Created 7/1/1996

Go to Brad De
Long's Home Page

**http://www.j-bradford-delong.net/Intro_Finance/BAonethirty8b.html**

**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((r
_{1}-E(r_{1}))(m-E(m))]/E[(m-E(m))^{2}] - r*
_{i}= r*_{f}+ beta_{i}(r*_{m}-r_{f})

**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 |
Mega |
Exciting |
Startup |

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...

Value additivity...

Project betas...

Measuring betas...

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