Last question -Business Mathematics: Amortization of Loans

[Ivanthedumbfck]

1. A lease agreement valued at $33,000 requires payment of $4,300 every three months in advance. The payments are deferred for three years and money is worth 10% compounded quarterly.
   

a.) How many lease payments are to be made under the contract?
b.) What is the size of the final lease payment?


I got A.)

n1= 3x4=12
n2=?

I=0.10/4=0.025

FV(defer)= 33000(1+0.025)^12=44381.33
PV(new)= PMT[1-(1+i)^-n]/i

44381.33=4300[1-(1+0.025)^-n2]/0.025
n2= 12 quarter payments

However, I messed up on b.) and i need to get some coffee to keep me awake .

b.) FV(original debt)=PV(1+I)^N
=44381.33(1+0.025)^12= 59687.95

FV(Payments Made)= PMT[(1+i)^n-1]/i
4500[(1+0.025)^12-1]/0.025
=59320.88
Final lease payment= FV(original debt)- FV(Payments Made)=367.07

Correct answer is 3211.27

 

[Ishuda]

You have to watch out for when the payments start.

If they start immediately, as in this problem, then you need to subtract 1 from the computed factor and make it for a pay period less. That is, when starting payments immediately, for the second payment you have interest on Principle minus payment: Letting P be principle (the $44381.33), p be payment (the $4300), and x = 1 plus interest (the 1.05) you have a balance owing after the nth payment
P x^n-1 - p (1 + x + x² + ... + x^n-1) = P x^n-1 - p x^n-1 - p ( 1 + x + x2 + ... + x^n-2)
= P x^n-1 - p x^n-1 - p \(\displaystyle \frac{x^{n-1} - 1}{x - 1}\) = x^n-1 { P - p [\(\displaystyle \frac{1 - x^{-(n-1)}}{x - 1}\) - 1]}

Whereas, if you made the first payment at the end of the first period you have interest on the complete balance for the initial period and a balance owing after the nth payment of
P xⁿ - p (1 + x + x² + x^n-1) = P xⁿ - p \(\displaystyle \frac{x^n - 1}{x - 1}\) = xⁿ {P - p [\(\displaystyle \frac{1 - x^{-n} }{x - 1}\)]}

Setting the balance to zero gives two different formulas, the one for starting immediate payments
Immediate: P = p [\(\displaystyle \frac{1 - x^{-(n-1)}}{x - 1}\) - 1]
and the other for starting payment at the end of the first period
First Period: P = p [\(\displaystyle \frac{1 - x^{-n} }{x - 1}\)]

 

[Ivanthedumbfck]

Hey guys, sry about replying so late. Man, last 2 weeks, i was so busy that i forgot to shave. I have a brown Santa Clause beard right now. I managed to solve the question, here is my solution:

1. A lease agreement valued at $33,000 requires payment of $4,300 every three months in advance. The payments are deferred for three years and money is worth 10% compounded quarterly.
   

a.) How many lease payments are to be made under the contract?
b.) What is the size of the final lease payment?

a.)
n1= 3x4=12
n2=?

I=0.10/4=0.025

FV(defer)= 33000(1+0.025)^12=44381.33
PV(new)= PMT[1-(1+i)^-n]/i

44381.33=4300[1-(1+0.025)^-n2]/0.025
n2= 12 quarter payments

b.)

FV(Original Debt)= 44381.33(1.025)^(12-1)=58232.15
FV(Payment Made)= 4300[(1.025^11-1)]/0.025 x (1.025)= 55020.878
Final Payment= 58232.15- 55020.878= 3211.27