The Equity Premium Puzzle: Investments in equities (like stocks) yield substantially higher returns than investment in bonds. By itself, this is no surprise€”because stocks are riskier than bonds. What is a surprise when viewed through the usual model of risk is the size of the premium that equities provide to investors. In a typical year, for instance, bonds might give investors a safe rate of return of 2% while stocks might give a return of between 6 and 8%. Economists who have tried to explain this €œequity premium€ simply in terms of risk aversion have concluded that the level of risk aversion necessary to explain the premium is simply far beyond what anyone can take seriously. Risk aversion alone therefore cannot explain the equity premium€”which raises an €œanomaly€ known as the equity premium puzzle.
A: Consider the equity premium puzzle with the lens of reference-based preferences. In particular, suppose you are investing $1,000 and you know you can get a 2% return on this over 1 year by investing your money in bonds. Alternatively, you can invest the $1,000 in a stock and expect to lose $100 with probability 0.1 and gain $100 with probability 0.9.
(a) What is the expected rate of return from a stock investment? What does this imply is the equity premium?
(b) Suppose you thought that investors had reference-based preferences. What do you think their reference point might be when comparing the two investments?
(c) Can you use the concept of €œloss aversion€ to explain how behavioral economics might have an explanation for the equity premium puzzle?
(d) In your explanation in (c), you almost certainly thought of the investor as having a 1-year horizon. Suppose investors are in it €œfor the long run€ €” facing a 10% chance of a loss on their stocks each year. Do you think your behavioral economics explanation that relies on reference-based preferences and loss aversion can still explain the equity premium puzzle?
B: Consider prospect theory (which you implicitly used in part A) a little more closely. Suppose that an investor bases his decision on a 1-year investment horizon and evaluates risky gambles relative to a reference point that is equal to the amount he invests. Suppose he invests $1,000 €”which then becomes his reference point. If invested in risk-free bonds, the $1,000 will be worth $1,022.54 one year from now. If he invests the same amount in stocks, his investment will be worth $900 with probability 0.12 and $1,100 with probability 0.88. The utility of any amount x is evaluated using the function

Where r is the reference point, and the utility of a gamble that results in x1 with probability δ and x2 with probability (1ˆ’δ) is given by U = δ u (x1,r )+(1ˆ’δ)u(x2 ,r ).
(a) We discussed four features of prospect theory in the text: (1) reference-dependence, (2) loss aversion, (3) diminishing sensitivity and (4) probability weighting. Which of these are we modeling here and which are we not?
(b) What is the expected return on investing $1,000 in stocks? What is the equity premium?
(c) What utility will this investor get from investing $1,000 in bonds?
(d) What utility will he get from investing $1,000 in stocks?
(e) If this is a typical investor, is the equity premium explained by our version of prospect theory?
(f) Suppose you are a young investor who is investing for retirement in 30 years. For all practical purposes, you can in this case be almost certain that an investment in stocks will result in an average rate of return equal to the expected rate of return. Re-calculate the average annual utility from investing $1,000 in bonds versus investing $1,000 in stocks for such an investor.
(g) If all investors were like this young investor, could our prospect theory still explain the equity premium puzzle?

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