Financial economics/Tutorials

The Capital Asset Pricing Model

The rate of return, r,  from an equity asset is given by

r = r_f  β(r_m  - r_f)

'

where

r_f  is the risk-free rate of return

r_m  is the equity market rate of return

(and r_m - r_f is known as the equity risk premium)

and β is the covariance of the asset's return with market's return divided by the variance of the market's return.

(for a proof of this theorem see David Blake Financial Market Analysis page 297 McGraw Hill 1990)

The Arbitrage Pricing Model

The rate of return on the ith asset in a portfolio of n assets, subject to the influences of factors j=1 to k is given by

${r_{i}}={r_{f}}+\sum _{j=1}^{k}{y_{j}}{beta_{ij}}$

where

${beta_{ij}}={\frac {Cov(r_{i},d_{j})}{Var(d_{j})}}$

and

$y_{j}$ is the weighting multiple for factor $j$

$Cov(r_{i},d_{j})$ is the covariance between the return on the ith asset and the jth factor,

$Var(d_{j})$ is the variance of the jth factor

Gambler's ruin

If q is the risk of losing one throw in a win-or-lose winner-takes-all game in which an amount c is repeatedly staked, and k is the amount with which the gambler starts, then the risk, r, of losing it all is given by:

r  =  (q/p)^(k/c)

where p  =  (1 - q),  and q  ≠  1/2

(for a fuller exposition, see Miller & Starr Executive Decisions and Operations Research Chapter 12, Prentice Hall 1960)