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Created 7/1/1996
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Business Administration 130

Brealey and Myers, chapter 7: Risk and Return

http://www.j-bradford-delong.net/Intro_Finance/BAonethirty7.html

 


Basics:


Introduction to Risk and Return

"The opportunity cost of capital depends on the risk of the project": I've been saying this for three and a half weeks now. But what does it mean? What is the risk of a project? Why should the appropriate cost of capital vary depending on how risky the project is?

Let's start with risk.

 Portfolio

 Average real return
(inflation-adjusted)

 Average risk premium
(vis-a-vis Treasury bills)

 Treasury bills

 0.6%/year

 0

 Treasury bonds

 2.1%/year

 1.4%/year

 Corporate bonds

 2.7%/year

 2.0%/year

 Stocks (S&P 500)

 8.9%/year

 8.3%/year

 Small stocks

 13.9%/year

 13.2%/year

Arithmetic average returns--not compound annual rates of return. Difference? Compound annual have you reinvesting last year's gains (or losses) in the market. Hence a bad year--you not only lose money; you also lose your capital and can take less advantage of future good years.

 

But over 69 years, 1926-1995: Average inflation factor 9.347

 Portfolio

 Cumulative real wealth
(inflation-adjusted)

 Average inflation factor

 Treasury bills

 1.46

 9.347

 Treasury bonds

 3.09

 Corporate bonds

 4.55

 Stocks (S&P 500)

 96.99

 Small stocks

 340.17

So, why not invest in small stocks? Why doesn't everyone, all the time, invest in nothing but small stocks? Because they are risky. Start investng in a portfolio of small stocks in 1926; by 1933--even with compounded and reinvested dividends--you have lost70% of your real wealth. Invest in a porfolio of small stocks in 1967, and by 1974 you find that you have lost 70% of your real wealth has well.

Yes, but aren't there offsetting gains that make these risks worthwhile? Maybe. The point is that when the value of your portfolio falls you become poor--and when you become poor, the money that you don't have matters a lot to you. By contrast, when you are rich your marginal dollar matters much less.

r(market) = r(free)+normal risk premium. We assume that the risk-free rate is more likely to vary than is the risk premium.


Standard statistical measures of risk:

Variance = Expected value of [r(market realized) - r(market expected)]^2

Standard deviation = sqrt(variance)

HH +40%

HT +10%

TH +10%

TT -20%

Variance=.25*(.3)^2+.25*(.3)^2 = .045

Standard deviation=sqrt(.045)=.212=21.2%


Estimating risk:

Suppose we flip coins: THTTTTHTTHTHTTHTHTHH

or: 10%, -20%, -20%, 10%, 10%, 10%, -20%, 10%, 10%, 30%

average:3%, Std Dev: 17.03%

Our estimates are only estimates, and are far from being perfect...


Diversification and risk:

 "State of the World"

Universal
Utility

Mega
Manufacturing

Simple
Service

Exciting
Exports

Startup
Semiconductor

 HH

 +20%

 +40%

 +20%

 +40%

 +20%

 HT

 +0%

 +10%

 +5%

 +50%

 -20%

 TH

 +10%

 +10%

 +5%

 -30%

 -20%

 TT

 -10%

 -20%

 -10%

 -20%

 +20%

$100 in Mega Manufacturing: Expected Return=10% ($10); Std Dev=21.2%

$50 in Mega, $50 in Universal Utility: Expected Return=7.5% ($7.5); Std Dev=16.0%

$25 in Mega, $25 in Simple, $25 in Exciting, $25 in Startup: Returns are (30%, 11.25%, -8.75%, 12.5%) Expected Return=6.25% ($6.25), Std Dev=14.6%

[Average all five: $20 in each; 28%, 9%, -5%, -10%; expected return 5.5%, std dev 12.7%]

Why does diversification reduce risk? Because stocks do not all move together...


Beta

Calculating portfolio risk; suppose we invest x1 in security 1 and x2 in security 2.

 

What is the return on the entire portfolio? That part is easy: it is just the weighted average of the individual returns.

What is the risk of the entire portfolio. We set it out on a 2 x 2 grid: the risk--the variance-- is the sum of these four boxes

 security 1

 security 2

 security 1

 x12s12

 x1x2s12

 security 2

 x1x2s12

 x22s22

E(x(1)r(1)+x(2)r(2))^2 = E(x(1)r(1))^2 + E(x(1)r(1)x(2)r(2))+E(x(1)r(1)x(2)r(2))+E(x(2)r(2))^2

This formula easily generalizes