Joe must pay liabilities of 1,000 due 6 months from now and another 1,000 due one year from now. There are two available investments:

Bond I: A 6-month bond with face amount of 1,000, an 8% nominal annual coupon rate convertible semiannually, and a 6% nominal annual yield rate convertible semiannually;

Bond II: A one-year bond with face amount of 1,000, a 5% nominal annual coupon rate convertible semiannually, and a 7% nominal annual yield rate convertible semiannually.

Calculate the amount of each bond that Joe should purchase to exactly match the liabilities.

• Bond I: 1, Bond II: 0.97561
• Bond I: 0.93809, Bond II: 1
• Bond I: 0.97561, Bond II: 0.94293
• Bond I: 0.93809, Bond II: 0.97561
• Bond I: 0.98345, Bond II: 0.97561

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

Step 1: Understanding Liabilities and Investment Options

Joe faces two distinct liabilities:

• A payment of $1,000 due 6 months from now (at [math]t=0.5[/math] years).
• A payment of $1,000 due one year from now (at [math]t=1[/math] year).

To meet these obligations, Joe has access to two investment bonds:

 Bond I: 6-Month Bond

• Face Amount: $1,000
• Nominal Annual Coupon Rate: 8%, convertible semiannually. This means a 4% coupon rate per 6-month period.
• Maturity: 6 months ([math]t=0.5[/math] years).

 Bond II: 1-Year Bond

• Face Amount: $1,000
• Nominal Annual Coupon Rate: 5%, convertible semiannually. This means a 2.5% coupon rate per 6-month period.
• Maturity: 1 year ([math]t=1[/math] year).

Step 2: Calculating Cash Flows for a Single Unit of Each Bond

Before determining the number of units to purchase, we calculate the cash flows provided by one unit (with a face amount of $1,000) of each bond at the relevant time points.

 Bond I (6-Month Bond):

The coupon rate per 6-month period is [math]\frac{8\%}{2} = 4\% = 0.04[/math]. This bond matures at [math]t=0.5[/math] years. The cash flow at [math]t=0.5[/math] years (6 months) includes the coupon payment and the face value:

[[math]]CF_{I, 0.5} = (1000 \times 0.04) + 1000 = 40 + 1000 = 1040[[/math]]

 Bond II (1-Year Bond):

The coupon rate per 6-month period is [math]\frac{5\%}{2} = 2.5\% = 0.025[/math]. This bond provides a coupon payment at [math]t=0.5[/math] years and a final coupon plus face value at [math]t=1[/math] year. Cash flow at [math]t=0.5[/math] years (6 months):

[[math]]CF_{II, 0.5} = 1000 \times 0.025 = 25[[/math]]

Cash flow at [math]t=1[/math] year:

[[math]]CF_{II, 1} = (1000 \times 0.025) + 1000 = 25 + 1000 = 1025[[/math]]

A summary of the cash flows per unit of each bond is presented in the table below:

Cash Flows per Unit of Bond

Bond Type	Cash Flow at [math]t=0.5[/math] (6 months)	Cash Flow at [math]t=1[/math] (1 year)
Bond I	$1,040	$0
Bond II	$25	$1,025

Step 3: Addressing the Liability at [math]t=1[/math] Year

The liability due at [math]t=1[/math] year is $1,000. Based on the cash flow analysis in Step 2, only Bond II provides a cash flow at [math]t=1[/math] year. Therefore, Bond II must be used to cover this specific liability. One unit of Bond II provides $1,025 at [math]t=1[/math]. Let [math]N_{II}[/math] be the number of units of Bond II Joe should purchase. To meet the $1,000 liability at [math]t=1[/math]:

[[math]]N_{II} \times 1025 = 1000[[/math]]

Solving for [math]N_{II}[/math]:

[[math]]N_{II} = \frac{1000}{1025} \approx 0.97561[[/math]]

Thus, Joe needs to purchase approximately 0.97561 units of Bond II.

Step 4: Calculating the Remaining Liability at [math]t=0.5[/math] Years

The total liability at [math]t=0.5[/math] years is $1,000. When Joe purchases [math]N_{II} = 0.97561[/math] units of Bond II to cover the [math]t=1[/math] liability, these units also generate a cash flow at [math]t=0.5[/math] years (from Bond II's first coupon payment). The cash flow generated by [math]N_{II}[/math] units of Bond II at [math]t=0.5[/math] years is:

[[math]]CF_{II, 0.5}^{\text{from } N_{II}} = N_{II} \times CF_{II, 0.5} = 0.97561 \times 25 \approx 24.39025[[/math]]

This amount partially covers the liability at [math]t=0.5[/math]. The remaining liability that still needs to be covered (which must be met by Bond I) is:

[[math]]\text{Remaining Liability}_{0.5} = \text{Total Liability}_{0.5} - CF_{II, 0.5}^{\text{from } N_{II}}[[/math]]

[[math]]\text{Remaining Liability}_{0.5} = 1000 - 24.39025 = 975.60975[[/math]]

Step 5: Addressing the Remaining Liability at [math]t=0.5[/math] Years

The remaining liability at [math]t=0.5[/math] years is $975.60975. This amount must be covered by purchasing Bond I. One unit of Bond I provides $1,040 at [math]t=0.5[/math] years (as calculated in Step 2). Let [math]N_I[/math] be the number of units of Bond I Joe should purchase. To meet the remaining liability at [math]t=0.5[/math]:

[[math]]N_I \times 1040 = 975.60975[[/math]]

Solving for [math]N_I[/math]:

[[math]]N_I = \frac{975.60975}{1040} \approx 0.93809[[/math]]

Therefore, Joe needs to purchase approximately 0.93809 units of Bond I.

Step 6: Conclusion

Based on the calculations, to exactly match the given liabilities, Joe should purchase:

• Bond I: Approximately 0.93809 units
• Bond II: Approximately 0.97561 units

This result aligns with Option D.

Key Insights

• When performing liability matching, identify unique cash flow sources first; if only one asset provides cash flow at the latest liability date, it must be used to cover that liability.
• Proceed backward from the latest liability: calculate the units of the bond needed for the latest liability, then account for any cash flows it generates at earlier dates.
• The cash flows from bonds purchased for later liabilities reduce the amount needed from other bonds for earlier liabilities.
• Liability matching often requires purchasing fractional units of assets to achieve exact coverage.

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