Jon is to receive 10 annual payments where the first payment of 10,000 will be paid in one year.
Payments in years 2-10 will increase by an annual rate of 5%.
The present value of these payments using an annual effective interest rate of 8% is X.
Calculate X.

To calculate the present value $X$ of the payments Jon will receive, we need to discount each future payment back to the present using the given 8% annual effective interest rate.

Here’s how to approach the problem step by step:

### Step 1: Define the payments
Jon will receive 10 payments. The first payment is 10,000, and subsequent payments increase by 5% annually. The payment stream is thus:

- Year 1: $ P_1 = 10,000 $
- Year 2: $ P_2 = 10,000 \times (1.05) $
- Year 3: $ P_3 = 10,000 \times (1.05)^2 $
- And so on, until Year 10: $ P_{10} = 10,000 \times (1.05)^9 $

### Step 2: Discount the payments
The present value $X$ is the sum of the discounted values of each of the 10 payments. The discount factor for each year is based on the annual effective interest rate of 8%. The discount factor for a payment in year $n$ is $ \frac{1}{(1.08)^n} $.

Thus, the present value $X$ is:

\[
X = \frac{10,000}{1.08} + \frac{10,000 \times 1.05}{(1.08)^2} + \frac{10,000 \times (1.05)^2}{(1.08)^3} + \cdots + \frac{10,000 \times (1.05)^9}{(1.08)^{10}}
\]

### Step 3: Write it as a summation
The above expression can be simplified into a summation formula:

\[
X = \sum_{n=1}^{10} \frac{10,000 \times (1.05)^{n-1}}{(1.08)^n}
\]

Now, let's compute this summation to find the present value $X$. I will calculate it for you.The present value $X$ of Jon's 10 annual payments is approximately \$81,835.54, given the 8% annual effective interest rate.

Would you like more details or have any questions?

Here are five related questions you might find interesting:

1. How would the present value change if the interest rate were higher, say 10%?
2. What if the payment increases by 6% annually instead of 5%?
3. How would you calculate the future value of these payments after 10 years?
4. Can you determine how much Jon would receive in total over the 10 years without discounting?
5. How would the present value be affected if Jon receives payments semi-annually instead of annually?

**Tip:** When dealing with increasing payments, calculating present value using a summation of discounted cash flows is crucial to account for both growth and time value of money.

Jon is to receive 10 annual payments where the first payment of 10,000 will be paid in one year. Payments in years 2-10 will increase by an annual rate of 5%. The present value of these payments using an annual effective interest rate of 8% is X. Calculate X.

Learn how to calculate the present value of Jon's 10 annual payments, starting at $10,000 with a 5% annual increase, discounted at an 8% interest rate. This solution covers time value of money, discounted cash flow, and geometric series concepts, suitable for finance and actuarial 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