Of Risk and Myopia

Of all of finance's mysteries, the most enduring is the so-called "equity-risk-premium puzzle"—why are stocks priced so low and, thus, their returns so high? In the 20th century, stocks rewarded more than 6% in excess of riskless T-bills. Surely, in an efficient market, investors would have priced stocks higher, with consequently lower returns.

This may now have already happened. With stocks yielding about 1.5% and long-term real earnings growth in the 2% range, stocks are priced for a long-term real return of about 3.5%. Until a short while ago, T-bills offered about a 3% real yield, but recent events have inflated the price of safety to the point where their real yield is zero. Thus, the expected equity risk premium is about 3.5%.

Richard Thaler and Shlomo Benzarti, in their classic article "Myopic Loss Aversion and the Equity Premium Puzzle" (Quarterly Journal of Economics, 1995), recalled a story told by Paul Samuelson, who offered a colleague a coin toss with a gain of $200 for heads and a loss of $100 for tails. Even though the coin toss had an expected return of $50 ([$200 x 0.5]-[$100 x 0.5]), his colleague turned him down. The reason: he would feel more pain with the loss of $100 than with the pleasure of a $200 gain. The colleague quickly added that he'd be happy to accept 100 such coin tosses, where in order to lose, he would have to flip less than 34 heads, the odds of which are less than one in a thousand.

This is, of course, highly illogical. Thaler and Benzarti realized that the key ingredient here was the frequency of evaluation: although Samuelson's colleague would happily accept 100 coin tosses, he could not bear to watch them being made singly.

To repeat: the risk tolerance of an investor is determined largely by how often he checks his portfolio. This is nothing new. Benjamin Graham commented in The Intelligent Investor that holders of obscure mortgage bonds happily held onto them through the depths of the Depression until they eventually recovered their value because they were highly illiquid and not often quoted. On the other hand, holders of frequently-quoted corporate bonds (far less risky but priced daily in the papers) panicked and sold after their initial drop. The largest financial holding of most families is their house—it's a good thing we don't see its value printed every day in the financial section.

Using a clever interpretation of Kahneman and Tversky's prospect theory, Thaler and Benzarti determined that stock investors behaved as if their time-horizon were about one year. Unfortunately, their methodology is a bit obscure. Recently, I came across a much more facile explanation of the risk-aversion-myopia phenomenon in Nassim Nicholas Taleb's Fooled by Randomness. (I highly recommend this delightful book to anyone interested in the role of chance in the financial markets. But be forewarned; many will find the author's ego a bit much. If literary self-absorption annoys you, better pass.)

The Taleb paradigm is stunning in its simplicity. Here's the short version: Over relatively brief periods, return increases proportionally with time but risk increases more slowly, as the square root of time. This means that the risk/return ratio becomes increasingly favorable over long time horizons.

But to start with, over short horizons, the risk/return ratio is extremely unfavorable. Imagine that stocks have an annualized return of 10% and a standard deviation of 20%. Over a time horizon of one day (1/260th of a year), the return calculates out to 0.038% and the standard deviation to 1.24% (20%/sqrt[260]). According to the laws of probability, the investor will see a positive return only 51.2% of the time.

Now, assume that a loss hurts twice as much as a gain. The "utility" of our "daily investor" is thus –0.464 (that is, .512 – [.488 x 2]). With a 2:1 loss-gain utility function, 1.0 denotes perfect happiness; zero, ambivalence; and minus 2.0, perfect misery.) Even worse, our daily investor, like most investors, cares nothing for historical data. If he is trading actively enough, he may not even be able to determine reliably whether he is winning more often than he is losing. Since the average loss hurts more than the average gain, he soon loses heart and sells.

In the below table, I've calculated the return, standard deviation, probability of a positive return, and utility over varying time horizons: