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 in 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

$12/31/20224534\mathrm{.}23199\mathrm{.}56$

$12/31/20214766\mathrm{.}18211\mathrm{.}50$

$12/31/20203756\mathrm{.}07107\mathrm{.}70$

$12/31/20193230\mathrm{.}78161\mathrm{.}74$

$12/31/20182506\mathrm{.}85157\mathrm{.}04$

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

EPS $\text{'}22$:

EPS $\text{'}21$:

EPS $\text{'}20$:

EPS $\text{'}19$:

b$)$ For the following years, compute the P$/$E ratio of the S&P $500$ index.

P$/$E $\text{'}22$:

P$/$E $\text{'}21$:

P$/$E $\text{'}20$:

c$)$ Forthe years $2022$ to $2019,$ attribute the change in index from one year to another to multiple expansion and earnings growth.