Joe must pay liabilities of 1,000 due one year from now and another 2,000 due three years from now. There are two available investments:

Bond I: A one-year zero-coupon bond that matures for 1000. The yield rate is 6% per year

Bond II: A two-year zero-coupon bond with face amount of 1,000. The yield rate is 7% per year.

At the present time the one-year forward rate for an investment made two years from now is 6.5%

Joe plans to buy amounts of each bond. He plans to reinvest the proceeds from Bond II in a one-year zero-coupon bond. Assuming the reinvestment earns the forward rate, calculate the total purchase price of Bond I and Bond II where the amounts are selected to exactly match the liabilities.

• 2584
• 2697
• 2801
• 2907
• 3000

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Step 1: Understand the Liabilities and Available Instruments

Joe faces two distinct liabilities that must be met:

Liabilities to be Met

Time ([math]t[/math])	Amount
[math]t=1[/math] year	$1,000
[math]t=3[/math] years	$2,000

To meet these liabilities, Joe has access to the following financial instruments and a specific reinvestment opportunity:

Available Instruments and Reinvestment Opportunity

Instrument/Opportunity	Details	Yield/Rate
Bond I	One-year zero-coupon bond, matures for $1,000	6% per year
Bond II	Two-year zero-coupon bond, face amount $1,000	7% per year
Reinvestment Opportunity	Proceeds from Bond II (maturing at [math]t=2[/math]) will be reinvested for one year (from [math]t=2[/math] to [math]t=3[/math])	6.5% one-year forward rate ([math]f_{2,3}[/math])

The objective is to calculate the total current purchase price of Bond I and Bond II required to exactly match these liabilities.

Step 2: Calculate the Purchase Price of Bond I for the [math]t=1[/math] Liability

The first liability is $1,000 due at [math]t=1[/math]. Bond I is a one-year zero-coupon bond that matures for $1,000. This bond is perfectly suited to directly match the first liability. The purchase price of Bond I is its present value, calculated by discounting its maturity value ($1,000) at its yield rate of 6% over one year:

[[math]]PV_{ text{Bond I}} = \frac{1000}{(1 + 0.06)^1} = \frac{1000}{1.06}[[/math]]

[[math]]PV_{ text{Bond I}} \approx 943.3962[[/math]]

Step 3: Determine the Required Value of Bond II at [math]t=2[/math] for the [math]t=3[/math] Liability

The second liability is $2,000 due at [math]t=3[/math]. Bond II, however, matures at [math]t=2[/math]. To cover the [math]t=3[/math] liability, the proceeds from Bond II at [math]t=2[/math] will be reinvested for one year at the one-year forward rate of 6.5% for an investment made two years from now ([math]f_{2,3}[/math]). First, we need to determine the amount of money ([math]X[/math]) that must be available from Bond II's maturity at [math]t=2[/math]. This amount [math]X[/math] must grow to $2,000 by [math]t=3[/math] at the 6.5% forward rate.

[[math]]X \times (1 + 0.065) = 2000[[/math]]

Solving for [math]X[/math]:

[[math]]X = \frac{2000}{1.065}[[/math]]

[[math]]X \approx 1877.9343[[/math]]

Therefore, Bond II must provide approximately $1,877.93 at its maturity at [math]t=2[/math] to cover the subsequent liability.

Step 4: Calculate the Purchase Price of Bond II

Bond II is a two-year zero-coupon bond with a yield rate of 7% per year. As determined in Step 3, it needs to provide $1,877.9343 at its maturity at [math]t=2[/math]. The purchase price of Bond II is its present value, calculated by discounting the required maturity value ($1,877.9343) back to [math]t=0[/math] at its yield rate of 7% over two years:

[[math]]PV_{ text{Bond II}} = \frac{1877.9343}{(1 + 0.07)^2} = \frac{1877.9343}{(1.07)^2}[[/math]]

[[math]]PV_{ text{Bond II}} \approx 1640.2608[[/math]]

Step 5: Calculate the Total Purchase Price

The total purchase price required to exactly match both liabilities is the sum of the purchase prices of Bond I and Bond II.

[[math]] text{Total Purchase Price} = PV_{ text{Bond I}} + PV_{ text{Bond II}}[[/math]]

Substituting the values calculated in Step 2 and Step 4:

[[math]] text{Total Purchase Price} = 943.3962 + 1640.2608[[/math]]

[[math]] text{Total Purchase Price} = 2583.6570[[/math]]

Rounding to two decimal places, the total purchase price is approximately $2,583.66. The correct option provided is A.

Key Insights

• Liability matching often requires a combination of different financial instruments with varying maturities to precisely cover future obligations.
• Zero-coupon bonds are effective tools for precise liability matching because they provide a single, known payment at a specific future date, simplifying the cash flow management.
• When an asset matures before its corresponding liability, the proceeds from the asset must be reinvested. The appropriate forward interest rate is crucial for determining the necessary amount at the asset's maturity to cover the later liability.
• The present value of an investment is calculated by discounting its future value (or the required future proceeds) back to the present using its specific yield rate over the relevant investment period.

This article was generated by AI and may contain errors. If permitted, please edit the article to improve it.

Solution: A

Bond I provides the cash flow at time one. Because 1000 is needed, one unit of the bond should be purchased, at a cost of 1000/1.06 = 943.40.

Bond II must provide 2000 at time three. Therefore, the amount to be reinvested at time two is 2000/1.065 = 1877.93. The purchase price of the two-year bond is 1877.93/1.07² = 1,640.26.

The total price is 2583.66.

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.