As a mid-size company, you have a pension plan which pays out $10 million a year forever. The first payment is exactly one year from now. The term structure is currently flat at 5%.

Suppose that the pension plan is fully funded (i.e., the value of your assets equal the value of pension liabilities). You want to invest all your assets in bonds to avoid any interest rate risk. What should the duration of your bond portfolio be?

• 20
• 20.4
• 20.8
• 21.2
• 21.6

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: B

Using the perpetuity formula

[[math]] \begin{gathered} P V_{\text {Liability }}=\frac{10 M}{r}=\frac{10 M}{0.05}=200 M \\ P V_{\text {Liability }}=\frac{10 M}{r}=\frac{10 M}{0.049}=204.0816 M \end{gathered} [[/math]]

The value of the liabilities would increase by [math]4.0816 \mathrm{M}[/math].

[[math]] \begin{aligned} P_{\text {new }}=P_{\text {old }}-P_{\text {old }} \times M D \times \Delta y & \\ \rightarrow M D & =\frac{P_{\text {old }}-P_{\text {new }}}{P_{\text {old }} \times \Delta y} \\ & =\frac{200-204.0816}{200 \times-0.001}=20.4082 \end{aligned} [[/math]]

You should match the modified duration to neutralize first order interest rate risk.

MD=20.4082

References

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