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.

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

Solution: D

Because only Bond II provides a cash flow at time 1, it must be considered first. The bond provides 1025 at time 1 and thus 1000/1025 = 0.97561 units of this bond provides the required cash. This bond then also provides 0.97561(25) = 24.39025 at time 0.5. Thus Bond I must provide 1000 – 24.39025 = 975.60975 at time 0.5. The bond provides 1040 and thus 975.60975/1040 = 0.93809 units must be purchased.

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