What is annual interest received on an investment of a value of 11000 Rand with a nominal interest rate of 9% and 4 compounding periods, with one payment period?
The correct answer and explanation is:
Let’s break down the problem carefully:
Problem Recap:
- Principal (P) = 11,000 Rand
- Nominal interest rate (annual) = 9% = 0.09
- Compounding periods per year (m) = 4 (quarterly compounding)
- One payment period = 1 year (meaning we want to find the interest earned in one year)
What is being asked?
Annual interest received means the actual interest amount earned over one year on the investment.
Step 1: Understand the nominal interest rate and compounding periods
The nominal interest rate of 9% compounded quarterly means that the interest is not just applied once at 9% per year, but rather divided into 4 parts, each applied every quarter (every 3 months).
So the periodic interest rate (per quarter) is: i = \frac{0.09}{4} = 0.0225 \quad \text{(or 2.25% per quarter)}
Step 2: Calculate the effective annual interest rate (EAR)
The effective annual interest rate (EAR) takes compounding into account. It is calculated as: EAR=(1+rm)m−1EAR = \left(1 + \frac{r}{m}\right)^m – 1
Where:
- rr = nominal annual rate (0.09)
- mm = number of compounding periods (4)
EAR=(1+0.0225)4−1=(1.0225)4−1EAR = \left(1 + 0.0225\right)^4 – 1 = (1.0225)^4 – 1
Calculate (1.0225)4(1.0225)^4: (1.0225)4≈1.09308(1.0225)^4 \approx 1.09308
Therefore, EAR=1.09308−1=0.09308=9.308%EAR = 1.09308 – 1 = 0.09308 = 9.308\%
Step 3: Calculate the annual interest amount earned
The actual amount of interest earned over one year is: Interest=P×EAR=11000×0.09308=1023.88 Rand\text{Interest} = P \times EAR = 11000 \times 0.09308 = 1023.88 \text{ Rand}
Final Answer:
The annual interest received on the investment is 1,023.88 Rand.
Explanation (300 words):
When an investment has a nominal interest rate with multiple compounding periods per year, the interest does not simply accumulate once at the nominal rate. Instead, the interest is applied multiple times, causing the investment to grow faster due to compounding.
In this case, the nominal interest rate is 9% per year, compounded quarterly (4 times a year). This means the 9% annual rate is split into four equal parts of 2.25% applied every three months. Each quarter, the interest earned is added to the principal, so the next quarter’s interest is calculated on the new, larger amount. This process is called compound interest.
To find the actual annual interest earned (or effective interest), we calculate the Effective Annual Rate (EAR), which accounts for the impact of compounding. The formula to find EAR from the nominal rate and compounding frequency is: EAR=(1+rm)m−1EAR = \left(1 + \frac{r}{m}\right)^m – 1
Plugging in the values (9% nominal rate and 4 compounding periods), the EAR comes out to approximately 9.31%. This is slightly higher than the nominal 9% due to quarterly compounding.
Finally, multiplying the EAR by the principal amount (11,000 Rand) gives the actual interest earned over one year, which is about 1,023.88 Rand. This amount reflects the growth of the investment including the effects of compounding, giving a more accurate measure of the return than simply using the nominal rate.
Thus, when dealing with nominal rates and multiple compounding periods, it’s crucial to use EAR to understand how much interest will truly be earned over the period.