Question

The following situations require the application of the time value of money:

Use the appropriate present or future value table:

FV of $1, PV of $1, FV of Annuity of $1 and PV of Annuity of $1

1. On January 1, 2017, $16,000 is deposited. Assuming an 8% interest rate, calculate the amount accumulated on January 1, 2022, if interest is compounded (a) annually, (b) semiannually, and (c) quarterly. Round your answers to the nearest dollar.

		Future Value
	a. Annual compounding	$
	b. Semiannual compounding	$
	c. Quarterly compounding	$

2. Assume that a deposit made on January 1, 2017, earns 8% interest. The deposit plus interest accumulated to $20,000 on January 1, 2022. How much was invested on January 1, 2017, if interest was compounded (a) annually, (b) semiannually, and (c) quarterly? Round your answers to the nearest dollar.

		Present Value
	a. Annual compounding	$
	b. Semiannual compounding	$
	c. Quarterly compounding	$

3.

Steve Jones has decided to start saving for his son's college education by depositing $2,000 at the end of every year for 15 years. A bank has agreed to pay interest at the rate of 4% compounded annually.

Use the appropriate present or future value table:

FV of $1, PV of $1, FV of Annuity of $1 and PV of Annuity of $1

Required:

How much will Steve have in the bank immediately after his 15th deposit? Round your answer to the nearest dollar. Use the full factor when calculating your results.
$

Answer 1

1

The formula for computing Future value of compound interest = FV = PV( 1+ r / t ) ^tn
Where, FV = Future Value of deposit
PV= Principal amount
r = Rate of interest
n = No. of years / periods.
t = No. of times compounded
Now, the above function is for annual/semi annual/quarterly/weekly compounding, for example if it is compounded semi annually (then 't' can be taken as 2 times in a year) and quarterly (then 't' can be taken as 4 times in a year compounding),

Particulars	No. of times compounded	Future Value
a) Annual compounding	1 (Since it is annual, therefore compounding done only once)	FV = $16000 X (1 + 0.08/1 ) ^ 1 x 5 = $23509
b) semi annual compounding	2 (Since it is semi annual, therefore compounding done 2 times after every 6 months)	FV = $16000 X (1 + 0.08/2) ^ 2 x 5 =$23684
c) Quarterly compounding	4 (Since it is quarter, therefore it is done 4 times after every 3 months)	FV = $16000 X (1 + 0.08/4) ^ 4 X 5 =$23775

where Principal amount = $16000
Rate of interest = 8 % or 0.08
Time = 5 years ( Jan 1, 2017 To Jan 1, 2022)

2

The formula for computing Present value of compound interest= PV = FV( 1+ r / t ) ^-tn
Where, FV = Future Value of deposit
PV= Principal amount
r = Rate of interest
n = No. of years / periods.
t = No. of times compounded
It is the same as above formula of FUTURE VALUE, the only difference is that 'tn' becomes negative '-' due to cross multiplication

Particulars	Future Value
a) annual compounding	PV = $20000 X (1+ 0.08/1) ^ -(1 x 5) = $13612
b) semi annual compounding	PV = $20000 X (1+ 0.08/2) ^ -(2 x 5) = $13511
c) Quarterly compounding	PV = $20000 X (1+ 0.08/4) ^ -(4 x 5) = $13459

where Future Value = $20000
Rate of interest = 8 % or 0.08
Time = 5 years ( Jan 1, 2017 To Jan 1, 2022)

3

If steve starts depositing $2000 every year's end for 15 years in bank giving 4% interest compounded annually
Then immediately after last deposit, the balance will be as follows
FV = [ P X {(1+r)^n - 1} / r ] + $2000
where, FV = Future value
P = Periodic deposit amount = $2000
r = rate of interest = 4% or 0.04
n = No. of years = 14 years
(There reason we are not taking n= 15 years is because 1st deposit was made at year end, in last deposit only previous 14 deposits would have gotten annual compounding, the final deposit will be added as it is to derive the bank balance immediately after his 15th deposit)

FV =[$2000 x { (1+0.04)^14 - 1} / 0.04] + $2000
Hence, bank balance after 15th deposit = $36584 + $2000 => $38584 (Ans)