As of 12/31/2013, an insurance company has a known obligation to pay 1,000,000 on 12/31/2017. To fund this liability, the company immediately purchases 4-year 5% annual coupon bonds totaling 822,703 of par value. The company anticipates reinvestment interest rates to remain constant at 5% through 12/31/2017. The maturity value of the bond equals the par value.

Consider two reinvestment interest rate movement scenarios effective 1/1/2014. Scenario A has interest rates drop by 0.5%. Scenario B has interest rates increase by 0.5%.

Determine which of the following best describes the insurance company’s profit or (loss) as of 12/31/2017 after the liability is paid.

• Scenario A – 6,610, Scenario B – 11,150
• Scenario A – (14,760), Scenario B – 14,420
• Scenario A – (18,910), Scenario B – 19,190
• Scenario A – (1,310), Scenario B – 1,320
• Scenario A – 0, Scenario B – 0

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

Step 1: Understand the Problem Setup and Key Variables

The insurance company has a liability of $1,000,000 due on 12/31/2017 (which is 4 years from 12/31/2013). To fund this, they purchase 4-year, 5% annual coupon bonds with a par value of $822,703. The maturity value of the bond equals its par value. The initial anticipated reinvestment interest rate is 5% per year. We need to analyze two scenarios for interest rate movements effective 1/1/2014:

• Scenario A: Interest rates drop by 0.5%.
• Scenario B: Interest rates increase by 0.5%.

The goal is to determine the company's profit or loss as of 12/31/2017 after the liability is paid under each scenario.

Key Information Summary:

Key Problem Parameters

Parameter	Value
Liability Due Date	12/31/2017 (4 years from 12/31/2013)
Liability Amount	$1,000,000
Bond Par Value	$822,703
Bond Coupon Rate	5% per annum
Bond Maturity Term	4 years
Bond Maturity Value	Equals Par Value ($822,703)
Initial Anticipated Reinvestment Rate	5%

Step 2: Calculate Annual Coupon Payments

The bonds have a 5% annual coupon rate on a par value of $822,703. The annual coupon payment is calculated as:

[[math]]\text{Annual Coupon Payment} = \text{Par Value} \times \text{Coupon Rate}[[/math]]

[[math]]\text{Annual Coupon Payment} = \$822,703 \times 0.05 = \$41,135.15[[/math]]

For the purpose of consistency with the provided solution and options, we will use the rounded value of $41,135 for the annual coupon payments. These coupon payments will be reinvested annually until the liability is due at [math]t=4[/math].

Step 3: Analyze Scenario A (Interest Rates Drop)

In Scenario A, interest rates drop by 0.5% from the anticipated 5%. Therefore, the new reinvestment rate for the coupons is [math]5\% - 0.5\% = 4.5\%[/math]. The total funds available at 12/31/2017 will consist of two parts:

1. The accumulated value of the four annual coupon payments ($41,135 each), reinvested at 4.5% per year.
2. The maturity value of the bond, which is its par value of $822,703, received at [math]t=4[/math].

1. Accumulated Value of Coupons: The future value of an ordinary annuity of $41,135 per year for 4 years at 4.5% is given by [math]41,135 s_{\overline{4}|0.045}[/math]. Using the future value annuity factor for [math]i=0.045[/math] and [math]n=4[/math]:

[[math]]s_{\overline{4}|0.045} = \frac{(1+0.045)^4 - 1}{0.045} \approx 4.2782[[/math]]

Accumulated Coupons = [math]41,135 \times 4.2782 = \$175,984.057[/math] For consistency with the provided solution, we will use $175,984.

2. Total Accumulated Value at 12/31/2017:

[[math]]\text{Total Accumulated Value}_A = \text{Accumulated Coupons} + \text{Bond Maturity Value}[[/math]]

[[math]]\text{Total Accumulated Value}_A = \$175,984 + \$822,703 = \$998,687[[/math]]

3. Profit or Loss in Scenario A:

[[math]]\text{Profit/(Loss)}_A = \text{Total Accumulated Value}_A - \text{Liability Amount}[[/math]]

[[math]]\text{Profit/(Loss)}_A = \$998,687 - \$1,000,000 = (\$1,313)[[/math]]

This indicates a loss of $1,313 in Scenario A.

Step 4: Analyze Scenario B (Interest Rates Increase)

In Scenario B, interest rates increase by 0.5% from the anticipated 5%. Therefore, the new reinvestment rate for the coupons is [math]5\% + 0.5\% = 5.5\%[/math]. Similar to Scenario A, the total funds available at 12/31/2017 will consist of the accumulated value of the coupon payments and the bond's maturity value.

1. Accumulated Value of Coupons: The future value of an ordinary annuity of $41,135 per year for 4 years at 5.5% is given by [math]41,135 s_{\overline{4}|0.055}[/math]. Using the future value annuity factor for [math]i=0.055[/math] and [math]n=4[/math]:

[[math]]s_{\overline{4}|0.055} = \frac{(1+0.055)^4 - 1}{0.055} \approx 4.3423[[/math]]

Accumulated Coupons = [math]41,135 \times 4.3423 = \$178,621.1405[/math] For consistency with the provided solution, we will use $178,621.

2. Total Accumulated Value at 12/31/2017:

[[math]]\text{Total Accumulated Value}_B = \text{Accumulated Coupons} + \text{Bond Maturity Value}[[/math]]

[[math]]\text{Total Accumulated Value}_B = \$178,621 + \$822,703 = \$1,001,324[[/math]]

3. Profit or Loss in Scenario B:

[[math]]\text{Profit/(Loss)}_B = \text{Total Accumulated Value}_B - \text{Liability Amount}[[/math]]

[[math]]\text{Profit/(Loss)}_B = \$1,001,324 - \$1,000,000 = \$1,324[[/math]]

This indicates a profit of $1,324 in Scenario B.

Step 5: Conclude Profit or Loss for Each Scenario

Based on the calculations:

Summary of Profit / (Loss)

Scenario	Reinvestment Rate	Accumulated Coupons	Maturity Value	Total Assets	Liability	Profit/(Loss)
A	4.5%	$175,984	$822,703	$998,687	$1,000,000	($1,313)
B	5.5%	$178,621	$822,703	$1,001,324	$1,000,000	$1,324

Comparing these results to the given options:

• Scenario A – ($1,310)
• Scenario B – $1,320

Our calculated values are ($1,313) and $1,324, which are very close to option D due to minor rounding differences in the annuity factor. Therefore, option D best describes the insurance company's profit or loss.

Key Insights

• Reinvestment Risk: This problem highlights reinvestment risk, where changes in interest rates affect the accumulation of coupon payments, impacting the final value of assets available to meet a liability.
• Components of Accumulated Value: The total accumulated value consists of two parts: the future value of reinvested coupon payments and the bond's maturity value.
• Impact of Rate Changes: When interest rates drop, the accumulated value of reinvested coupons decreases, leading to a potential shortfall. Conversely, when rates rise, the accumulated value increases, potentially leading to a surplus.
• Immunization: While the company aims to fund a future liability, it is not perfectly immunized against interest rate changes. A perfect immunization strategy would result in a zero profit/loss regardless of rate movements (within reasonable bounds).
• Approximation in Practice: Actuarial problems often involve rounded intermediate values (like [math]s_{\overline{n}|i}[/math] factors), leading to final answers that may slightly differ from exact calculations but align with the closest provided option.

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

Solution: D

Under either scenario, the company will have 822,703(0.05) = 41,135 to invest at the end of each of the four years. Under Scenario A these payments will be invested at 4.5% and accumulate to

[[math]]41,135 s_{\overline{4}|0.055} = 41,135(4.2782) = 175,984.[[/math]]

Adding the maturity value produces 998,687 for a loss of 1,313. Note that only answer D has this value. The Scenario B calculation is

[[math]]41,135s_{\overline{4}|0.055}=41,135(4.3423)=178,621+822,703-1,000,000=1,324. [[/math]]

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