What minimum amount of money earning $7.55 \%$ compounded semiannually will sustain withdrawals of $\$ 2,100$ at the beginning of every month for 12 years? (Do not round intermediate calculations and round your final answer to 2 decimal places.)
The minimum amount
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Answer
The minimum amount of money earning 7.55% compounded semiannually that will sustain withdrawals of $2,100 at the beginning of every month for 12 years is approximately $34,006.16.
Step 1 ：Given that the payment per period (PMT) is $2,100, the annual interest rate (r) is 7.55%, the number of compounding periods per year (n) is 2 (since it's compounded semiannually), and the number of years (t) is 12.
Step 2 ：We need to convert the interest rate from percentage to decimal by dividing it by 100. So, r = 0.0755.
Step 3 ：We are asked to find the present value (PV) of an annuity due, where payments are made at the beginning of each period. The formula for the present value of an annuity due is: \[PV = PMT \times \left(1 - (1 + r/n)^{-nt}\right) / (r/n) \times (1 + r/n)\]
Step 4 ：Substituting the given values into the formula, we get: \[PV = 2100 \times \left(1 - (1 + 0.0755/2)^{-2*12}\right) / (0.0755/2) \times (1 + 0.0755/2)\]
Step 5 ：Solving the above expression, we find that PV = 34006.1574946278
Step 6 ：Rounding to two decimal places, we get the final answer: \(\boxed{34006.16}\)
Step 7 ：The minimum amount of money earning 7.55% compounded semiannually that will sustain withdrawals of $2,100 at the beginning of every month for 12 years is approximately $34,006.16.
