Calculating Payment

[lgubarik]

A businessman receives a $15,000 loan on Jan 1st. He is charged a 6% interest per year. His loan is for three years. He is expected to make monthly payments except for every April. Interest is still accruing even though no payment is made in April. What are his monthly payments?

This question confuses me because if I use the standard formula to calculate payments it doesn't account for the fact that interest is still accruing even though no payment is made every April. monthly payment= [rate+ (rate/((1+rate)^months)-1) - principle. If i put 36 months, the calculation assumes I make every payment, if I put 33 payments(which is skipping every April) it doesn't calculate the interest for April.

 

[lgubarik]

By the way I tried to answer it using my financial calculator by putting PV=15000 Fv=0 I/Y=.5 N=36 and the payment comes out=456.33. But that is the wrong answer as it doesn't account for the fact that 3 payments are being missed.

 

[Denis]

Since you seem to be learning this stuff, I wonder why you're given such a tough one.

I'll do my best to explain the "how"; keep in mind this isn't a classroom.

First, I'm assuming a loan of $15,000 with the 1st payment due 1 month after loan is made.
So 3 payments are made, 4th skipped, 11 payments are made, 16th skipped,
11 payments are made, 28th skipped, finally 8 payments are made.

Easiest way (if there is an easiest way!) is treat this as 2 accounts:
1: 15,000 deposited and earning interest at 6% annual compounded monthly for 36 months:
this will accumulate to 15000(1.005^36) = ~17,950.21
2: in another account, the payments are deposited when made, same interest rate:
this account must end up with the above ~17,950.21

The 2nd account then can be "looked at" this way (p = payment amount):
1: FV of 1st 3 payments: p(1.005^3 - 1) / .005; this will then accumulate
for 33 months: p(1.005^3 - 1)(1.005^33) / .005 : let a = (1.005^3 - 1)(1.005^33) / .005

2: FV of next 11 payments: p(1.005^11 - 1) / .005; this will then accumulate
for 21 months: p(1.005^11 - 1)(1.005^21) / .005 : let b = (1.005^11 - 1)(1.005^21) / .005

3: FV of next 11 payments: p(1.005^11 - 1) / .005; this will then accumulate
for 9 months: p(1.005^11 - 1)(1.005^9) / .005 : let c = (1.005^11 - 1)(1.005^9) / .005

4: FV of last 8 payments: p(1.005^8 - 1) / .005 : let d = (1.005^8 - 1) / .005

OK; then pa + pb + pc + pd = ~17,950.21
p(a + b + c + d) = ~17,950.21
p = ~17,950.21 / (a + b + c + d) = ~498.38

That payment is correct: I checked it by doing an actual amortization schedule.

If you're told the payment is different, then it has to do with the beginning of the loan:
the fact that the dates are the 1st of the months may mean something different
from what I assumed.

If the payment formula I'm using is Greek to you, then you're really not ready
for this type of stuff; or your teacher "forgot?" to bring it up; or you missed classes ????? :wink:

 

[Denis]

For your "education", schedule will look like:

Month Payment  Interest  Balance 
 0                       15000.00
 1   -498.38    75.00    14576.62
 2   -498.38    72.88    14151.12
 3   -498.38    70.76    13723.50
 4       .00    68.62    13792.12 (months 16 and 28 will be similar)
 5   -498.38    68.96    13362.70
.....
 35  -498.38     4.94      495.90
 36  -498.38     2.48         .00