XYZ Insurer is pricing a 10-year pure endowment benefit of $100,000 on an individual age 65.

i. Mortality follows the Standard Ultimate Life Table.
ii. The current effective interest rate is 4%.

If the interest rate changes from 4% to 5%, what will be the resulting change in the present value of this benefit?

To determine the change in the present value of the pure endowment benefit, we need to calculate the present value (PV) of the benefit at both interest rates (4% and 5%) and then find the difference between the two values. Here’s the step-by-step approach:

### Step 1: Understanding the Pure Endowment
A pure endowment pays a benefit only if the individual survives until the end of the term. In this case, the benefit is $100,000, payable after 10 years if the individual, currently age 65, survives to age 75.

The present value of a pure endowment benefit is calculated as:
\[
PV = 100,000 \times v^{10} \times \, _{10}p_{65}
\]
where:
- $ v = \frac{1}{1+i} $ is the discount factor,
- $ _{10}p_{65} $ is the probability that the individual survives from age 65 to age 75, which we get from the Standard Ultimate Life Table.

### Step 2: Discount Factors
We need to calculate the discount factors for both interest rates, 4% and 5%.

At 4% interest:
\[
v_{4\%} = \frac{1}{1 + 0.04} = \frac{1}{1.04} \approx 0.96154
\]

At 5% interest:
\[
v_{5\%} = \frac{1}{1 + 0.05} = \frac{1}{1.05} \approx 0.95238
\]

### Step 3: Survival Probability from the Standard Ultimate Life Table
Using the Standard Ultimate Life Table, we can find the probability of survival from age 65 to age 75, $ _{10}p_{65} $. Let’s assume:
\[
_{10}p_{65} = 0.75
\]
(You would need to verify this from the table, but for the purpose of this example, we'll use 0.75.)

### Step 4: Calculate the Present Values

At 4% interest:
\[
PV_{4\%} = 100,000 \times (0.96154)^{10} \times 0.75
\]
First, calculate $ 0.96154^{10} $:
\[
0.96154^{10} \approx 0.665
\]
Now calculate the present value:
\[
PV_{4\%} = 100,000 \times 0.665 \times 0.75 = 100,000 \times 0.49875 = 49,875
\]

At 5% interest:
\[
PV_{5\%} = 100,000 \times (0.95238)^{10} \times 0.75
\]
First, calculate $ 0.95238^{10} $:
\[
0.95238^{10} \approx 0.614
\]
Now calculate the present value:
\[
PV_{5\%} = 100,000 \times 0.614 \times 0.75 = 100,000 \times 0.4605 = 46,050
\]

### Step 5: Calculate the Change in Present Value
The change in the present value is the difference between the two present values:
\[
\Delta PV = PV_{4\%} - PV_{5\%} = 49,875 - 46,050 = 3,825
\]

### Final Answer:
The resulting change in the present value of the benefit is a **decrease of $3,825** when the interest rate increases from 4% to 5%.

---

Do you want any further details or clarification?

Here are five related questions for deeper exploration:
1. How does a pure endowment differ from other types of insurance policies?
2. How does the choice of mortality table impact the calculation of a pure endowment's present value?
3. What if the interest rate dropped to 3% instead of rising to 5%? How would that affect the present value?
4. How is the survival probability calculated from the Standard Ultimate Life Table?
5. How would the present value change if the benefit was paid annually instead of in a lump sum?

**Tip:** Always check the accuracy of survival probabilities and other table-based factors, as slight differences can significantly affect the result in insurance calculations.

XYZ Insurer is pricing a 10-year pure endowment benefit of $100,000 on an individual age 65. If the interest rate changes from 4% to 5%, what will be the resulting change in the present value of this benefit?

Explore how a change in interest rate from 4% to 5% affects the present value of a $100,000 10-year pure endowment benefit for an individual aged 65. This problem covers present value calculations, survival probabilities from mortality tables, and discount factor concepts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation