Wilma saves 12,000 at the end of every six months for 10 years. Assume 10% compounded semiannually and find the present value
To find the present value of the savings, we need to calculate the present value of each individual savings amount and sum them up.
The formula to calculate the present value of a future value compounded semiannually is:
PV = FV / (1 + r/n)^(n*t)
Where:
PV = Present Value
FV = Future Value
r = Annual interest rate (in decimal form)
n = Number of compounding periods per year
t = Number of years
In this case, we have:
FV = $12,000 (the savings amount every six months)
r = 10% (0.10 in decimal form)
n = 2 (as it is compounded semiannually)
t = 10 (number of years)
Now, substituting the values into the formula, we can calculate the present value for each individual savings amount:
PV_1 = $12,000 / (1 + 0.10/2)^(2 * 1)
PV_2 = $12,000 / (1 + 0.10/2)^(2 * 2)
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.
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PV_20 = $12,000 / (1 + 0.10/2)^(2 * 20)
After calculating the present value for each individual savings amount, we can sum them up to find the total present value.
A=biannual payment = 12,000
APR=10% => i=10%/2=0.05
n=number of periods (each six-months) = 10*2=20
Present value
P=A*((1+i)^n-1)/(i(1+i)^n)
You can substitute values and compute.
P is the same as the amount borrowed in a mortgage with a semi-annual payment of 12,000.
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