# What is the theory behind diversification? Why does it work?

We are told always that diversification in a portfolio is good. "Don't put all your eggs in one basket." But is there mathematical theory behind diversification to back up the advice?

At its simplest level it's an application of basic statistics/probability:

Suppose you have n independent and identically distributed assets with the return on asset i denoted R_i which has mean m and variance s^2 (same for all assets). You can easily weaken these assumptions but I make them to simplify the exposition [Square brackets show a numerical example with n=20, m=8%, s=2%]

if you invest in one of these assets you expect to get a return of m [8%] with standard deviation s [2%] (so you expect with probability 95% (approx) to get a return between m-2*s and m+2*s. [between 4% and 12%]

Now suppose you split your money equally among the n-assets. Your return is now

R = (1/n)\Sum{i=1}^n R_i

E(R) = (1/n)\Sum{i=1}^n E(R_i) = (1/n)\Sum{i=1}^n m = m [8%]

the variance of your return is

Var(R) = Var( (1/n)\Sum{i=1}^n R_i ) = (1/n^2)\Sum{i=1}^n Var(R_i) = n * s^2 / n^2) = s^2/n

So, the standard deviation is SD(R) = Sqrt(V(R)) = s/Sqrt(n) [2%/Sqrt(20) = 0.44%]

Now, with 95% probability we get a return between E(R)-2*SD(R) and E(R)+2*SD(R) [between 7.12% and 8.88%]. This interval is smaller than when we invested in the single asset, so in effect with this portfolio we are achieving the same return m [8%] but with lower variance (risk) [0.44% instead of 2%]. This is the result of diversification.

You can assume the assets are not independent (and most book expositions of this topic do indeed do that). In that case the calculation is modified because the variance of the portfolio now depends on the correlation between returns, as does the reduction in variance caused by the diversification. If assets are negatively correlated the result of the diversification will be more reduction in risk and vice versa. You can also assume the assets are not identically distributed and the above analysis does not change too much.

You might look for some references on CAPM (Capital Asset Pricing Model) or portfolio theory but broadly these are based on what I have described above - finding the portfolio with minimum variance for a given return by investing proportionally in treasury bonds and risky assets.

Diversification is a good method of risk management. Different types of investments do better in different situations and economic climates. Invest all your money at the wrong time in a single product and you could lose everything. You could also technically make a great deal of money, but actions such as these are the actions of speculators, not investors.

In theory, the idea is that diversified assets will perform differently in different circumstances, spreading your risk around.

Whether that still functions in practice is a decent question, as the "truth" of most probability based arguments for diversification rely on the different assets being at least somewhat uncorrelated.