Question: Activity: Simple and compound interest, effective interest rate, present and future value of annuities, amortization of loans and sinking value of annuities and how they can apply these mathematical concepts to take financial decisions.
Feedback: The feedback will be provided by the online facilitator in the discussion forum.
1. A person deposits $400 at the end of every month into a savings account which earns an interest of 10% per year, compounding quarterly. How much amount will he receive at the end of 10 years?
2. After making a down payment of $2,000 for a bike, the buyer paid $500 monthly for 3 years with interest charged at 8% per year, compounding monthly. What is the price of the bike?
3. Find the periodic payment required to amortize a loan of $20,000 over 25 years with interest charged at the rate of 8% per year compounding quarterly.
4. Find the effective interest rate corresponding to a normal rate of 8% per year compounding
i. annually
ii. semi-annually
iii. quarterly
iv. monthly
### Problem 1: Future value of monthly deposits with quarterly compounding interest
1. **Problem statement:** A person deposits $400 at the end of every month into a savings account which earns an interest of 10% per year, compounding quarterly. Find the amount he will receive at the end of 10 years.
2. **Formula used:** The future value of an annuity with periodic deposits and compound interest is given by:
$$FV = P \times \frac{(1 + i)^n - 1}{i}$$
where:
- $P$ = periodic payment
- $i$ = interest rate per compounding period
- $n$ = total number of compounding periods
3. **Important notes:**
- Interest rate is 10% per year compounded quarterly, so quarterly interest rate $i_q = \frac{0.10}{4} = 0.025$.
- Deposits are monthly, but compounding is quarterly, so we must align periods.
- Number of years = 10, so total quarters $n = 10 \times 4 = 40$ quarters.
4. **Adjusting monthly deposits to quarterly periods:**
- Each quarter has 3 months, so total deposits per quarter = $400 \times 3 = 1200$.
- We treat $1200$ as the payment per quarter.
5. **Calculate future value:**
$$FV = 1200 \times \frac{(1 + 0.025)^{40} - 1}{0.025}$$
Calculate powers:
$$ (1 + 0.025)^{40} = 1.025^{40} \approx 2.685 \quad \text{(using a calculator)}$$
So,
$$FV = 1200 \times \frac{2.685 - 1}{0.025} = 1200 \times \frac{1.685}{0.025} = 1200 \times 67.4 = 80,880$$
6. **Answer:** The amount received at the end of 10 years is approximately **80,880**.
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**Note:** Only the first problem is solved as per instructions.