PROBLEM 4.31
Portfolio Optimization Analysis
Problem Statement
A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term government and corporate bond fund, and the third is a T-bill money market fund that yields a sure rate of 5.5%. The probability distributions of the risky funds are:
| Expected return | Standard deviation | |
|---|---|---|
| Stock fund | 15% | 32% |
| Bond fund | 9% | 23% |
The correlation between the fund returns is 0.15.
Questions
a
Tabulate and draw the investment opportunity set of the two risky funds. Use investment proportions for the stock fund of 0 to 100% in increments of 20%. What expected return and standard deviation does your graph show for the minimum variance portfolio?
b
Draw a tangent from the risk-free rate to the opportunity set. What does your graph show for the expected return and standard deviation of the optimal risky portfolio?
c
What is the reward-to-variability ratio of the best feasible CAL?
d
Suppose now that your portfolio must yield an expected return of 12% and be efficient, that is, on the best feasible CAL.
i. What is the standard deviation of your portfolio?
ii. What is the proportion invested in the T-bill fund and each of the two risky funds?
i. What is the standard deviation of your portfolio?
ii. What is the proportion invested in the T-bill fund and each of the two risky funds?
e
If you were to use only the two risky funds and still require an expected return of 12%, what would be the investment proportions of your portfolio? Compare its standard deviation to that of the optimal portfolio in the previous problem. What do you conclude?
Solutions
a
We have, Investment proportion of stock fund (wS)
$$w_S = \frac{\sigma_B^2 - \text{Cov}_{SB}}{\sigma_S^2 + \sigma_B^2 - 2\text{Cov}_{SB}} = \frac{(23)^2 - 110.4}{(32)^2 + (23)^2 - 2 \times 110.4} = \frac{418.6}{1,332.2} = 0.3142$$
Working notes: CovSB = ρSB × σS × σB = 0.15 × 32% × 23% = 110.4
Investment proportion of bond fund (wB) = 1 - ws = 1 - 0.3142 = 0.6858
Calculation of expected value and standard deviation of rate of return on minimum variance portfolio:
Expected return on minimum variance portfolio
E(rmin) = wS E(rS) + wB E(rB) = 0.3142 × 15 + 0.6858 × 9 = 10.8852% ≈ 10.89%
Expected return on minimum variance portfolio
E(rmin) = wS E(rS) + wB E(rB) = 0.3142 × 15 + 0.6858 × 9 = 10.8852% ≈ 10.89%
Standard deviation of returns on minimum variance portfolio:
$$\sigma_P = \sqrt{W_S^2 \times \sigma_S^2 + W_B^2 \times \sigma_B^2 + 2W_SW_B\text{Cov}_{SB}}$$ $$= \sqrt{(0.3142)^2 \times (32)^2 + (0.6858)^2 \times (23)^2 + 2 \times 0.3142 \times 0.6858 \times 110.4}$$ $$= \sqrt{101.0916 + 248.5001 + 47.576}$$ $$= \sqrt{397.1687} = 19.93\% ≈ 19.04\%$$
$$\sigma_P = \sqrt{W_S^2 \times \sigma_S^2 + W_B^2 \times \sigma_B^2 + 2W_SW_B\text{Cov}_{SB}}$$ $$= \sqrt{(0.3142)^2 \times (32)^2 + (0.6858)^2 \times (23)^2 + 2 \times 0.3142 \times 0.6858 \times 110.4}$$ $$= \sqrt{101.0916 + 248.5001 + 47.576}$$ $$= \sqrt{397.1687} = 19.93\% ≈ 19.04\%$$
| Proportion in stock fund |
Proportion in bond fund |
Expected return |
Standard Deviation |
|
|---|---|---|---|---|
| 0.00 | 1.0000 | 9% | 23% | |
| 0.20 | 0.8000 | 10% | 20% | |
| 0.3142 | 0.6858 | 10.89% | 19.04% | Minimum variance portfolio |
| 0.40 | 0.6000 | 11% | 20% | |
| 0.60 | 0.4000 | 13% | 23% | |
| 0.6466 | 0.3534 | 12.88% | 25.34% | Tangency portfolio |
| 0.8000 | 0.2000 | 14% | 27% | |
| 1.0000 | 0.0000 | 15% | 32% |
| The graph approximates the points: | E(r) | σ |
| Minimum variance portfolio | 10.89% | 19.04% |
b
Calculation of portfolio risk and return at various weights
| Tangency portfolio | 12.88% | 25.34% |
c
The reward-to-variability ratio (Sharpe ratio) of the optimal CAL is:
$$\text{Sharpe ratio} = \frac{E(r_p) - r_f}{\sigma_p} = \frac{12.88 - 5.5}{25.34} = \frac{7.38}{25.34} = 0.3162$$
d
Expected return, E(rp) = 12%
i. The equation for the CAL is:
$$E(r_c) = r_f + \frac{E(r_p) - r_f}{\sigma_p} \times \sigma_c$$ Or, 12% = 5.5 + 0.3162 σc
Or, 6.5% = 0.3162 σc
Or, σc = 20.5566%
$$E(r_c) = r_f + \frac{E(r_p) - r_f}{\sigma_p} \times \sigma_c$$ Or, 12% = 5.5 + 0.3162 σc
Or, 6.5% = 0.3162 σc
Or, σc = 20.5566%
ii. The mean of the complete portfolio as a function of the proportion invested in the risky portfolio (y) is:
E(rc) = rf + [E(rp) - rf] y
Or, 12% = 5.5 + (12.88 - 5.5) y
Or, 6.5% = 7.38% y
Or, y = 0.8808 Or, 88.08%
Therefore the weight of risky portfolio is 88.08%.
Weight of risk free asset = 1 - y = 1 - 0.8808 = 0.1192 Or, 11.92%
Proportion of stocks in complete portfolio (Ws) = 0.6466 × 0.8808 = 0.5695
Proportion of bonds in complete portfolio (WB) = 0.3534 × 0.8808 = 0.3113
E(rc) = rf + [E(rp) - rf] y
Or, 12% = 5.5 + (12.88 - 5.5) y
Or, 6.5% = 7.38% y
Or, y = 0.8808 Or, 88.08%
Therefore the weight of risky portfolio is 88.08%.
Weight of risk free asset = 1 - y = 1 - 0.8808 = 0.1192 Or, 11.92%
Proportion of stocks in complete portfolio (Ws) = 0.6466 × 0.8808 = 0.5695
Proportion of bonds in complete portfolio (WB) = 0.3534 × 0.8808 = 0.3113
We have,
E(rp) = Ws × E(rs) + WB × E(rB)
Or, 12 = Ws × 15 + (1 - Ws) × 9
Or, 3 = Ws × 6
Or, Ws = 0.50
$$\sigma_p = \sqrt{W_s^2 \times \sigma_s^2 + W_B^2 \times \sigma_B^2 + 2W_sW_B\text{Cov}_{SB}}$$ $$= \sqrt{(0.50)^2 \times (32)^2 + (0.50)^2 \times (23)^2 + 2 \times 0.50 \times 0.50 \times 110.4}$$ $$= \sqrt{256 + 132.25 + 55.20}$$ $$= \sqrt{443.45} ≈ 21.06\%$$
E(rp) = Ws × E(rs) + WB × E(rB)
Or, 12 = Ws × 15 + (1 - Ws) × 9
Or, 3 = Ws × 6
Or, Ws = 0.50
$$\sigma_p = \sqrt{W_s^2 \times \sigma_s^2 + W_B^2 \times \sigma_B^2 + 2W_sW_B\text{Cov}_{SB}}$$ $$= \sqrt{(0.50)^2 \times (32)^2 + (0.50)^2 \times (23)^2 + 2 \times 0.50 \times 0.50 \times 110.4}$$ $$= \sqrt{256 + 132.25 + 55.20}$$ $$= \sqrt{443.45} ≈ 21.06\%$$
The efficient portfolio with a mean of 12% has a standard deviation of only 21.06%. This is considerably greater than the standard deviation of 20.5566% achieved using CAL.
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