# Risk Premiums and Diversification

In his recent post on employee stock option valuation, Patri said this about the relationship between risk and return:

Now, this analysis was done on average. Clearly the risk of holding onto the option is much higher than the risk involved in an index fund. Yet I think one can reasonably argue that having to chop one’s interest-bearing capital in half is rather a high price to pay for diversification. Furthermore, we can assume that the return on the individual stock is not just the market rate of return, but the risk-adjusted market rate of return, so it already includes some extra compensation for its riskiness.

I can't find it now, but earlier this year I made a similar claim in a comment thread here at Catallarchy---that risky investments tend to have higher average returns. Having had some time to think about this, I realize that I was wrong, or at least dramatically oversimplifying the issue. I know I'm not breaking any new ground here, but I figure that if Patri and I are both confused about something, chances are that some other people might be, too, so I'd like to share my thoughts on this topic.

But before I explain why I now think this is wrong, I want to discuss briefly the logic behind it and why it seemed right. The law of diminishing marginal utility tells us that the utility gain from receiving $X is less than the utility loss from losing $X. One way to model this behavior is to assume a utility function such that the utility of having $X is the square root of X. Suppose I have $500. If I have this utility function, what's the most I'd be willing to pay for a 50% chance to win another $100?

The math is a bit tricky, but the answer is $48.75. If I win, I have $551.25, and my utility gain is 1.12. If I lose, I have $451.25, and my utility loss is 1.12. The expected utility of this bet is zero. But my expected return is positive, specifically $1.25, or about 2.6%.

Suppose that we change the terms of the bet. Now I'm buying a 25% chance to win $200. How much am I willing to pay now? About $46.58. If I win, I have $653.42, and my utility gain is about 3.2. If I lose, I have $453.42, and my utility loss is about 1.07. Since I'm three times more likely to lose, the expected utility of this bet is zero. But my expected return is $3.42, or about 7.3%. In order to convince me to take a riskier bet, you have to offer me a higher return.

The problem with this reasoning is that it doesn't take into account the ability to eliminate risk through diversification. The calculations above are for one-shot games. But if I can play the game as many times as I want, then I can eliminate my risk, and the amount that I'm willing to pay increases.

So the real connection between risk and reward is this: Greater risk requires greater reward *if and only if that risk cannot be eliminated through diversification*.

If a company is working on a project which has a high probability of failure, but a potentially huge payoff, then buying only that one stock is a risky investment. But that risk can be eliminated. If I divide my money between many such stocks, I can dramatically reduce my risk. And just as I'm willing to pay more money per game for a repeated game than I am for a one-shot game, I'm willing to pay more per share if I buy many different stocks than I am if I buy just one. As I and many other investors bid up these stocks, the risk premium disappears.

Well...not quite. No matter how much you diversify, you can't eliminate all risk. The problem is that stocks are correlated---their movement is not independent. Just as a rising tide lifts all boats, a falling tide drops them, and the tides on Wall Street are deep and irregular. In a bear market, all of your stocks may fall. Or perhaps a few will rise, but not enough to make up for others which sink deeper than you expected. So there's still a premium for the risk inherent in the market as a whole, because it can't be diversified away. This is part of the reason why stocks have historically yielded higher average returns than government bonds.

To go back to Patri's example, you don't get a risk premium for investing in just one stock, because you don't have to. You could diversify the risk away, and since other investors are bound to be doing just that, the stock is priced accordingly. There's no risk premium for taking risks that other investors don't have to make, so an employee with a significant portion of his portfolio tied to his employer's stock, whether as an unexercised option or as an unvested stock grant, has to take on extra risk with no chance of extra reward.

For discussion: Are there subsets of the market for which risk above and beyond the market risk cannot be diversified away, and which therefore must have higher risk premiums?