Question

Suppose you have $250,000 of loan. The terms of the loan are that the yearly interest is 6% compounded quarterly. You are to make equal quarterly payments of such magnitude as to repay this loan over 30 years.

(Keep all your answers to 2 decimal places, e.g. XX.12.)

(a) How much are the quarterly payments?

(b)  After 5 years' payments, what principal remains to be paid?

(c) How much interest is paid in the first quarter of the 6^th year?

(d) How much is the total interest paid over the 30 years?

(e) If you have a lump sum payment of $20,000 at the end of 5 years, and maintain the same level of quarterly payment, when will you pay off your loan, i.e. how many years in total will you pay off the loan?

Answer 1

(a) How much are the quarterly payments?

Present value of the loan= 250000

Interest ,r=6% p.a. ie. 1.5% per quarter or 0.015

n= 30 yrs. =30*4=120 quarterly compounding periods

Now,

Using PV of ordinary,qtrly.period-end annuity formula,

PV=Quarterly Pmt.*(1-(1+r)^-n)/r

& substituting the known values,

250000=Q.pmt.*(1-1.015^-120)/0.015

Solving the above, we get the quarterly pmt. As

4504.63

(b) After 5 years' payments, what principal remains to be paid?

Formula to be used:

FV=(PV*(1+r)^n)-(P*(((1+r)^n-1)/r))

where,

FV=Future value of the Remaining balance----??

PV=Present value of the original balance-- 250000

P=periodic pmt,ie. Here,the Quarterly pmt.--4504.63

r= interest rate/pmt.,ie.0.015/qtr.

n= no.of compounding periods, here, 5yrs.*4=20

Now, putting in the values,

FV=(250000*(1+0.015)^20)-(4504.63*((1+0.015)^20-1)/0.015))

232550.19

So,After 5 years' payments, what principal remains to be paid=

232550.19

(c) How much interest is paid in the first quarter of the 6th year?

Principal balance as at end of the 5th year (from b above) is

232550.19

So,interest is paid in the first quarter of the 6th year=

232550.19*1.5%=

3488.25

(Answer)

(d) How much is the total interest paid over the 30 years?

Total quarterly payments= 4504.63*120=

540555.60

PV of the loan= 250000

So, total interest paid=

540555.60-250000=

290555.60

e. If a lumpsum of $ 20000 is paid at end of 5 yrs.

the principal balance becomes

232550.19-20000=

212550.19

& if we

maintain the same level of quarterly payment,

we need to use the same formula & find n

212550.19=4504.63*(1-1.015^-n)/0.015

Solving for n in an online equation solver,we get

n= 82.63

quarters

ie.82.63/4=

20.66

Yrs.

ie.

Total no.of yrs. To pay off the loan = 5+20.66=25.66 or 26 yrs.