Tuesday, 2 August 2011

Proof that One Should Not Put All the Eggs in the Same Basket

It is common sense that a simple form of risk mitigation is through portfolio diversification. This principle has been popularized as “do not put all the eggs in the same basket!” That is, do not invest all your Euros in a single investment. However, a few discordant voices claim that one should put all the eggs in a just a couple of baskets, so that they can be properly monitored.

The view proclaiming the advantage of diversification is based on the use of a statistical measure of dispersion called variance. It has the unique property that the variance of (a+b) is smaller than the variance of (a) plus the variance of (b). Since modern portfolio theory adopted variance as a proxy to measure risk it follows that the more securities one adds to a portfolio the smaller its risk. In fact, in most markets a substantial degree of risk reduction can be achieved by holding between 25 and 50 assets. So, a simple rule of thumb (for an equally weighted portfolio) is that investors should invest no more than 2 or 4% of their capital on each exposure.

In practice this simple rule has three major drawbacks: a) it is difficult to adopt by small investors without enough funds to buy economically at least 25 different assets, and is also hard to follow by those with large amounts of money without incurring excessive exposures to a single entity; b) exposures to 25 or 50 investments are hard to monitor by a single person; and c) most importantly, the expected risk reduction is strongly dependent on the degree of correlation between the different assets and correlations are extremely difficult to forecast.

On logical grounds it is clearly questionable to prove a property by choosing to identify it with something that has the property we wish to prove. In addition, it has also three more theoretical limitations: a) price and total return volatilities are not necessarily correlated with other important sources of risk, namely bankruptcy and market delisting; b) for some investment strategies volatility is more useful as a measure of opportunity rather than risk; and c) it does not fully accounts for the reduced returns caused by its implicit bias towards cash-like exposures.

So it seems worthwhile to add an alternative proof that reinforces the case for the superiority of diversification. Marginalist theories tell us that we should invest up to the point where the expected marginal efficiency of capital equals the risk free rate of return. This may be so at the aggregate level. But, when assessing individual investment opportunities, it implies that investors should invest as much as possible on the exposure with the highest expected returned, followed by the second and so on until they invest all their funds. For most investors this rule would mean investing their entire capital in the single most promising investment opportunity.

Yet, with the schedules of the marginal efficiency of capital sloping downwards (the normal textbook case), investing up to the point where it meets the free risk rate of return is not an optimal solution. To verify this imagine the case of an investor considering opportunities A and B with linear schedules sloping downwards. The first Euro invested in opportunity A has an expected return of 30% while the last Euro invested earns the same as the risk free rate of return, which is 10%. The first Euro invested in the second best opportunity B earns only 20% while the last earns the risk free return. As a result the expected total return for A is 20% and for B it is only 15%. Naturally, ignoring risk or assuming identical levels of risk, investors would put all their money into opportunity A.

This would not be an optimal solution, since by switching the last Euro invested in opportunity A to B it would earn 20% instead of 15%. Likewise one may improve the total return by switching the penultimate Euro and so on until it is no longer worthwhile. This point will be reached when the integrals of the two marginal efficiency curves in the two defined ranges are equal. In the numerical example given above an investor would be able to invest up to two thirds of his capital in opportunity B and still achieve a total return of 20%. However, his optimal allocation would be to invest two thirds into A and one third in B to achieve an expected total return of 21.67%.

Note that our proof of the supremacy of diversification was given under conditions of certainty. Should we wish to add risk we may consider a worst case scenario where the investment opportunity with the highest return has also the highest risk. We may, for instance, assume that the expected return of the first Euro invested in A might be missed by ten percentage points while that of B may be missed by only 2 percentage points. With these revised marginal efficiency curves the return for A would be 15% and for B would be 14%. Under this scenario investors could allocate up to 89% of their capital to B and still achieve a total return of 15%. But now the optimal capital allocation to A would be only 56%, with an expected total return for both assets of 16.78%.

Our simple model may be extended to more than two assets and to incorporate the effects of leverage and still prove the case for diversification. Its theoretical value is that it proves the case for diversification with or without risk (uncertainty). Its main practical drawback is the reliance on the marginal efficiency of capital curves which are not as easy to estimate as the volatilities. Still we believe that it will be enough to convince the few remaining skeptics about the superiority of diversification.

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