Question

​ (Future value)  You are hoping to buy a house in the future and recently received an inheritance of ​$18000. You intend to use your inheritance as a down payment on your house.

a.  If you put your inheritance in an account that earns 7 percent interest compounded​ annually, how many years will it be before your inheritance grows to ​$31000​?

b.  If you let your money grow for 9.75 years at 7 percent​, how much will you​ have?

c.  How long will it take your money to grow to ​$31000 if you move it into an account that pays 5 percent compounded​ annually? How long will it take your money to grow to ​$31000 if you move it into an account that pays 13 percent​?

d.  What does all this tell you about the relationship among interest​ rates, time, and future​ sums?

Answer 1

a.

Required number of periods can be computed using relation of PV and FV as:

PV = FV/(1+r) n

PV = Present amount = $ 18,000

FV = Future amount = $ 31,000

r = Periodic rate = 0.07

n = Number of periods

$ 18,000 = $ 31,000/ (1+0.07) n

$ 18,000 = $ 31,000/ (1.07) n

(1.07) n = $ 31,000/ $ 18,000

(1.07) n = 1.7222222222

Taking log of both sides and solving for n, we get:

n x log 1.07 = log 1.7222222222

n x 0.029383777685 = 0.23608918873

n = 0.23608918873/0.029383777685 = 8.0346778845431 or 8.03 periods

It will take 8.03 years to grow the inheritance to $ 31,000

b.

FV = PV x (1+r) n

PV = $ 18,000; r = 0.07; n = 9.75

FV = $ 18,000 x (1+0.07)9.75

    = $ 18,000 x (1.07)9.75

   = $ 18,000 x 1.93415748349984

   = $ 34,814.8347029972 or $ 34,814.83

The fund will grow to $ 34,814.83 in 9.75 years

c)

PV = FV/(1+r) n

PV = $ 18,000; FV = $ 31,000; r = 5 %;

$ 18,000 = $ 31,000/ (1+0.05) n

$ 18,000 = $ 31,000/ (1.05) n

(1.05) n = $ 31,000/ $ 18,000

(1.05) n = 1.7222222222

Taking log of both sides and solving for n, we get:

n x log 1.05 = log 1.7222222222

n x 0.02118929907 = 0.23608918873

n = 0.23608918873/ 0.02118929907 = 11.1419064854419 or 11.14 periods

It will take 11.14 years to grow the inheritance to $ 31,000 at interest rate of 5 %

r = 13%;

$ 18,000 = $ 31,000/ (1+0.13) n

$ 18,000 = $ 31,000/ (1.13) n

(1.13) n = $ 31,000/ $ 18,000

(1.13) n = 1.7222222222

Taking log of both sides and solving for n, we get:

n x log 1.13 = log 1.7222222222

n x 0.053078443483 = 0.23608918873

n = 0.23608918873/ 0.053078443483 = 4.44792976654665 or 4.45 periods

It will take 4.45 years to grow the inheritance to $ 31,000 at interest rate of 13 %

d.

If interest rate increases, less time required to grow the investment and vice-versa.

Hence interest rate and time of investment is directly proportional to the future sum.