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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?

18

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

your expected return is

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.

7

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.

Spreading your investments appropriately lets you maximize your growth opportunities while limiting your risk.

4

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.

This article suggests that might not be true. Specifically:

The correlations we note among industry sectors are profoundly and dysfunctionally high.

and

Gold and silver traders have gotten too used to the negative correlation trade with stocks. This is, in fact, an unusual relationship for precious metals tostocks. The correlation should actually be zero.

2

Years ago I wrote an article Risk, Reward, Coin Flipping which explains from a 'game theory' perspective how diversifying works to minimize standard deviation in one's returns. It's long and tedious, not easy to summarize, but it holds up well, I'm pleased with how the analogy does its job.

Update - the above is too "link-only", written over 5 years ago. The article I wrote offers a mathematical approach via an understandable example of coin flipping. With just 2 options, a 'head' is a 10% loss, while a 'tail' is a 30% gain. This actually represents the market fairly well as it results in a 10% average gain and 28% standard deviation for just 2 flips. The article shows how by 'diversifying', choosing to make multiple smaller bets, the average 10% stays the same, but the standard deviation is brought down dramatically, 7.6% when we use a sample experiment with 7 coins.

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