Created 7/1/1996
Go to Brad De
Long's Home Page
http://www.jbradforddelong.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 
Average risk premium 
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 returnsnot compound annual rates of return. Difference? Compound annual have you reinvesting last year's gains (or losses) in the market. Hence a bad yearyou not only lose money; you also lose your capital and can take less advantage of future good years.
But over 69 years, 19261995: Average inflation factor 9.347
Portfolio 
Cumulative real wealth 
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 1933even with compounded and reinvested dividendsyou 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 poorand 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 riskfree 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 
Mega 
Simple 
Exciting 
Startup 
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 x_{1} in security 1 and x_{2} 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 riskthe variance is the sum of these four boxes

security 1 
security 2 
security 1 
x_{1}^{2}s_{1}^{2} 
x_{1}x_{2}s_{12} 
security 2 
x_{1}x_{2}s_{12} 
x_{2}^{2}s_{2}^{2} 
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
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