Question

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&P’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 period’s earnings per share (EPS₁) can be rewritten as EPS₀(1+g). Then divide the price by EPS₀to obtain the P/E ratio. 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 Yale—Nobel Laureate in Economics in 2013—used similar calculations in his best-selling book “Irrational Exuberance”, published in 2000, right before the burst of the Dot-com bubble.

1. 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.)

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?

2. Backing out expected returns, cont'd. 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.)

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?

3. 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.)

What was the average expected growth rate of earnings over this period, based on the historical average P/E ratio of 16?

4. Backing out expected growth rates, cont'd. 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.)

What was the expected growth rate of earnings as of 8/1999, when the S&P's P/E ratio was 36?

Answer 1

Part (1):

Given, P/E=16 and g=7.2% or, 0.72 and Pay out ratio= 50%

Let us denote EPS as E for simplicity

Therefore, E0=2D0.  

P/E= 16 Or, P/2D0 = 16. P=2D0 *16   Or, P= 32D0 …..(1)

Also, P= D1 /(r-g)

Therefore, P= D0*1.072/(r-0.072)……(2)

Combining the above equations,

32D0 = 1.072 D0/(r-0.072)

r-0.072 = 1.072 D0/32D0= 0.0335

Therefore, required rate of return = 0.0335+0.072 = 0.1055 Or, 10.55%

Part (2):

If P/E was 36,

36*2D0 = 1.072 D0/(r-0.072) Or, 72D0 = 1.072 D0/(r-0.072)

r-0.072 = 1.072 D0/72D0= 0.014889

Therefore, required rate of return = 0.014889+0.072 = 0.0868889 Or, 8.68889%

Part (3):

Given, P/E=16 and r=10.55 or, 0.1055 and Pay out ratio= 50%

Then,

P= 32D0 …..(1)

P= D0(1+g)/(0.1055-g)……(2)

32D0 = D0(1+g)/(0.1055-g)

(0.1055-g)= (1+g)/32

3.376-32g =1+g

3.376 = 1+g+32g= 1+33g Or, 33 g= 2.376

Therefore, growth rate (g) = 2.376/33 = 0.072 Or, 2.76%

Part (4)

If the PE is 36,

72D0 = D0(1+g)/(0.1055-g)

(0.1055-g)= (1+g)/72

7.596-72g =1+g

7.596 = 1+g+72g= 1+73g Or, 73 g= 6.596

Growth rate (g) = 6.596/73= 0.090356 Or, 9.0356%