A financial analyst is assessing the interest rate sensitivity of a fixed-payment structured settlement. The current present value of this settlement, based on prevailing market rates, is $150,000. The Macaulay duration of the settlement's cash flows, given the current market discount rate, is 10.0 years. The current annual effective market discount rate used for valuation is 4.0%. The analyst needs to estimate the immediate impact on the present value of the structured settlement if the market discount rate were to decrease by 25 basis points (0.25%).

Using this information, calculate the approximate change in the present value of the structured settlement.

• -$3,750.00
• -$3,605.77
• $3,605.77
• $3,750.00
• $3,906.25

Step 1: Understand the Goal and Given Information

The objective is to calculate the approximate change in the present value (PV) of a structured settlement due to a change in the market discount rate. We are provided with the following information:

• Current Present Value ([math]PV[/math]): [math]$150,000[/math]
• Macaulay Duration ([math]MacDur[/math]): [math]10.0[/math] years
• Current Annual Effective Market Discount Rate ([math]i[/math]): [math]4.0\% = 0.04[/math]
• Change in Market Discount Rate ([math]\Delta i[/math]): A decrease of [math]25[/math] basis points, which is [math]-0.25\% = -0.0025[/math].

Step 2: Recall the Approximation Formula

The approximate change in the present value of a financial instrument can be estimated using its modified duration. The formulas required are:

1. Modified Duration: The modified duration ([math]ModDur[/math]) is calculated from the Macaulay duration as:
   [[math]]ModDur = \frac{MacDur}{1+i}[[/math]]
2. Approximate Change in Present Value: The approximate change in present value ([math]\Delta PV[/math]) is given by:

[[math]]\Delta PV \approx -PV \times ModDur \times \Delta i[[/math]]

Step 3: Calculate the Modified Duration

Using the given Macaulay duration and the current market discount rate, we calculate the modified duration:

[[math]]ModDur = \frac{10.0}{1 + 0.04} = \frac{10.0}{1.04} \approx 9.6153846[[/math]]

Step 4: Calculate the Approximate Change in Present Value

Now, we apply the formula for the approximate change in present value, substituting the known values:

[[math]]\Delta PV \approx -PV \times ModDur \times \Delta i[[/math]]

[[math]]\Delta PV \approx -\$150,000 \times 9.6153846 \times (-0.0025)[[/math]]

Since [math]\Delta i[/math] is a decrease, it is negative, resulting in a positive change in PV (an increase).

[[math]]\Delta PV \approx \$150,000 \times 9.6153846 \times 0.0025[[/math]]

[[math]]\Delta PV \approx \$150,000 \times 0.02403846[[/math]]

[[math]]\Delta PV \approx \$3,605.769[[/math]]

Rounding to two decimal places, the approximate change in the present value of the structured settlement is [math]$3,605.77[/math].

Key Insights

• The approximation of change in present value using duration is a first-order approximation and assumes a linear relationship between interest rates and bond prices, which is generally valid for small interest rate changes.
• Macaulay duration measures the weighted average time until a bond's cash flows are received, while modified duration measures the percentage change in a bond's price for a 1% change in yield.
• To use the duration approximation for price changes, Macaulay duration must first be converted to Modified Duration by dividing by [math](1 + i)[/math], where [math]i[/math] is the periodic yield.
• A decrease in interest rates typically leads to an increase in the present value (price) of a bond, and vice-versa, as reflected by the negative sign in the approximation formula [math]\Delta PV \approx -PV \times ModDur \times \Delta i[/math].
• Basis points must be converted to decimal form (e.g., 25 basis points = 0.0025) before being used in calculations.

Answer was verified by AI

Answer was verified numerically