What amount can be borrowed at 9.14% p.a., compounded half-yearly, if it is to be repaid by regular half-yearly payments, the first payment being R4000 (payable at the end of the first half-year) and the subsequent payments decreasing by R400 per half-year until a final payment of R400 is made? (Give your answer rounded to 2 decimal places. If you think the answer is R1012.3456, type in 1012.35)

We need to calculate the present value of a series of decreasing payments, where payments are made half-yearly, and the interest rate is compounded half-yearly as well.

### Key Information:
- **Interest rate (i):** 9.14% p.a. compounded half-yearly, so the half-yearly rate is:
  \[
  i = \frac{9.14\%}{2} = 4.57\%
  \]
  or \( i = 0.0457 \) in decimal form.
- **Payment amounts:**
  - First payment = R4000
  - Second payment = R3600
  - Third payment = R3200
  - ... Decreasing by R400 until the final payment of R400.

### Step-by-Step Process:

The general formula for the present value (PV) of a payment made at the end of period \(n\), with decreasing payments, is:

\[
PV = P_1 \times v + P_2 \times v^2 + P_3 \times v^3 + \cdots + P_n \times v^n
\]
Where:
- \(P_1, P_2, \ldots, P_n\) are the payment amounts.
- \(v = \frac{1}{1+i}\) is the discount factor.

We can summarize the payments as follows:
- Payment 1: R4000
- Payment 2: R3600
- Payment 3: R3200
- ...
- Last Payment: R400

#### 1. Calculate the discount factor:
\[
v = \frac{1}{1 + 0.0457} = \frac{1}{1.0457} \approx 0.9563
\]

#### 2. Calculate the present value of each payment:

\[
PV = 4000 \times v + 3600 \times v^2 + 3200 \times v^3 + \cdots + 400 \times v^{10}
\]

Now I will calculate each term and sum the present values of the payments.The amount that can be borrowed is approximately **R18,507.18**.

Would you like further details or have any questions? Here are some related questions that could expand on this:

1. How does compounding frequency affect the total amount that can be borrowed?
2. What is the effect of increasing or decreasing the interest rate on the amount borrowed?
3. How would the calculation change if payments increased rather than decreased?
4. How can the present value formula be applied to different types of loans?
5. What if the payments were made monthly instead of half-yearly?

**Tip:** The more frequently interest is compounded, the higher the effective interest rate will be over time.

What amount can be borrowed at 9.14% p.a., compounded half-yearly, if it is to be repaid by regular half-yearly payments, the first payment being R4000 (payable at the end of the first half-year) and the subsequent payments decreasing by R400 per half-year until a final payment of R400 is made?

Learn how to calculate the amount that can be borrowed at 9.14% p.a., compounded half-yearly, with half-yearly payments starting at R4000 and decreasing by R400 per period. This detailed solution uses time value of money concepts, including the present value of decreasing payments and compound interest.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation