Question

1. Derek decides to buy a new car. The dealership offers him a choice of paying $571.00 per month for 5 years (with the first payment due next month) or paying some amount today. He can borrow money from his bank to buy the car. The bank requires a 6.00% interest rate. What is the most that he would be willing to pay today rather than making the payments?

2. Derek plans to buy a $33,258.00 car. The dealership offers zero percent financing for 49.00 months with the first payment due at signing (today). Derek would be willing to pay for the car in full today if the dealership offers him $____ cash back. He can borrow money from his bank at an interest rate of 5.50%.

3. Derek borrows $31,725.00 to buy a car. He will make monthly payments for 6 years. The car loan has an interest rate of 5.53%. After a 14.00 months Derek decides to pay off his car loan. How much must he give the bank?

Answer 1

Part 1:

Max amount of car = PV of EMIs.

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
Cash Flow	$               571.00
Int Rate	0.5000%
Periods	60

PV of Annuity = Cash Flow * [ 1 - [(1+r)^-n]] /r
= $ 571 * [ 1 - [(1+0.005)^-60]] /0.005
= $ 571 * [ 1 - [(1.005)^-60]] /0.005
= $ 571 * [ 1 - [0.7414]] /0.005
= $ 571 * [0.2586]] /0.005
= $ 29535.3
Max amount that can be paid for car today is $ 29535.30

Part 2:

Instalment = Price / No. of Instalments

= 33258 / 50

= $ 665.16

Max Price that can be paid with 5.5%:

PV of Annuity Due:

Annuity is series of cash flows that are deposited at regular intervals for specific period of time. Here cash flows are happened at the begining 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 Due = Cash Flow + [ Cash Flow * [ 1 - [(1+r)^-(n-1)]] /r ]
r - Int rate per period
n - No. of periods

Particulars	Amount
Cash Flow	$               665.16
Int Rate	0.458%
Periods	50

PV of Annuity Due = [ Cash Flow + Cash Flow * [ 1 - [(1+r)^-(n-1)]] / r ]
= [ $ 665.16 + $ 665.16 * [ 1 - [(1+0.0046)^-49] ] / 0.0046 ]
= [ $ 665.16 + $ 665.16 * [ 1 - [(1.0046)^-49] ] / 0.0046 ]
= [ $ 665.16 + $ 665.16 * [ 1 - [0.7993] ] / 0.0046 ]
= [ $ 665.16 + $ 665.16 * [0.2007] ] / 0.0046 ]
= [ $ 665.16 + $ 29132.7 ]
= $ 29797.86

Cash back to be offered = Price - PV of annuity due

= $ 33258 - $ 29797.86

= $ 3460.14

Part 3:

EMI :
EMI or Instalment is sum of money due as one of several equal payments for loan/ Mortgage taken today, spread over an agreed period of time.

EMI = Loan / PVAF (r%, n)
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

Particulars	Amount
Loan Amount	$             31,725.00
Int rate per Month	0.4608%
No. of Months	72

EMI = Loan Amount / PVAF (r%, n)
Where r is Int rate per Month & n is No. of Months
= $ 31725 / PVAF (0.0046 , 72)
= $ 31725 / 61.1549
= $ 518.76

Loan Outstanding after 14 Months:

Particulars	Amount
Loan Amount	$    31,725.00
Int rate per Month	0.4608%
No. of Months	72
Outstanding Bal after	14
EMI	$         518.76
Payments Left	58

Outstanding Bal = Instalment * [ 1 - ( 1 + r )^ - n ] / r
= $ 518.76 * [ 1 - ( 1 + 0.004608 ) ^ - 58 ] / 0.004608
= $ 518.76 * [ 1 - ( 1.004608 ) ^ - 58 ] / 0.004608
= $ 518.76 * [ 1 - 0.765926 ] / 0.004608
= $ 518.76 * [ 0.234074 ] / 0.004608
= $ 26351.61

r = Int Rate per period
n = Balance No. of periods

Loan oustanding after 14 Months is $ 26351.61