问一道题：NO.PZ2016082406000050

Assume that you have entered into a fixed-for-floating interest rate swap that starts today and ends in six years. Assume that the duration of your position is proportional to the time to maturity. Also assume that all changes in the yield curve are parallel shifts, and that the volatility of interest rates is proportional to the square root of time. When would the maximum potential exposure be reached?

选项：

In two months

In two years

In six years

In four years and five months

解释：

ANSWER: B

Exposure is a function of duration, which decreases with time, and interest rate volatility, which increases with the square root of time. Define T as the original maturity and k as a constant. This gives $\sigma{(V_t)}=k{(T-t)}\sqrt t$. Taking the derivative with respect to t gives a maximum $t=\frac T3$. This gives $t=\frac63=2$years.

这里不是说duration会随着时间增长而增加吗

那个1/3的结论不是说到期duration的效果减少吗，那是不是没有摊销效应