Created 7/1/1996
Go to Brad DeLong's
Home Page
Business Administration 130:
Using the CAPM
http://www.jbradforddelong.net/Intro_Finance/BAonethirty9.html
Basics:
 Present value of a perpetuity: C/r
 Present value of a growing (or shrinking) perpetuity: C/(rg)
 Present value of C dollars t years from now: C/[(1+r)^{t}]
 Present value of a Cdollar tyear annuity: C[(1/r)(1/[r(1+r)^{t}])
 beta = [E((r_{1}r*_{1})(r_{m}r*_{m})]/[(r_{m}r*_{m})^{2}]
 r*_{i} = r*_{f} + beta_{i}(r*_{m}r_{f})
 Expected return of a portfolio with N securities, a share 1/N invested
in each security:
 Standard deviation of a portfolio with N securities, a share 1/N invested
in each security:
Using the CAPM:
So what does the CAPMall of this adjusting of discount rates for "systematic"
"market" riskhave to do with what we did in the first three
or four weeks: this evaluating individual investment projects by calculating
their NPVs?
One possibility: use the "company cost of capital" to assess
and value individual investment projects.
A better thing to do: value each individual investment project as
if it were a little minifirm.
Companies should discount cash flows depending on their systematic riskinessnot
on whether their expected return exceeds the company costofcapital.
Measuring betas:
 Look at past "stuff"at the past fluctuations in the company's
stock price relative to the market.
 Alphasdrift in price over the recent past (ought to be ironed down
to competitive levels by the market)
 Rsquared: the proportion of the total risk of the company that is
systematic market risk.
 Residual standard deviation: how much idiosyncratic risk is associated
with the security.
 Standard errors of alpha and beta; these things are estimates,
after all; estimates are only estimates: AT&T's beta varies from 0.54
to 0.26 to 0.67 to 0.96 over successive five year periods; does its "true"
beta vary that much? Probably not (but maybe). HP's estimated beta varies
from 1.27 to 1.39 to 1.36 to 1.65; does its "true" beta vary
that much? Probably not.
 Adjusted beta: if your estimated beta is low, its error is probably
negative. A cryptic sentence from Brealey and Myers: "the Bayesian
statistics needed to understand adjusted betas are beyond the scope of
this book." Pooh. Actually, you have two sources of informationfirst,
that this thing is a stock, drawn from a distribution with an average beta
of one; second, the estimated beta; a good forecast will use both pieces
of information.
 Industry betas: a way of using more information than is found in the
simple firm beta.
 Betas of portfolios are subject to much less sampling error than are
betas of firms.
Brealey and Myers wonder why they spend so much time on firm and industry
betas given that what matters is the project beta. I find their excuses
unconvincing... (musical "perfect pitch" metaphor)
Capital structure and company beta:
riskfree rate=5%; market required rate=10%
Unleveraged Firm

Initial Price 
Market Goes Down 
Market Stays Constant 
Market Goes Up 
Value of Firm 
$1,000 
$900 
$1,100 
$1,300 
Value of Bonds 
$0 
$0 
$0 
$0 
Value of Stocks (beta=1; 10 percent expected return) 
$1,000 
$900 
$1,100 
$1,300 
Slightly Leveraged Firm

Initial Price 
Market Goes Down 
Market Stays Constant 
Market Goes Up 
Value of Firm 
$1,000 
$900 
$1,100 
$1,300 
Value of Bonds 
$500 
$525 
$525 
$525 
Value of Stocks (beta=2; 15% expected return) 
$500 
$375 
$575 
$775 
LeveragedtotheGills Firm

Initial Price 
Market Goes Down 
Market Stays Constant 
Market Goes Up 
Value of Firm 
$1,000 
$900 
$1,100 
$1,300 
Value of Bonds 
$850 
$892.50 
$892.50 
$892.50 
Value of Stocks (beta=6 2/3; 38.3% expected return) 
$150 
$7.50 
$207.50 
$407.50 
LeveragedBeyondtheGills FirmLeft as an Exercise for the
Student

Initial Price 
Market Goes Down 
Market Stays Constant 
Market Goes Up 
Value of Firm 
$1,000 
$900 
$1,100 
$1,300 
Value of Bonds 
?? 
$900 
$1,100 
$1,100 
Value of Stocks (beta=6 2/3; 38.3% expected return) 
?? 
$0 
$0 
$200 
Gearing; the effect of financial structure and leverage on security betas;
firm beta as weighted average of debt beta and equity beta.
What If There Is No Beta Book Out There?
 Distinguish between risk and systematic risk
 A dry oil well is not systematic risk
 FDA approval or nonapproval of a drug is not systematic risk.
 Expropriation by the Cubans is not systematic risk: "Not all takeovers
start with a tender offer"
 Start thinking hard about systematic risk
 Avoid fudge factors in the discount rates (adjust expected cash flows
instead)
 Think about the determinants of asset betas
 Cyclicality: a lot of market risk is business cycle risk
 Operating leverage (high fixed costs, like airlines)
Examples:
 A project costs $100,000 and offers you a betaoftwo expected return
of $150,000 in one year; risk premium = 8.5%; the Wall Street Journal
reports that the riskless rate is 5.0%; the appropriate discount factor
is thus 22%; 150,000/(1.22)= $123,000 (approximately)thus the project
has a net present value of $23,000
 A company is financed 40% by riskfree debt; the interest rate is 10%;
the expected market return is 18%; and the stock's beta is 0.5; what is
the company cost of capital? Answer: A unit investment in the company
has an expected return of 0.4 x 10% + 0.6 x (10% + 0.5 x 8%) = 12.4%, which
is the company's cost of capital.
 An oil copmpany is drilling new wells; 20% of new wells are dry; 32%
produce 1000 barrels a day; 48% produce 5000 barrels a day; price of oil
of $18 a barrel. Answer: Expected oil production of 2,720 barrels
a day or $49,000 a dayor $17,870,000 a year; the geologist who proposes
to discount this cash flow at 30% because of the risk of dry holes is wrong
(an economist who proposed to discount this cash flow at less than the
riskless rate because oil wells are a negativebeta investment is probably
right).
Send email to Brad DeLong
Return to Brad DeLong's Home
Page
Return to Business Administration 130 Home
Page
ÿ