Copyright ⃝c Copyright University of Wales 2021. All rights reserved. Economics of Finance
Tutorial 7
1. Suppose an investor decides to construct a portfolio consisting of a risk-free asset that pays 6 percent and a stock index fund that has an expected rate of return of 12 percent and a standard deviation of 20 percent. Let x denote the proportion invested in the stock index fund. This implies that the proportion (1 − x) is invested in the risk-free asset. (In your answers use 6 for 6 percent etc).
(a) Find the equation for the efficient frontier. Graph the efficient frontier in (ep,sp) space where ep is the expected return on the portfolio and sp the standard deviation of the return on the portfolio.
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Let eIF and sIF denote the expected rate of return and the standard deviation of the stock index fund. Let eRF be the risk-free rate. Then, the expected return of the portfolio is
ep =xeIF +(1−x)eRF =12x+6(1−x). (1) Notice, that the variance of a linear combination of two random variables y and z is given by
V ar(ay + bz) = a2V ar(y) + b2V ar(z) + 2abCov(y, z).
The variance of the return on the risk-free asset is zero. The covariance of the return on the risk-free
asset with any risky return is zero as well. Therefore, the variance of the portfolio is vp =x2(sIF)2
while its standard deviation is
sp = |x|sIF = 20|x|, (2) where |x| denotes absolute value. Solving (2) for x and substituting it into equation (1) yields the
following equation for the efficient frontier:
ep =6+0.3sp. The graph of the frontier is the blue solid line in this figure.
(b) Suppose the expected utility of the investor is given by
Eu = ep − 0.025s2p. (3)
What does expected utility imply in terms of preferences? What is the interpretation of the coeffi- cient in front of s2p in equation (3)?
Given this mean-variance expected utility, an investor prefers higher expected return, but dislikes risk (variance). The coefficient in front of s2p indicates the rate at which the Investor is willing to trade expected value (or return) for variance. Thus, the risk tolerance t, the reciprocal of this value, is the rate at which the Investor is willing to trade variance for expected return.
(c) What is the investor’s optimal portfolio, assuming that he/she is an expected utility maximiser? (In other words, find the investor’s optimal choice for x). Show graphically the optimal portfolio choice in (ep, sp) space.
Consider the set of all portfolios that provide the level of the expected utility k. In (ep,sp) space, these portfolios will satisfy the following equation:
ep = k + 0.025s2p. The slope of this indifference curve is given by
dep =0.05sp. dsp
Since the equation for the efficient frontier is ep = 6 + 0.3sp, its slope is dep =0.3.
At the optimum the two slopes will be equated. Hence, it follows that 0.05sp = 0.3 or sp = 6. From the equation (2) it follows that 20x = 6 or x = 0.3. From the equation of the frontier we obtain ep = 6 + 0.3 · 6 = 7.8. See the figure above for a graphical exposition.
(d) Consider the portfolio with ep = 8.4 and sp = 8. Show that this portfolio is efficient. Demonstrate that this portfolio is not optimal in the sense of maximizing the investor’s expected utility. (Hint: Calculate the certainty equivalent of this portfolio and the one in part (c) and compare the two). Show graphically the certainty equivalent of this portfolio and the one in part (c) on the same graph in (ep, sp) space.
Since the equation for the efficient frontier is ep = 6 + 0.3sp it must be the case that an efficient portfolio with a standard deviation of 8 percent command the expected return of 6 + 0.3 · 8 = 8.4 percent. This is the case for the portfolio in question.
The utility level our investor will obtain from this portfolio is:
Eud) =ep −0.025s2p =8.4−0.025·(8)2 =6.8
Notice from the utility function in (3) that a risk-free asset with a 6.8 percent rate of return will make our investor equally happy as this risky portfolio. Hence, the certainty equivalent of this portfolio is 6.8.
The utility level our investor will obtain from the portfolio in part (c) is: Euc) =ep −0.025s2p =7.8−0.025·(6)2 =6.9
Hence, the certainty equivalent of this portfolio is 6.9. Since, there is another efficient portfolio with a higher certainty equivalent for our investor, the portfolio with ep = 8.4 and sp = 8 is not optimal. See the figure for a graphical exposition.
(e) Suppose the expected utility of the investor is given by
Eu = ep − 0.01s2p. (4)
What is the investor’s optimal portfolio in this case? Compare your answer to the optimal portfolio you found in part (c). Comment
Consider the set of all portfolios that provide the level of the expected utility k. In (ep,sp) space, these portfolios will satisfy the following equation:
ep = k + 0.01s2p. The slope of this indifference curve is given by
dep =0.02sp. dsp
As before, the slope of the efficient frontier is 0.3 At the optimum the two slopes must be equated: 0.02sp = 0.3. Therefore sp = 15. From the equation (2) it follows that 20x = 15 or x = 0.75. From the equation of the frontier we obtain ep = 6 + 0.3 · 15 = 10.5.
The new investor is less risk averse. His risk tolerance t = 100, while the tolerance of the investor in part b) is t = 40. Notice, that the new investor is 2.5 time more risk tolerant, he/she invested 2.5 times more wealth into risky asset, and his/her optimal portfolio has 2.5 higher standard deviation than that of the original investor. See the figure for a graphical exposition.
(f) Is there a relationship between the standard deviation of the optimal portfolio you found in part (c) and (e) and the investors degree of risk tolerance. If so, can that relationship be described precisely in mathematical terms?
If the efficient portfolio is linear in mean/standard deviation space, there is a one-to-one mapping between risk undertaken and risk tolerance, assuming efficient investment strategies are utilised. Let the efficient frontier be:
e = a + bs.
2 (e−a)2 v=s= b2 .
The rate at which v can be substituted for e along the frontier is thus: dv 2 (e − a) 2 (e − a) 2
de=b2 =b b=bs.
For an investment strategy to be optimal, this must equal the investor’s risk tolerance t:
It follows that, compared with an “average investor”: an Investor with twice the risk tolerance should take twice the risk; an Investor with half the risk tolerance should take half the risk.
2. Consider the following two stocks: stock 1 has expected return e1 = 6 and standard deviation s1 = 16; stock 2 has expected return e2 = 10 and standard deviation s2 = 20. Assume that the correlation between the returns on the two stocks is 0.75, that is, the correlation coefficient r12 = 0.75.
(a) Write down an equation for the expected return (ep) and the variance (vp) of the return on the portfolio as a function of x2 only, where x2 is the proportion of invested wealth in stock 2.
In general, the expected return of the portfolio as a function of x2 is
ep =(1−x2)e1 +x2e2 =e1 +(e2 −e1)x2,
while the variance of the returns is
vp = (1 − x2)2v1 + x2v2 + 2(1 − x2)x2s1s2r12.
Substituting the given values yields
ep =6+4×2,
vp =(1−x2)2 ·162 +x2 ·202 +2(1−x2)x2·16·20·0.75
= 16 11×2 − 2×2 + 16.0
(b) Find the minimum variance portfolio. What is the expected return and variance of this portfolio?
Differentiating vp with the respect to x2 and equating the result to zero yields dvp =16(22×2 −2)=0,
from which it follows that x2 = 1 . Therefore x1 = 10 ,
11 11 −2 sp =
11 +16.0 =16·1511 =254.55 254.55 = 15.955
12 1 10
ep = 6 + 4 1 = 6.3636 11
(c) Is Stock 1 on the efficient frontier? Draw a graph in expected return-standard deviation space to illustrate your answer.
Figure 1: Minimum variance portfolio
Stock 1 is not efficient because the minimum variance portfolio has a higher expected return and a smaller standard deviation (see Figure 1).
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