The document provides information to calculate the mean and standard deviation of historical percentage returns (HPR) for the stock market given different economic states and their probabilities. [1] The mean HPR is calculated as the weighted average of returns based on probabilities, which equals 14%. [2] Variance is calculated by taking the difference of each return from the mean, squaring it, and weighting it by the probabilities. [3] Standard deviation is the square root of variance, which equals 24.0249% or approximately 24.0% rounded.

1. Essentials of Investments
BODIE, KANE, MARCUS, 8TH EDITION
Problem + Solution for Chapter 5, Problem 5

2. Compute mean and standard deviation of
the market, given:
State of the
Probability HPR
economy
Suppose this is your expectation of the Boom 0.3 44%
return on the entire stock market
Normal
Use equations 5.6.-5.8 0.4 14
Growth
Pages 114-115 of the textbook
Recession 0.3 -16

3. Expected Return = E(r)
Expected return of the stock (s) is the mean value, or average of the return (r).
The probabilities are weighted based on probability (p).

4. Mean = Probability * Return
State of the
Probability HPR
economy
All probabilities add up to equal 1,
or 100% Boom 0.3 44%
.3 + .4 + .3 = 1
Normal
0.4 14
Growth
30% + 40% + 30% = 100%
Recession 0.3 -16

5. Calculate!
State of the
Probability HPR
economy
.3 * 44 = 13.2
.4 * 14 = 5.6 Boom 0.3 44%
.3 * -16 = -4.8 Normal
0.4 14
Growth
13.2 + 5.6 + -4.8 = 14%
Recession 0.3 -16

6. Expected Return E(r)
14% is the E(r), or mean

7. Equation for mean
E(r) = ∑ p(s) r(s)
Expected (r)eturn is equal to the sum of the (s)tock’s
(p)robabilities multiplied by the (r)eturns

8. Compute
State of the
Probability HPR
Remember:Compute mean and economy
standard deviation of the HPR
Boom 0.3 44%
given:
Normal
Mean = 14 Growth
0.4 14
Standard Deviation Recession 0.3 -16

9. σ =standard deviation
To calculate standard deviation, you must first calculate
variance.

10. σ 2= variance
Variance is equal to the square of standard deviation.
Calculate variance using E(r) or Expected Return, which we have
already calculated

11. Finding variance - Part 1
State of the Probability HPR
economy p(s) r(s)
Simply subtract the expected return from
the HPR for each row, and square that value Boom 0.3 44%
[r(s) - E(r)]2
Normal
0.4 14
Growth
Remember: We calculated E(r) = 14
Recession 0.3 -16

12. Compute!
HPR
r(s) - E(r) [r(s) - E(r)]2
r(s)
Subtract E(r) from each HPR to find 44 44-14 = 30 30 2 =900
the difference.
Then, square this number. 14 14-14 = 0 0 2 =0
-16 -16 - 14 = -32 -32 2 =1024

13. Weight the Variance - Part 2
State of
Probability HPR Variance
the
p(s) r(s) Var(r)
economy
Now the variance must be weighted as well Boom 0.3 44% 900
You’ll use the same probabilities p(s) as
Normal
before 0.4 14 0
Growth
Recession 0.3 -16 1024

14. Equation for Variance
σ 2 = ∑ p(s) [r(s) - E(r)]2

15. Compute!
Variance = .3 * (900) + .4 * (0) + .3 (1024)
Variance = 270 + 0 + 307.2 = 577.2

16. From Variance to Standard Deviation
Simply find the square root of the variance.
σ 2 = 577.2
σ = √577.2
σ = 24.0249

17. Final Answer
Mean = 14.0%
Standard Deviation = 24.0%