Problem 3 The P/E ratio for the S&P 500 describes the value of the index divided by the earnings of the company in the index, weighted according to market capitalization. The P/E ratio changes over time for two reasons:

1. Multiple Expansion: Investors are willing to pay a higher price for the same level of earnings.

2. Earnings Growth: Firms increase their earnings by reinvesting into and growing their business. Find below a table of the price levels and earnings per share of the S&P 500 index.

Date Index Value EPS

Dec 31, 2021 4766.18 211.50

Dec 31, 2020 3756.07 107.70

Dec 31, 2019 3230.78 161.74

Dec 31, 2018 2506.85 157.04

Dec 31, 2017 2673.61 132.83

a) For each year, compute the earnings growth rate, i.e., the change in EPS year over year.

b) For each year, compute the P/E ratio of the S&P 500 index.

c) Attribute the change in index from one year to another to multiple expansion and earnings growth.

Help: If E0= 10, E1 = 12, P0 = 120, P1 = 156, then earnings growth is E1/E0 1 = 20%. If there was no multiple expansion, then the value of the index should go up by 20%. However, the index went up by P1/E1 1 = 30%. This means that 10% of the increase in index level can be attributed to multiple expansion. We can check: The P/E ratio in year 0 is P0/E0 = 12 and the P/E ratio in year 1 is P1/E1= 13, which is bigger than in year 0 Note that earnings growth can be negative, and also multiples can shrink from year to year.