ECO00032M Investment and Portfolio Management 2020
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ECO00032M
MSc Degree Examinations 2020
DEPARTMENT OF ECONOMICS & RELATED STUDIES
Investment and Portfolio Management
Answer any THREE questions
1.
(a) [25 marks] Suppose that you have quadratic utility:
U = E[rP ] - AσP(2) .
Consider four investment opportunities:
Investment Expected return E[r] Standard deviation, σ
1 |
0.12 |
0.30 |
2 |
0.15 |
0.50 |
3 |
0.21 |
0.16 |
4 |
0.24 |
0.21 |
On the basis of the utility formula above, which investment would you select if you were risk averse with A = 4? Which investment would you select if you were risk neutral?
Answer: We choose the investment with the highest utility value, Investment 3.
Investment Expected return E[r] Standard deviation, σ Utility, U
1 |
0.12 |
0.30 |
-0.0600 |
2 |
0.15 |
0.50 |
-0.3500 |
3 |
0.21 |
0.16 |
0.1588 |
4 |
0.24 |
0.21 |
0.1518 |
When investors are risk neutral, then A = 0; the investment with the highest utility is Investment 4 because it has the highest expected return.
(b) [25 marks] Given $100, 000 to invest, what is the expected risk premium (in percent) of investing in equities versus risk-free T-bills on the basis of the following table?
Equity T-bills
Probability Profit/Loss
0.60 $50, 000
0.40 -$30, 000
1.00
$5, 000
Answer: The expected risk premium in dollars is:
Dollarriskpremium = (0.6 × $50, 000) + [0.4 × (-$30, 000)] - $5, 000 = $13, 000
Hence, the risk premium in percent is:
= 13%
25 marks You manage an equity fund with an expected risk premium of 10% and an expected standard deviation of 14%. The rate on Treasury bills is 6%. Your client chooses to invest $60 , 000 of her portfolio in your equity fund and $40, 000 in a T-bill money market fund. What is the expected return (in percent and in dollars) and standard deviation of return on your client’s portfolio?
Answer:
Expectedreturnforequity = T - billrate + Riskpremium
= 6% + 10% = 16%
Expectedrateofreturnoftheclient′ sportfolio = (0.6 × 16%) + (0.4 × 6%) = 12% Expectedreturnoftheclient′ sportfolio = 0.12 × $100, 000 = $12, 000
(which implies expected total wealth at the end of the period = $112 , 000) Standarddeviationofclient′ soverallportfolio = 0.6 × 14% = 8.4%
(c) [25 marks] Suppose that you have found the optimal risky portfolio P with expected return E[rP ] and standard deviation σP .
● Show that any portfolio consisting of a proportion y (y < 1) invested in the portfolio P and (1 - y) invested at the risk-free rate rf has the same Sharpe ratio as the risky portfolio alone. Answer: In the expected return-standard deviation plane all portfolios that are constructed from the same risky and risk-free funds (with various proportions) lie on a line from the risk-free rate through the risky fund. The slope of the capital allocation line (CAL) is the same everywhere; hence the reward-to-volatility (Sharpe ratio is the same for all of these portfolios. Formally, if you invest a proportion y in a risky fund with expected return E[rP ] and standard deviation σP and the remainder (1 - y) in a risk-free asset with a sure rate rf , then the portfolio’s expected return and standard deviation are
E[rC ] = rf + y(E[rP ] - rf )
σC = yσP
and therefore the Sharpe ratio of this portfolio is:
SC = = = = SP
● Now suppose that the borrowing rate is higher than the lending risk-free rate, r f(B) > r f(L) . What is the capital allocation line now? Plot it in the expected return-standard deviation space.
Answer: The CAL will be now kinked at point P . To the left of P the investor is lending at r f(L) and to the right of P , where y > 1, the investor is borrowing at
rf(B) > rf(L).ftbpF4.7011in2.9542in0ptcalk inked.jpg
2.
(a) [40 marks] Assume that security returns are generated by the single-index model Ri = αi + βi RM + ei ,
where Ri is the excess return for security i and RM is the market excess return. The risk free rate is 2%. Suppose also that there are three securities, A, B and C, characterized by the following data:
Security βi E[Ri] σ(ei )
A |
0.8 |
10% |
25% |
B |
1.0 |
12% |
10% |
C |
1.2 |
14% |
20% |
● If σM = 20%, calculate the variance of returns of securities A, B and C . Answer:
σ 2 = β 2 σ M(2) + σ2 (e)
σA(2) = (0.82 × 202 ) + 252 = 881or0.0881
σ B(2) = (1.02 × 202 ) + 102 = 500
σC(2) = (1.22 × 202 ) + 202 = 976
● Now assume that there is an infinite number of assets with return characteristics identical to those of A, B and C. If one forms a well-diversified portfolio composed only of type A, B or C stocks, what would be the mean and variance of these portfolio’s excess returns?
Answer: If there are an infinite number of assets with identical characteristics, then a well-diversified portfolio of each type will have only systematic risk since the nonsystematic risk will approach zero with large n. Each variance is simply 2 × market variance:
Well - diversifiedσA(2) ≈ 256
Well - diversifiedσ B(2) ≈ 400
Well - diversifiedσ C(2) ≈ 576
● Is there an arbitrage opportunity in this market? Analyze it graphically.
Answer: Although the expected excess return-beta co-ordinates of these securities lie on a straight line, it seems that they do not lie on the security market line. For excess returns the SML should cross the vertical axis at zero, while a line crossing the security co-ordinates is too flat. Therefore, we can conclude that there are some arbitrage opportunities here. Assuming that the security B is fairly priced, the graphical representation would like as in the following figure.ftbpF3.7178in2.6792in0ptfigs ml.jpg
(b) [30 marks] What are the following “effects” considered efficient market anomalies? Are there rational explanations for any of these effects?
● P/E (price-earnings) effect.
● Book-to-market effect.
● Momentum effect.
● Small-firm effect.
Answer: The following effects seem to suggest predictability within equity markets and thus disprove the efficient market hypothesis. However, consider the following:
P/E effect: Multiple studies suggest that “value” stocks (measured often by low P/E multiples) earn higher returns over time than “growth” stocks (high P/E multiples). This could suggest a strategy for earning higher returns over time. However, another rational argument may be that traditional forms of CAPM (such as Sharpe’s model) do not fully account for all risk factors that affect a firm’s price level. A firm viewed as riskier may have a lower price and thus P/E multiple. Book-to-market effect: The book-to-market effect suggests that an investor can earn excess returns by investing in companies with high book value (the value of a firm’s assets minus its liabilities divided by the number of shares outstanding) to market value. A study by Fama and French suggests that book-to-market value reflects a risk factor that is not accounted for by traditional one variable CAPM. For example, companies experiencing financial distress see the ratio of book to market value increase. Thus a more complex CAPM that includes
book-to-market value as an explanatory variable should be used to test market anomalies. Momentum: Stock price momentum can be positively correlated with past performance (short to intermediate horizon) or negatively correlated (long horizon). Historical data seem to imply statistical significance to these patterns. Explanations for this include a bandwagon effect or the behaviorists’ (see Chapter 12) explanation that there is a tendency for investors to underreact to new information, thus producing a positive serial correlation. However, statistical significance does not imply economic significance. Several studies that included transaction costs in the momentum models discovered that momentum traders tended to not outperform the efficient market hypothesis strategy of buy and hold.
Small-firm effect: The small-firm effect states that smaller firms produce better returns than larger firms. Since 1926, returns from small firms outpace large firm stock returns by about 1% per year. The measure of systematic risk according to Sharpe’s CAPM is the stock’s beta (or sensitivity of returns of the stock to market returns). If the stock’s beta is the best explanation of risk, then the small-firm effect does indicate an inefficient market. Dividing the market into deciles based on their betas shows an increasing relationship between betas and returns. Fama and French show that the empirical relationship between beta and stock returns is flat over a fairly long horizon (1963–1990). Breaking the market into deciles based on sizes and then examining the relationship between beta and stock returns within each size decile exhibits this flat relationship. This implies that firm size may be a better measure of risk than beta and the size-effect should not be viewed as an indicator that markets are inefficient. Heuristically this makes sense, as smaller firms are generally viewed as risky compared to larger firms and perceived risk and return are positively correlated.
In general all these ‘anomalies’ might indicate market inefficiency but they can be also explained by risk-based stories, since they can represent systematic factors not captured by the CAPM.
(c) [30 marks] Suppose you want to test the CAPM model: E[Ri] = βi E[RM ],
where Ri is the excess return for portfolio i and RM is the market excess return. Describe the Fama and MacBeth (1973) two-pass testing procedure, assuming that you have observations on the excess returns of N portfolios over T periods and the excess returns on the market portfolio. Answer: First, for each portfolio we estimate the beta coefficients as the slope of a first-pass time series regression equation:
Ri,t = ai + bi RM,t + ei,t .
This will give us N estimates of bi . In the second step, for a sequence of T periods, Fama and MacBeth proposed to estimate for each period the cross-sectional regression:
Ri,t = γ0 + γ1i + ηt ,
where i are estimates from the first-pass regressions. Having obtained a series of T estimates of γ0 and γ1 , we can now calculate their mean values and associated t-statistics. If the model holds, we should not be able to reject the null hypotheses that γ0 = 0 and γ1 = RM .
[For extra points] Actually, Fama and MacBeth extended the usual test of the CAPM by including extra terms in the second pass regressions:
Ri,t = γ0 + γ1i + γ2i(2) + γ2 (ei ) + ηt ,
where (ei ) are the standard deviations of the first-pass residuals. The term γ2 measures potential nonlinearity of return and γ3 measures the explanatory power of nonsystematic risk (ei ). According to the CAPM, γ2 = γ3 = 0.
3.
(a) [40 marks] Suppose that Nodett is a firm that is all-equity financed and has total assets of $100 million. It pays corporate taxes at the rate of 40% of taxable earnings. The following table shows Nodett’s earnings before interest and taxes (EBIT) and net profits under three scenarios representing phases of the business cycle.
Scenario EBIT
Bad year |
5 |
Normal year |
10 |
Good year |
15 |
● Find Nodett’s return on assets (ROA) and return on equity (ROE) in particular years. Answer:
ROA =
ROE =
EBIT
Assets
Netprofit
Equity
Since Nodett is all-equity financed, its Equity = Assets. The net profits are (1 - 0.40) × EBIT . Thus:
Scenario EBIT ROA Net profits ROE
Bad year |
5 |
5% |
3 |
3% |
Normal year |
10 |
10% |
6 |
6% |
Good year |
15 |
15% |
9 |
9% |
● Mordett is a company with the same assets as Nodett but it has a debt-to equity ratio of 1 .0 and an interest rate of 9%. What would its net profit and ROE be in particular years. Answer: A debt-to-equity ratio of 1.0 implies that Mordett will have $50 million of debt of $50 million of equity. Interest expense will be 0 .09 × $50 million, or $4.5 million per year. Mordett’s net profits and ROE over the business cycle will therefore be
(1 - 0.40) × (EBIT - $4.5)
Hence:
Scenario EBIT Net profits ROE
Bad year |
5 |
0.3 |
0.6% |
Normal year |
10 |
3.3 |
6.6% |
Good year |
15 |
6.3 |
12.6% |
(b) [30 marks] Consider two firms producing smartphones. One uses a highly automated robotics process, whereas the other uses workers on an assembly line and pays overtime when there is heavy production demand.
● Which firm will have higher profits in a recession and which in a boom? Explain.
● Which firm’s stock will have a higher beta? Why?
Answer: The robotics process entails higher fixed costs and lower variable (labor) costs. Therefore, this firm will perform better in a boom and worse in a recession. For example, costs will rise less rapidly than revenue when sales volume expands during a boom.
Because the more automated firm’s profits are more sensitive to the business cycle, the robotics firm will have the higher beta.
(c) [30 marks] Use the DuPont decomposition and the following data to find return on equity ROE .
2022-01-08