Derivation of the Capital Asset Pricing Model - CAPM

E(xi) expected return on the asset ir risk free rateE(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).

[1]

[2]

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

Define the function C as follows

[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.

[4.1]

[4.2]

[4.3]

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[4.4]

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

[5.1]

[5.2]

…

[5.3]

[5.4]

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

[6]

The term in the bracket on the left-hand side is a variance of the optimal portfolio o

2. Substituting it for the bracketed expression on the LHS and multiplying the RHS expression through we obtain:

[7]

Thus the standard deviation of optimal portfolio is

[8]

At a specific point where we obtain:

[9]

and hence

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[10]

where m denotes market portfolio, which is optimal for all investors, and m

is a standard deviation of the market portfolio.

Expression defines the slope of the market line.

Reciprocal of Lagrange multiplier 1/ measures the price of a unit of risk or required increase in expected return when one unit of risk is added to the portfolio.

Now we can derive the equilibrium relationship between an individual asset’s expected return and its risk:

Risk reflects the standard deviation of returns on the asset itself and also the covariance with the returns of all other risky assets in the market.

We can use the set of equations [4.x] to derive the general relationship among the expected returns of all shares and their risk.

In general, the i-th equation of [4.x] at the point can be rewritten as:

[11

Solving this equation for the expected return of i-th asset E(xi) we obtain:

[12]

Substituting for from the equation [10], we obtain

[13]

Now, recall, that by definition, the covariance of the i-th asset with the market portfolio can be written as

[14]

Thus the expected return on any risky asset can be written as:

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[15]

or alternatively

[16]

where . [17]

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