Question

  You are hoping to buy a house in the future and recently received an inheritance of ​$24,000. You intend to use your inheritance as a down payment on your house.

a.  If you put your inheritance in an account that earns 9 percent interest compounded​ annually, how many years will it be before your inheritance grows to ​$31,000​?

b.  If you let your money grow for 9.5 years at 9 percent​, how much will you​ have?

c.  How long will it take your money to grow to ​$31,000 if you move it into an account that pays 4 percent compounded​ annually? How long will it take your money to grow to ​$31,000 if you move it into an account that pays 12 percent​?

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

Answer 1

Part A:

Future Value:
FV = PV (1+r)^n
Where r is Int rate per period
n - No. of periods

Particulars	Amount
Present Value	$    24,000.00
Future Value	$    31,000.00
Int Rate per period	9.0000%

Future Value = Cash Flow * ( 1 + r )^n
$ 31000 = $ 24000 ( 1 + 0.09 ) ^ n
( 1 + 0.09 ) ^ n = $ 31000 / $ 24000
1.09 ^ n = 1.2917
Take Log on Both sides
Log ( 1.09 ^ n ) = Log ( 1.2917 )
Log ( a^b ) = b * Log (a)
n * Log ( 1.09 ) = Log ( 1.2917 )
n * 0.0374 = 0.1112
n = 2.9698 i.e Almost 3 Years

Part B:

Particulars	Amount
Present Value	$    24,000.00
Int Rate	9.0000%
Periods	9

Future Value = Present Value * ( 1 + r )^n
= $ 24000 ( 1 + 0.09) ^ 9
= $ 24000 ( 1.09 ^ 9)
= $ 24000 * 2.1719
= $ 52125.44
Amount after 9.5 Years = 52125.44 * 1.045

= $ 54471.09

Part C1:

Particulars	Amount
Present Value	$    24,000.00
Future Value	$    31,000.00
Int Rate per period	4.0000%

Future Value = Cash Flow * ( 1 + r )^n
$ 31000 = $ 24000 ( 1 + 0.04 ) ^ n
( 1 + 0.04 ) ^ n = $ 31000 / $ 24000
1.04 ^ n = 1.2917
Take Log on Both sides
Log ( 1.04 ^ n ) = Log ( 1.2917 )
Log ( a^b ) = b * Log (a)
n * Log ( 1.04 ) = Log ( 1.2917 )
n * 0.017 = 0.1112
n = 6.5255

Part C2 :

Particulars	Amount
Present Value	$    24,000.00
Future Value	$    31,000.00
Int Rate per period	12.0000%

Future Value = Cash Flow * ( 1 + r )^n
$ 31000 = $ 24000 ( 1 + 0.12 ) ^ n
( 1 + 0.12 ) ^ n = $ 31000 / $ 24000
1.12 ^ n = 1.2917
Take Log on Both sides
Log ( 1.12 ^ n ) = Log ( 1.2917 )
Log ( a^b ) = b * Log (a)
n * Log ( 1.12 ) = Log ( 1.2917 )
n * 0.0492 = 0.1112
n = 2.2583

Part D:

FV = PV ( 1 + r)^n

Thus There is direct relation between FV & Int Rate and TIme Period.

I.e as the time period & int rate is increased, FV will also increase.