Prof. Rushen Chahal
CHAPTER 6
Risk and Rates of Return CHAPTER ORIENTATION This chapter introduces the concepts that underlie the valuation of securities and their rates of return. We are specifically concerned with common stock, preferred stock, and bonds. We also look at the concept of the investor's expected rate of return on an investment.
CHAPTER OUTLINE I.
II.
The relationship between risk and rates of return A.
Data have been compiled by Ibbotson and Sinquefield on the actual returns for various portfolios of securities from 1926-2002.
B.
The following portfolios were studied. 1.
Common stocks of small firms
2.
Common stocks of large companies
3.
Long-term corporate bonds
4.
Long-term U.S. government bonds
5.
U.S. Treasury bills
C.
Investors historically have received greater returns for greater risk-taking with the exception of the U.S. government bonds.
D.
The only portfolio with returns consistently exceeding the inflation rate has been common stocks.
Effects of Inflation on Rates of Return A.
When a rate of interest is quoted, it is generally the nominal or, observed rate. The real rate of interest represents the rate of increase in actual purchasing power, after adjusting for inflation.
144
Prof. Rushen Chahal B.
III. III.
Consequently, the nominal rate of interest is equal to the sum of the real rate of interest, the inflation rate, and the product of the real rate and the inflation rate.
Term Structure of Interest Rates The relationship between a debt security’s rate of return and the length of time until the debt matures is known as the term structure of interest rates or the yield to maturity.
IV.
Expected Return A.
The expected benefits or returns to be received from an investment come in the form of the cash flows the investment generates.
B.
Conventionally, we measure the expected cash flow, X , as follows: N X = ∑ XiP(Xi) i where
N
=
the number of possible states of the economy.
Xi
=
the cash flow in the ith state of the economy.
P(Xi) = V.
the probability of the ith cash flow.
Riskiness of the cash flows A.
Risk can be defined as the possible variation in cash flow about an expected cash flow.
B.
Statistically, risk may be measured by the standard deviation about the expected cash flow.
C.
Risk and diversification 1.
Total variability can be divided into: a.
The variability of returns unique to the security (diversifiable or unsystematic risk)
b.
The risk related to market movements (nondiversifiable or systematic risk)
2.
By diversifying, the investor can eliminate the "unique" security risk. The systematic risk, however, cannot be diversified away.
3.
The market rewards diversification. We can lower risk without sacrificing expected return, and/or we can increase expected return without having to assume more risk.
4.
Diversifying among different kinds of assets is called asset allocation. Compared to diversification within the different asset classes, the benefits received are far greater through effective asset allocation.
5.
Risk and being patient
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Prof. Rushen Chahal
VI.
a.
An investor in common stocks must often wait longer to earn the higher returns than those provided by bonds.
b.
The capital markets reward us not just for diversifying, but also for being patient. The returns tend to converge toward toward the average as we lengthen our holding period.
6.
The characteristic line tells us the average movement in a firm's stock price in response to a movement in the general market, such as the stock market. The slope of the characteristic characteristic line, which has come to be called beta, is a measure of a stock's systematic or market risk. The slope of the line is merely the ratio of the "rise" of the line relative to the "run" of the line.
7.
If a security's beta equals one, a 10 percent increase (decrease) in market returns will produce on average a 10 percent increase (decrease) in security returns.
8.
A security having a higher beta is more volatile and thus more risky than a security having a lower beta value.
9.
A portfolio's beta is equal to the average of the betas of the stocks in the portfolio.
Required rate of return A.
The required rate of return is the minimum rate necessary to compensate an investor for accepting the risk he or she associates with the purchase and ownership of an asset.
B.
Two factors determine the required rate of return for the investor:
C.
1.
The risk-free rate of interest which recognizes the time value of money.
2.
The risk premium which considers the riskiness (variability of returns) of the asset and the investor's inv estor's attitude toward risk.
Capital asset pricing model-CAPM 1.
The required rate of return for a given security can be expressed as Required risk-free market - risk-free = + beta x rate rate rate return or k j = k rf + β j (k m - k rf )
2.
Security market line a.
Graphically illustrates the CAPM.
b.
Designates the risk-return trade-off existing in the market, where risk is defined in terms of beta according to the CAPM equation.
146
Prof. Rushen Chahal ANSWERS TO END-OF-CHAPTER QUESTIONS 6-1.
Data have been compiled by Ibbotson and Sinquefield on the actual returns for the following portfolios of securities from 1926-2002. 1.
U.S. Treasury bills
2.
U.S. government bonds
3.
Corporate bonds
4.
Common stocks for large firms
5.
Common stocks for small firms
Investors historically have received greater returns for greater risk-taking with the exception of the the U.S. government government bonds. Also, the only portfolio with with returns consistently exceeding the inflation rate has been common stocks. 6-2
When a rate of interest interest is quoted, it is generally the nominal or, observed rate. The real rate of interest represents the rate of increase in actual purchasing power, after adjusting for inflation. inflation. Consequently, the nominal rate rate of interest is is equal to the sum of the real rate of interest, the inflation rate, and the product of the real rate and the inflation rate.
6-3
The relationship between a debt security’s rate of return and the length of time until the debt matures is known as the term structure of interest rates or the yield to maturity. In most cases, longer terms to maturity command higher returns or yields.
6-4.
(a)
The investor's required rate of return is the minimum rate of return necessary to attract an investor to purchase or hold a security.
(b)
Risk is the potential variability in returns on an investment. Thus, the greater the uncertainty as to the the exact outcome, the greater is is the risk. Risk may be measured in terms of the standard deviation or by the variance term, which is simply the standard deviation squared.
(c)
A large standard deviation of the returns indicates greater riskiness associated with an investment. However, whether the the standard deviation is large relative relative to the returns has to be examined with respect to other investment opportunities. Alternatively, probability probability analysis analysis is a meaningful approach to capture greater understanding of the significance of a standard deviation figure. However, we have chosen not to incorporate incorporate such an analysis analysis into our explanation of the valuation process.
(a)
Unique risk is the variability in a firm's stock price that is associated with the specific firm and not the result of some broader influence. An employee strike is an example of a company-unique influence.
(b)
Systematic risk is the variability in a firm's stock price that is the result of general influences within the industry or resulting from overall market or
6-5.
147
Prof. Rushen Chahal economic influences. A general change in interest rates charged by banks is an example of systematic risk. 6-6.
Beta indicates the responsiveness of a security's returns to changes in the market returns. Beta is multiplied by the market risk premium and added to the risk-free risk-free rate of return to calculate a required rate of return.
6-7.
The security security market line is a graphical representation of the risk-return trade-off that exists in the market. The line indicates the minimum acceptable rate of return for investors given the level of risk. Since the security market line results results from actual market transactions, the relationship not only represents the risk-return preferences of investors in the market but also represents the investors' available opportunity set.
6-8.
The beta for a portfolio is equal to the weighted average of the individual stock betas, weighted by the percentage invested in each stock.
6-9.
If a stock has a great amount of variability about its its characteristic characteristic line (the graph of the stock's returns against the market's returns), then it has a high amount of unsystematic or company-unique company-unique risk. If, however, the stock's returns closely closely follow the market movements, then there is little unsystematic risk.
SOLUTIONS TO END-OF-CHAPTER PROBLEMS Solutions to Problems Set A 6-1A. k rf rf = .045 + .073 + (.045 x .073) k rf rf = .1213 or 12.13% = nominal rate of interest 6-2A. k rf rf = .064 + .038 + (.064 x .038) k rf rf = .1044 or 10.44% = nominal rate of interest
148
Prof. Rushen Chahal 6-3A. (A) Probability P(k i)
(B) Return (k i)
(A) x (B) Expected Return k
.15 .30 .40 .15
-1% 2 3 8
-.15% 0.60% 1.20% 1.20% 2.85%
k =
Weighted Deviation (k i - k )2P(k i)
2
=
2.223% 0.217% 0.009% 3.978% 6.427%
=
2.535%
No, Pritchard should not invest in the security. The level of risk is excessive for a return which is less than the rate offered on treasury bills. 6-4A. Common Stock A: (A) Probability P(k i)
(B) Return (k i)
(A) x (B) Expected Return k
Weighted Deviation (k i - k )2P(k i)
0.3 0.4 0.3
11% 15 19
3.3% 6.0 5.7 15.0%
4.8% 0.0 4.8 9.6%
k =
2 = =
3.10%
Common Stock B (A) Probability P(k i)
(B) Return (k i)
(A) x (B) Expected Return k
Weighted Deviation (k i - k )2P(k i)
0.2 0.3 0.3 0.2
-5% 6 14 22
-1.0% 1.8 4.2 4.4
41.472% 3.468 6.348 31.752
k =
9.4%
2 = 83.04% = 9.11%
Common Stock A is better. It has a higher expected return with less risk. risk.
149
Prof. Rushen Chahal 6-5A. Common Stock A: (A) Probability P(k i)
(B) Return (k i)
(A) x (B) Expected Return k
Weighted Deviation (k i - k )2P(k i)
0.2 0.5 0.3
- 2% 18 27
-0.4% 9.0 8.1
69.9% 0.8 31.8
k =
2 = =
16.7%
102.5% 10.12%
Common Stock B: (A) Probability P(k i)
(B) Return (k i)
(A) x (B) Expected Return k
0.1 0.3 0.4 0.2
4% 6 10 15
0.4% 1.8 4.0 3.0 9.2%
k =
Common Stock A k = 16.7% = 10.12%
Weighted Deviation (k i - k )2P(k i)
2 = =
2.704% 3.072 0.256 6.728 12.76% 3.57%
Common Stock B k = 9.2% = 3.57%
We cannot say which investment is "better." It would depend on the investor's attitude toward the risk-return tradeoff. 6-6A. (a)
Required rate = Risk-free + Beta Market Risk of return rate Premium = 6 % + 1.2 (16% - 6%) = 18%
(b)
The 18 percent "fair rate" compensates the investor for the time value of money and for assuming risk. However, only nondiversifiable risk is being considered, which is appropriate.
6-7A. Eye balling the characteristic line for the problem, the rise relative relative to the run is about 0.5. That is, when the S & P 500 return is eight percent Aram's expected return would be about four percent. Thus, the beta is also approximately approx imately 0.5 (4 ÷ 8). 150
Prof. Rushen Chahal 6-8A.
A B C D
Risk-Free Rate
+
6.75% 6.75% 6.75% 6.75%
+ + + +
Required 6-9A.` Rate of Return
Expected Market - Risk-Free x Return Rate (12% (12% (12% (12%
-
6.75%) 6.75%) 6.75%) 6.75%)
=
Risk-Free Rate
=
7.5% + (11.5% - 7.5%) x 0.765
=
10.56%
+
x x x x
Beta
=
1.50 0.82 0.60 1.15
= = = =
Required Rate of Return 14.63% 11.06% 9.90% 12.79%
(Market Return - Risk-Free Rate) X Beta
6-10A. If If the expected market return is 12.8 percent p ercent and the risk premium is 4.3 percent, the riskless rate of return return is 8.5 percent (12.8% - 4.3%). Therefore; Tasaco
=
8.5% + (12.8% - 8.5%) x 0.864 = 12.22%
LBM
=
8.5% + (12.8% - 8.5%) x 0.693 = 11.48%
Exxos
=
8.5% + (12.8% - 8.5%) x 0.575 = 10.97%
6-11A. Asman Time 1 2 3 4
Price $10 12 11 13
Return 20.00% -8.33 18.18
Salinas Price Return $30 28 -6.67% 32 14.29 35 9.38
A holding-period return indicates the rate of return you would earn if you bought a security at the beginning of a time period and sold it at the end of the period, such as the end of the month or o r year.
151
Prof. Rushen Chahal 6-12A.a.
Zemin k b (k b - k )2
Month 1 2 3 4 5 6 Sum
6.00% 3.00 1.00 -3.00 5.00 0.00 12.00
Average monthly return (Sum ÷ 6) Annualized average returns
16.00% 1.00 1.00 25.00 9.00 4.00 56.00
1.17%
24.00%
14.04% 11.20%
Standard deviation
b.
4.00% 2.00 -1.00 -1.00 -2.00 2.00 2.00 7.00
2.00%
Variance (Sum 5)
Required Rate of Return
Market k b (k b - k )2
3.35%
=
8.03% 0.69 4.69 10.03 0.69 0.69 24.82
4.97%
2.23%
Risk-Free + (Market (Market Return - Risk-Free Rate) X Beta Rate
=
8%
+
[(14% - 8%) X 1.54]
= 17.24%
c.
Zemin's historical return of 24 percent exceeds what we would consider a fair return of 17.24 percent, given the stock's systematic risk.
a.
The portfolio expected return, k p, equals a weighted average of the individual stock's expected returns.
6-13A.
k p
=
(0.20)(16%) + (0.30)(14%) + (0.15)(20%) + (0.25)(12%) + (0.10)(24%)
=
15.8%
152
Prof. Rushen Chahal b.
The portfolio beta, ßp, equals a weighted average of the individual stock betas ßp
=
(0.20)(1.00) + (0.30)(0.85) + (0.15)(1.20) + (0.25)(0.60) + (0.10)(1.60)
= c.
0.95
Plot the security market line and the individual stocks
25.00
5 3
20.00 P 15.00
1 M
2 4
10.00
5.00
0.00 0. 00
0. 50
1. 00
1. 50
2. 00
Beta
d.
A "winner" may be defined as a stock that falls above the security market line, which means these stocks are expected to earn a return exceeding what should be expected given their beta or systematic risk. In the above graph, these stocks include 1, 3, and 5. "Losers" would be those stocks falling falling below the security market line, which are represented by stocks 2 and 4 ever so slightly.
e.
Our results are less than certain because we have problems estimating the security market market line with with certainty. certainty. For instance, we have difficulty in specifying the market portfolio.
153
Prof. Rushen Chahal 6-14A a. Market k t
Month
Price
Jul-02 Aug-02 Sep-02 Oct-02 Nov-02 Dec-02 Jan-03 Feb-03 Mar-03 Apr-03 May-03 Jun-03 Jul-03
1328.72 1320.41 1282.71 1362.93 1362.9 3 1388.91 1469.25 1469.2 5 1394.46 1366.42 1498.58 1452.43 1420.60 1454.60 1430.83
Sum
2
(k t - k )
-0.63% -2.86% 6.25% 1.91% 5.78% -5.09% -2.01% 9.67% -3.08% -2.19% 2.39% -1.63%
0.0002 0.0013 0.0031 0.0001 0.0026 0.0034 0.0007 0.0080 0.0014 0.0008 0.0003 0.0005
8.52%
0.0225
Price
Mathews k t (k t - k )2
34.50 41.09 37.16 38.72 38.34 41.16 49.47 56.50 65.97 63.41 62.34 66.84 66.75
19.10% -9.56% 4.20% -0.98% 7.36% 20.19% 14.21% 16.76% -3.88% -1.69% 7.22% -0.13% 72.79%
0.0170 0.0244 0.0003 0.0050 0.0002 0.0199 0.0066 0.0114 0.0099 0.0060 0.0001 0.0038 0.1048
b) Average monthly return Standard deviation
0.71%
6.07% 4.52%
9.76%
c) ws ws 25.00% Mathe
20.00%
15.00%
10.00%
5.00% Market Index
0.00% -10.00%
-5.00%
0.00%
-5.00%
-10.00%
-15.00%
154
5.00%
10.00%
15.00%
Prof. Rushen Chahal d.
Mathews returns seem to correlate to the market returns during the majority of the year, but show sh ow great volatility.
6-15A Stock 1 (A) Probability P(k i) 0.15 0.40 0.30 0.15
(B) Return (k i)
(A) x (B) Expected Return k
2% 7 10 15 k =
0.30% 2.80 3.00 2.25 8.35%
Weighted Deviation (k i - k )2P(k i) 6.048% 0.729 0.817 6.633 14.227%
2 = =
3.77%
Stock 2 (A) Probability P(k i) 0.25 0.50 0.25
(B) Return (k i)
(A) x (B) Expected Return k
-3% 20 25 k
=
-0.75% 10.00 6.25 15.50%
Weighted Deviation (k i - k )2P(k i) 85.56% 10.13 22.56 118.25%
2 = =
10.87%
Stock 3 (A) Probability P(k i) 0.10 0.40 0.30 0.20
(B) Return (k i)
(A) x (B) Expected Return k
-5% 10 15 30 k =
-0.50% 4.00 4.50 6.00 14.00%
Weighted Deviation (k i - k )2P(k i)
2 = =
36.1% 6.4 0.3 51.2 94.0% 9.7%
We cannot say which investment is "better." It would depend on the the investor's attitude toward the risk-return tradeoff.
155
Prof. Rushen Chahal 6-16A Risk-Free + Rate H T P W
5.5% 5.5% 5.5% 5.5%
+ + + +
Expected Market - Risk-Free x Return Rate (11% (11% (11% (11%
-
5.5%) 5.5%) 5.5%) 5.5%)
x x x x
Beta
=
0.75 1.40 0.95 1.25
= = = =
6-17A Time 1 2 3 4
Williams Price Return $33 27 -18.18% 35 29.63 39 11.43 11 .43
Davis Price $19 15 14 23
Return -21.05% -6.67 64.29 64.2 9
6-18A (a)
Required rate = Risk-free + Beta Market Risk of return rate Premium = 5 % + 1.2 (9% - 5%) = 9.8%
(b)
Required rate = Risk-free + Beta Market Risk of return rate Premium = 5 % + 0.85 (9% - 5%) = 8.4%
(c)
If beta is 1.2: Required rate of return
= 5 % + 1.2 (12% - 5%) = 13.4%
If beta is 0.85: Required rate of return
= 5 % + 0.85 (12% - 5%) = 10.95%
156
Required Rate of Return 9.63% 13.20% 10.73% 12.38%
Prof. Rushen Chahal SOLUTION TO INTEGRATIVE PROBLEM 1.
Holding-period returns for Market, Reynolds Computer, and Andrews
Price 01May 01May June July Aug Sept Oct Nov Dec 02Jan 02Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec 03Jan 03Jan Febr Mar Apr May Sum
2.
Market k t (k t - k )2
1090.82 1133.84 3.94% 1120.67 -1.16% 957.28 -14.58% 1017.01 6.24% 1098.67 8.03% 1163.63 5.91% 1229.23 5.64% 1279.64 4.10% 1238.33 -3.23% 1286.37 3.88% 1335.18 3.79% 1301.84 -2.50% 1372.71 5.44% 1328.72 -3.20% 1320.41 -0.63% 1282.71 -2.86% 1362.93 6.25% 1388.91 1.91% 1469.25 5.78% 1394.46 -5.09% 1366.42 -2.01% 1498.58 9.67% 1452.43 -3.08% 1420.60 -2.19% 30.07%
Average Monthly Return Standard Deviation
0.0007 0.0006 0.0251 0.0025 0.0046 0.0022 0.0019 0.0008 0.0020 0.0007 0.0006 0.0014 0.0018 0.0020 0.0004 0.0017 0.0025 0.0000 0.0021 0.0040 0.0011 0.0071 0.0019 0.0012 .0689
Reynolds Computer Price k t (k t - k )2
Price
20.60 23.20 27.15 25.00 32.88 32.75 30.41 36.59 50.00 40.06 40.88 41.19 34.44 37.00 40.88 48.81 41.81 40.13 43.00 51.00 38.44 40.81 53.94 50.13 43.13
24.00 26.72 20.94 15.78 18.09 21.69 23.06 28.06 26.03 26.44 28.06 36.94 36.88 37.56 23.25 22.88 24.78 27.19 26.56 24.25 32.00 35.13 44.81 30.23 34.00
12.62% 17.03% -7.92% 31.52% -0.40% -7.15% 20.32% 36.65% -19.88% 2.05% 0.76% -16.39% 7.43% 10.49% 19.40% -14.34% -4.02% 7.15% 18.60% -24.63% 6.17% 32.17% -7.06% -13.96% 106.62%
0.0067 0.0158 0.0153 0.0733 0.0023 0.0134 0.0252 0.1037 0.0592 0.0006 0.0014 0.0434 0.0009 0.0037 0.0224 0.0353 0.0072 0.0007 0.0201 0.0845 0.0003 0.0769 0.0132 0.0339
Andrews k t (k t - k )2 11.33% -21.63% -24.64% 14.64% 19.90% 6.32% 21.68% -7.23% 1.58% 6.13% 31.65% -0.16% 1.84% -38.10% -1.59% 8.30% 9.73% -2.32% -8.70% 31.96% 9.78% 27.55% -32.54% 12.47% 77.95%
1.25%
4.44%
3.25%
5.47%
16.93%
18.60%
157
0.0065 0.0619 0.0778 0.0130 0.0277 0.0009 0.0340 0.0110 0.0003 0.0008 0.0806 0.0012 0.0002 0.1710 0.0023 0.0026 0.0042 0.0031 0.0143 0.0824 0.0043 0.0591 0.1281 0.0085 .7958
Prof. Rushen Chahal 3.
Reynolds vs Market
0.4
0.3
0.2
t e k r a M
0.1
0
-0.2
-0.1
0
0. 1
0 .2
-0.1
-0.2
-0.3 Reynolds Andrews Andrews vs. Market
0.4 0.3 0.2 0.1 s w e r d -0.2 n A
0 -0.1
0 -0.1 -0.2 -0.3 -0.4 -0.5 Market
158
0.1
0.2
Prof. Rushen Chahal 4
Reynolds’s returns have a great amount of volatility with some correlation to the market returns.
The same can be said of Andrews. The returns show a great great amount of volatility that followed the market returns only part of the time. 5.
Monthly returns of a portfolio of equal amounts of Reynolds and Andrews. Monthly 2001 June July August September October November December 2002 January February March April May June July August September October November December 2003 January February March April May Average return Standard deviation
159
Returns 11.98% -2.32% -16.27% 23.08% 9.74% -0.41% 21.02% 14.70% -9.16% 4.09% 16.20% -8.28% 4.65% -13.81% 8.90% -3.00% 2.84% 2.43% 4.95% 3.66% 7.97% 29.87% -19.80% -0.75% 3.84% 12.29%
Prof. Rushen Chahal 6. Reynolds and Andrews 40.00%
30.00%
s w e r d n A % 0 5 s d l o n y e R-20.00% % 0 5
20.00%
10.00%
0.00% -10.00%
0.00%
10.00%
20.00%
-10.00%
-20.00%
-30.00% Market
We see in this new graph where both stocks are included as a single portfolio that the relationship of the stocks with the market approximates an average of the relationships taken alone. Note the reduction in volatility that occurs when risk is diversified diversified even between just two stocks.
160
Prof. Rushen Chahal 7.
Monthly holding-period returns for long-term government bonds Annual Monthly (k i - k )2 Return Return 2001 June 5.70% 0.48% 0.000000% July 5.68% 0.47% 0.000001% August 5.54% 0.46% 0.000004% September 5.20% 0.43% 0.000023% October 5.01% 0.42% 0.000041% November 5.25% 0.44% 0.000020% December 5.06% 0.42% 0.000036% 2002 January 5.16% 0.43% 0.000027% February 5.37% 0.45% 0.000012% March 5.58% 0.47% 0.000003% April 5.55% 0.46% 0.000004% May 5.81% 0.48% 0.000000% June 6.04% 0.50% 0.000005% July 5.98% 0.50% 0.000003% August 6.07% 0.51% 0.000006% September 6.07% 0.51% 0.000006% October 6.26% 0.52% 0.000016% November 6.15% 0.51% 0.000009% December 6.35% 0.53% 0.000022% 2003 January 6.63% 0.55% 0.000050% February 6.23% 0.52% 0.000014% March 6.05% 0.50% 0.000005% April 5.85% 0.49% 0.000000% May 6.15% 0.51% 0.000009% Average Monthly Return
0.48%
Standard Deviation
0.04%
161
Prof. Rushen Chahal 8.
Monthly portfolio returns when portfolio consists of equal amounts invested in Reynolds, Andrews, and long-term government bonds. Monthly Returns 8.14% -1.39% -10.69% 15.53% 6.63% -0.13% 14.15% 9.94% -5.95% 2.88% 10.95% -5.36% 3.27% -9.04% 6.10% -1.83% 2.07% 1.79% 3.48% 2.63% 5.49% 20.08% -13.04% -0.33% 65.36%
2001 June July August September October November December 2002 January February March April May June July August September October November December 2003 January February March April May Sum Average Monthly Return
(k i - k )2 0.0029 0.0017 0.0180 0.0164 0.0015 0.0008 0.0131 0.0052 0.0075 0.0000 0.0068 0.0065 0.0000 0.0138 0.0011 0.0021 0.0000 0.0001 0.0001 0.0000 0.0008 0.0301 0.0248 0.0009 0.1542
2.72%
Std. Dev..
8.19%
162
Prof. Rushen Chahal 9.
Comparison of average returns and standard deviations Average Returns Reynolds Andrews Government security Reynolds & Andrews Reynolds, Andrews, & government security Market
Standard Deviations
4.44% 3.25% 0.48% 3.84% 2.72%
16.93% 18.60% 0.04% 12.29% 8.19%
1.25%
5.47%
From the findings above, we see that higher average returns are associated with higher risk (standard deviations), and that by diversification we can reduce risk, possibly without reducing the average return. 10.
11.
Based on the standard deviations, Andrews has more risk than Reynolds, 18.60 percent standard deviation versus 16.93 percent standard deviation. However, when we only consider systematic risk, Andrews is slightly less risky--Reynolds's beta is 1.96 compared compared to to Andrews’ Andrews’ beta beta of 1.49. (The betas betas given here here for Reynolds and Andrews come from from financial services who calculate calculate firms' betas. These are not consistent with the graphs above where we see Andrews' returns as being more responsive to the general market. market. We are seeing the problem problem of using only 24 months of returns as we have done.) Required Risk-Free Rate of = + (Market Return - Risk-Free Rate) X Beta Rate Return Market Return = 1.25 % Average Monthly Return X 12 Months = 15%. (The average returns for the market over a two-year period may be high or low relative to the longer-term past, and as a result should not be considered as “typical” investor expectations. For instance, instance, if we used information from Ibbotson & Sinquefield for the years 1926-2002, the market risk premium — market market return less risk-free rate — was was 8.4 percent, and not the 19 percent that we use below. The point: Do not think two years fairly captures what we can expect in the future?) Reynolds: 23.64% = 6% + (15% - 6%) X 1.96 Andrews: 19.41% = 6% + (15% - 6%) X 1.49 And if we used the market premium of 8.4 percent: Reynolds: 22.46% = 6% + 8.4% X 1.96 Andrews: 18.52% = 6% + 8.4% X 1.49
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Prof. Rushen Chahal Solutions to Problem Set B 6-1B. k rf rf = .05 + .07 + (.05 x .07) k rf rf = .1235 or 12.35% = nominal rate of interest 6-2B. k rf rf = .03 + .05 + (.03 x .05) k rf rf = .0815 or 8.15% = nominal rate of interest 6-3B. (A) Probability P(k i)
(B) Return (k i)
.15 .30 .40 .15
-3% 2 4 6
(A) x (B) Expected Return k
k =
-0.45% 0.60 1.60 0.90 2.65%
Weighted Deviation (k i - k )2P(k i)
2 = =
4.788 0.127 0.729 1.683 7.327% 2.707%
No, Gautney should not invest in the security. The security’s security’s expected rate of return is less than the rate offered o ffered on treasury bills.
6-4B. Security A: (A) Probability P(k i) 0.2 0.5 0.3
(B) Return (k i)
(A) x (B) Expected Return k
Weighted Deviation (k i - k )2P(k i)
- 2% 19 25
-0.4% 9.5 7.5 16.6%
69.19% 2.88 21.17 93.24%
k =
164
2 = =
9.66%
Prof. Rushen Chahal Security B: (A) Probability P(k i)
(B) Return (k i)
0.1 0.3 0.4 0.2
5% 7 12 14
(A) x (B) Expected Return k
Weighted Deviation (k i - k )2P(k i)
0.5% 2.1 4.8 2.8 k =
10.2%
Security A k = 16.6%
Security B k = 10.2%
= 9.66%
= 3.16%
2.704% 3.072 1.296 2.888
2 = =
9.96% 3.16%
We cannot say which investment is is "better." It would depend on the investor's attitude toward the risk-return tradeoff. 6-5B. Common Stock A: (A) Probability P(k i)
(B) Return (k i)
(A) x (B) Expected Return k
0.2 0.6 0.2
10% 13 20
2.0% 7.8 4.0 13.8%
k =
Weighted Deviation (k i - k )2P(k i) 2.89% 0.38 7.69 = 10.96%
2 = 3.31%
Common Stock B (A) Probability P(k i) 0.15 0.30 0.40 0.15
(B) Return (k i)
(A) x (B) Expected Return k
Weighted Deviation (k i - k )2P(k i)
6% 8 15 19
0.9% 2.4 6.0 2.85 12.15%
5.67% 5.17 3.25 7.04 21.13%
k =
165
2 = =
4.60%
Prof. Rushen Chahal Common Stock A is better. It has a higher higher expected return with less less risk. 6-6B. (a)
Required rate = Risk-free + Beta Market Risk of return rate Premium = 8 % + 1.5 (16% - 8%) = 20%
(b)
The 20 percent "fair rate" compensates the investor for the time value of money and for assuming risk. However, only nondiversifiable risk is being considered, which is appropriate.
6-7B. Eye balling the characteristic line for the problem, the rise relative relative to the run is about 1.75. That is, when the S & P 500 return return is four percent percent Bram's expected return would be about seven percent. Thus, the beta is also approximately 1.75 (7 ÷ 4). 6-8B.
A B C D
Risk-Free Rate
+
6.75% 6.75% 6.75% 6.75%
+ + + +
Required 6-9B. Rate of Return
Expected Market - Risk-Free Return Rate (12% (12% (12% (12%
-
6.75%) 6.75%) 6.75%) 6.75%)
x
Beta
=
Required Rate of Return
x x x x
1.40 0.75 0.80 1.20
= = = =
14.10% 10.69% 10.95% 13.05%
=
Risk-Free + (Market (Market Return - Risk-Free Rate) X Beta Rate
=
7.5% + (10.5% - 7.5%) x 0.85
= 10.05% 6-10B. If the expected market return is 12.8 percent and the risk premium is 4.3 percent, the riskless rate of return return is 8.5 percent (12.8% - 4.3%). Therefore; Dupree
= 8.5% + (12.8% - 8.5%) x 0.82 = 12.03%
Yofota
= 8.5% + (12.8% - 8.5%) x 0.57 = 10.95%
MacGrill = 8.5% + (12.8% - 8.5%) x 0.68 = 11.42% 6-11B. O'Toole Time Price Return 1 $22 2 24 9.09% 3 20 -16.67% 4 25 25.00%
Baltimore Price Return $45 50 11.11% 48 -4.00% 52 8.33%
166
Prof. Rushen Chahal A holding-period return indicates the rate of return you would earn if you bought a security at the beginning of a time period and sold it at the end of the period, such as the end of the month or o r year,
167
Prof. Rushen Chahal 6-12B. (a)
Sugita k t (k t - k )2 1.80% 0.01% -0.50 -0.50 5.68 2.00 0.01 -2.00 15.08 5.00 9.71 5.00 9.71 11.30 40.20
Month 1 2 3 4 5 6 Sum Average monthly return (Sum ÷ 6) Annualized average returns
Market k t (k t - k )2 1.50% 0.06% 1.00 0.06 0.00 1.56 -2.00 10.56 4.00 7.56 3.00 3.06 7.50 22.86
1.88%
1.25%
22.60%
15.00%
Variance (Sum ÷ 5) Standard deviation
8.04%
4.58%
2.84%
2.14%
b. Required Risk-Free Rate of = + (Market (Market Return - Risk-Free Rate) X Beta Rate Return =
8%
+
[(15% - 8%) X 1.18] = 16.26%
c.
Sugita's historical return of 22.6 percent exceeds what we would consider a fair return of 16.26 percent, given the stock's systematic risk.
a.
The portfolio expected return, k p, equals a weighted average of the individual stock's expected returns. (0.10)(12%) + (0.25)(11%) + (0.15)(15%) + (0.30)(9%) + k p =
6-13B
(0.20)(14%) =
11.7%
168
Prof. Rushen Chahal
b.
The portfolio beta, ßp, equals a weighted average of the individual stock betas ßp
=
(0.10)(1.00) + (0.25)(0.75) + (0.15)(1.30) + (0.30)(0.60) + (0.20)(1.20)
= c.
0.90
Plot the security market line and the individual stocks 16.00
3
14.00 12.00
P
2
1
5
M
10.00 4
8.00 6.00 4.00 2.00 0.00 0. 00
0. 20
0. 40
0. 60
0. 80
1. 00
1. 20
1. 40
Beta
d.
A "winner" may be defined as a stock that falls above the security market line, which means these stocks are expected to earn a return exceeding what should be expected given their beta or systematic risk. In the above graph, these stocks include 1, 2, 3, and 5. "Losers" would be those stocks falling below the security market line, that being stock 4.
e.
Our results are less than certain because we have problems estimating the security market market line with with certainty. certainty. For instance, we have difficulty in specifying the market portfolio.
169
Prof. Rushen Chahal 6-14B a) Month Jul-02 Aug-02 Sep-02 Oct-02 Nov-02 Dec-02 Jan-03 Feb-03 Mar-03 Apr-03 May-03 Jun-03 Jul-03
Price 1328.72 1320.41 1282.71 1362.93 1388.91 1469.25 1394.46 1366.42 1498.58 1452.43 1420.60 1454.60 1430.83
Sum b)
Average Monthly Return Standard deviation
Market k t (k t - k )2 -0.63% -2.86% 6.25% 1.91% 5.78% -5.09% -2.01% 9.67% -3.08% -2.19% 2.39% -1.63%
0.0002 0.0013 0.0031 0.0001 0.0026 0.0034 0.0007 0.0080 0.0014 0.0008 0.0003 0.0005
8.52%
0.0225
0.71%
21.00 19.50 17.19 16.88 18.06 24.88 22.75 26.25 33.56 43.31 43.50 43.50 43.63
-7.14% -11.85% -1.80% 6.99% 37.76% -8.56% 15.38% 27.85% 29.05% 0.44% 0.00% 0.30% 88.42%
(k t - k )2 0.0211 0.0 369 0.0369 0.0084 0.0000 0.0924 0.0254 0.0064 0.0419 0.0470 0.0048 0.0054 0.0050 0.2948
7.37% 4.52%
170
Price
Hilary’s k t
16.37%
Prof. Rushen Chahal c)
50.00%
Hilary's 40.00%
30.00%
20.00%
10.00% Market 0.00% -10.00%
-5.00%
0.00%
5.00%
10.00%
15.00%
-10.00%
-20.00%
d.
The Hilary’s returns for th e last six months of 2002 and the first six months of 2003 were partially correlated, but with a lot of the variance in the stock’s returns, clearly not explained by the market — as would be expected. — as
171
Prof. Rushen Chahal 6-15B Stock A (A) Probability P(k i) 0.10 0.30 0.40 0.20
(B) Return (k i)
(A) x (B) Expected Return k
Weighted Deviation (k i - k )2P(k i)
-4% 2 13 17
-0.40% 0.60 5.20 3.40 8.80%
16.384% 13.872 7.056 13.448 50.76%
k =
2 = =
7.125%
Stock B (A) Probability P(k i) 0.13 0.40 0.27 0.20
(B) Return (k i)
(A) x (B) Expected Return k
4% 4% 10 19 23
0.52% 4.00 5.13 4.60 k = 14.25%
Weighted Deviation (k i - k )2P(k i)
2 = =
13.658% 7.225 6.092 15.31 42.285% 6.503%
Stock C (A) Probability P(k i) 0.20 0.25 0.45 0.10
(B) Return (k i)
(A) x (B) Expected Return k
Weighted Deviation (k i - k )2P(k i)
-2% 5 14 25
-0.40% 1.25 6.30 2.50 9.65%
27.145% 5.406 8.515 23.562 64.628%
k =
2 = =
8.039%
Stock B has a higher expected rate of return with less risk than Stocks A and C.
172
Prof. Rushen Chahal 6-16B
K G B U
Risk-Free Rate
+
5.5% 5.5% 5.5% 5.5%
+ + + +
Expected Market - Risk-Free x Beta = Return Rate (11% (11% (11% (11%
-
5.5%) 5.5%) 5.5%) 5.5%)
x x x x
1.12 1.30 0.75 1.02
= = = =
6-17B Time 1 2 3 4
Watkins Price $40 45 43 49
Fisher Return 12.50% -4.44 13.95
Price $27 31 35 36
6-18B (a)
Required rate = Risk-free + Beta Market Risk of return rate Premium = 4% + 0.95 (7% - 4%) = 6.85%
(b)
Required rate = Risk-free + Beta Market Risk of return rate Premium = 4 % + 1.25 (7% - 4%) = 7.75%
(c)
If beta is 0.95: Required rate of return
= 4 % + 0.95 (10% - 4%) = 9.7%
If beta is 1.25: Required rate of return
= 4 % + 1.25 (10% - 4%) = 11.5%
173
Return 14.81% 12.90 2.86
Required Rate of Return 11.66% 12.65% 9.63% 11.11%
