An insurance company is offering a new policy to its customers. Typically, the policy is bought by a parent or grandparent for a child at the child’s birth. The details of the policy are as follows: The purchaser (say, the parent) makes the following six payments to the insurance company:

First birthday:

$ 860

Second birthday:

$ 860

Third birthday:

$ 960

Fourth birthday:

$ 960

Fifth birthday:

$ 1,060

Sixth birthday:

$ 1,060

After the child’s sixth birthday, no more payments are made. When the child reaches age 65, he or she receives $120,000.

If the relevant interest rate is 9 percent for the first six years and 5 percent for all subsequent years, what is the value of the policy at the child's 65th birthday?

**Note: Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.**

To determine the value of the insurance policy at the child's 65th birthday, we can break down the problem into two parts:

1. **The present value of the payments made during the first six years, considering the 9% interest rate.**
2. **How this value grows over the subsequent years until the child turns 65, given a 5% interest rate.**

Let's follow these steps:

### Step 1: Discount each payment to the child's birth (time zero) using the 9% interest rate

For the payments, we are given the amounts and the payment years (birthdays). The first payment is made at the end of the first year, so we will discount each payment back to the child's birth using the formula for present value (PV):

\[
PV = \frac{C}{(1 + r)^t}
\]

Where:
- $ C $ is the payment,
- $ r $ is the interest rate (9% = 0.09),
- $ t $ is the number of years until the payment.

We'll calculate the present value of each of the six payments:

1. **First payment at age 1 (860):**
   \[
   PV_1 = \frac{860}{(1 + 0.09)^1}
   \]

2. **Second payment at age 2 (860):**
   \[
   PV_2 = \frac{860}{(1 + 0.09)^2}
   \]

3. **Third payment at age 3 (960):**
   \[
   PV_3 = \frac{960}{(1 + 0.09)^3}
   \]

4. **Fourth payment at age 4 (960):**
   \[
   PV_4 = \frac{960}{(1 + 0.09)^4}
   \]

5. **Fifth payment at age 5 (1,060):**
   \[
   PV_5 = \frac{1,060}{(1 + 0.09)^5}
   \]

6. **Sixth payment at age 6 (1,060):**
   \[
   PV_6 = \frac{1,060}{(1 + 0.09)^6}
   \]

### Step 2: Sum the present values to get the total present value at the child's birth

We will now sum the present values from Step 1 to determine the total present value at the child's birth (age 0).

### Step 3: Compound the total present value from age 0 to age 65

Once we have the total present value at the child's birth, we need to calculate how much this value will grow by the time the child turns 65, given a 5% interest rate. The total number of years from the child's birth to age 65 is 65 years. However, for the first 6 years, the payments are being made, so we compound for the remaining 59 years at a 5% interest rate.

The future value (FV) of the policy at age 65 is given by the formula:

\[
FV = PV \times (1 + r)^{t}
\]

Where:
- $ PV $ is the present value at the child's birth,
- $ r $ is the interest rate (5% = 0.05),
- $ t $ is the number of years (59 years).

Let's compute the exact value.The value of the policy at the child's 65th birthday, after compounding the payments at the given interest rates, is approximately **$75,698.57**.

Would you like more details or have any further questions on this? Here are 5 related questions to explore:

1. How would the policy value change with different interest rates?
2. What is the impact of delaying one of the payments by a year?
3. How does the present value formula apply in other financial contexts?
4. Can we calculate the breakeven point for the total payments made?
5. What if the policy was structured to include yearly payments after age 65?

**Tip:** When calculating future values, always ensure that you account for the correct time period and corresponding interest rate.

An insurance company is offering a new policy to its customers. Typically, the policy is bought by a parent or grandparent for a child at the child’s birth. The details of the policy are as follows: The purchaser (say, the parent) makes the following six payments to the insurance company:

First birthday: $860
Second birthday: $860
Third birthday: $960
Fourth birthday: $960
Fifth birthday: $1,060
Sixth birthday: $1,060

After the child’s sixth birthday, no more payments are made. When the child reaches age 65, he or she receives $120,000.

If the relevant interest rate is 9 percent for the first six years and 5 percent for all subsequent years, what is the value of the policy at the child's 65th birthday?

Solve the insurance policy value problem with payments made over the first six years at 9% interest and compounding at 5% until the child turns 65. This problem covers present value, future value, and compound interest calculations. Ideal for high school and college-level finance students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation