EIFM Seminar 8 – week commencing Nov 26^{th} 2021
Question 1: Consider two assets: Unilever stock and Burberry stock. Unilever stock either provides a rate of return of 15% or 5%, with equal probability. Burberry stock either provides a return of 20% or 10%, with equal probability. You decide to hold a portfolio by investing half of your money in Unilever stock and the other half in Burberry stock.
Calculate the expected return and standard deviation of Unilever stock.
Calculate the expected return and standard deviation of Burberry stock.
If Unilever stock and Burberry stock’s returns are perfectly negatively correlated, what will be the expected return and standard deviation of your portfolio?
If Unilever stock and Burberry stock’s returns are independent of each other, what will be the expected return and standard deviation of your portfolio?
Answer:
Unilever stock
R^{e} = 0.5×0.15+0.5×(0.05) = 0.05 = 5%
Burberry stock
R^{e} = 0.5×0.2+0.5×(0.1) = 0.05 = 5%
Portfolio (perfectly negatively correlated)
R^{e} = 0.5×0.025+0.5×0.075=0.05= 5%
Portfolio (independent)
R^{e} = 0.25×0.175+0.25×0.025+0.25×0.075+0.25×(0.075)=0.05= 5%
Question 2: Consider historical data showing that the average annual rates of return on the S&P 500 portfolio over the past 80 years has averaged roughly 8% more than the Treasury bill return and that the S&P 500 standard deviation has been about 20% per year. Assume these values are representative of investors’ expectations for future performance and that the current Tbill rate is 5%.
Calculate the expected return and variance of portfolios invested in Tbills and the S&P 500 index with weights as follows.
Calculate the utility levels of each portfolio of part a) for an investor with A = 2. What do you conclude?
Repeat part b) for an investor with A = 3. What do you conclude?
Answer:
The portfolio expected return and variance are computed as follows:

(1)
W_{Bills}
(2)
r_{Bills}
(3)
W_{Index}
(4)
r_{Index}
r_{Portfolio}
(1)×(2)+(3)×(4)
_{Portfolio}
(3) × 20%
^{ 2}_{ Portfolio} 0.0
5%
1.0
13.0%
13.0% = 0.130
20% = 0.20
0.0400
0.2
5%
0.8
13.0%
11.4% = 0.114 16% = 0.16 0.0256
0.4
5%
0.6
13.0%
9.8% = 0.098 12% = 0.12 0.0144
0.6
5%
0.4
13.0%
8.2% = 0.082
8% = 0.08 0.0064
0.8
5%
0.2
13.0%
6.6% = 0.066 4% = 0.04 0.0016
1.0
5%
0.0
13.0%
5.0% = 0.050 0% = 0.00 0.0000
Computing utility from U = E(r) – 0.5 × Aσ^{2} = E(r) – σ^{2}, we arrive at the values in the column labeled U(A = 2) in the following table:

W_{Bills}
W_{Index}
r_{Portfolio}
_{Portfolio}
^{2}_{Portfolio} U(A = 2)
U(A = 3)
0.0
1.0
0.130
0.20
0.0400
0.0900
.0700
0.2
0.8
0.114
0.16
0.0256
0.0884
.0756
0.4
0.6
0.098
0.12
0.0144
0.0836
.0764
0.6
0.4
0.082
0.08
0.0064
0.0756
.0724
0.8
0.2
0.066
0.04
0.0016
0.0644
.0636
1.0
0.0
0.050
0.00
0.0000
0.0500
.0500
The column labeled U(A = 2) implies that investors with A = 2 prefer a portfolio that is invested 100% in the market index to any of the other portfolios in the table.
The column labeled U(A = 3) in the table above is computed from:
U = E(r) – 0.5Aσ^{2} = E(r) – 1.5σ^{2}
The more risk averse investors prefer the portfolio that is invested 40% in the market, rather than the 100% market weight preferred by investors with A = 2.
Question 3: You estimate that a passive portfolio, that is, one invested in a risky portfolio that mimics the S&P 500 stock index, yields an expected rate of return of 13% with a standard deviation of 25%. You manage an active portfolio with expected return 18% and standard deviation 28%. The riskfree rate is 8%. Your client ponders whether to switch the 70% that is invested in your fund to the passive portfolio. Explain to your client the disadvantage of the switch.
Answer: Sharpe ratio of the index fund
Sharpe ratio of the active fund = (0.180.08)/0.28 = 0.3571
CML plots the portfolios made of the market index fund and riskfree asset. CAL plots the portfolios made of the active fund and the riskfree asset. The latter offers more expected return for a given level of risk. Your fund allows an investor to achieve a higher mean for any given standard deviation than would a passive strategy, i.e., a higher expected return for any given level of risk.
Question 4: Stocks offer an expected rate of return of 18%, with a standard deviation of 22%. Gold offers an expected return of 10% with a standard deviation of 30%. In light of the apparent inferiority of gold with respect to both mean return and volatility, would anyone hold gold? If so, demonstrate graphically why one would do so.
Answer:
Even though it seems that gold is dominated by stocks, gold might still be an attractive asset to hold as a part of a portfolio. If the correlation between gold and stocks is sufficiently low, gold will be held as a component in a portfolio, specifically, the optimal tangency portfolio.