           w w 
j   i
i   j    ij

 p  wi2 i2  w 2 2  2wi w j ij
j  j

 p  wi2 i2  w 2 2  2wi w j  ij  i j
j  j

  p1  (0.5 2  0.15 2 )  (0.5 2  0.4 2 )  (2  0.5  0.5  0.6  0.15  0.4)
 0.0636  0.2522 or 25.2%
  p 2  (0.12  0.15 2 )  (0.9 2  0.4 2 )  (2  0.1  0.9  0.6  0.15  0.4)
 0.1363  0.3692 or 36.9%
rp1
 0.595
 p1
rp 2
 0.515
 p2
Portfolio 1 offers the highest ‘reward-to-volatility’ ratio. End of answer.

(b) Discuss how can one calculate the minimum variance portfolio.
See EG p.75.

2. Explain why the Markowitz efficient frontier must be concave.
See SAB p.175 onwards.

3. In terms of the Markowitz model, explain, using words and graphs, how an investor
goes about identifying his or her optimal portfolio. What specific information does an
investor need to identify this portfolio?
See lecture notes. A good answer might contain:

2. Representation of investor preferences using indifference curves.
3. An explanation of the MEF.
4. Superimposing indifference curves on the MEF to identify the optimal portfolio.

4. Are there any problems with implementing the Markowitz model in practice?
See SAB p.176 and 177 "The trouble with optimisers."

5.What does the efficient set look like if riskfree borrowing is permitted but no
lending is allowed?