Question 1195213: A debtor owes Ryan the following sums, due without interest: $1000 due at the end of 2 years and $2000 due at the end of 5 years. What equivalent single payment would Ryan be willing to accept at the end of 4 years, if money is worth {8%, m=4} to him?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! not sure what you mean by m = 4.
it could be that the money is compounded 4 times a year.
i'll do both compounded once a year and compounded 4 times a year to show you what the difference is and how to calculate both ways.
the formula used is f = p * (1 + r) ^ n
f is the future value
p is the present value
r is the interest rate per time period (years in this example).
n is the number of time periods (years in this example).
the easiest way to do this is to get the present value of both existing payments at 2 years and 5 years and then get the future value for the sum of that present value at 4 years.
the result will be as shown below:
first calculation:
1000 = p * 1.08 ^ 2
solve for p to get:
p = 1000 / (1.08 ^ 2) = 857.3388203.
second calculation:
2000 = p * 1.08 ^ 5
solve for p to get:
p = 2000 / (1.08 ^ 5) = 1361.166394
sum of first p and second p = 2218.505214.
third calculation:
f = 2218.505214 * 1.08 ^ 4 = 3018.251852.
payments of 1000 in 2 years and 2000 in 5 years are equivalent to 3018.251852 in 4 years.
that would be your solution, assuming money is compounded once a year.
if you compound 4 times a year, then the procedure is the same, except that you would divide the interest rate per year by 4 to get the interest rate per quarterly time period and you would multiply the number of years by 4 to get the number of quarterly time periods.
your first calculation would become:
p = 1000 / 1.02 ^ 8 = 853.4903712.
your second calculation would become:
p = 2000 / 1.02 ^ 20 = 1345.942666.
the sum of those present values would become equal to 2199.433037..
your third calculation would become:
f = 2199.433037 * 1.02 ^ 16 = 3019.350233.
note that:
the present value of something for 2 years and then getting the future value of that present value for 4 years is equivalent to getting the future value of that something for 2 years for an additional 2 years.
the equivalent calculations are shown below:
calculation 1:
p = 2000 / 1.08 ^ 2 = 1714.677641 * 1.08 ^ 4 = 2332.8.
calculation 2:
f = 2000 * 1.08 ^ 2 = 2332.8.
let me know if you have any questions.
theo
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