# >Finance

Created 7/1/1996
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### The Capital Asset Pricing Model [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-E(r1))(m-E(m))]/E[(m-E(m))2]
• r*i = r*f + betai(r*m-rf)

Portfolio Risk-Return Combinations:

If stocks are perfectly correlated, the risk-return combinations from diversifying across them are a straight line.

If stocks are uncorrelated, the risk-return combinations look like a bunch of parabolas...

Efficient frontiers: why choose an inefficient portfolio? Keep adding stocks:

These are the sets of returns available from:

 "State of the World" Universal Utility Mega Manufacturing Exciting Exports Startup Semiconductor HH +20% +40% +40% +20% HT +0% +10% +50% -20% TH +10% +10% -30% -20% TT -10% -20% -20% +20%

Universal-- 5%, 11.2%

*Mega-- 10%, 21.2%

Exciting-- 10%, 35.4%

Startup--0%, 20%

Add in a "riskfree" rate of zero...

Choose the portfolio S that just touches the line that goes through the riskfree-rate point and lies entirely to one side of the feasible portfolio set.

Then borrow (and lend) until you get to the risk-return characteristics you want; simply combine the "best" efficient risky portfolio with the risk-free rate...

Capital asset pricing model... :

Plot beta on the horizontal axis; expected return on the vertical axis; start with the riskfree rate. Add the diversified market. Draw the security market line. Everything must yield the expected return that places it on the market line for its beta.

Relationship between beta-return diagram and risk-return diagram: risk return diagram adds in idiosyncratic risk...

Proofs of the CAPM:

Start with the market portfolio M, yielding return r*(m) and variance V(m).

Add a small amount e of security S to the portfolio by selling e of the market:

return = (1-e)r*(m) + er*(s)

variance = Variance[(1-e)M+eS]=Expected Value[(1-e)^2M^2+2e(1-e)MS+e^2S^2]

=V(m)[(1-e)^2+2e(1-e)beta(S)+e^2beta(S)^2]

=V(m)[(1-e)+ebeta(S)]^2

Suppose this new risk, variance point is not on the security market line...

If it is above the security market line, then it is a new super-efficient portfolio--and everyone should stampede into it.

If it is below the security market line, then it is an _inefficient_ portfolio--and no one should want to hold even a small amount e of stock S.

An invest or can always obtain an expected risk premium of beta(r(m)-r(f)) by holding a mixture of the market portfolio and a risk-free rate investment. So in well functioning markets nobody will hold a stock that offers an expected risk premium of less than beta(r(m)-r(f))--and everyone would stampede into holding a stock that offered more...

Is the CAPM Valid?

A fine prescriptive theory; is it a good descriptive theory?

Pretty good--certainly good enough that it's hard to make money outguessing it...

Small firms have done better since the mid-1960s than the CAPM would predict; firms with a high market-to-book ratio have done better than the CAPM would predict. CAPM pretty sensitively dependent to everyone diversifying...

Other Theories:

Consumption CAPM; risk premium oddly large

APT--a bunch of different kinds of undiversifiable macroeconomic risk; oil price risk, inflation risk, recession risk, and so forth. APT doesn't tell you what the macroeconomic risk factors are.

One such list:

• Slope of yield curve
• Level of short-run interest rate
• Exchange rate
• Real GDP
• Inflation
• Market minus the effects of the first five...

Start on capital budgeting...

Project betas...

Measuring betas...