Harry Markowitz in his Portfolio Analysis introduced into financial analysis the power of diversification. It was a powerful theoretical breakthrough that allowed statistical method to be applied. The Capital Assets Pricing Model of William Sharpe showed that the correlation of the performance of different stocks limited the reduction in risk that could be achieved by diversification. Sharpe introduced the term market risk to denote the risk that could not be reduced by diversification. In the stock market the profits of companies and the prices of stocks may be correlated due to their dependence upon the overall performance of the economy as represented by the level of GDP. The performance of some groups of companies may to correlated due to a mutual dependence on the price of petroleum. Another set of companies may have their performance tied to the price of steel. The price of steel may also be correlated with the price of petroleum thus tying together the performances of the two groups of companies.

The concepts of the reduction of risk can be usefully applied to other markets such as home mortgages. This gave rise to the practise of securitization in which a pool of thousands of mortgages is created and then shares of this pool and its income are sold. In such securitization the risks of default are spread. However, such securitization does not reduce the risk of defaults associated with the price level of real estate. That is the topic of the analysis which follows.

Let r_i be the rate of return of the i-th asset and

r_i = a_i + b_ip + u_i for i=1,…, n

where a_i and b_i are constants, p is a market variable and u_i is a random variable. It is assumed that u_i and u_j are independent for i≠j. For simplicity let p be a random variable with a mean value of zero. Thus p is the anomaly of the market variable. It is also assumed that p and all of the u_i variables are independent of each other. These preceeding assumptions are imply that the expected value of r_i equal to a_i.

Let x_i be the share of an investment portfolio being held in the i-th asset. The rate of return r₀ on the portfolio is then

r₀ = Σx_ir_i
and thus its expected value is
E(r₀) = Σx_ia_i

The variance of a random variable is the expected value of its deviation from its expected value. The variance of the rate of return on the portfolio Var(r₀) is then given by

Var(r₀) = Σx_i²Var(r_i) + Var(p)(Σx_ix_jb_ib_j)

This relationship holds only because all of the covariances among the variables are zero.

For the simple case in which x_i=1/n, Var(r_i)=σ² and b_i=1 the above relationship reduces to

Var(r₀) = (n/n²)σ² + Var(p)[n²/n²]
which further reduces to
Var(r₀) = (1/n)σ² + Var(p)

Thus in the limit as n→∞, Var(r₀) → Var(p).

(To be continued.)