Question
At the end of each quarter a payment of K500 is made into an account which bearing interest compounded quarterly. If the annuity accrues to K7500 over three years, determined the nominal interest rate.
Answers
GPT 3.5
To determine the nominal interest rate, we need to use the formula for the future value of an annuity:
\[FV = P \times \left(1 + \frac{r}{n}\right)^{nt} - P\]
Where:
FV = Future value of the annuity (K7500)
P = Quarterly payment (K500)
r = nominal interest rate
n = number of compounding periods per year (4 for quarterly compounding)
t = number of years (3 years)
Substitute the values into the formula:
\[7500 = 500 \times \left(1 + \frac{r}{4}\right)^{4*3} - 500\]
Simplify the equation:
\[7500 = 500 \times \left(1 + \frac{r}{4}\right)^{12} - 500\]
\[8000 = \left(1 + \frac{r}{4}\right)^{12}\]
Taking the 12th root of both sides:
\[1 + \frac{r}{4} = \sqrt[12]{8000}\]
\[1 + \frac{r}{4} \approx 1.1089\]
\[\frac{r}{4} \approx 0.1089\]
\[r \approx 0.4356\]
Therefore, the nominal interest rate is approximately 43.56%.
\[FV = P \times \left(1 + \frac{r}{n}\right)^{nt} - P\]
Where:
FV = Future value of the annuity (K7500)
P = Quarterly payment (K500)
r = nominal interest rate
n = number of compounding periods per year (4 for quarterly compounding)
t = number of years (3 years)
Substitute the values into the formula:
\[7500 = 500 \times \left(1 + \frac{r}{4}\right)^{4*3} - 500\]
Simplify the equation:
\[7500 = 500 \times \left(1 + \frac{r}{4}\right)^{12} - 500\]
\[8000 = \left(1 + \frac{r}{4}\right)^{12}\]
Taking the 12th root of both sides:
\[1 + \frac{r}{4} = \sqrt[12]{8000}\]
\[1 + \frac{r}{4} \approx 1.1089\]
\[\frac{r}{4} \approx 0.1089\]
\[r \approx 0.4356\]
Therefore, the nominal interest rate is approximately 43.56%.