Let P(0, t) be the current price of a zero-coupon bond that will pay 1 at time t. Let X be the value at time n of an investment of 1 made at time m, where m < n. Assume all investments earn the same interest rate.

Determine X.

• [[math]]\frac{P(0, m)}{P(0, n)}-1[[/math]]
• [[math]]\frac{P(0, n)}{P(0, m)}+1[[/math]]
• [[math]]\frac{P(0, m)}{P(0, n)}+1[[/math]]
• [[math]]\frac{P(0, m)}{P(0, n)}[[/math]]
• [[math]]\frac{P(0, n)}{P(0, m)}[[/math]]

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: D.

[[math]] \begin{aligned} & P(0, m)=(1+i)^{-m} \\ & P(0, n)=(1+i)^{-n} \\ & X=\frac{(1+i)^{-m}}{(1+i)^{-n}}=(1+i)^{-m+n} \\ & X=\frac{P(0, m)}{P(0, n)}\end{aligned} [[/math]]

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.