This question uses the Cyclically Adjusted Price-Earnings Ratio (CAPE) for the S&P

500 (this measure was popularized by Nobel-prize winning economist Robert Shiller).

It is available at http://www.multpl.com/shiller-pe/. Click on \Table" and \By

Month" underneath the chart to get the historical monthly values (or go directly to

http://www.multpl.com/shiller-pe/table?f=m) .The CAPE divides the price of the S&P 500 by the last ten year's average of in

ation-adjusted earnings. In this question we gure out what earnings growth rate was priced

in by the CAPE at different historical times, assuming a constant future earnings

growth.

To adjust for the fact that the denominator of the CAPE is not E1 but an average

over the past ten years earnings, we approximate the denominator \E" in the CAPE

with E_-4, the midpoint of the ten-year interval. As usual, E0 denotes the earnings of

the year that just ended and E1 denotes the earnings at the end of this year (year 1).

Thus, approximate the CAPE by

P0

E_-4, where E_-4 * (1 + g)5 = E1.

In the following assume that investors demand an average return R of 7.5%, unless the

problem species otherwise. This is approximately the historical real (i.e., in

ation-

adjusted) average return of the S&P 500. In addition, assume the plowback ratio is

25%.

(a) Solve for the value of g priced in by the average monthly value of the CAPE

since 1950 (not the whole series going back to the 1800s). You can calculate this

average by copying the table of monthly values of the CAPE into Excel.

2

(b) What value of g was implied by the maximum value of the CAPE, assuming

investors required R of 7.5% at that time? (Note: we're using the real (in

ation-

adjusted) value of R, so the value of g we're solving for is also real.)

(c) Now assume that in fact the `true' g was equal to the value you calculated in part

(a) at the time that the CAPE was at its maximum value. What expected return

did the S&P500 price in at that time?

(d) Suppose that when the CAPE was at its maximum value, investors suddenly

realized that they had been too optimistic and decreased their estimates of g by

2%. What realized return would the S&P500 have experienced as a result of this

sudden change of heart?