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:

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

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.

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 $E_0=10,E_1=12,P_0=120,P_1=156,$ then earnings growth is $\frac{E_1}{E_0}-1=20\%.$ If

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there was no multiple expansion, then the value of the index should go up by $20\%.$ However, the

index went up by $\frac{P_1}{E_1}-1=30\%.$ This means that $10\%$ of the increase in index level can be

attributed to multiple expansion. We can check: The $\frac{P}{E}$ ratio in year $0$ is $\frac{P_0}{E_0}=12$ and the $\frac{P}{E}$

ratio in year $1$ is $\frac{P_1}{E_1}=13,$ which is bigger than in year $0$

Note that earnings growth can be negative, and also multiples can shrink from year to year.