Diversification: Still the Only Free Lunch in All of Finance

Posted by Alex Frey (@alexhfrey )

Diversification Is A Free Lunch I don't know if it is really true that there is no such thing as a free lunch in life, but I will say that most of the supposedly "free meals" that I have consumed have had some side effects that I would have preferred to avoid. 

In the investing world, there is an important exception. Owning a diversified portfolio truly one way to get something for nothing.

When 1 + 1 = 3

The true benefits of diversification are a bit difficult to grasp, because they counter many people's intuitions. In most examples outside of investing, diversifying is a way to insure average outcomes, but it doesn't create any value. For instance, if you spend twelve hours betting on a roulette wheel, splitting your bet size over five wheels every spin will reduce the size of your swings, but it will not make your money last significantly longer than putting the total amount on just one wheel every time. No value is really created.

The primary reason that investing is different is that returns (should be) are measured in risk-adjusted terms, and when you equally combine two assets that are not perfectly correlated, the resulting return is the average of the two asset's returns, while the resulting risk is less than the average of the two asset's risks. So if one stock is expected to go up 8% and the other up 10%, your portfolio will be expected to go up 9%. But if one stock is expected to have a "risk" of 20% and the other stock is expected to have a "risk" of 25%, your resulting risk will be less than 22.5% (here we have deliberately left out just what the unit of "risk" is). In risk-adjusted terms, value has been created.

To see how this happens in more concrete terms, imagine that you are given the option of investing in two stocks. Each is expected to go up 8% a year on average, but has a 10% chance of going down 50% in any given year. Assume that a bad year for one stock is no more or less likely than normal to be a bad year for the other stock. 

If you put all your money into one of the stocks, you will make 8% on average, but lose 50% in one out of every ten years. Losing 50% can really hurt, as it will take about 9 years of 8% returns to make up for it, due to the nature of the compound arithmetic.  

But if you invest in both stocks, something "funny" happens. While you will still make the same average return (8%), your chance of losing 50% of your money in any one year goes down to 1%, meaning you will suffer this fate in only about 1 out of every 100 years rather than 1 out of every 10 years. So you now have lower risk for the same degree of returns. 

Here's the math:

Two Investments - Possibility Matrix
Stock A Outcome Stock B Outcome Probability of Outcome Return
Up Up .9 * .9 = .81 8%
Up Down .9 * .1 = .09 .5 * (8%) + .5* (-50%) = -21%
Down Up .1 * .9 = .09 .5 * (-50%) + .5* (8%) = -21%
Down Down .1 * .1 = .01 -50%

Diversification can Increase Returns Too

If you have a high tolerance for risk, you should still care about diversification, because it is usually possible to easily tradeoff risk and return. 

To see how this can work in practice, imagine that in the example above you were an investor who is only comfortable accepting a 1% chance of losing more than 25% of your money. If you were to only invest in one of the assets, you would have to keep 50% of your money in cash, which right now is going to get you 0% returns. So in a good year, your return would be 4% (0 * 50% + 8 * 50%) and in a bad year you would return -25% (50% x - 50 + 50% x 0). By contrast, if you split your money into both assets, you would meet your risk requirements while still earning 8% a year on average. More return -- without any additional risk.

Implications of the Free Lunch

Understanding the "free lunch" of diversification has some wide-ranging circumstances. A big one is that if you want to hold the optimal portfolio, what you should search for is not just the assets that are going to have the best returns, but the assets that are going to have good returns and be uncorrelated with your existing portfolio.

A good example is adding Treasury Bonds to an all-stock portfolio. Bonds have achieved much lower returns than stocks over time, but because they are uncorrelated with stocks, they can significantly reduce risk. So even if you anticipate Treasury Bonds to have very low returns in the future (as many do), it may still make sense to include at least some in a portfolio as a diversifier. 

Another implication is that holding a concentrated portfolio of individual stocks is risky. By definition, this will result in a portfolio that is not as diversified as it could be. But a direct consequence of this is that there is likely a portfolio out there that will offer the same expected returns while also having much lower risk (or uncertainty of returns). To hold this kind of portfolio, you not only have to be right that your stocks are mis-priced, but the size of the mis-pricing has to be greater than the benefits that you are losing out on by not being diversified.

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By Alex Frey