Chapter 12, Problem 8QP

Risk Premiums [LO2, 3] Refer to Table 12.1 in the text and look at the period from 1970 through 1975.

a. Calculate the arithmetic average returns for large-company stocks and T-bills over this period.

b. Calculate the standard deviation of the returns for large-company stocks and T-bills over this period.

c. Calculate the observed risk premium in each year for the large-company stocks versus the T-bills. What was the average risk premium over this period? What was the standard deviation of the risk premium over this period?

d. Is it possible for the risk premium to be negative before an investment is undertaken? Can the risk premium be negative after the fact? Explain.

Summary Introduction

To determine: The arithmetic average for large-company stocks and Treasury bills.

Introduction:

Arithmetic average return refers to the returns that an investment earns in an average year over different periods.

Answer to Problem 8QP

The arithmetic average of large company stocks is 5.55 percent, and the arithmetic average of Treasury bills is 6.04 percent.

Explanation of Solution

Given information:

Refer to Table 12.1 in the chapter. Extract the data for large-company stocks and Treasury bills from 1970 to 1975 as follows:

Year	Large Company Stock Return	Treasury Bill Return	Risk Premium
1970	3.94%	6.50%	−2.56%
1971	14.30%	4.36%	9.94%
1972	18.99%	4.23%	14.76%
1973	–14.69%	7.29%	–21.98%
1974	–26.47%	7.99%	–34.46%
1975	37.23%	5.87%	31.36%
Total	33.30%	36.24%	–2.94%

The formula to calculate the arithmetic average return:

$\text{Arithmetic average}\left(\overline{X}\right)=\frac{{\displaystyle \sum _{i=1}^N{X}_i}}{N}$

Where,

“X_i” refers to each of the observations from X₁ to X_N (as “i” goes from 1 to “N”)

“N” refers to the number of observations

Compute the arithmetic average for Large-company stocks:

The total of observations is 33.30%. There are 6 observations.

$\begin{array}{c}\text{Arithmetic average}\left(\overline{X}\right)=\frac{{\displaystyle \sum _{i=1}^N{X}_i}}{N}\\ =\frac{33.30\%}{6}\\ =5.55\%\end{array}$

Hence, the arithmetic average of large-company stocks is 5.55 percent.

Compute the arithmetic average for Treasury bill return:

The total of observations is 36.24%. There are 6 observations.

$\begin{array}{c}\text{Arithmetic average}\left(\overline{X}\right)=\frac{{\displaystyle \sum _{i=1}^N{X}_i}}{N}\\ =\frac{36.24\%}{6}\\ =6.04\%\end{array}$

Hence, the arithmetic average of Treasury bills is 6.04 percent.

Summary Introduction

To determine: The standard deviation of large-company stocks and Treasury bills.

Introduction:

Standard deviation refers to the deviation of the observations from the mean.

Answer to Problem 8QP

The standard deviation of large-company stocks is 23.23 percent, and the standard deviation of Treasury bills is 1.53 percent.

Explanation of Solution

Given information:

Refer to Table 12.1 in the chapter. Extract the data for large-company stocks and Treasury bills from 1970 to 1975 as follows:

Year	Large Company Stock Return	Treasury Bill Return	Risk Premium
1970	3.94%	6.50%	−2.56%
1971	14.30%	4.36%	9.94%
1972	18.99%	4.23%	14.76%
1973	–14.69%	7.29%	–21.98%
1974	–26.47%	7.99%	–34.46%
1975	37.23%	5.87%	31.36%
Total	33.30%	36.24%	–2.94%

The formula to calculate the standard deviation:

$SD(R)=\sigma =\sqrt{\frac{{\displaystyle \sum _{i=1}^N{(X_i-\overline{X})}^2}}{N-1}}$

Where,

“SD (R)” refers to the variance

“X̅” refers to the arithmetic average

“X_i” refers to each of the observations from X₁ to X_N (as “i” goes from 1 to “N”)

“N” refers to the number of observations

Compute the squared deviations of large company stocks:

Large company stocks
Actual return (A)	Average return (B)	Deviation (A)–(B)=(C)	Squared deviation (C)2
0.0394	0.0555	-0.0161	0.00026
0.1430	0.0555	0.0875	0.00766
0.1899	0.0555	0.1344	0.01806
-0.1469	0.0555	-0.2024	0.04097
-0.2647	0.0555	-0.3202	0.10253
0.3723	0.0555	0.3168	0.10036
Total of squared deviation ${\displaystyle \sum _{i=1}^N{(X_i-\overline{X})}^2}$			0.26983

Compute the standard deviation:

$\begin{array}{c}SD(R)=\sigma =\sqrt{\frac{{\displaystyle \sum _{i=1}^N{(X_i-\overline{X})}^2}}{N-1}}\\ =\sqrt{\frac{0.26983}{6-1}}\\ =0.2323\text{ or }23.23\%\end{array}$

Hence, the standard deviation of Large company stocks is 23.23 percent.

Compute the squared deviations of Treasury bill:

Treasury bills
Actual return (A)	Average return (B)	Deviation (A)–(B)=(C)	Squared deviation (C)2
0.065	0.0604	0.0046	0.00002116
0.0436	0.0604	-0.0168	0.00028224
0.0423	0.0604	-0.0181	0.00032761
0.0729	0.0604	0.0125	0.00015625
0.0799	0.0604	0.0195	0.00038025
0.0587	0.0604	-0.0017	0.00000289
Total of squared deviation ${\displaystyle \sum _{i=1}^N{(X_i-\overline{X})}^2}$			0.0011704

Compute the standard deviation:

$\begin{array}{c}SD(R)=\sigma =\sqrt{\frac{{\displaystyle \sum _{i=1}^N{(X_i-\overline{X})}^2}}{N-1}}\\ =\sqrt{\frac{0.0011704}{6-1}}\\ =0.0153\text{ or }1.53\%\end{array}$

Hence, the standard deviation of Treasury bills is 1.53 percent.

Summary Introduction

To determine: The arithmetic average and the standard deviation of observed risk premium.

Introduction:

Arithmetic average return refers to the returns that an investment earns in an average year over different periods. Standard deviation refers to the deviation of the observations from the mean.

Answer to Problem 8QP

The arithmetic average is (0.49 percent), and the standard deviation is 25.42 percent.

Explanation of Solution

Given information:

Refer to Table 12.1 in the chapter. Extract the data for large-company stocks and Treasury bills from 1970 to 1975 as follows:

Year	Large Company Stock Return	Treasury Bill Return	Risk Premium
1970	3.94%	6.50%	−2.56%
1971	14.30%	4.36%	9.94%
1972	18.99%	4.23%	14.76%
1973	–14.69%	7.29%	–21.98%
1974	–26.47%	7.99%	–34.46%
1975	37.23%	5.87%	31.36%
Total	33.30%	36.24%	–2.94%

The formula to calculate the arithmetic average return:

$\text{Arithmetic average}\left(\overline{X}\right)=\frac{{\displaystyle \sum _{i=1}^N{X}_i}}{N}$

Where,

“X_i” refers to each of the observations from X₁ to X_N (as “i” goes from 1 to “N”)

“N” refers to the number of observations

The formula to calculate the standard deviation:

$SD(R)=\sigma =\sqrt{\frac{{\displaystyle \sum _{i=1}^N{(X_i-\overline{X})}^2}}{N-1}}$

Where,

“SD (R)” refers to the variance

“X̅” refers to the arithmetic average

“X_i” refers to each of the observations from X₁ to X_N (as “i” goes from 1 to “N”)

“N” refers to the number of observations

Compute the arithmetic average for risk premium:

The total of observations is (2.94%). There are 6 observations.

$\begin{array}{c}\text{Arithmetic average}\left(\overline{X}\right)=\frac{{\displaystyle \sum _{i=1}^N{X}_i}}{N}\\ =\frac{\left(2.94\%\right)}{6}\\ =\left(0.49\%\right)\end{array}$

Hence, the arithmetic average of risk premium is (0.49 percent).

Compute the squared deviations of risk premium:

Risk premium
Actual return (A)	Average return (B)	Deviation (A)–(B)=(C)	Squared deviation (C)2
-0.0256	0.0604	-0.086	0.0074
0.0994	0.0604	0.039	0.00152
0.1476	0.0604	0.0872	0.0076
-0.2198	0.0604	-0.2802	0.07851
-0.3446	0.0604	-0.405	0.16403
0.3136	0.0604	0.2532	0.06411
Total of squared deviation ${\displaystyle \sum _{i=1}^N{(X_i-\overline{X})}^2}$			0.32317

Compute the standard deviation:

$\begin{array}{c}SD(R)=\sigma =\sqrt{\frac{{\displaystyle \sum _{i=1}^N{(X_i-\overline{X})}^2}}{N-1}}\\ =\sqrt{\frac{0.32317}{6-1}}\\ =0.2542\text{ or }25.42\%\end{array}$

Hence, the standard deviation of risk premium is 25.42 percent.

Summary Introduction

To determine: Whether the risk premium can be negative before and after investment.

Explanation of Solution

The risk premium cannot be negative before investment because investors require compensation for assuming the risk. They will invest if the stock compensates for the risk. The risk premium can be negative after investment if the nominal returns are very low when compared to the risk-free returns.