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%.

Suppose that you invest the $18 million in 1-year Treasury bills (i.e., 1-year zero-coupon bond) and in a Treasury bond with modified duration of 20. What % of your investments do you allocate to 1 year T-bills in order to avoid interest rate risk of your portfolio, which includes both assets and liabilities?

• 7%
• 9%
• 11%
• 13%
• 15%

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: D

[[math]] \begin{gathered} P V=\frac{20}{(1+5 \%)^{10}}+\frac{30}{(1+5 \%)^{30}}=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]]

[math]\mathrm{y}=5 \%[/math] because the yield curve is flat When rates drop by [math]0.25 \%[/math], the PV of liabilities will go up by [math]0.25 \% * M D * P V=[/math] 0.7881 million

First, you calculate the desired MD of your assets. We want:

[[math]] M D_{\text {asset }} * P V_{\text {asset }}=M D_{\text {liabilites }} * P V_{\text {liabilities }} [[/math]]

Therefore, [math]M D_{\text {asset }}=\frac{16.4 * 19.22}{18}=17.51[/math] Now we can determine the allocation of our portfolio. Suppose we invest a fraction of [math]\mathrm{x}[/math] of our portfolio into 1-year bond and the rest into treasury bond, then the MD of our portfolio will be:

[[math]] M D_{\text {portfolio }}=x * M D_{1 y r b o n d}+(1-x) * M D_{\text {tbond }}=x * \frac{1}{1+5 \%}+(1-x) * 20 [[/math]]

Equating [math]M D_{\text {portfolio }}=17.51[/math], we get [math]\mathrm{x}=13.05 \%[/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.