You manage a pension fund, which provides retired workers with lifetime annuities. The fund must pay out $1 million per year to cover these annuities. Assume for simplicity that these payments continue for 20 years and then cease. The interest rate is 4% (flat term structure). You plan to cover this obligation by investing in 5- and 20-year maturity Treasury zero coupon bonds.

You decide to minimize the funds exposure to changes in interest rates. How much should you invest in the 5- and 20- year bonds? What will be the par value of your holdings of each bond?

• 5.9M in the five year bond and 12.4M in the 20 year bond
• 6.9M in the five year bond and 11.4M in the 20 year bond
• 7.9M in the five year bond and 10.4M in the 20 year bond
• 8.9M in the five year bond and 9.4M in the 20 year bond
• 11.9M in the five year bond and 8.4M in the 20 year bond

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: E

[[math]]D=\frac{\sum_{i=1}^{20} \frac{i}{1.04^i}}{\sum_{i=1}^{20} \frac{1}{1.04^i}}=9.21[[/math]]

years.

Need to match the duration and also the value of investment today should be equal to the total liabilities. So have the following two equations:

[[math]] V_5 * 5+V_{20} * 20=D *\left(V_5+V_{20}\right) [[/math]]

[math]V_5+V_{20}=\$ 13.59 M[/math] Annuity formula Solving gives [math]V_5=\$ 9.78 M[/math] and [math]V_{20}= \$3.81 M [/math]

[[math]] \begin{aligned} & P_5=V_5 *(1.04)^5=\$ 11.9 M \\ & P_{20}=V_{20} *(1.04)^{20}=\$ 8.36 M \end{aligned} [[/math]]

References

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