Chapter 3.2, Problem 42E

a)

To determine

To find standard deviation for each type of investment

Answer to Problem 42E

Standard deviation for each type of investments:

Stocks: 15.39

Bills: 2.11

Bonds: 8.58

Explanation of Solution

Formula:

Mean:

  $\overline{x}=\frac{{\displaystyle \sum _{i=1}^n{X}_i}}{n}$

Population standard deviation:

  ${\sigma }^{}=\sqrt{\frac{{\displaystyle \sum {(x_i-\overline{x})}^2}}{n}}$

Calculation:

Stocks:

Creating table for finding standard deviation:

$x$	$x-\overline{x}$	${(x-\overline{x})}^2$
26.01	18.46	340.77
22.64	15.09	227.71
16.1	8.55	73.10
25.22	17.67	312.23
-6.18	-13.73	188.51
-7.1	-14.65	214.62
-16.76	-24.31	590.98
25.32	17.77	315.77
3.15	-4.4	19.36
-0.61	-8.16	66.59
16.29	8.74	76.39
6.43	-1.12	1.25
-33.84	-41.39	1713.13
18.82	11.27	127.01
11.02	3.47	12.04
5.53	-2.02	4.08
7.26	-0.29	0.08
26.5	18.95	359.10
7.52	-0.03	0.00
-2.23	-9.78	95.65

Here n = 20

Values of Xi are return from stock.

Putting all values in formula of mean,

  $\overline{x}=\frac{151.09}{20}=7.55$

From table,

  ${\displaystyle \sum {(x_i-\overline{x})}^2}=4738.39$

Put all values in the formula of population standard deviation,

  $\sigma =\sqrt{\frac{4738.39}{20}}=15.39$

Bills:

Creating table for finding standard deviation:

$x$	$x-\overline{x}$	${(x-\overline{x})}^2$
5.02	2.66	7.06
5.05	2.69	7.22
4.73	2.37	5.60
4.51	2.15	4.61
5.76	3.40	11.54
3.67	1.31	1.71
1.66	-0.70	0.49
1.03	-1.33	1.78
1.23	-1.13	1.28
3.01	0.65	0.42
4.68	2.32	5.37
4.64	2.28	5.18
1.59	-0.77	0.60
0.14	-2.22	4.94
0.13	-2.23	4.99
0.03	-2.33	5.44
0.05	-2.31	5.35
0.07	-2.29	5.26
0.05	-2.31	5.35
0.21	-2.15	4.64

Here n = 20

Values of Xi are return from bills.

Putting all values in formula of mean,

  $\overline{x}=\frac{47.26}{20}=2.36$

From table,

  ${\displaystyle \sum {(x_i-\overline{x})}^2}=88.83$

Put all values in the formula of population standard deviation,

  $\sigma =\sqrt{\frac{88.83}{20}}=2.11$

Bonds:

Creating table for finding standard deviation:

$x$	$x-\overline{x}$	${(x-\overline{x})}^2$
1.43	-4.30	18.52
9.94	4.21	17.69
14.92	9.19	84.38
-8.25	-13.98	195.55
16.66	10.93	119.38
5.57	-0.16	0.03
15.12	9.39	88.10
0.38	-5.35	28.67
4.49	-1.24	1.55
2.87	-2.86	8.20
1.96	-3.77	14.24
10.21	4.48	20.03
20.1	14.37	206.38
-11.12	-16.85	284.06
8.46	2.73	7.43
16.04	10.31	106.21
2.97	-2.76	7.64
-9.1	-14.83	220.05
10.75	5.02	25.16
1.28	-4.45	19.84

Here n = 20

Values of Xi are return from bonds.

Putting all values in formula of mean,

  $\overline{x}=\frac{114.68}{20}=5.73$

From table,

  ${\displaystyle \sum {(x_i-\overline{x})}^2}=1473.11$

Put all values in the formula of population standard deviation,

  $\sigma =\sqrt{\frac{1473.11}{20}}=8.58$

Here, Population standard deviation for each type of investments:

Stocks: 15.39

Bills: 2.11

Bonds: 8.58

There is highest population standard deviation for Stocks, which leading high risk.

There are is low standard deviation for Bills, which leading least risk

b)

To determine

To justify results with finance theory

Explanation of Solution

Given:

Bills are short-term loans and Bonds are long-term loans to the U.S. government.

As per finance theory, Long term loans are riskier than short term loans.

Justification:

Population standard deviation of Bills = 2.11

Population standard deviation of Bonds = 8.58

Here, Population standard deviation of Bonds is greater than Bills. That means Loans from Bonds are riskier than Bills. This result agrees with finance theory

c)

To determine

To find mean for each type of investment

Explanation of Solution

Given:

As per finance theory, the more risk of an investment has, higher their mean must be.

Formula:

Mean:

  $\overline{x}=\frac{{\displaystyle \sum _{i=1}^n{X}_i}}{n}$

Calculation:

Stocks:

Here n = 20

Values of Xi are return from stock.

Putting all values in formula of mean,

  $\overline{x}=\frac{151.09}{20}=7.55$

Bills:

Here n = 20

Values of Xi are return from bills.

Putting all values in formula of mean,

  $\overline{x}=\frac{47.26}{20}=2.36$

Bonds:

Here n = 20

Values of Xi are return from bonds.

Putting all values in formula of mean,

  $\overline{x}=\frac{114.68}{20}=5.73$

From all calculations,

Mean for each type of investment:

Stocks:7.55

Bills:2.36

Bonds:5.73

There is highest mean for Stocks, which leading high risk.

There are is low mean for Bills, which leading least risk.

The results from population standard deviation and finance theory are same as from mean of an investment. Hence, this result agrees with finance theory