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

Black-Scholes option pricing theorem

The fair price,P, of a call option on a security is given by:

$P=CN(d_{1})-Xe^{-rt}N(d_{2})\,$

where:

C is the current price of the security;

$N(d_{1})$ is the cumulative probability distribution for the standard normal variate from -∞ to $d_{1}$;

X is the exercise price (see options definition);

r is the risk-free interest rate;

t is the time to expiry of the option;

$d_{1}$ and $d_{2}$ are given by the equations:

$d_{1}={\frac {\ln(C/X)+(r+\sigma ^{2}/2)t}{\sigma {\sqrt {t}}}}$;

$d_{2}=d_{1}-\sigma {\sqrt {t}}$;

and

$\sigma$ is the standard deviation (or volatility) of the price of the asset.

The underlying assumptions are that:

• Dividend payments are not included;
• Options cannot be exercise before the stipulated date;
• Markets are efficient;
• No commissions are paid;
• Volatility is constant;
• The interest rate is constant; and,
• Returns are log-normally distributed.

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)