The problem asks to find the present value of a 5-year annuity due with periodic cash flows of $500 each year, given an interest rate of 8 percent. The final answer should be rounded to the nearest whole number.

Applied MathematicsFinancial MathematicsPresent ValueAnnuity DueCompound InterestTime Value of Money
2025/7/25

1. Problem Description

The problem asks to find the present value of a 5-year annuity due with periodic cash flows of $500 each year, given an interest rate of 8 percent. The final answer should be rounded to the nearest whole number.

2. Solution Steps

Since this is an annuity due, the payments are made at the beginning of each period. The formula for the present value of an annuity due is:
PV=PMT1(1+i)ni(1+i)PV = PMT * \frac{1 - (1 + i)^{-n}}{i} * (1 + i)
Where:
PVPV is the present value of the annuity due
PMTPMT is the periodic payment
ii is the interest rate per period
nn is the number of periods
In this case:
PMT=500PMT = 500
i=0.08i = 0.08
n=5n = 5
Plugging in the values:
PV=5001(1+0.08)50.08(1+0.08)PV = 500 * \frac{1 - (1 + 0.08)^{-5}}{0.08} * (1 + 0.08)
PV=5001(1.08)50.08(1.08)PV = 500 * \frac{1 - (1.08)^{-5}}{0.08} * (1.08)
PV=50010.680580.081.08PV = 500 * \frac{1 - 0.68058}{0.08} * 1.08
PV=5000.319420.081.08PV = 500 * \frac{0.31942}{0.08} * 1.08
PV=5003.992751.08PV = 500 * 3.99275 * 1.08
PV=5004.31217PV = 500 * 4.31217
PV=2156.085PV = 2156.085
Rounding to the nearest whole number, the present value is 21562156.

3. Final Answer

2156

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