Question

As a pension fund manager in 2020, you are given $67,556,416.88 to manage so that the fund will have at least $100,000,000 available in 2030, 10 years from now. The yield curve is flat, and the current interest rate for all maturities is 4%. The only bonds that you can buy are annual coupon Treasury bonds with 1-year maturities and bonds issued in 2020 that mature in 2040 (currently a maturity of 20 years). All bonds are issued at par. 1. How many of each Treasury bond should you buy today?

$_________________1-Year

$_________________20-Year

2. If all interest rates increase by 1% (to 5%) in year (2021) and then remain at 5%, how much will the fund have in 2030? Reinvest all coupon payments in one-year bonds. (The answer should be fairly close to the goal amount if you selected the portfolio correctly. If you set up the Excel spreadsheet right, you can see that you will have the goal amount regardless of the new interest rate.)

$_______________________

3. Very briefly, why was the amount not equal to exactly $100,000,000? (I’m looking for a key word.)

Answer 1

(1) Let us first look at 1-year maturity bonds. In case a total of $67,556,416.88 is invested in 1-year bond in 2020, the cash-in hand should be based on coupon rate of 4% at the end of the year 2021. The entire money is re-invested at the end of 2021 at 4% coupon rate and so on until total cash in-hand is $100,000,000. In case the bond is issued n times,

On solving this, we obtain n = 10. Thus, 10 1-year bonds should be purchased (until 2029), and 0 20-years bond need to be purchased.

(2) In case the interest rate increases to 5%, total cash in-hand at the end of year 2030 by investing in 1-year bonds from the beginning: -

In case initial cash in-hand is invested in 20-year bond, and coupon payments are re-invested in 1-year bonds thereafter: -

Coupon payment at the end of every year: -

Henceforth, total cash in-hand at the end of year 2030: -

On solving,

On solving,

Hence, investing in 20-year bond in the beginning shouldn't make the objective because one would not get the principal payment back by 2030. The idea should be to invest in 1-year bond in the beginning, and to re-invest the cash (principal plus interest) until 2029 to get more than $100,000,000 in hand by 2030. At 4%, the cash in-hand is exactly $100,000,000. So, it doesn't matter if interest rate increases to 5% or not.

(3) In case of 4%, the amount in-hand is exactly $100,000,000. However, it assumes that markets are frictionless, and there is no transaction costs which is practically not true. Because of transaction costs, the amount in-hand may be slightly less than $100,000,000.