Unit 2

You manage a risky portfolio with an expected rate of return of 18% and a standard deviation of 28%. The T-bill rate is 8%. Your client chooses to invest 70% of a portfolio in your fund and 30% in an essentially risk-free money market fund. What is the expected value and standard deviation of the rate of return on his portfolio?

Expected return = (0.7 × 18%) + (0.3 × 8%) = 15% Standard deviation = 0.7 × 28% = 19.6%

You manage a risky portfolio with an expected rate of return of 18% and a standard deviation of 28%. The T-bill rate is 8%. Suppose that your risky portfolio includes the following investments in the given proportions: Stock A 25% Stock B 32% Stock C 43% What are the investment proportions of your client's overall portfolio, including the position in T-bills?

Investment proportions: 30.0% in T-bills 0.7 × 25% = 17.5% in Stock A 0.7 × 32% = 22.4% in Stock B 0.7 × 43% = 30.1% in Stock C

You manage a risky portfolio with an expected rate of return of 18% and a standard deviation of 28%. The T-bill rate is 8%. Draw the CAL of your portfolio on an expected return-standard deviation diagram. What is the slope of the CAL? Show the position of your client on your fund's CAL.

Slope = 0.3571 Portfolio is at E(r) = between 15 & 20% SD = under 30% Client is at E(r) = 15% SD = 20%

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 rate of 8%. The probability distribution of the risky funds is as follows: Expected Return Standard Deviation Stock fund (S) 20% 30% Bond fund (B) 12% 15% The correlation between the fund returns is .10. What are the investment proportions in the minimum-variance portfolio of the two risky funds, and what is the expected value and standard deviation of its rate of return?

The parameters of the opportunity set are: E(rS) = 20%, E(rB) = 12%, σS = 30%, σB = 15%, ρ = 0.10 From the standard deviations and the correlation coefficient we generate the covariance matrix [note that Cov(rs, rb) = ρ * σs * σB Bonds Stocks Bonds 225 45 Stocks 45 900 The minimum-variance portfolio is computed: Wmin (S) = (σ^2B - Cov(rs, rB) / (σ^2s + σ^2B - 2* COV) (225 - 45) / (900 + 225 - (2 * 45) = 0.1738 Wmin (B) = 1 - Wmin (S) >> 1 - 0.1739 The minimum variance portfolio mean and SD are: E(r min) = (0.1739 * .20) + (0.8261 * .12) = 0.1339 or 13.39% σmin = [(0.17392 ´ 900) + (0.82612 ´ 225) + (2 ´ 0.1739 ´ 0.8261 ´ 45)]1/2 = 13.92%

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 rate of 8%. The probability distribution of the risky funds is as follows: Expected Return Standard Deviation Stock fund (S) 20% 30% Bond fund (B) 12% 15% The correlation between the fund returns is .10. 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%. 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 portfolio? C. Solve numerically for the proportions of each asset and for the expected return and standard deviation of the optimal risky portfolio. D. What is the Sharpe ratio of the best feasible CAL?

https://www.chegg.com/homework-help/questions-and-answers/pension-fund-manager-considering-three-mutual-funds-first-stock-fund-second-long-term-gove-q22314035 B. The above graph indicates that the optimal portfolio is the tangency portfolio with expected return approximately 15.6% and standard deviation approximately 16.5%. C. The mean and standard deviation of the optimal risky portfolio are: E(rP) = (0.4516 × .20) + (0.5484 × .12) = .1561 = 15.61% σp = [(0.45162 ´ 900) + (0.54842 ´ 225) + (2 ´ 0.4516 ´ 0.5484 × 45)]1/2 = 16.54% D. (.1561 - 0.08) / .1654 = 0.4601

You manage a risky portfolio with an expected rate of return of 18% and a standard deviation of 28%. The T-bill rate is 8%. Your client's degree of risk aversion is A = 3.5. a. What proportion, y, of the total investment should be invested in your fund? b. What is the expected value and standard deviation of the rate of return on your client's optimized portfolio?

A. y* = (E9rp) - Rf)/A * (SDp)^2 = (0.18 - 0.08)/(3.5 * 0.28^2) = 0.10/0.2744 = 0.3644 Therefore, the client's optimal proportions are 36.44% invested in the risk portfolio & 63.56% in T-Bills B. E(rc) = 0.08 + 0.10 * y* = 0.08 + (0.3644 * 0.1) = 0.1164 or 11.64% SD c = 0.3644 * 28 = 10.203%

You manage a risky portfolio with an expected rate of return of 18% and a standard deviation of 28%. The T-bill rate is 8%. Suppose that your client decides to invest in your portfolio a proportion y of the total investment budget so that the overall portfolio will have an expected rate of return of 16%. a. What is the proportion y? b. What are your client's investment proportions in your three stocks and the T-bill fund? c. What is the standard deviation of the rate of return on your client's portfolio?

A. E(rc) = Rf + y * (E(rp) - Rf) = 8 + y * (18 - 8) If the expected return for the portfolio is 16% then, 16% = 8% + 10% * y >>> y = (.16 - .08)/.10 = 0.8 Therefore, in order to have a portfolio with expected rate of return equal to 16%, the client must invest 80% of total funds in the risky portfolio in 20% in T-Bills B. Client's Investment Proportions: 20% in T-Bills 0.8 * 25% = 20% in Stock A 0.8 * 32% = 25.6% in Stock B 0.8 * 43% = 34.4% in Stock C C. Stand Dev c = 0.8 * Stand Dev p = 0.8 * 28% = 22.4%

You manage a risky portfolio with an expected rate of return of 18% and a standard deviation of 28%. The T-bill rate is 8%. Suppose that your client prefers to invest in your fund a proportion y that maximizes the expected return on the complete portfolio subject to the constraint that the complete portfolio's standard deviation will not exceed 18%. a. What is the investment proportion, y? b. What is the expected rate of return on the complete portfolio?

A. Stand Dev c = y * 28% If you client prefers a standard deviation of at most 18% then, y = 18/28 = 0.6429 or 64.29% B. E(rc) = 0.08 + .1 * y = 0.08 + ().6429 * -.1) = 14.429%

The Fisher equation tells us that the real interest rate approximately equals the nominal rate minus the inflation rate. Suppose the inflation rate increases from 3% to 5%. Does the Fisher equation imply that this increase will result in a fall in the real rate of interest? Explain.

The Fisher equation predicts that the nominal rate will equal the equilibrium real rate plus the expected inflation rate. Hence, if the inflation rate increases from 3% to 5% while there is no change in the real rate, then the nominal rate will increase by 2%. On the other hand, it is possible that an increase in the expected inflation rate would be accompanied by a change in the real rate of interest. While it is conceivable that the nominal interest rate could remain constant as the inflation rate increased, implying that the real rate decreased as inflation increased, this is not a likely scenario.

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 rate of 8%. The probability distribution of the risky funds is as follows: Expected Return Standard Deviation Stock fund (S) 20% 30% Bond fund (B) 12% 15% The correlation between the fund returns is .10. If you were to use only the two risky funds, and still require an expected return of 14%, what would be the investment proportions of your portfolio? Compare its standard deviation to that of the optimized portfolio in Problem 9. What do you conclude?

Using only the stock and bond funds to achieve a portfolio expected return of 14%, we must find the appropriate proportion in the stock fund (wS) and the appropriate proportion in the bond fund (wB = 1 − wS) as follows: 0.14 = 0.20 × wS + 0.12 × (1 − wS) = 0.12 + 0.08 × wS Þ wS = 0.25 So the proportions are 25% invested in the stock fund and 75% in the bond fund. The standard deviation of this portfolio will be: σP = [(0.252 ´ 900) + (0.752 ´ 225) + (2 ´ 0.25 ´ 0.75 ´ 45)]1/2 = 14.13% This is considerably greater than the standard deviation of 13.04% achieved using T-bills and the optimal portfolio.

You manage a risky portfolio with an expected rate of return of 18% and a standard deviation of 28%. The T-bill rate is 8%. What is the reward-to-volatility (Sharpe) ratio (S) of your risky portfolio? Your client's?

Your Reward-to-volatility (Sharpe) Ratio S = (.18 - .08)/.28 = 0/3571 Client's RTV = (.15 - .08) / .196 = 0.3571 SHARPE RATIO = (Expected Return - Rf) / Standard Dev

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 rate of 8%. The probability distribution of the risky funds is as follows: Expected Return Standard Deviation Stock fund (S) 20% 30% Bond fund (B) 12% 15% The correlation between the fund returns is .10 . You require that your portfolio yield an expected return of 14%, and that it be efficient, on the best feasible CAL. a. What is the standard deviation of your portfolio? b. What is the proportion invested in the T-bill fund and each of the two risky funds

a. If you require that your portfolio yield an expected return of 14%, then you can find the corresponding standard deviation from the optimal CAL. The equation for this CAL is: E(rc) = Rf + ((E(rp) - Rf) / σp) * σc = 0.08 + 0.4601 * σc If E(rC) is equal to 14%, then the standard deviation of the portfolio is 13.04%. B. To find the proportion invested in the T-bill fund, remember that the mean of the complete portfolio (i.e., 14%) is an average of the T-bill rate and the optimal combination of stocks and bonds (P). Let y be the proportion invested in the portfolio P. The mean of any portfolio along the optimal CAL is: E(rc) = (1 - y) * Rf + y * E(rp) = Rf + y * (E(rp) - Rf) = 0.08 + y * (.1561 - 0.08) Setting E(rC) = 14% we find: y = 0.7884 and (1 − y) = 0.2119 (the proportion invested in the T-bill fund). To find the proportions invested in each of the funds, multiply 0.7884 times the respective proportions of stocks and bonds in the optimal risky portfolio: Proportion of stocks in complete portfolio = 0.7884 ´ 0.4516 = 0.3560 Proportion of bonds in complete portfolio = 0.7884 ´ 0.5484 = 0.4323