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博迪投资学第八版Chap005

Chapter 5

Risk and Return: Past and Prologue

1. The 1% VaR will be less than -30%. As percentile or probability of a return declines so does the magnitude of that return. Thus, a 1 percentile probability will produce a smaller VaR than a 5 percentile probability.

2. The geometric return represents a compounding growth number and will artificially inflate the annual performance of the portfolio.

3. No. Since all items are presented in nominal figures, the input should also use nominal data.

4. Decrease. Typically, standard deviation exceeds return. Thus, a reduction of 4% in each will artificially decrease the return per unit of risk. To return to the proper risk return relationship the portfolio will need to decrease the amount of risk free investments.

5. E(r) = [0.3 ? 44%] + [0.4 ? 14%] + [0.3 ? (–16%)] = 14%

σ2 = [0.3 ? (44 – 14)2] + [0.4 ? (14 – 14)2] + [0.3 ? (–16 – 14)2] = 540 σ = 23.24%

The mean is unchanged, but the standard deviation has increased.

a. The holding period returns for the three scenarios are: Boom:

(50 – 40 + 2)/40 = 0.30 = 30.00%

Normal: (43 – 40 + 1)/40 = 0.10 = 10.00% Recession: (34 – 40 + 0.50)/40 = –0.1375 = –13.75%

E(HPR) = [(1/3) ? 30%] + [(1/3) ? 10%] + [(1/3) ? (–13.75%)] = 8.75% σ2(HPR) = [(1/3) ? (30 – 8.75)2] + [(1/3) ? (10 – 8.75)2] + [(1/3) ? (–13.75 – 8.75)2]

= 319.79

σ = 79.319= 17.88%

b. E(r) = (0.5 ? 8.75%) + (0.5 ? 4%) = 6.375%

σ = 0.5 ? 17.88% = 8.94%

a.Time-weighted average returns are based on year-by-year rates of return.

Year Return = [(capital gains + dividend)/price]

2007-2008 (110 – 100 + 4)/100 = 14.00%

2008-2009 (90 – 110 + 4)/110 = –14.55%

2009-2010 (95 – 90 + 4)/90 = 10.00%

Arithmetic mean: 3.15%

Geometric mean: 2.33%

Time Cash flow Explanation

0 -300 Purchase of three shares at $100 per share

1 -208 Purchase of two shares at $110,

plus dividend income on three shares held

2 110 Dividends on five shares,

plus sale of one share at $90

3 396 Dividends on four shares,

plus sale of four shares at $95 per share

Dollar-weighted return = Internal rate of return = –0.1661% 396 | | | |

110|

|| Date:1/1/071/1/081/1/091/1/10 ||

|208

300

a.E(r m) – r f = AσP2 = 4 ? (0.20)2 = 0.16 = 16.0%

b.0.09 = AσP2 = A ? (0.20)2? A = 0.09/0.04 = 2.25

c.Increased risk tolerance means decreased risk aversion (A), which results in a

decline in risk premiums.

9.For the period 1926 – 2008, the mean annual risk premium for large stocks over T-

bills is 9.34%

E(r) = Risk-free rate + Risk premium = 5% + 7.68% =12.68%

10. In the table below, we use data from Table 5.2. Excess returns are real returns since the

risk free rate incorporates inflation.

Large Stocks: 7.68%

Small Stocks: 13.51%

Long-Term T-Bonds: 1.85%

T-Bills: 0.66 % (table 5.4)

11.

a.The expected cash flow is: (0.5 ? $50,000) + (0.5 ? $150,000) = $100,000

With a risk premium of 10%, the required rate of return is 15%. Therefore, if

the value of the portfolio is X, then, in order to earn a 15% expected return: X(1.15) = $100,000 ? X = $86,957

b.If the portfolio is purchased at $86,957, and the expected payoff is $100,000, then

the expected rate of return, E(r), is:

957

, 86

$957 ,

000

100

= 0.15 = 15.0%

The portfolio price is set to equate the expected return with the required rate of return.

c.If the risk premium over T-bills is now 15%, then the required return is:

5% + 15% = 20%

The value of the portfolio (X) must satisfy:

X(1.20) = $100, 000 ? X = $83,333

d.For a given expected cash flow, portfolios that command greater risk premia must

sell at lower prices. The extra discount from expected value is a penalty for risk.

12.

a.E(r P) = (0.3 ? 7%) + (0.7 ? 17%) = 14% per year

σP = 0.7 ? 27% = 18.9% per year

Security Investment Proportions

T-Bills 30.0% Stock A 0.7 ? 27% = 18.9% Stock B 0.7 ? 33% = 23.1% Stock C 0.7 ? 40% = 28.0%

c.Your Reward-to-variability ratio = S =

277

17-

= 0.3704

Client's Reward-to-variability ratio =

187

14-

= 0.3704 d.

13.

a.Mean of portfolio = (1 – y)rf + y rP = rf + (rP – rf )y = 7 + 10y

If the expected rate of return for the portfolio is 15%, then, solving for y:

15 = 7 + 10y ? y =

107

15-

= 0.8

Therefore, in order to achieve an expected rate of return of 15%, the client must invest 80% of total funds in the risky portfolio and 20% in T-bills. b.

Security Investment Proportions

T-Bills 20.0%

Stock A 0.8 ? 27% = 21.6%

Stock B 0.8 ? 33% = 26.4%

Stock C 0.8 ? 40% = 32.0%

c.σP = 0.8 ? 27% = 21.6% per year

14.

a.Portfolio standard deviation = σP = y ? 27%

If the client wants a standard deviation of 20%, then:

y = (20%/27%) = 0.7407 = 74.07% in the risky portfolio.

b.Expected rate of return = 7 + 10y = 7 + (0.7407 ? 10) = 14.407%

15.

a.Slope of the CML =257

13-

= 0.24 Slope of my fund :(17-7)/27=0.3704

See the diagram on the next page.

b.My fund allows an investor to achieve a higher expected rate of return for any

given standard deviation than would a passive strategy, i.e., a higher expected return for any given level of risk.

16.

a.With 70% of his money in my fund's portfolio, the client has an expected rate of

return of 14% per year and a standard deviation of 18.9% per year. If he shifts

that money to the passive portfolio (which has an expected rate of return of 13%

and standard deviation of 25%), his overall expected return and standard

deviation would become:

E(r C) = r f + 0.7(r M- r f)

In this case, r f = 7% and r M = 13%. Therefore:

E(r C) = 7 + (0.7 ? 6) = 11.2%

The standard deviation of the complete portfolio using the passive portfolio

would be:

σC = 0.7 ?σM = 0.7 ? 25% = 17.5%

Therefore, the shift entails a decline in the mean from 14% to 11.2% and a decline

in the standard deviation from 18.9% to 17.5%. Since both mean return and

standard deviation fall, it is not yet clear whether the move is beneficial. The

disadvantage of the shift is apparent from the fact that, if my client is willing to

accept an expected return on his total portfolio of 11.2%, he can achieve that

return with a lower standard deviation using my fund portfolio rather than the

passive portfolio. To achieve a target mean of 11.2%, we first write the mean of

the complete portfolio as a function of the proportions invested in my fund

portfolio, y:

E(r C ) = 7 + y(17 - 7) = 7 + 10y

Because our target is: E(r C ) = 11.2%, the proportion that must be invested in my fund is determined as follows:

11.2 = 7 + 10y ? y =

2.11-= 0.42 The standard deviation of the portfolio would be:

σC = y ? 27% = 0.42 ? 27% = 11.34%

Thus, by using my portfolio, the same 11.2% expected rate of return can be achieved with a standard deviation of only 11.34% as opposed to the standard deviation of 17.5% using the passive portfolio.

b. The fee would reduce the reward-to-variability ratio, i.e., the slope of the CAL.

Clients will be indifferent between my fund and the passive portfolio if the slope of the after-fee CAL and the CML are equal. Let f denote the fee:

Slope of CAL with fee =

27f 717--= 27

10-

Slope of CML (which requires no fee) =

13-= 0.24 Setting these slopes equal and solving for f:

10-= 0.24 10 - f = 27 ? 0.24 = 6.48 f = 10 - 6.48 = 3.52% per year

17. Assuming no change in tastes, that is, an unchanged risk aversion, investors perceiving higher risk will demand a higher risk premium to hold the same portfolio they held before. If we assume that the risk-free rate is unaffected, the increase in the risk premium would require a higher expected rate of return in the equity market.

18. Expected return for your fund = T-bill rate + risk premium = 6% + 10% = 16% Expected return of client’s overall portfolio = (0.6 ? 16%) + (0.4 ? 6%) = 12% Standard deviation of client’s overall portfolio = 0.6 ? 14% = 8.4%

19. Reward to variability ratio 71.014

deviation Standard premium Risk ===

20.

Excess Return (%)

Average

Standard Deviation

Sharpe Measure

1926–200813.5137.81 0.36 1926–195520.0249.25 0.41 1956–198412.1832.31 0.38 1985–2008

6.77

25.44 0.27

a. In three of the four time frames presented, small stocks provide worse ratios

than large stocks.

b. Small stocks show a declining trend in risk, but the decline is not stable.

21. Geometric return data is used from table 5.2 and geometric inflation data from table 5.4. Standard deviations are from the excess return data in table 5.2.

Average

Inflation

Real Return

1926–20089.34 3.02 6.1%1926–19559.66 1.368.2%1956–19849.52 4.8 4.5%1985–2008

8.68

2.91

5.6%

Average Standard

Deviation (%)Sharpe Measure

1926–2008 6.1%20.880.291926–19558.2%25.40.321956–1984 4.5%17.580.261985–2008

5.6%

18.23

0.31

Real Returns - Large Cap

Risk Return Ratio - Large Cap

22.

Average

Inflation

Real Return

1926–200811.43 3.028.2%1926–195511.32 1.369.8%1956–198413.81 4.88.6%1985–2008

8.56

2.91

5.5%

Average Standard

Deviation (%)Sharpe Measure

1926–20088.2%37.810.221926–19559.8%49.250.201956–19848.6%32.310.271985–2008

5.5%

25.44

0.22

Real Returns - Small Cap

Risk Return Ratio - Large Cap

23.

Comparison

The combined markets index represents the Fama-French market factor (Mkt). It is

better diversified than the S&P 500 index since it contains approximately ten times as

many stocks. The total market capitalization of the additional stocks, however, is

relatively small compared to the S&P 500. As a result, the performance of the value

weighted portfolios is expected to be quite similar, and the correlation of the excess

returns very high. Even though the sample contains 82 observations, the standard

deviation of the annual returns is relatively high, but the difference between the two

indices is very small. When comparing the continuously compounded excess returns we see that the difference between the two portfolios is indeed quite small, and the

correlation coefficient between their returns is 0.99. Both deviate from the normal

distribution as seen from the negative skew and positive kurtosis. Accordingly, the VaR (5% percentile) of the two is smaller than what is expected from a normal distribution

with the same mean and standard deviation. This is also indicated by the lower

minimum excess return for the period. The serial correlation is also small and

indistinguishable across the portfolios.

As a result of all this, we expect the risk premium of the two portfolios to be similar, as we find from the sample. It is worth noting that the excess return of both portfolios has

a small negative correlation with the risk-free rate. Since we expect the risk-free rate to

be highly correlated with the rate of inflation, this suggests that equities are not a

perfect hedge against inflation. More rigorous analysis of this point is important, but

beyond the scope of this question.

CFA 1

Answer: V(12/31/2007) = V(1/1/1991) ? (1 + g)7 = $100,000 ? (1.05)7 = $140,710.04 CF 2

Answer: i and ii. The standard deviation is non-negative.

CFA 3

Answer: c. Determines most of the portfolio’s return and volatility over time.

CFA 4

Investment 3. For each portfolio: Utility = E(r) – (0.5 ? 4 ?σ2)

Investment E(r) σU

1 0.1

2 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

We choose the portfolio with the highest utility value.

CFA 5

Answer: Investment 4. When an investor is risk neutral, A = 0 so that the portfolio with the highest utility is the portfolio with the highest expected return.

CFA 6

Answer: b

CFA 7

E(r X) = [0.2 ? (–20%)] + [0.5 ? 18%] + [0.3 ? 50%)] = 20%

E(r Y) = [0.2 ? (–15%)] + [0.5 ? 20%] + [0.3 ? 10%)] = 10%

CFA 8

σX2 = [0.2 ? (–20 – 20)2] + [0.5 ? (18 – 20)2] + [0.3 ? (50 – 20)2] = 592

σX = 24.33%

σY = [0.2 ? (–15 – 10)2] + [0.5 ? (20 – 10)2] + [0.3 ? (10 – 10)2] = 175

σY = 13.23%

CFA 9

E(r) = (0.9 ? 20%) + (0.1 ? 10%) = 19%

CFA 10

The probability is 0.50 that the state of the economy is neutral. Given a neutral

economy, the probability that the performance of the stock will be poor is 0.30, and the probability of both a neutral economy and poor stock performance is:

.30 ? 0.50 = 0.15

CFA 11

E(r) = [0.1 ? 15%] + [0.6 ? 13%] + [0.3 ? 7%)] = 11.4%