You manage a pension fund, and your liabilities consist of two payments as follows:

Time	Payment
10 Years	$20 million
20 Years	$30 million

Your assets are $18 million. The term structure is currently flat at 5%.

Compute an approximate change in the present value of your liabilities, using duration, when interest rates fall by 0.25%.

• 0.8M
• 0.85M
• 0.9M
• 0.95M
• 1M

References

Lo, Andrew W.; Wang, Jiang. "MIT Sloan Finance Problems and Solutions Collection Finance Theory I" (PDF). alo.mit.edu. Retrieved November 30, 2023.

Solution: A

[[math]]\begin{gathered}P V=\frac{20}{(1+5 \%)^1 0}+\frac{20}{(1+5 \%)^3 0}=12.28+6.94=19.22 \text { million } \\ D=\frac{12.28 * 10+6.94 * 30}{19.22}=17.22 \\ M D=\frac{D}{1+y}=16.40\end{gathered}[[/math]]

When rates drop by 0.25%, the PV of liabilities will go up by

0.25% ∗ MD ∗ PV = 0.7881 million

References

Lo, Andrew W.; Wang, Jiang. "MIT Sloan Finance Problems and Solutions Collection Finance Theory I" (PDF). alo.mit.edu. Retrieved November 30, 2023.