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Question 1168746: Becky needs another $1,100 in her vehicle fund to purchase the car she wants. Her parents offer to loan her the money, but want to teach her about the time value of money. They offer to have her repay the loan in the future using money from freelance work. They agree that she will repay $475 at each of her next three booked engagements. These events are 3, 9 and 16 months from now. Assume a 5% cost of capital. Assume there is no risk of default, and that compounding is monthly. What is the NPV of the loan from her parents' perspective?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! net present value to time period 0 is what you are looking for as far as i can tell.
3 months from then is time period 3.
9 months from then is time period 9.
16 months from then is time period 16.
the cost of capital is 5% per year compounded monthly.
divide 5% by 12 to get the cost of capital is (5/12)% per month.
divide that by 100 to get the cost of capital is 5/1200 per month.
5/12 is the percent.
5/12 is the percent.
5/1200 is the rate.
the equation used rate, not percent.
to find the net present value, use the time value formula of f = p * (1 + r) ^ n
f is the future value
p is the present value which is the same as the net present value in this problem.
r is the interest rate per time period.
n is the number of time periods.
one time period is one month.
since you want the present value, solve the time value formula for p.
start with f = p * (1 + r) ^ n
divide both sides of the equation by (1 + r) ^ n to get:
f / (1 + r) ^ n = p
switch sides in the formula to get:
p = f / (1 + r) ^ n
you will use this formula 3 times.
one for 3 months out.
one for 9 months out.
one for 16 months out.
3 months out formula is p = 475 / (1 + 5/1200) ^ 3
solve for p to get:
p = 469.1116377.
9 months out formula is p = 475 / (1 + 5/1200) ^ 9
solve for p to get:
p = 457.5529944.
16 months out formula is p = 475 / (1 + 5/1200) ^ 16
solve for p to get:
444.4273687.
add them up to get:
p = 1371.092001.
that's the net present value.
they gave her 1100 in time period 0.
they received back the equivalent of 1371.092001 in time period 0.
i also solved the problem in excel, using cash flow analysis techniques.
it looks like this.
the sum of the payments was 3 * 475 = 1425.
the sum of the present value of the payments was 1371.092001.
the sum of the future value of the payments was 1465.410877.
you would probably want to round to the nearest penny.
it appears from the present value that her parents received, that they got back more than they gave out.
they gave her 1100 in time period 0 dollars.
they got back 1371 in time period 0 dollars.
the future value of the loan and the future value of the payments tells pretty much the same story.
the future value of the 1100, if invested at the same interest rate, would be worth 1175.67017 dollars in 16 months.
the future value of the payments that she made and that they invested at the same interest rate were worth 1465.410877 in 16 months.
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