$[$ How to predict market risk? $]$ J$.$P$.$ Morgan proposes a simple statistical model

to predict downside market risk for financial instruments $($stock index, foreign

exchange, portfolio$).$ Suppose $Y_t$ is the percentage change of a foreign exchange

rate from time $t-1$ to time $t.$ Then the J$.$P$.$ Morgan RiskMetrics model is

$Y_{t}N({}_t,{}_t^2)$

where ${}_t$ is the mean of $Y_t$ predicted using the information available at time $t-1,$

and ${}_t^2$ is the variance of $Y_t$ predicted using the information available at time $t-1.$

The value at risk $($VaR$)$ at the $\%$ level for $Y_t$ is defined as the value $V_t$ which is

predicted at time $t-1$ such that

Now suppose that $Y_t$ is the percentage of Japanese Yen per dollar from time $t-1$

to time $t.$ We assume that ${}_t=0$ and ${}_t^2=0\mathrm{.}4561$

$(1)$ Please find the value at risk $V_t$ at the $10\%$ level for $Y_t.$

$(2)$ Suppose a manager is conservative so he would like to have a VaR at the level

$$ smaller than $10\%.$ Is the VaR at this new level larger or smaller than the VaR

at the $10\%$ level? Explain.

$(3)$ Suppose the Japanese yen becomes more volatile so that the conditional vari$-$

ance ${}_t^2$ becomes larger. How will the VaR at the $10\%$ level will change? Explain.