Financial Markets and Institutions (2-downloads)

deviation. The standard deviation of returns on an asset is calculated as follows.

First you need to calculate the expected return, R

e

; then you subtract the expected

return from each return to get a deviation; then you square each deviation and mul-

tiply it by the probability of occurrence of that outcome; finally, you add up all these

weighted squared deviations and take the square root. The formula for the stan-

dard deviation,  , is thus:

(2)

The higher the standard deviation,  , the greater the risk of an asset.

␴

␴ ⫽ 2p

1

1R

1

⫺ R

e

2

2

⫹ p

2

1R

2

⫺ R

e

2

2

⫹ p ⫹ p

n

1R

n

⫺ R

e

2

2

␴

What is the standard deviation of the returns on the Fly-by-Night Airlines stock and Feet-

on-the Ground Bus Company, with the same return outcomes and probabilities described

above? Of these two stocks, which is riskier?

Solution

Fly-by-Night Airlines has a standard deviation of returns of 5%.

where

p

1

=

probability of occurrence of return 1 

= = 

0.50

R

1

=

return in state 1 

= 15% = 0.15

p

2

=

probability of occurrence of return 2 

= = 

0.50

R

2

=

return in state 2 

= 5% = 0.05

R

e

=

expected return 

= (.50)(0.15) + (.50)(0.05) = 0.10

1

2

1

2

R

e

⫽ p

1

R

1

⫹ p

2

R

2

␴ ⫽ 2p

1

1R

1

⫺ R

e

2

2

⫹ p

2

1R

2

⫺ R

e

2

2

E X A M P L E   4 . 2 Standard Deviation

---

Chapter 4 Why Do Interest Rates Change?

67

2

Diversification, the holding of many risky assets in a portfolio, reduces the overall risk an investor

faces. If you are interested in how diversification lowers risk and what effect this has on the price of an

asset, you can look at an appendix to this chapter describing models of asset pricing that is on the

book’s Web site at 

www.pearsonhighered.com/mishkin_eakins

.

Thus,

Feet-on-the-Ground Bus Company has a standard deviation of returns of 0%.

where

p

1

=

probability of occurrence of return 1 

= 1.0

R

1

=

return in state 1 

= 10% = 0.10

R

e

=

expected return 

= (1.0)(0.10) = 0.10

Thus,

Clearly, Fly-by-Night Airlines is a riskier stock because its standard deviation of returns

of 5% is higher than the zero standard deviation of returns for Feet-on-the-Ground Bus

Company, which has a certain return.

⫽ 20 ⫽ 0 ⫽ 0%

␴ ⫽ 211.02 10.10 ⫺ 0.102

2

R

e

⫽ p

1

R

1

␴ ⫽ 2p

1

1R

1

⫺ R

e

2

2

␴ ⫽ 21.502 10.00252 ⫹ 1.5020.00252 ⫽ 20.0025 ⫽ 0.05 ⫽ 5%

␴ ⫽ 21.502 10.15 ⫺ 0.102

2

⫹ 1.502 10.05 ⫺ 0.102

2

A risk-averse person prefers stock in the Feet-on-the-Ground (the sure thing) to

Fly-by-Night stock (the riskier asset), even though the stocks have the same expected

return, 10%. By contrast, a person who prefers risk is a risk preferer or risk lover.

Most people are risk-averse, especially in their financial decisions: Everything else

being equal, they prefer to hold the less risky asset. Hence, holding everything else