Derivation of the Capital Asset Pricing Model - CAPM

[pic]

E(xi) expected return on the asset i r risk free rate
E(xm) expected return on the market portfolio (S&P index)
(i i-th asset’s systematic risk (a proportion of market risk)

Optimal investment proportions - each individual investor on the market attempts to reach the highest feasible market line. The Market line can be found by minimising the standard deviation (o for any given portfolio‘s expected return E(xo).

[pic] [1]

[pic] [2]

where pi is the proportion of the portfolio invested in i-th asset.

Define the function C as follows

[pic] [3]

where ( is a Lagrange multiplier, and the expression in brackets equals zero.
We are trying to find the optimal proportion of each asset, which minimises the risk of optimal portfolio. The Market line can be found analytically by differentiating equation [3] with respect to each pi and with respect to Lagrange multiplier and setting the first derivatives equal to zero. This yields n+1 equations.

[pic] [4.1]

[pic] [4.2]
[pic] [4.3]

[pic] [4.4]

Let’s now take the set of equations [4.x] and multiply them by p1, p2 etc.
We obtain:

[pic] [5.1]

[pic] [5.2]

…
[pic] [5.3]

[pic] [5.4]

If we sum up the equations [5.1] to [5.3], we obtain

[pic] [6]

The term in the bracket on the left-hand side is a variance of the optimal portfolio (o2. Substituting it for the bracketed expression on the LHS and multiplying the RHS expression through we obtain:

[pic] [7]

Thus the standard deviation of optimal portfolio is

[pic] [8]

At a specific point where [pic] we obtain:

[pic] [9]

and hence

[pic] [10]

where m denotes market portfolio, which is optimal for all investors, and (m is a standard deviation of the market portfolio.

Expression [pic] defines