A lease valued at ?$23,000 requires payments of ?$2,011 at the beginning of every three months. If money is worth 10% compounded quarterly, what is the size of the final lease? payment?

The size of the final payment is ?$

.

?(Round the final answer to the nearest cent as needed. Round all intermediate values to six decimal places as? needed.)

 

We shall first use the debt repayment formula at beginning of period to find the number of payment or number of quarters.

Then we take the residual decimal and multiply with the periodic payment to determine the last payment.

The formula is;

PV = pmt(1-(1+i)^-n)x(1+i)/i

PV = 23000, pmt = 2011, i = 10%/4=0.1/4=0.025

Substituting we get;

23000 = 2011(1-(1+0.025)^-n)x(1+0.025)/0.025

23000 = 2011(1-(1.025)^-n)x(1.025)/0.025

23000 x 0.025/(2011x1.025) = 1 - (1.025)^-n

575/(2011x1.025) = 1 - (1.025)^-n

(1.025)^-n = 1 - [575/(2011x1.025]

taking log on both side to cause -n to drop according to the logarithms property logaⁿ = nloga

log(1.025)^-n = log(1 - [575/(2011x1.025])

-nlog(1.025) = log(1 - [575/(2011x1.025])

-n = log(1 - [575/(2011x1.025])/log(1.025)

n = -log(1 - [575/(2011x1.025])/log(1.025)

n = 13.244922

Residual is 0.244922

So final payment = 0.244922 x 2011

= 492.538142

= $492.54