Question

Amortization schedule with periodic payments.

  Moulton Motors is advertising the following deal on a used Honda​ Accord: ​ "Monthly payments of ​$233.92 for the next 48 months and this beauty can be​ yours!" The sticker price of the car is $9,400. If you bought the​ car, what interest rate would you be paying in both APR and EAR​terms? What is the amortization schedule of the first six​ payments?

Answer 1

PV of Annuity:

Annuity is series of cash flows that are deposited at regular intervals for specific period of time. Here cash flows are happened at the end of the period. PV of annuity is current value of cash flows to be received at regular intervals discounted at specified int rate or discount rate to current date.

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

Particulars	Amount
PV Annuity	$         9,400.00
Time Period	48.00
Cash Flow	$            233.92

PV of Annuity = Cash flow * PVAF(r%, n)
PVAF(r%, n ) = PV of Annuity / Cash Flow
= $ 9400 / $ 233.92
= 40.1847

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

The Rate at which PVAF for 48 Periods will be equal to 40.1847 will be the answer.
PVAF(0.75%48) = 40.1848
PVAF(0.8%48) = 39.7284

Required Rate = 0.75 % + [ [ 40.1848 - 40.1847 ] / [ 40.1848 - 39.7284 ] ] * 0.05 %
= 0.75 % + [ [ 0.0001 ] / [ 0.4564 ] ] * 0.05 %
= 0.75 % + [ 0.0002 ] * 0.05 %
= 0.75 % + 0.00001 %
= 0.75001 %
APR = Monthly Rate * 12

= 0.75% * 12

= 9%

Effective Annual Rate = ( 1 + r ) ^ n - 1
r = Int Rate per period
n = No.of periods per anum

Particulars	Amount
Ret period	0.7500%
No. of periods	12.0000

EAR = [ ( 1 + r ) ^ n ] - 1
= [ ( 1 + 0.0075 ) ^ 12 ] - 1
= [ ( 1.0075 ) ^ 12 ] - 1
= [ 1.0938 ] - 1
= 0.0938
I.e EAR is 9.38 %