Joe must pay liabilities of 2000 due one year from now and another 1000 due two years from now. He exactly matches his liabilities with the following two investments:

Mortgage I: A one year mortgage in which X is lent. It is repaid with a single payment at time one. The annual effective interest rate is 6%.

Mortgage II: A two-year mortgage in which Y is lent. It is repaid with two equal annual payments. The annual effective interest rate is 7%.

Calculate X + Y.

• 2600
• 2682
• 2751
• 2825
• 3000

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

Step 1: Understand the Liabilities and Investment Structure

Joe faces specific financial obligations and plans to meet them with two types of mortgages.

Liabilities:

Liabilities Schedule

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

Available Investments:

• Mortgage I: A one-year loan of [math]X[/math], repaid with a single payment at time [math]t=1[/math]. The annual effective interest rate is 6%.
• Mortgage II: A two-year loan of [math]Y[/math], repaid with two equal annual payments. The annual effective interest rate is 7%.

The goal is to calculate the total amount lent, [math]X + Y[/math], to exactly match these liabilities.

Step 2: Determine the Loan Amount for Mortgage II (Y)

We begin by analyzing Mortgage II because it is the only investment that provides a cash flow at time [math]t=2[/math], which is required to meet the $1,000 liability at that time.

Annual Payment Calculation: Since Mortgage II provides equal annual payments and is the only source of funds at [math]t=2[/math], its payment at [math]t=2[/math] must exactly cover the $1,000 liability. Let [math]P[/math] be the equal annual payment from Mortgage II. Thus, [math]P = $1,000[/math]. The loan amount [math]Y[/math] for Mortgage II is the present value of these two equal annual payments of $1,000, discounted at the mortgage's interest rate of 7%. The present value of an annuity of 2 payments of 1 at 7% is [math]a_{\overline{2}|0.07}[/math].

[[math]]Y = P \cdot a_{\overline{2}|0.07}[[/math]]

First, calculate the present value annuity factor:

[[math]]a_{\overline{2}|0.07} = \frac{1 - (1 + 0.07)^{-2}}{0.07} = \frac{1 - (1.07)^{-2}}{0.07}[[/math]]

[[math]]a_{\overline{2}|0.07} = \frac{1 - 0.87343878}{0.07} = \frac{0.12656122}{0.07} \approx 1.8080174[[/math]]

Now, calculate the loan amount [math]Y[/math]:

[[math]]Y = $1,000 \times 1.8080174 = $1,808.0174[[/math]]

Rounded to two decimal places, [math]Y \approx $1,808.02[/math].

Cash Flows from Mortgage II: Since the annual payment is $1,000, Mortgage II provides $1,000 at [math]t=1[/math] and $1,000 at [math]t=2[/math].

Mortgage II Cash Flows

Time ([math]t[/math])	Cash Flow from Mortgage II
1 year	$1,000
2 years	$1,000

Step 3: Determine the Loan Amount for Mortgage I (X)

Now we consider the liability at time [math]t=1[/math]. The total liability at [math]t=1[/math] is $2,000. From Step 2, we know that Mortgage II provides $1,000 at [math]t=1[/math]. Therefore, Mortgage I must cover the remaining liability at [math]t=1[/math].

Remaining Liability at [math]t=1[/math]:

[[math]]\text{Remaining Liability} = \text{Total Liability at } t=1 - \text{Cash Flow from Mortgage II at } t=1[[/math]]

[[math]]\text{Remaining Liability} = $2,000 - $1,000 = $1,000[[/math]]

Loan Amount Calculation: Mortgage I is a one-year loan of [math]X[/math] repaid with a single payment at [math]t=1[/math]. This single payment must be $1,000. The payment amount is [math]X(1 + 0.06)[/math].

[[math]]X(1 + 0.06) = $1,000[[/math]]

[[math]]X(1.06) = $1,000[[/math]]

[[math]]X = \frac{$1,000}{1.06} \approx $943.3962[[/math]]

Rounded to two decimal places, [math]X \approx $943.40[/math].

Cash Flows from Mortgage I:

Mortgage I Cash Flows

Time ([math]t[/math])	Cash Flow from Mortgage I
1 year	$1,000

Step 4: Calculate Total Loan Amount (X + Y)

Finally, we sum the loan amounts for Mortgage I ([math]X[/math]) and Mortgage II ([math]Y[/math]) to find the total amount Joe lent.

[[math]]\text{Total Loan Amount} = X + Y[[/math]]

[[math]]\text{Total Loan Amount} = $943.3962 + $1,808.0174 = $2,751.4136[[/math]]

The total loan amount is approximately $2,751.41. This matches option C ($2,751) when rounded to the nearest dollar, consistent with typical exam answer formats.

Key Insights

• Prioritize Unique Liabilities: When matching multiple liabilities with different investment options, it is often strategic to first address the liabilities that can only be met by a specific investment or a specific characteristic of an investment (e.g., a cash flow at a unique future time).
• Backwards Induction for Liability Matching: For problems involving matching future liabilities, it is often efficient to work backward from the latest liability to the earliest, determining the required investment amounts at each stage.
• Present Value Application: The present value concept is crucial for determining the initial principal of a loan based on its future cash flows (payments) and the applicable interest rate.
• Mortgage/Annuity Formulas: Understanding how to calculate loan payments or principal amounts using annuity formulas (e.g., [math]a_{\overline{n}|i}[/math]) is fundamental for these types of problems.

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

Solution: C

Because only Mortgage II provides a cash flow at time two, it must be considered first. The mortgage provides [math]Y/a_{\overline{2}|0.07} = 0.553092Y[/math] at times one and two. Therefore, 0.553092Y = 1000 for Y = 1808.02. Mortgage I must provide 2000 – 1000 = 1000 at time one and thus X = 1000/1.06 = 943.40. The sum is 2751.42.

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