Las Revueltas

Often we might want to consider something similar to the stock portfolio problem, where a person might decide how much to invest in several activities at once. For example, suppose that an investor can divide her wealth between two assets. One is a safe asset that returns exactly the amount invested with certainty. Thus, if an investor invests x in the safe asset, she will be able to obtain x at the time she sells this asset. The second is a risky asset, such that every dollar invested in the asset will be worth z at the time of sale, where z is a random variable. Suppose that the expected value of z is μ > 1. Then, the investor with wealth w who wishes to maximize her expected utility of investing in the safe and risky assets will solve

max

wz E u w − wz + wzz

6 12 where w is wealth to be invested and wz is the amount of wealth invested in the risky

asset. The solution to this problem is approximated by:

wz= μ − 1 σ2_R

A

, 6 13

where RA is the coefficient of absolute risk aversion and the value σ2= E x − μ 2 is commonly called the variance of z. The variance measures how dispersed the values of z you on if you promise to let me make 100 such bets.” The willingness to take on 100

such bets but not one suggests that the colleague’s preferences are not independent of

segmentation. Rather, he appears willing to take on many undesirable bets in violation of the basic rational choice model.

It is notable that the colleague cites the pain of loss from $100 dollars dominating the gain of $200, suggesting that loss aversion plays some role. Moreover, it is clear that the desirability of each individual gamble depends on whether the decision is made independently or jointly with the decision to take on many other similar gambles. Samuelson saw this as potential evidence that behavior did not conform to expected utility or that people have difficulty in understanding how concepts such as diversifying risk really works. In this case, the probability of loss is much smaller with 100 gambles (less than 1 percent) than with 1 gamble (50 percent), but the potential magnitude of the losses is much larger ($10,000 vs. $100). The increase in potential losses must dominate the reduced probability of losses under the expected utility model.

Thus, diversifying investments can reduce the probability of a loss, but if the invest- ments are not desirable to begin with, diversification cannot magically make a bad

gamble good. Consider a group of two of Samuelson’s gambles. In this case, the col-

league faces a 0.25 probability of losing $200, a 0.50 probability of gaining $100, and a 0.25 probability of gaining $400. I suspect that many readers at this point might consider this gamble much more attractive than the original gamble. Nonetheless, in almost all reasonable cases, turning down the original gamble should lead a rational decision maker to reject this latter gamble under expected utility theory. Why then does grouping the gambles together seem to make them more attractive?

are—how far away from the mean we believe the value of z may be. The higher the value ofσ2, the riskier the investment. Thus, the higher is the aversion to risk, the lower the investment in the risky asset. Similarly, the riskier the asset (the higher the variance), the lower the investment in the risky asset. As well, increasing the expected payout of the risky asset relative to the safe asset increases investment in the risky asset. (The interested reader can see the derivation in the Advanced Concept box, The Portfolio Problem.)

EXAMPLE 6.2 The Equity Premium Puzzle

The return from investing in stocks has historically been far superior to the return from investing in bonds. For example, from 1871 until 1990 the return on stocks was about 6.5 percent per year, whereas the return on bonds was around 1 percent. Over any long- term investment horizon (more than 30 years), stocks overwhelmingly outperform bonds—often by more than seven times—leading one to wonder why anyone would ever invest in bonds. Bonds are often used as a relatively risk-free investment, though they still contain significant risk. Bonds pay a fixed nominal rate of return so long as the

issuing body remains obligated. This issuing body may be afirm or a government. Thus,

if the issuing body goes bankrupt it might default. As well, inflation can erode the rate of return over the life of the bond. Nonetheless, if we were to think about this problem in terms of the portfolio problem presented above, the rate of return,μ, would be about 7. Further, the commonly assumed range for the level of relative risk aversion is about 1 to 3. Thus, absolute risk aversion is between 1 and 3 divided by the total amount of wealth, a truly small number. If we assume 3, then we could rewrite the formula (equation 6.13) as

wz= 3w_σ2 , 6 14

where wzis the optimal amount of money to invest in stocks,σ2is the variance in return of

the stock portfolio, and w is the total amount of wealth.

The variance of the stock portfolio depends on the time horizon. If we consider the long horizon on a representative portfolio of stocks, this variance is actually quite low, well below 3. In this case, the optimal investment is greater than the total wealth to be invested, leaving no room for bonds. More generally, economists estimate that the level of relative risk aversion necessary to induce one to purchase any bonds must be about 30, ten times the reasonable upper bound used by economists. To put this in per- spective, consider a gamble with 0.50 probability of ending with $100,000 of wealth and a 0.50 probability of ending with $50,000 of wealth. The investor must be willing to pay at least $50,000 for this gamble, because this is the lowest possible outcome. An investor with relative risk aversion of 30 would have a certainty equivalent for this gamble of just $51,209 despite the expected value of $75,000. Thus, a 50 percent chance of winning an extra $50,000 is worth only $1,209 to her. This is an astounding and unreasonable level of risk aversion. Thus, one might reasonably conclude that stock prices reflect a violation of the rational model of decision under uncertainty.