# Finance

Created 7/1/1996
Go to

### Using the CAPM

Basics:

• Present value of a perpetuity: C/r
• Present value of a growing (or shrinking) perpetuity: C/(r-g)
• Present value of C dollars t years from now: C/[(1+r)t]
• Present value of a C-dollar t-year annuity: C[(1/r)-(1/[r(1+r)t])
• beta = [E((r1-r*1)(rm-r*m)]/[(rm-r*m)2]
• r*i = r*f + betai(r*m-rf)
• 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 CAPM--all of this adjusting of discount rates for "systematic" "market" risk--have 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 mini-firm.

Companies should discount cash flows depending on their systematic riskiness--not on whether their expected return exceeds the company cost-of-capital.

Measuring betas:

• Look at past "stuff"--at the past fluctuations in the company's stock price relative to the market.
• Alphas--drift in price over the recent past (ought to be ironed down to competitive levels by the market)
• R-squared: 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). H-P'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 information--first, 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:

risk-free 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

Leveraged-to-the-Gills 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

Leveraged-Beyond-the-Gills Firm--Left 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 non-approval 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 beta-of-two 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 risk-free 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 day--or \$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 negative-beta investment is probably right).