Question - Calculating Accumulated Value of Monthly Payments with Semi-annual Compounding
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Solution:
The question provided in your image asks to calculate the accumulated value after two years of monthly payments of $10,000, made at the end of each month, if the interest rate is 4% compounded semi-annually.Firstly, since the compounding frequency is semi-annual, but payments are monthly, we need to determine the equivalent monthly interest rate. However, since the interest rate is compounded semi-annually, we will have to treat the account as if it were two separate accounts: one that accumulates during the first six months of the year at a monthly rate corresponding to the semi-annual rate, and another that accumulates during the last six months at the same monthly rate.Since the annual interest rate is 4%, the semi-annual interest rate is 2%. To find the equivalent monthly interest rate (i), we use the formula for converting a semi-annual rate to a monthly rate when compounding is not monthly:$$ 1 + i = (1 + \text{semi-annual rate})^{(1/6)} $$$$ 1 + i = (1 + 0.02)^{(1/6)} $$Calculate the effective monthly rate:$$ i = (1.02)^{(1/6)} - 1 $$Once we have the monthly rate, we can calculate the future value of an ordinary annuity (payments are at the end of the period) using the future value formula. For an ordinary annuity, the future value FV is calculated using the formula:$$ FV = P \times \frac{((1 + i)^n - 1)}{i} $$Where:- P is the payment amount per period- i is the interest rate per period- n is the total number of paymentsHere, P = $10,000, i is the monthly rate we calculated, and n = 24 (since there are 24 months in two years).Upon finding the effective monthly rate i, you would plug the values into the future value formula to calculate the accumulated value after two years. Please use a financial calculator or software to compute the effective monthly rate and subsequently the future value, as the calculations involve several decimal places and exponents.
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