The PIE Ratio and the SSP 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 rm. 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 PIE (price-eamings} 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 1f'l968 to 122000. the SP's average PIE 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 mess to 122000) and that it does not change in any of the scenarios considered. Hint: Use the perpemity-version of the DOM to express the price as a function of the next dividend (DIVI). the cost of capital (r). and the growth rate (Q) of expected eamings (and hence dividends given the constant payout ratio). Then realize that next period's earnings per share [EPS.) can be rewritten as EPSn('l+g). Then divide the price by EPSD to obtain the PE ratio. Now you have an expression linking the PE ratio to r. g. and the dividend payout ratio [DIVIEPS]. From this expression, you can answer the following. Note: Robert J. Shiller of YaleNobel Laureate in Economics in 2013used similar calculations in his best-selling book 'lrrational Exuberanoe". 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 PE ratio must reect variation in the expected return.) 5) What was the average expected return on the market (is. r) over this period. based on the historical average PIE ratio of 16? 6) What was the expected return on the market (i.e. r as of 8l1999. when the 3&P's PIE 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 PE ratio must reect 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 PIE ratio of 16? 3) What was the expected growth rate of earnings as of $1999. when the S&F"s PIE ratio was 36