#### Question Details

The P/E Ratio and the S&P 500

The Dividend Discount Model (DDM) can be used to think about an entire market index such as

the S&P 500 in the same way it is used to think about an individual firm. In this problem we use

the DDM with a constant dividend growth rate and constant discount rates to think about the

valuation of the U.S. stock market overall during a particularly interesting period. As of August

1999, the value-weighted average P/E (price-earnings) ratio for the U.S. stock market (or, more

precisely, for the S&P 500 Index) was at a historical high of 36. In contrast, over the period from

1/1968 to 12/2000, the S&Pâ€™s average P/E ratio was 16.

For the following problems, assume the dividend payout ratio on the S&P 500 Index is 50%

(which is its historical average from 1/1968 to 12/2000) and that it does not change in any of the

scenarios considered.

Hint: Use the perpetuity-version of the DDM to express the price as a function of the next

dividend (DIV1), the cost of capital (r), and the growth rate (g) of expected earnings (and hence

dividends given the constant payout ratio). Then realize that next periodâ€™s earnings per share

(EPS1) can be rewritten as EPS0(1+g). Then divide the price by EPS0 to obtain the P/E ratio.

2

Now you have an expression linking the P/E ratio to r, g, and the dividend payout ratio

(DIV/EPS). From this expression, you can answer the following.

Note: Robert J. Shiller of Yaleâ€”Nobel Laureate in Economics in 2013â€”used similar

calculations in his best-selling book â€œIrrational Exuberanceâ€, published in 2000, right before the

burst of the Dot-com bubble.

Backing out expected returns. First, suppose that, over the entire period, the expected

growth rate of earnings was a constant 7.2%. (Note that, if the expected growth rate and the

payout ratio are constant, variation over time in the P/E ratio must reflect variation in the

expected return.)

5) What was the average expected return on the market (i.e. r) over this period, based on the

historical average P/E ratio of 16?

6) What was the expected return on the market (i.e. r as of 8/1999, when the S&P's P/E ratio

was 36?

Backing out expected growth rates. Next suppose instead that, over the entire period, the

expected return on the market was a constant 10.55%. (Note that, if the expected return and the

payout ratio are constant, variation over time in the P/E ratio must reflect variation in the

expected earnings growth rate.)

7) What was the average expected growth rate of earnings over this period, based on the

historical average P/E ratio of 16?

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