Problem $3$: Asset pricing

There are two possible future states, boom and recession. They occur with probability $0\mathrm{.}5$

each. Let $c_t$ and $c_{t+1}$ denote an investor's consumption in period's $t$ and $t+1.$ Assume that:

$\frac{c_{t+1}}{c_t}=\{\begin{array}{ccc}1\mathrm{.}04& if\text{a}\text{b}\text{o}\text{o}\text{m}\text{o}\text{c}\text{u}\text{r}\text{s}& \\ 0\mathrm{.}99& if\text{a}\text{r}\text{e}\text{c}\text{e}\text{s}\text{s}\text{i}\text{o}\text{n}\text{o}\text{c}\text{c}\text{u}\text{r}\text{s}\text{.}& \end{array}$

A$)$ An investor with utility $logc($and a coefficient of relative risk aversion $=1)$ and

discount factor $0\mathrm{.}9$ buys a riskless asset. What is the risk$-$free rate of return?

B$)$ A risky asset pays out $$1\mathrm{.}10$ in the boom and $$0\mathrm{.}90$ in the recession. How much would

the investor pay for it$?$ What is its risk premium? $($Hint: remember that the risky

return on an asset satisfies $1+r=\frac{D}{P},$ where $D$ is the risky payout and $P$ is the price.

Use Slide $30$ or $31,$ Lecture $14.$ Remember you can move variables that are certain

through expectations. What variable is known by an investor when she buys an asset?$)$