Winnie is close to retirement. He has accumulated a lot of assets $($$$800,000)$ which he has invested

in stocks. There are three possible scenarios: $1)$ His wealth could increase sharply to $$1\mathrm{.}44$ million

with a probability of $0\mathrm{.}2,2)$ his wealth could go up to $$810,000$ with a probability of $0\mathrm{.}6,$ or $3)$ it

could decrease to $$490,000.$ Winnie's utility over wealth is $U(W)=W^{\frac{1}{2}}.$

$($a$)$ What is Winnie's measure of absolute risk aversion? Is the risk$-$aversion of a $$1000$ bet greater

at a wealth of $$100,000$ or at a wealth of $$1,000,000?$ What is Winnie's measure of relative risk

aversion? Is the risk$-$aversion of a bet of $10\%$ of his wealth greater at a wealth of $$100,000$ or

at a wealth of $$1,000,000?(2)$

$($b$)$ What is the expected value of Winnie's wealth? What is Winnie's expected utility? $(2)$

$($c$)$ If Winnie invested all his assets in government bonds instead, which always pay the same

amount, then how much would they have to pay so that Winnie is indifferent between investing

in stocks and government bonds? $(2)$

$($d$)$ What is the risk premium that Winnie is willing to pay in order to avoid the risk of investing

in stocks? $(2)$