9.1 a, Capital gains = $38 - $37 = $1 per share
Chapter 9: Capital Market Theory: An Overview
9.1 a. Capital gains = $38 - $37 = $1 per share
b. Total dollar returns = Dividends + Capital Gains
= $1,000 + ($1*500) = $1,500
On a per share basis, this calculation is $2 + $1 = $3 per share
c. On a per share basis, $3/$37 = 0.0811 = 8.11%
On a total dollar basis, $1,500/(500*$37) = 0.0811 = 8.11%
d. No, you do not need to sell the shares to include the capital gains in the computation of the returns. The capital gain is included whether or not you realize the gain. Since you could realize the gain if you choose, you should include it.
9.2 Purchase Price = $10,400/200 = $52.00
a. Total dollar return = $600 + 200($54.25 - $52) =$1,050
b. Capital gain = 200($54.25-52) = $450
c. Percentage Return = $1050/$10400 = 10.10%
d. Dividend Yield = $600/(200*52) = 5.77%
9.3 [pic]
9.4 The expected holding period return is:
[pic]
9.5 You can find the nominal returns, I, on each of the securities in the text. The inflation rate, (, for the period is also in the text. It is 3.2%. The real return, r, is (1+I)/(1+()-1. An approximation for the real rate is r = i - (. Notice that the approximation is good when the nominal interest rate is close to the inflation rate.
| |Nominal |Real |Approximation |
|a. Common Stocks |12.2% |8.7% |9.2% |
|b. L/T Corp. Bonds |5.7% |2.4% |2.5% |
|c. L/T Govt. Bonds |5.2% |1.9% |2.0% |
|d. U.S. T-Bills |3.7% |0.5% |0.5% |
9.6 E(R) = T-Bill rate + Average Excess Return
= 6.2% + (12.4% -3.9%)
= 14.7%
9.7 Suppose the two companies’ stock price 2 years ago were [pic]
2 years ago 1 year ago Today
Koke [pic] 1.1[pic] 1.1*0.9*[pic]= 0.99[pic]
Pepsee [pic] 0.9[pic] 0.9*1.1*[pic]= 0.99 [pic]
Both stocks have the same prices, but their prices are lower than 2 years ago.
9.8 Five-year Holding Period Return
= (1-0.0491)((1+0.2141)((1+0.2251)((1+0.0627)((1+0.3216)-1
= 98.64%
9.9 Risk Premium = 5.7% - 3.7% = 2%
Expected Return on the market long term corporate bonds
= 4.36% + 2% = 6.36%
9.10 [pic]
b. R [pic][pic] [pic][pic]
-0.026 -0.185 0.03423
-0.010 -0.169 0.02856
0.438 0.279 0.07784
0.047 -0.112 0.01254
0.164 0.005 0.00003
0.301 0.142 0.02016
0.199 0.040 0.00160
Total 0.17496
[pic]
Note, because the data are historical data, the appropriate denominator in the calculation of the variance is N-1.
9.11 a. Common Treasury Realized
Stocks Bills Risk Premium
-7 32.4% 11.2% 21.2% -6 -4.9 14.7 -19.6
-5 21.4 10.5 10.9
-4 22.5 8.8 13.7
-3 6.3 9.9 -3.6
-2 32.2 7.7 24.5
Last 18.5 6.2 12.3
b. The average risk premium is 8.49%.
[pic]
c. Yes, it is possible for the observed risk premium to be negative. This can happen in any single year. The average risk premium over many years should be positive.
9.12 a.
| | |Return if | |
|Economic State |Prob. (P) |State Occurs |P(Return |
|Recession |0.2 |0.05 |0.010 |
|Moderate Growth |0.6 |0.08 |0.048 |
|Rapid Expansion |0.2 |0.15 |0.030 |
| | |Expected Return = |0.088 |
b.
| | |[pic] |[pic] |
|Return if State Occurs |[pic] |[pic] |P([pic] |
|0.05 |-0.038 |0.001444 |0.0002888 |
|0.08 |-0.008 |0.000064 |0.0000384 |
|0.15 |0.062 |0.003844 |0.0007688 |
| | |Variance = |0.0010960 |
Standard deviation = [pic]
9.13 a.
| | |Return if State occurs | |
|Economic State |Prob.(P) | |P(Return |
|Recession |0.3 |0.02 |0.006 |
|Moderate Growth |0.4 |0.05 |0.020 |
|Rapid Expansion |0.3 |0.10 |0.030 |
| | |Expected Return = |0.056 |
b.
| | |[pic] |[pic] |
|Return if State occurs |[pic] |[pic] |P([pic] |
|0.02 |-0.036 |0.001296 |0.0003888 |
|0.05 |-0.006 |0.000036 |0.0000144 |
|0.10 | 0.044 |0.001936 |0.0005808 |
| | |Variance = |0.0009840 |
Standard deviation = [pic]= 0.03137 = 3.137%
9.14 a.[pic]
b.
[pic]
9.15 a.
[pic]
b. [pic] [pic]
-0.0175 0.00031
-0.0025 0.00001
+0.0325 0.00106
-0.0175 0.00031
0.00169
Variance of [pic]
Standard Deviation of [pic]
[pic]
-0.04 0.0016
-0.02 0.0004
0.01 0.0001
0.05 0.0025
0.0046
Variance of [pic]
Standard Deviation of[pic]
16. [pic]= Average Return on the Small Company Stocks.
[pic]= Average Return on the Market Index.
[pic] = Variance in the Returns of the Small Company Stocks.
SS = Standard Deviation in the Returns of the Small Company Stocks.
[pic]= Variance in the Returns of the Market Index.
[pic]= Standard Deviation in the Returns of the Market Index.
a. [pic]= [pic]
[pic]=[pic]
b. Small Company Stocks Market Index
[pic] [pic] [pic] [pic]
0.3228 0.10419984 0.2416 0.05837056
0.1848 0.03415104 0.4876 0.23775376
-0.5042 0.25421764 -0.7404 0.54819216
0.1558 0.02427364 0.1676 0.02808976
-0.1592 0.02534464 -0.1564 0.02446096
Total = 0.89686720
[pic]
Note, because the data are historical returns, the appropriate denominator in the calculation of the variance is N-1.
17. Let Rcs = The Returns on Common Stocks (in %)
Let Rss = The Returns on Small Stocks (in %)
Let Rcb = The Returns on Long-term Corporate Bonds (in %)
Let Rgb = The Returns on Long- term Government Bonds (in %)
Let Rtb = The Returns on Treasury Bills (in %)
Let – over a variable denote its average value
|Year |Rcs |Rss |Rch-0.1405 |Rgb |Rtb |
|1980 |0.3242 |0.3988 |-0.0262 |-0.0395 |0.1124 |
|1981 |-0.0491 |0.1388 |-0.0096 |0.0185 |0.1471 |
|1982 |0.2141 |0.2801 |0.4379 |0.4035 |0.1054 |
|1983 |0.2251 |0.3967 |0.0470 |0.0068 |0.0880 |
|1984 |0.0627 |-0.0667 |0.1639 |0.1543 |0.0985 |
|1985 |0.3216 |0.2466 |0.3090 |0.3097 |0.0772 |
|1986 |0.1847 |0.0685 |0.1985 |0.2444 |0.0616 |
|Total |1.2833 |1.4628 |1.1205 |1.0977 |0.6902 |
|Average |0.1833 |0.2090 |0.1601 |0.1568 |0.0986 |
| |__ |__ |__ |__ |__ |
|Year |Rcs- Rcs |Rss- Rss |Rch- Rch |Rgb- Rgb |Rtb- Rtb |
|1980 |0.1409 |0.1898 |-0.1863 |-0.1963 |0.0138 |
|1981 |-0.2324 |-0.0702 |-0.1697 |-0.1383 |0.0485 |
|1982 |0.0308 |0.0711 |0.2778 |0.2467 |0.0068 |
|1983 |0.0418 |0.1877 |-0.1131 |-0.1500 |-0.0106 |
|1984 |-0.1206 |-0.2757 |0.0038 |-0.0025 |-0.0001 |
|1985 |0.1383 |0.0376 |0.1489 |0.1529 |-0.0214 |
|1986 |0.0014 |-0.1405 |0.0384 |0.0876 |-0.0370 |
| |__ |__ |__ |__ |__ |
|Year |(Rcs- Rcs)2 |(Rss- Rss)2 |(Rch- Rch)2 |(Rgb- Rgb)2 |(Rtb- Rtb)2 |
|1980 |0.0198 |0.0360 |0.0347 |0.0385 |0.0002 |
|1981 |0.0540 |0.0049 |0.0288 |0.0191 |0.0024 |
|1982 |0.0009 |0.0051 |0.0772 |0.0609 |0.0000 |
|1983 |0.0017 |0.0352 |0.0128 |0.0225 |0.0001 |
|1984 |0.0146 |0.0760 |0.0000 |0.0000 |0.0000 |
|1985 |0.0191 |0.0014 |0.0222 |0.0234 |0.0005 |
|1986 |0.0000 |0.0197 |0.0015 |0.0077 |0.0014 |
|Total |0.1102 |0.1784 |0.1771 |0.1721 |0.0045 |
Because these data are historical data, the proper divisor for computing the variance is N-1. Thus, the variance of the returns of each security is the sum of the squared deviations divided by six.
|Var ( [pic]) = 0.018372 |SD ( [pic]) = 0.1355 |
|Var ( [pic]) = 0.029734 |SD ( [pic]) = 0.1724 |
|Var ( [pic]) = 0.029522 |SD ( [pic]) = 0.1718 |
|Var ( [pic]) = 0.02868 |SD ( [pic]) = 0.16935 |
|Var ( [pic]) = 0.00075 |SD ( [pic]) = 0.02747 |
9.18 a. The average return on small company stocks is
[pic](6.85-9.30+22.87+10.18-21.56+44.63)%/6 = 8.95%
The average return on T-bills is:
[pic]
b.
|Small Company Stock |T-Bills |
|[pic] |Rs -[pic] |[pic] |[pic] |RT -[pic] |[pic] |
| 0.0685 |-0.020950 |0.000439 |0.0616 |-0.004667 |0.000022 |
|-0.0930 |-0.182450 |0.033288 |0.0547 |-0.011567 |0.000134 |
| 0.2287 | 0.139250 |0.019391 |0.0635 |-0.002767 |0.000008 |
| 0.1018 | 0.012350 |0.000153 |0.0837 | 0.017433 |0.000304 |
|-0.2156 |-0.305050 |0.093056 |0.0781 | 0.011833 |0.000140 |
| 0.4463 | 0.356850 |0.127342 |0.0560 |-0.010267 |0.000105 |
| |Total |0.273667 | |Total |0.000713 |
| |Var = |5.47% | |Var = |0.01% |
| |Std. = |23.40% | |Std. = |1.19% |
| | |
c. Returns on T-bills are lower than small stock returns but their variance is much smaller.
9.19 The range with 95% probability is: [pic]
( ( 17.5-2( 8.5, 17.5+2(8.5(
( ( 0.5%, 34.5%(
9.20 a. Expected Return on the Market:
= 0.25(-8.2%)+0.5(12.3%)+0.25(25.8%)
= 10.55%
Expected Return on T-Bills: = 3.5%
b. Expected Premium = 0.25(-8.2-3.5)+0.5(12.3-3.5)+0.25(25.8-3.5) = 7.05%
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