Efficient Frontier

William J. Bernstein

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 Investorthat 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:

Time Horizon

Return

Standard Deviation

Positive Return Probability

UtilityOne Hour

0.005%

0.438%

50.42%

-0.487

One Day

0.038%

1.240%

51.20%

-0.464

One Week

0.183%

2.773%

52.64%

-0.421

One Month

0.797%

5.773%

55.49%

-0.335

One Year

10.000%

20.000%

69.15%

0.075

Ten Years

159.400%

63.200%

99.41%

0.982

100 Years

1,377,961%

200.000%

100.00%

1.000

I've displayed this phenomena graphically with four "Taleb Plots." Note that in all plots, the annualized return is 0.1 and the annualized standard deviation is 0.2.

The first diagram, shown below, demonstrates how at very short time horizons, risk (standard deviation) dwarfs return, with the crossover occurring at about three years:

The next plot shows the risk/reward ratio:

Here's the probability of a successful outcome:

And finally, the utility to the investor with a loss/gain emotional ratio of 2.0:

The key, then, to a successful investing career is not just buy and hold, but

buy and forget,something not easily done in our information-saturated society. Taleb's solution involves forcibly ignoring the news (and its frequent pricing). I imagine he must have some difficulty carrying that out, since he runs a hedge fund (Empirica Capital) that presumably is burdened by pesky folks called "clients," who may rudely want their positions calculated from time to time.So how do ordinary folks, who do not live at the bottom of a well and are not gifted with the iron constitutions of Mr. Taleb and his clients, manage this trick? The key is to improve your utility function. The easiest way to do this is not to commit 100% to equity. One of my favorite bromides from

The Motley Foolwent something like this: "Every penny you don't have invested in stocks will hurt you." Until, of course, the inevitable point comes when you lose most of your pennies.To be honest, I've derived no small pleasure from the collapse of the large-cap growth/tech sector over the past few years. The long-term investor must buy at some point, and lower is better than higher. Yes, the losses hurt, but such pain can be substantially mitigated by the availability of dry powder. Not to mention a markedly diminished density of smug neighbors.

One way of determining your optimal stock/bond allocation is to find that point where substantial market declines produce absolute equipoise: where the pain of loss is exactly counterbalanced by the pleasure of buying low. The only drawback to this method is that it takes decades to calibrate.

Which gets us to the last issue: whither the equity risk premium? Is it 3.5% and growing, as an increasing mass of investors, no longer playing with easy house money, find (as every generation does) that "risk" is the ugliest four-letter word in the financial dictionary? Or has the equity risk premium permanently shrunk as history replaces yesterday's demons with today's pale (but no less frightening) imitations?

If you know the answer, please write. Until then, here's my modern finance version of Pascal's wager: Unless you have a Taleb-like detachment from the here and now, prudent amounts of short-duration high-quality debt seem a no-lose proposition.

Copyright © 2002, William J. Bernstein. All rights reserved.

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