Chapter 10, Problem 8QP

a)

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. Variance refers to the average difference of squared deviations of the actual data from the mean or average. Standard deviation refers to the deviation of the observations from the mean.

Answer to Problem 8QP

The arithmetic average of large-company stocks is 3.24%, and the arithmetic average of Treasury bills is 6.55%.

Explanation of Solution

Given information:

Refer Table 10.1 in the chapter. Extract the data for large-company stocks and Treasury bills from 1973 to 1978 as follows:

Year	Large co. stock return (A)	T-bill return (B)	Risk premium (A)-(B)
1973	–14.69%	7.29%	–21.98%
1974	–26.47%	7.99%	–34.46%
1975	37.23%	5.87%	31.36%
1976	23.93%	5.07%	18.86%
1977	–7.16%	5.45%	–12.61%
1978	6.57%	7.64%	–1.07%
Total	19.41%	39.31%	–19.90%

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 the total of observations,

“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:

$\begin{array}{c}\text{Arithmetic average}\left(\overline{X}\right)=\frac{{\displaystyle \sum _{i=1}^N{X}_i}}{N}\\ =\frac{0.1941}{6}\\ =0.0324\text{or}\text{3}.24\%\end{array}$

Hence, the arithmetic average of large-company stocks is 3.24 %.

Compute the arithmetic average for Treasury bill return:

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

$\begin{array}{c}\text{Arithmetic average}\left(\overline{X}\right)=\frac{{\displaystyle \sum _{i=1}^N{X}_i}}{N}\\ =\frac{0.3931}{6}\\ =0.0655\text{or}6.55\%\end{array}$

Hence, the arithmetic average of Treasury bills is 6.55%.

b)

Summary Introduction

To determine: The standard deviation of 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. Variance refers to the average difference of squared deviations of the actual data from the mean or average. Standard deviation refers to the deviation of the observations from the mean.

Answer to Problem 8QP

The standard deviation of large-company stocks is 24.11%, and the standard deviation of Treasury bills is 1.24%.

Explanation of Solution

Given information:

Refer Table 10.1 in the chapter. The arithmetic average of Treasury bills is 6.55%.

Extract the data for large-company stocks and Treasury bills from 1973 to 1978 as follows:

Year	Large co. stock return (A)	T-bill return (B)	Risk premium (A)-(B)
1973	–14.69%	7.29%	–21.98%
1974	–26.47%	7.99%	–34.46%
1975	37.23%	5.87%	31.36%
1976	23.93%	5.07%	18.86%
1977	–7.16%	5.45%	–12.61%
1978	6.57%	7.64%	–1.07%
Total	19.41%	39.31%	–19.90%

The formula to calculate the standard deviation:

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

“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	Average return (B)	Deviation (A)–(B)=(C)	Squared deviation (C)2
(A)
−0.1469	0.0324	−0.1793	0.0321485
−0.2647	0.0324	−0.2971	0.0882684
0.3723	0.0324	0.3399	0.115532
0.2393	0.0324	0.2069	0.0428076
−0.0716	0.0324	−0.104	0.010816
0.0657	0.0324	0.0333	0.0011089
Total of squared deviation			0.05813
$\sqrt{\frac{{\displaystyle \sum _{i=1}^N{(X_i-\overline{X})}^2}}{N-1}}$

Compute the standard deviation:

$\begin{array}{c}SD(R)=\sigma =\sqrt{\frac{{\displaystyle \sum _{i=1}^N{(X_i-\overline{X})}^2}}{N-1}}\\ =\sqrt{\frac{0.2906814}{6-1}}\\ =0.24111\text{ or 24}\text{.111}\%\end{array}$

Hence, the standard deviation of large-company stocks is 24.111%.

Compute the squared deviations of Treasury bill:

Large-company stocks
Actual return	Average return (B)	Deviation (A)–(B)=(C)	Squared deviation (C)2
(A)
0.0729	0.0655	0.0074	0.00005
0.0799	0.0655	0.0144	0.00020736
0.0587	0.0655	−0.0068	0.00004624
0.0507	0.0655	−0.0148	0.00021904
0.0545	0.0655	−0.011	0.000121
0.0764	0.0655	0.0109	0.00011881
$\sqrt{\frac{{\displaystyle \sum _{i=1}^N{(X_i-\overline{X})}^2}}{N-1}}$			0.000154

Compute the standard deviation:

$\begin{array}{c}SD(R)=\sigma =\sqrt{\frac{{\displaystyle \sum _{i=1}^N{(X_i-\overline{X})}^2}}{N-1}}\\ =\sqrt{\frac{0.00077}{6-1}}\\ =0.0124\text{ or }1.24\%\end{array}$

Hence, the standard deviation of Treasury bills is 1.24%.

c)

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. Variance refers to the average difference of squared deviations of the actual data from the mean or average. Standard deviation refers to the deviation of the observations from the mean.

Answer to Problem 8QP

The arithmetic average is −3.32%, and the standard deviation is 24.92%

Explanation of Solution

Given information:

Refer Table 10.1 in the chapter. Extract the data for large-company stocks and Treasury bills from 1973 to 1978 as follows:

Year	Large co. stock return (A)	T-bill return (B)	Risk premium (A)-(B)
1973	–14.69%	7.29%	–21.98%
1974	–26.47%	7.99%	–34.46%
1975	37.23%	5.87%	31.36%
1976	23.93%	5.07%	18.86%
1977	–7.16%	5.45%	–12.61%
1978	6.57%	7.64%	–1.07%
Total	19.41%	39.31%	–19.90%

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 the total of observations,

“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:

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

“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:

$\begin{array}{c}\text{Arithmetic average}\left(\overline{X}\right)=\frac{{\displaystyle \sum _{i=1}^N{X}_i}}{N}\\ =\frac{-0.1990}{6}\\ =-0.032\text{or 3}\text{.32}\end{array}$

Hence, the arithmetic average of risk premium is −3.32%.

Compute the squared deviations of risk premium:

Risk premium
Actual return (A)	Average return (B)	Deviation (A)–(B)=(C)	Squared deviation
			(C)2
−0.2198	−0.0332	−0.1866	0.034820
−0.3446	−0.0332	−0.3114	0.096970
0.3136	−0.0332	0.3468	0.120270
0.1886	−0.0332	0.2218	0.049195
−0.1261	−0.0332	−0.0929	0.008630
−0.0107	−0.0332	0.0225	0.000506
$\sqrt{\frac{{\displaystyle \sum _{i=1}^N{(X_i-\overline{X})}^2}}{N-1}}$			0.062078

Compute the standard deviation:

$\begin{array}{c}SD(R)=\sigma =\sqrt{\frac{{\displaystyle \sum _{i=1}^N{(X_i-\overline{X})}^2}}{N-1}}\\ =\sqrt{\frac{0.31039}{6-1}}\\ =0.2492\text{ or }24.92\%\end{array}$

Hence, the standard deviation of risk premium is 24.92%.

d)

Summary Introduction

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

Introduction:

Arithmetic average return refers to the returns that an investment earns in an average year over different periods. Variance refers to the average difference of squared deviations of the actual data from the mean or average. Standard deviation refers to the deviation of the observations from the mean.

Explanation of Solution

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