Chapter 11

CC

2. Why diversification reduces portfolio risk as measured by the portfolio standard deviation is important and worth exploring in some detail. The key concept is correlation, which is the extent to which returns on two assets move together. If the returns on two assets tend to move up and down together, we say they are positively correlated. If they tend to move in opposite directions, we say they are negatively correlated. If there is no particular relationship between the two assets, we say they are uncorrelated.

3. Any portfolio that plots below the minimum variance portfolio is a poor choice because, no matter which one you pick, there is another portfolio with the same risk and a much better return. In the jargon of finance, we say that these undesirable portfolios are dominated or inefficient. Either way, we mean that given their level of risk, the expected return is inadequate compared to some other portfolio of equivalent risk. A portfolio that offers the highest return for its level of risk is said to be an efficient portfolio.

7. We know that our current portfolio is the minimum variance portfolio (or below). According to this as you add more of the lower risk asset, the standard deviation of your portfolio increases and the expected return decreases.

Q&P

9. a)

boom: E[Rp] = 0.25(0.30) + 0.50(0.45) + 0.25(0.33) = 0.3825good: E[Rp] = 0.25(0.12) + 0.50(0.10) + 0.25(0.15) = 0.1175poor: E[Rp] = 0.25(.01) + 0.50(–0.15) + 0.25(–0.05) = –0.0850bust: E[Rp] = 0.25(–0.06) + 0.50(–0.30) + 0.25(–0.09) = –0.1875E[Rp] = 0.20(0.3825) + 0.40(0.1175) + 0.30(–0.0850) + 0.10(–0.1875) = 0.0793

b)

15.

16.

Chapter 12

CC

7. It is possible for a risky asset to have a beta of zero. Such an asset’s return is simply uncorrelated with the overall market. Based on the CAPM, this asset’s expected return would be equal to the risk-free rate. It is also possible to have a negative beta; the return would be less than the risk-free rate. A negative beta asset would carry a negative risk premium because of its value as a diversification instrument. A negative beta asset can be created by shorting an asset with a positive beta. A portfolio with a zero beta can always be created by combining long and short positions.

9. If every asset has the same reward-to-risk ratio, the implication is that every asset provides the same risk premium for each unit of risk. In other words, the only way to increase your return is to accept more risk. Investors will only take more risk if the reward is higher, and a constant reward-to-risk ratio ensures this will happen. We would expect every asset in a liquid, well-functioning to have the same reward-to-risk ratio due to competition and investor risk aversion. If an asset has a reward-to-risk ratio that is lower than all other assets, investors will avoid that asset, thereby driving the price down, increasing the expected return and the reward-to-risk ratio. Similarly, if an asset has a reward-to-risk ratio that is higher than other assets, investors will flock to the asset, increasing the price, and decreasing the expected return and the reward-to-risk ratio.

Q&P

12.

14.

(E(RA) – Rf)/σA = (E(RB) – Rf)/σB

σA/σB = (E(RA) – Rf)/(E(RB) – Rf)

18.E[Rp] = 0.15 = wX(0.19) + wY(0.122) + (1 – wX – wY)(0.06)

βp = 0.9 = wX(1.5) + wY(1.1) + (1 – wX – wY)(0)

We get that wX = 0.86400 and wY = –0.36000

wRf = 0.49600

Investment in stock Y = 0.36000($100,000) = $36,000

21.Furhman Labs: E(R) = 5.0% + 1.5(11.5% – 5.0%) = 14.75% > 13.25% Overvalued

Garten Testing: E(R) = 5.0% + 0.8(11.5% – 5.0%) = 10.20% < 11.25% Undervalued

If the forecast return is less than the required rate of return, the security is overvalued. If the forecast return is greater than the required rate of return, the security is undervalued.

Chapter 13

CC

2. A common weakness of the Jensen and Treynor measures is that both require a beta estimate. Betas from different sources can differ substantially, and, as a result, what appears to be a positive alpha might just be due to a miss-measured beta.

3. Third common measure of investment performance that draws on capital asset pricing theory for its formulation is Jensen’s alpha, proposed by Michael C. Jensen. Jensen’s alpha is computed as the raw portfolio return less the expected portfolio return predicted by the by the capital asset pricing model (CAPM). Jensen’s alpha is easy to understand. It is simply the excess return above or below the security market line (SML), and, in this sense, it can be interpreted as a measure of by how much the portfolio “beat the market”. This interpretation can show portfolio with a positive, zero and negative alpha. A positive alpha is a good thing because the portfolio has a relatively high return given its level of systematic risk.

4. The Sharpe ratio has the advantage that no beta is necessary, and standard deviations can be calculated unambiguously. For doing things like evaluating mutual funds, The Sharpe ratio is probably the most frequently used. Furthermore, if a mutual fund is not very diversified, then its standard deviation would be larger, resulting in a smaller Sharpe ratio. Thus, Sharpe ratio, in effect, penalizes a portfolio for being undiversified.

Q&P

2. Annual standard deviation = = 24.64%2-month standard deviation = 24.64% x = 10.06%

7.On a standard normal distribution, the mean is zero and standard deviation is one. These standard deviations just represent z values on this standard curve. Thus, we have the following percentages for the values:

5% = –1.6452.5% = –1.961% = –2.326

12.Prob (R ≤ .15 – 2.32(.47)) = 1%

Prob (R ≤ –.943) = 1%

20.E(R) = (0.12 + 0.17) / 2 = 0.1450

σ = [(0.5 )(0.41 ) + (0.5 )(0.62 ) + 2(0.5)(0.5)(0.41)(0.61)(0.5)] = 0.4491

Prob(R ≤ (0.1450/12) – 1.645(0.4491)(1/12) ) = 5%

Prob(R ≤ –0.2012) = 5%

25.Expected Return A 10.00%Standard deviation A 21.00%Expected Return B 15.00%Standard deviation B 62.00%Correlation 0.30Risk-free Rate 4.00%Starting Weight of Asset A 88.21%

Weight of Asset A 88.21%Weight of Asset B 1-0.8821= 11.79%

Portfolio Expected Return = (Weight of Asset A x Expected Return A) + (Weight of Asset B x Expected Return B)= (88.21% x 10.00%) + (11.29% x 15.00%) = 8.82% + 1.69%= 10.51%

Portfolio Standard Deviation =

=

= 21.86%

Sharpe Ratio = (Portfolio Expected Return-Risk-free Rate)/ 21.86%= (10.51%-4.00%)/ 21.86%= 6.51%/ 21.86% = 0.0651/ 0.2186= 0.2978