Year capital gain total $ return (loss) 2001 -$10.30 -$6.86 2002 3.40 6.84 2003 -11.00 -7.56 2004 39.55 42.99 2005 25.75 29.46 TR for 2003 = ($3.44 + ($56.70 - $67.70)) / $67.70 = -.1117 or -11.2% TR for 2004 = ($3.44 + ($96.25 - $56.70)) / $56.70 = .7582 or 75.8%
NOTE: These two years were chosen specifically for their contrast. This is a good opportunity for instructors to point out how TRs for a company can fluctuate violently from year-to-year. This shows dramatically the risk of common stocks as well as the opportunities for large returns.
6-2. This investor would have a (short-term) capital gain, with a tax liability of $5000 - $4000 = $1000 (.28) = $280
6-3. Calculating Total Returns (TRs) for these assets: (a) TRps = (Dt + (PE - PB)) / PB
where Dt = the preferred dividend PE = ending price or sale price
PB = beginning price or purchase price TR = (5 + -7) / 70
= -2.86% (b) TRw = (Ct + PC) / PB
where Ct is any cash payments paid (there are none for a warrant) PC = price change during the period
TR = (0 + 2)/11
= 18.18% for the three month period (c) TRb = (It + PC) / PB
= (240* + 60) / 870
= 34.5% for the two year period.
*interest received is $120 per year (12% of $1000) for two years. Calculating Return Relatives (RRs) for these examples:
(a) a TR of -2.86% is equal to a RR of .9714 or (1.0+ [-.0286]) (b) a TR of 18.18% is equal to a RR of 1.1818
(c) a TR of 34.5% is equal to a RR of 1.345
6-4. Calculate future values using tables at end of text: @12% $100 (1.762) = $176.20 after 5 years
$100 (3.106) = $310.60 after 10 years $100 (9.646) = $964.60 after 20 years $100 (29.96) = $2996.00 after 30 years
Calculate present values using tables at end of text: @12% $100 (.567) = $56.70 after 5 years
$100 (.322) = $32.20 after 10 years $100 (.104) = $10.40 after 20 years $100 (.033) = $3.30 after 30 years
6-5. (a) The arithmetic rate of return is
[.3148 + (4.847) + 20.367 + 22.312 + 5.966 + 31.057] / 6 = 17.72
The geometric mean rate of return for the S&P 500 Composite Index for 1980-1985 (from Table 6-1) is:
G = (1.3148 x .95153 x 1.20367 x 1.22312 x 1.05966 x 1.31057)1/6 - 1.0 = (2.5579111)1/6 - 1.0
6-6. Refer to Equation 6-12 for the standard deviation formula.
NOTE: We use n-1 in the calculation.
_ _ Year TR(%),X X-X (X-X)2 1980 31.480 13.7575 189.2688 1981 -4.847 -22.5695 509.3823 1982 20.367 2.6445 6.9934 1983 22.312 4.5895 21.0635 1984 5.966 -11.7565 138.2153 1985 31.057 13.3345 177.8089 ─────── ───────── 106.335 1042.7322 _ X = 17.7225 1042.7322/5 = 208.5464 = variance (208.5464)1/2 = 14.44% 6-7. $100(1.3148)(.95153)(1.20367)(1.22312)(1.05966)(1.31057)(1.18539)(1.05665) (1.16339) (1.31229) = 4.89140 = the cumulative wealth index for this period. (4.89140)1/10 = 1.17204
1.17204 - 1.0 = .17204 or 17.204%
6-8. There are 85 years for the period Jan. 1920 through Dec. 2004.
Cumulative wealth = (1.10259)85 = $4,028.97
6-9. Cumulative wealth = (1.0608)84 = $142.30
6-10. (1.0532)84 = $98.06
NOTE: the data start at the beginning of 1920; therefore, there are 84 years, or (2003 - 1920) + 1
This provides practice for periods other than the 85 years from 1920 through 2004.
6-11. (26.965)1/83 = 1.0405; 1.0405 - 1.0 = .0405 or 4.05%
6-12. First, raise 3.00 to the 73rd power;
Second, divide nominal cumulative wealth by the cumulative inflation index.
$13,293.14 / 8.652 = $1,536.42 = inflation-adjusted CWI for small common stocks, 1926-1998.
6-13. (8.54/1)1/78 = 1.0279
1.0279 - 1.0 = .0279 or 2.79%
NOTE: For a problem such as this, always divide the ending value by the beginning value. There are 78 years here [(2003-1926) +1]
6-14. (1.0446)85 = 40.81 = cumulative wealth index for the yield component
From Figure 6-2, 4,029 is the cumulative wealth index value for stocks at the end of 2004.
4,029 / 40.81 = 98.73 = cumulative wealth index value for the capital gain or price change component.
NOTE: 40.81 X 98.73 = 4,029.17 (rounding errors account for the difference).
6-15. Obviously, we must put the two components of cumulative wealth on the same basis.
Converting the geometric mean for the yield component to cumulative wealth, we have (1.01)79 = 13.4852 NOTE: [(1998-1920) +1] = 79 years
Cumulative wealth index = 13.4852 X 6056.65 = $81,675.14.
The CWI for this (or any other financial asset) series is the product of the two components.
NOTE: The numbe rs here are made-up, and clearly not realistic. They are for illustration purposes only.
6-16. The two ways to calculate inflation-adjusted returns are:
1. 1.05316 / 1.02496 = 1.0275; (1.0275)84 = 9.7756 2. (1.05316)84 = 77.54; (1.02496)84 = 7.9320;
77.54 / 7.932 = 9.7756
6-17. Using a spreadsheet package, enter the 5 TRs from Table 6-1 for the years 1927-1931 as
Return Relatives. Round the returns to two decimal places. The program should calculate the geometric mean as -4.46%.
Knowing that the ending wealth index for 1931 is 0.79591, the same result can be obtained by calculating the geometric mean. Taking the fifth root of the wealth index using a calculator produces a result of .955, which is a geometric mean of -4.46% (after subtracting from 1.0 and multiplying by 100).
6-18. Any set of TRs that are identical will produce a geometric mean equal to the arithmetic
mean; for example, 10%, 10% and 10%, or any other set of three identical numbers.
6-19. The calculated results are:
Arithmetic Mean 15.77%
Standard Deviation 13.15%
Geometric Mean 15.07%
As we can see, the standard deviation for the shorter period was less than that of the entire period. This is because of the good years in the 1980s that were more similar than in a typical 10 or 11 year period. Also, there were only two negative years during this period, whereas the historical norm for many years was 3 negative years out of 10 (this did not occur in the 1990s).
6-20. Using a spreadsheet should verify that the standard deviation is calculated as 19%.
Changing the 1975 value from 36.92 to 26.92 changes the standard deviation from 19% to 17.48%. This is obviously because the dispersion is reduced. This value moves closer to the mean.
COMPUTATIONAL PROBLEMS
6-1. First, convert the TRs to Return Relatives: .909, .881, .779, 1.287, and 1.107.
Multiply these RRs together to obtain .8888, the cumulative wealth for the first 5 years. The cumulative wealth for the 1970s was (1.0588)10 = 1.7707. Divide this result by .8888 to obtain 1.9922. Take the 5th root of this result to obtain 1.1478. Subtract the 1.0 to obtain .1478 or 14.78%.
Thus, the geometric mean for the last 5 years must be 14.78% if the entire decade is to equal the performance of the 1970s.
6-2. Cumulative wealth for the first 5 years is .8888 (from Problem 6-1). Cumulative wealth for 10 years, given a geometric mean of 10.35, = (1.1035)10 = 2.6775.
If one of the next 5 years has a loss of 10%, the cumulative wealth for 6 years would be .8888 X .9 = .7999.
Therefore, divide 2.6775 by .7999 to obtain 3.3473.
Take the 4th root of 3.3473 to obtain 1.3526; subtract 1.0 to obtain 35.26%.
Therefore, the geometric mean of the remaining 4 years must be 35.26% in order for the decade to match the 20th Century geometric mean of 10.35%.
6-3. Knowing these two items, the geometric mean for the total return and the geometric mean for the dividend yield component, we can calculate the other component of total return. (a) The other component is the price change component.
(b) A total return index for common stocks of $4,028.97 (calculated as (1.10259)85) and a yield component index of 29.45, calculated as (1.0406)85, implies an ending wealth for the price change component of $138.81 (calculated as 4028.97 / 29.45).
6-4. The linkage between the geometric mean and the arithmetic mean is given, as an approximation, by Equation 6-12.
(1 + G)2 ≈ (1 + A.M.)2 - (S.D.)2
G = the geometric mean of a series of asset returns A. M. = the arithmetic mean of a series of asset returns
S. D. = the standard deviation of the arithmetic series of returns
Thus, if we know the arithmetic mean of a series of asset returns and the standard deviation of the series, we can approximate the geometric mean for this series. As the standard deviation of the series increases, holding the arithmetic mean constant, the geometric mean decreases.
Using the data given
(1 + G)2 ≈ (1.184395)2 - (.379058)2 (1 + G)2 ≈ 1.4028 - .1437
(1 + G)2 ≈ 1.2591
1 + G ≈ 1.1221; G = 12.21%
In this example, the very high standard deviation for this category of stocks results in a very low geometric mean annual return despite the high arithmetic mean. Variability matters!
Chapter 7: Portfolio Theory