CHAPTER 5: RISK AND RETURN: PAST AND PROLOGUE
1.
i and ii. The standard deviation is non-negative.
2.
c.
3.
Investment 4. For each portfolio: Utility = E(r) – (0.5  3  2 )
Determines most of the portfolio’s return and volatility over time.
Investment
1
2
3
4
E(r)
0.15
0.20
0.15
0.18

0.30
0.50
0.16
0.21
U
0.0150
-0.1750
0.1116
0.1139
We choose the portfolio with the highest utility value.
4.
Investment 2. When an investor is risk neutral, A = 0 so that the portfolio with the
highest utility is the portfolio with the highest expected return.
5.
b.
6.
V(12/31/2004) = V(1/1/1998)  (1 + g)7 = €100,000  (1.06)7 = €150,363.03
7.
E(r) = [0.35  40%] + [0.4  15%] + [0.25  (–20%)] = 15%
2 = [0.35  (40 – 15)2] + [0.4  (15 – 15)2] + [0.25  (–20 – 15)2] = 525.00
 = 22.91%
The mean has increased and the standard deviation has increased.
8.
E(rX) = [0.25  (–25%)] + [0.45  18%] + [0.3  60%)] = 19.85%
E(rY) = [0.25  (–20%)] + [0.45  20%] + [0.3  15%)] = 8.5%
5-1
9.
X2 = [0.25  (–25 – 19.85)2] + [0.45  (18 – 19.85)2] + [0.3  (60 – 19.85)2] = 988.03
X = 31.43%
Y2 = [0.25  (–20 – 8.5)2] + [0.45  (20 – 8.5)2] + [0.3  (15 – 8.5)2] = 275.25
Y = 16.59%
10. E(r) = (0.9  19.85%) + (0.1  8.5%) = 18.715%
11.
a.
The holding period returns for the three scenarios are:
Boom:
(35 – 25 + 1)/25 = 0.44 = 44.00%
Normal:
(25 – 25 + 0.50)/25 = 0.02 = 2.00%
Recession: (10 – 25 + 0.25)/25 = –0.59 = –59.00%
E(HPR) = [(1/3)  44%] + [(1/3)  2%] + [(1/3)  (–59.00%)] = −4.33%
2(HPR) = [(1/3)  (44 – (−4.33))]2 + [(1/3)  (2 – (−4.33))]2 + (1/3)  [(–59 − (−4.33))]2
= 1788.22
 = 1788.22 = 42.29%
b.
E(r) = (0.5  −4.33%) + (0.5  4%) = −0.165%
 = 0.5  42.29% = 21.145%
12. a.
E(rP) = (0.3  6%) + (0.7  15%) = 12.30% per year
P = 0.7  25% = 17.5% per year
b.
Security
T-Bills
Stock A
Stock B
Stock C
c.
0.7  27% =
0.7  33% =
0.7  40% =
Investment
Proportions
30.0%
18.9%
23.1%
28.0%
Your Reward-to-variability ratio = S =
Client's Reward-to-variability ratio =
5-2
15  6
= 0.360
25
12.30  6
= 0.360
17.5
d.
E(r)
%
P
15
CAL( slope=0.360)
12.3
client
6

17.5
13. a.
25
%
Mean of portfolio = (1 – y)rf + y rP = rf + (rP – rf )y = 6 + 9y
If the expected rate of return for the portfolio is 12%, then, solving for y:
12 = 6 + 9y  y =
12  6
= 0.667
9
Therefore, in order to achieve an expected rate of return of 12%, the client
must invest 66.7% of total funds in the risky portfolio and 33.3% in T-bills.
b.
Security
T-Bills
Stock A
Stock B
Stock C
c.
Investment
Proportions
33.3%
0.667  27% =
18.0%
0.667  33% =
22.0%
0.667  40% =
26.7%
P = 0.667  25% = 16.7% per year
5-3
14. a.
Portfolio standard deviation = P = y  25%
If the client wants a standard deviation of 20%, then:
y = (20%/25%) = 0.80 = 80.0% in the risky portfolio.
15.
16.
b.
Expected rate of return = 6 + 9y = 6 + (0.80  9) = 13.20%
a.
Slope of the CML =
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.
a.
With 70% of his money in my fund's portfolio, the client has an expected rate
of return of 12.3% per year and a standard deviation of 17.5% per year. If he
shifts that money to the passive portfolio (which has an expected rate of return
of 14% and standard deviation of 24%), his overall expected return and
standard deviation would become:
14  6
= 0.333
24
E(rC) = rf + 0.7(rM  rf)
In this case, rf = 6% and rM = 14%. Therefore:
E(rC) = 6 + (0.7  8) = 11.6%
The standard deviation of the complete portfolio using the passive portfolio
would be:
C = 0.7  M = 0.7  24% = 16.8%
Therefore, the shift entails a decline in the mean from 12.3% to 11.6% and a
decline in the standard deviation from 17.5% to 16.8%. 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 11.6%, 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 11.6%, we first
write the mean of the complete portfolio as a function of the proportions
invested in my fund portfolio, y:
E(rC) = 6 + y(15  6) = 6 + 9y
5-4
Because our target is: E(rC) = 11.6%, the proportion that must be invested in
my fund is determined as follows:
11.6 = 6 + 9y y =
11.6  6
= 0.62
9
The standard deviation of the portfolio would be:
C = y  25% = 0.62  25% = 15.50%
Thus, by using my portfolio, the same 11.6% expected rate of return can be
achieved with a standard deviation of only 15.50% as opposed to the standard
deviation of 16.8% 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 =
15  6  f 9  f
=
25
25
Slope of CML (which requires no fee) =
14  6
= 0.3333
24
Setting these slopes equal and solving for f:
9f
= 0.3333
25
9  f = 25  0.3333 = 8.33
f = 9  8.33 = 0.67% per year
17.
The probability is 0.45 that the state of the economy is neutral. Given a neutral
economy, the probability that the performance of the stock will be poor is 0.15,
and the probability of both a neutral economy and poor stock performance is:
0.45  0.15 = 0.0675
18.
E(r) = [0.1  20%] + [0.6  10%] + [0.3  5%)] = 9.5%
19.
a.
E(rP) – rf = ½AP2 = ½  4  (0.25) = 0.125 = 12.5%
b.
0.09 = ½AP2 = ½  A  (0.25)  A = 0.09/( ½  0.0625) = 2.88
c.
Increased risk tolerance means decreased risk aversion (A), which results in a
decline in risk premiums.
5-5
20.
a.
Time-weighted average returns are based on year-by-year rates of return.
Year
2002-2003
2003-2004
2004-2005
Return = [(capital gains + dividend)/price]
(20 – 110 + 4)/110 = 12.73%
(80 – 120 + 4.5)/120 = –29.58%
(90 – 80 + 4.5)/80 = 18.13%
Arithmetic mean: 0.43%
Geometric mean: −2.12%
b.
Time
0
1
Cash flow
-£330.00
-£228.00
2
£102.50
3
£378.00
Date:
1/1/02
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330
Explanation
Purchase of three shares at £110 per share
Purchase of two shares at £120,
plus dividend income on three shares held
Dividends on five shares,
plus sale of one share at £80
Dividends on four shares,
plus sale of four shares at £90 per share
1/1/03
|
|
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228
102.50
|
|
1/1/04
378
|
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1/1/05
Dollar-weighted return = Internal rate of return = –6.100%
21.
For the period 1926 – 2003, the mean annual risk premium for large stocks over
T-bills is: 12.25% – 3.79% = 8.46%
E(r) = Risk-free rate + Risk premium = 4% + 8.46% =12.46%
5-6
22.
a.
The expected cash flow is: (0.5  €60,000) + (0.5  €160,000) = €110,000
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) = €110,000  X = €95,652
b.
If the portfolio is purchased at €95,652, and the expected payoff is €110,000,
then the expected rate of return, E(r), is:
110,000  95,652
= 0.15 = 15.0%
95,652
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 16%, then the required return is:
5% + 16% = 21%
The value of the portfolio (X) must satisfy:
X(1.21) = €110, 000  X = €90,909
d.
23.
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.
In the table below, we use data from Table 5.3 and the approximation: r  R – i:
Large Stocks:
Small Stocks:
Long-Term T-Bonds:
T-Bills:
r  12.25%  3.12% =
r  18.43%  3.12% =
r  5.64%  3.12% =
r  3.79%  3.12% =
9.13%
15.31%
2.52%
0.67%
Next, we compute real rates using the exact relationship:
r
1 R
R i
1 
1 i
1 i
Large Stocks:
Small Stocks:
Long-Term T-Bonds:
T-Bills
24.
r = 0.0913/1.0312 = 8.85%
r = 0.1531/1.0312 = 14.85%
r = 0.0252/1.0312 = 2.44%
r = 0.0067/1.0312 = 0.65%
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.
5-7
25. b.
26. Expected return for your fund = T-bill rate + risk premium = 5% + 8% = 13%
Expected return of client’s overall portfolio = (0.6  13%) + (0.4  5%) = 9.8%
Standard deviation of client’s overall portfolio = 0.6  14% = 8.4%
27. Reward to variability ratio 
Risk premium
8

 0.57
Standard deviation 14
28.
Average Rate of Return, Standard Deviation
and Reward-to-Variability Ratio of the Risk Premiums
for Small Common Stocks over One Month Bills
for 1926-2003 and Various Sub-Periods
1926-1943
1944-1963
1964-1983
1984-2003
1926-2003
Risk Premium(%)
Mean
SD
18.40
60.00
17.84
29.86
13.61
35.30
9.08
25.69
14.64
38.72
Reward-toVariability Ratio
0.3066
0.5975
0.3855
0.3535
0.3780
Table 5.5 (for comparison)
1926-1943
1944-1963
1964-1983
1983-2003
1926-2003
Risk Premium(%)
Mean
SD
7.15
29.07
14.74
18.38
2.76
17.29
9.08
16.79
8.46
20.80
Reward-toVariability Ratio
0.2460
0.8018
0.1596
0.5408
0.4071
a.
For the entire period (1926-2003), small stocks had a lower reward-tovariability ratio (0.3780) than large stocks (0.4071). However, in two of the
four sub-periods, small stocks performed better than large stocks.
b.
In general, yes.
5-8
29.
Average Real Return, Standard Deviation
and Reward-to-Variability Ratio for Real Returns
for Large Stocks for 1926-2003 and Various Sub-Periods
Real Return(%)
1926-1943
1944-1963
1964-1983
1984-2003
1926-2003
Mean
8.16
13.39
3.51
10.96
9.03
SD
28.20
19.44
16.80
16.86
20.56
Reward-toVariability Ratio
0.2895
0.6887
0.2091
0.6496
0.4391
Except for the period 1944-1963, reward-to-variability ratios for real rates are
higher than those for excess returns.
30.
Average Real Return, Standard Deviation
and Reward-to-Variability Ratio for Real Returns
for Small Stocks for 1926-2003 and Various Sub-Periods
Real Return(%)
1926-1943
1944-1963
1964-1983
1984-2003
1926-2003
Mean
18.99
16.48
13.84
11.04
14.99
SD
57.93
30.15
33.77
24.59
37.48
Reward-toVariability Ratio
0.3278
0.5467
0.4099
0.4491
0.4000
As is the case with large stocks, reward-to-variability ratios of real rates for small
stocks are generally higher than those for excess returns. This suggests that both
portfolios have similar correlations with inflation.
5-9