Question

A firm would like to finance an investment by taking out a loan that requires 10 fixed annual payments, with the first payment due in one year. The bank will require a return of 6.5% on this loan. The firm plans to borrow $450,000 using this loan. [Use annuity formulas to answer the questions below.]

a. What will the firm’s annual payments be?

b. Check your calculation by computing the present value of those annual payments, using the rate of return given above. Is it equal to the loan amount?

c. What would the first annual payment be, if the firm wanted the annual payments to grow at a rate of 4% each year?

d. What would the last payment be?

Answer 1

Part A:

Particulars	Amount
Loan Amount	$         4,50,000.00
Int rate per Anum	6.5000%
No. of Years	10

Annual Instalemnt = Loan Amount / PVAF (r%, n)
Where r is Int rate per Anum & n is No. of Years
= $ 450000 / PVAF (0.065 , 10)
= $ 450000 / 7.1888
= $ 62597.11

PVAF = SUm [ PVF(r%, n) ]
PVF(r%, n) = 1 / ( 1 + r)^n
r = Int rate per period
n = No. of periods

How to calculate PVAF using Excel:
=PV(Rate,NPER,-1)
Rate = Disc Rate
NPER = No.of periods

Part B:

Loan AMount = PV of EMIs

PV of Annuity:

Annuity is series of cash flows that are deposited at regular intervals for specific period of time.

PV of Annuity = Cash Flow * [ 1 - [(1+r)^-n]] /r
r - Int rate per period
n - No. of periods

Particulars	Amount
Cash Flow	$          62,597.11
Int Rate	6.5000%
Periods	10

PV of Annuity = Cash Flow * [ 1 - [(1+r)^-n]] /r
= $ 62597.11 * [ 1 - [(1+0.065)^-10]] /0.065
= $ 62597.11 * [ 1 - [(1.065)^-10]] /0.065
= $ 62597.11 * [ 1 - [0.5327]] /0.065
= $ 62597.11 * [0.4673]] /0.065
= $ 450000

Part C:

Loan = [ CF1 / ( r - g ) ] * [ 1 - [ ( 1 + g ) / ( 1 + r ) ] ^ n ]
450000 = CF1 / ( 6.5% - 4% ) ] * [ 1 - [ ( 1 + 0.04 ) / ( 1 + 0.065) ] ^ 10 ]
450000 = CF1 / (2.5% ) ] * [ 1 - [ ( 1.04 ) / ( 1.065) ] ^ 10 ]
450000 = CF1 / (2.5% ) ] * [ 1 - [ 0.9765 ] ^ 10 ]
450000 = CF1 / (2.5% ) ] * [ 1 - 0.7886 ]
450000 = CF1 / (2.5% ) ] * [ 0.2114 ]
CF1 = 450000 * 2.5% / 0.2114
CF1 = $ 53207.76

r = Int rate

g = Growth rat

Part D:

10th Payment = CF1 * ( 1 + g )^9

= $ 53207.76 ( 1 + 0.04 )^ 9

=$ 53207.76 * (1.04^9)

= $ 53207.76 * 1.4233

= 75731.23