A share of DivCorp common stock does not pay a regular dividend, but

sometimes awards a special dividend of $$20$ per share at the end of each

year. The hypothetical $$marginal$$ investor has a subjective discount rate

of $\backslash$beta $=0\mathrm{.}975$ per year and power utility of consumption with risk aversion

coefficient $\backslash$gamma $=1.$ Assume that this investor$$s income is fixed at $$100,000$

per year in perpetuity. This investor believes that the probability that Di$-$

vCorp pays a dividend in a given year is always $\backslash$pi $=0\mathrm{.}35.$ The probability

that DivCorp pays a dividend in a given year is uncorrelated with the

probability that it pays a dividend in any other year. The next possible

special dividend date is one year from today.

What should be the market price of one share of DivCorp? $(7$ points$)$

Hint $1$: Start by writing out the FPE and see if you can plug in each cash

flow and the appropriate discount factor. In terms of discounting for im$-$

patience, payoffs one period away are multiplied by $\backslash$beta $,$ payoffs two periods

away are multiplied by $\backslash$beta $2,$ and so on$.$ Remember that an expectation is

also just a summation over outcomes, which can be probabilistic outcomes

that are realized simultaneously $($i$.$e$.,$ the result of coin flip$),$ outcomes that

are realized sequentially in time $($i$.$e$.,$ payments on a risk$-$free bond$),$ or a

combination of both $($i$.$e$.,$ a stream of risky cash flows that default with

some probability$).$

Hint $2$: You can translate the subjective discount rate $\backslash$beta $<=1$ to a net

return r $>=0$ using the relationship:

$\backslash$beta $=1$

$1+$ r $(1)$

Hint $3$: The value p of a $$perpetuity$,$ or a contract that provides a fixed

sum of cash X at regular intervals forever, is given by:

p $=$ X

$1+$ r $+$ X

$(1+$ r$)2+...=$ X

r $(2)$

Where r $>0$ is the discount rate expressed as a net return $($i$.$e$.,$ interest

rate$)$ per interval.

Hint $4$: Remember that a constant can move inside and outside the expec$-$

tation operator: E$[$aX$]=$ aE$[$X$]$ for a constant a and a random variable

X$.$ Any value that is known at time t $($when the expectation is evaluated$)$

can be treated as a constant.