Question 1a) Find the present value of $500 due in 3 years at 8.75% p.a. compounded quarterly.b) Find the future value of $2,000 invested for 5 years at 5.85 % p.a. compounded daily.c) Find the future value of $2,000 invested for 3 years at 5.85% p.a., compounded dailythen at 5.5% p.a., compounded continuously for the next 2 years. No furthercompounding after year 5.Note: round your answers to 2 decimal places.
Question
Question 1a) Find the present value of 2,000 invested for 5 years at 5.85 % p.a. compounded daily.c) Find the future value of $2,000 invested for 3 years at 5.85% p.a., compounded dailythen at 5.5% p.a., compounded continuously for the next 2 years. No furthercompounding after year 5.Note: round your answers to 2 decimal places.
Solution
a) To find the present value, we use the formula for the present value of a future sum:
PV = FV / (1 + r/n)^(nt)
where: FV = future value = $500 r = annual interest rate = 8.75% = 0.0875 n = number of times interest is compounded per year = 4 (quarterly) t = number of years = 3
So,
PV = 500 / (1 + 0.0875/4)^(4*3) PV = 500 / (1 + 0.021875)^(12) PV = 500 / 1.28748 PV = $388.48 (rounded to 2 decimal places)
b) To find the future value, we use the formula for the future value of an investment:
FV = PV * (1 + r/n)^(nt)
where: PV = present value = $2000 r = annual interest rate = 5.85% = 0.0585 n = number of times interest is compounded per year = 365 (daily) t = number of years = 5
So,
FV = 2000 * (1 + 0.0585/365)^(365*5) FV = 2000 * 1.31686 FV = $2633.72 (rounded to 2 decimal places)
c) For the first 3 years, we use the same formula as in part b. Then, for the next 2 years, we use the formula for continuous compounding:
FV = PV * e^(rt)
where: PV = present value = $2000 r = annual interest rate = 5.5% = 0.055 t = number of years = 2 e = Euler's number, approximately 2.71828
So,
FV = 2000 * (1 + 0.0585/365)^(365*3) FV = 2000 * 1.18723 FV = $2374.46 (rounded to 2 decimal places)
Then,
FV = 2374.46 * e^(0.055*2) FV = 2374.46 * 1.11352 FV = $2643.96 (rounded to 2 decimal places)
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