Suppose that a bank has position $X$ that has a normal probability density. The value at risk is known to be $V$ at the loss tolerance $h$. The bank plans to take some capital out of reserve, which would increase the VaR. The bank argues that the change would be acceptable if the required confidence level were only slightly increased.

(a) Develop a formula that expresses implicitly, in terms of the inverse standard normal distribution function $F_{N}^{-1}$, the amount that the loss tolerance must change to compensate for a change $\Delta$ in capital. [Express the result as the difference of two quantities.]

(b) Approximate the difference in (a) in terms of a derivative, and use the fact that $\frac{\mathrm{d}}{\mathrm{d} h} F_{N}^{-1}(h)=1 / f(x)$, where $x=F_{N}^{-1}(h)$ and where $f$ is the probability density of a standardized normal random variable.

(b) Using the fact that $h \approx 0$, find the amount of the required change as a fraction of $\sigma$.

(d) For $h=1 %$ and $\bar{X}=-100, \sigma=20$, find the required new value of $h$ for an addition of -10 to $\bar{X}$.

(e) Verify the result of part (d) explicitly (without the approximation).