How much to repay the loan?

SydneyRoverP6B

SydneyRoverP6B

Well-Known Member
Staff member
A person borrows $5000 at an interest rate of 18% where the interest is compounded monthly on the balance owing. The loan is to be repaid in equal monthly instalments. What should the amount of each instalment be, in order to pay off the loan in 4 years?

Ron.
 
SydneyRoverP6B said:
A person borrows $5000 at an interest rate of 18% where the interest is compounded monthly on the balance owing. The loan is to be repaid in equal monthly instalments. What should the amount of each instalment be, in order to pay off the loan in 4 years?

Ron.
Click to expand...

Exorbitant! :p
 
Hello Richard,

Your answer of $144.70 is unfortunately not correct, but you are close. Can you please explain what you did to achieve that figure?

I agree the interest rate is excessive based on the figures of today.

Ron.
 
I just put it into a spread sheet.

Used 5,000 less the payment (a) then added (18/12)% to the remainder.

Then drop the answer down to the next row under the 5,000 and start again

The spread sheet does it instantly and you can just change the value of 'a' until you have nothing left at the end of 48 paymets - took about 5 minutes, but I was left with a 31p. £144.71 overpays by 39p

Richard
E1
5000 =A1-E$1 =B1+(B1*0.015) a
=C1 =A2-E$1 =B2+(B2*0.015)

Now copy line 2 down to line 48 and you are left with the results above.

Richard


$144.704435 is pretty close
 
I got 146.87...

based on 5000*1.015^48 = x* (1.015^47+1.015^46+1.015^45....etc)

x is the repayment amount

Rich.
 
Hello Richard,

I did mine all by hand,...Let the monthly instalment be $P and the amount owing after n months be $A(n)

So after 1 month,...A(1) = 5000 X 1.015 - P
after 2 months..A(2) = A(1) X 1.015 - P
= (5000 X 1.015 - P)1.015 - P
= 5000 X 1.015^2 - 1.015P - P
= 5000 X 1.015^2 - P(1 + 1.015)
after 3 months..A(3) = A(2) X 1.015 - P
= 5000 X 1.015^3 - P(1 + 1.015 + 1.015^2)
the pattern is now clear,....

so after 48 months..A(48) = 5000 X 1.015^48 - P(1 + 1.015 + 1.015^2 +...+ 1.015^47)

The expression within the parentheses is a geometric series with first term a = 1, common ratio r = 1.015 and n, the number of terms = 48.

Re arranging the expression with the sum of the geometric series expressed in closed form and then solving for P gives,...$P = 5000 X 1.015^48 X 0.015/(1.015^48 - 1)
So the monthly payment is $146.87.

I think where you may have gone astray is that you need to add the interest prior to making the first payment. If you do yours again and make this change, I expect you will obtain the same answer as I have.

Ron.
 
Hello Rich,

Yes indeed....give that man a cookie.. :D Well done!! and well done to you too Richard... :D

Ron.
 
Hello Rich,

You did indeed...glad you liked it... :D

I saw an interview just recently with Baroness Professor Susan Greenfield, and she spoke about how vitally important it is for all of us to use our brains, so rest assured I shall keep the problems coming!

Ron.
 
SydneyRoverP6B said:
I think where you may have gone astray is that you need to add the interest prior to making the first payment. If you do yours again and make this change, I expect you will obtain the same answer as I have.

Ron.
Click to expand...

Yes, that works 8)

I didn't know to do that as I try not to borrow any money - The only way to borrow money from a bank is to categorically prove that you don't need to borrow it!
 
Darn it, you cheats got before me! :p

$5000, with an annual interest of a <reasonable> rate of 18%... which you all seem to agree is (18/12)% per month, 18/12 = 1.5, so 1.5% interest per month.

the first month: $5000 + .015 x $5000 - $P = 1.015 x $5000 - P

the second month: 1.015 x (1.015 x $5000 - P) - P = (1.015^2) x $5000 - P x (1 + 1.015)

the third month: 1.015 x (1.015 x (1.015 x $5000 - P) - P) - P = (1.015^3) x $5000 - P x (1 + 1.015 + 1.015^2)

the fourth month: 1.015 x (1.015 x (1.015 x (1.015 x $5000 - P) - P) - P) - P = (1.015^2) x $5000 - P x (1 + 1.015 + 1.015^2 + 1.015^3)

the nth month: (1.015^n) x $5000 - P x (1 + 1.015 + 1.015^2 + 1.015^3 + ..... + 1.015^(n-1))

(1 + 1.015 + 1.015^2 + 1.015^3 + ..... + 1.015^(n-1)) = (1.015^n - 1) / (1.015 - 1) = (1.015^n - 1) / 0.015 = (200/3) x (1.015^n - 1)

n = 48, so 1.015 ^ 48 = 2.0434782893129897282153675440472

((1.015 ^ 48) - 1) = 1.0434782893129897282153675440472

(200/3) x ((1.015 ^ 48) - 1) = 69.565219287532648547691169603145

so,...

the 48th month,

(1.015^n) x $5000 - P x (1 + 1.015 + 1.015^2 + 1.015^3 + ..... + 1.015^(n-1)) = 2.0434782893129897282153675440472 x 5000 - 69.565219287532648547691169603145 x P

At 48 months, the amount needed is zero, so...

2.0434782893129897282153675440472 x 5000 - 69.565219287532648547691169603145 x P = 0

OR...

P = (2.0434782893129897282153675440472 x 5000) / 69.565219287532648547691169603145

P = 146.87499804081104738512338666692

Rounded off to $146.87.

However, I would rather pay $146.88, to ensure the debt is completely clear!

Good problem! :)

(Are you sure that it's as simple as dividing by 12 to get the compound interest per month?.. I've a gut feeling it's not that simple in real borrowing!)
 
Hello Darth,

Well done,... :D glad that you liked the problem too!!

I am pretty sure that if given an annual compound interest rate, and then a monthly or quarterly rate is required, then just divide accordingly and of course the number of periods must also be adjusted to reflect the change. In many instances though, the % rates will be different for different time periods, so a monthly rate may not be just as simple as dividing an annual rate by 12. In such a case performing some calculations before taking out a loan etc will enable the person to make an informed decision as to which product suits their financial situation most appropriately.

Ron.
 
My brain hurt when i started trying to think about thinking about it, so i gave up and settled on an answer of "$Lots"... :)

Definately an interesting one though, amazing how many people have no idea of what they will actually end up paying in terms of interest rates on loans. I was talking to a friend of my youngest sister a while back who thought that when she signed up for a car loan over 5 years at 13% that the 13% was a total across the whole period, not monthly compound. Seemed quite shocked at what she would actually end up paying over the life of the loan. I was more amazed she had signed up for something with no idea what it actually meant, just han't read the paperwork properly or bothered to think about it....

Definately good to have these things to work the brain on from time to time (even if i gave up).

Cheers,

Al

ps. I presume you're the same Ron who posts similar porblems on the Aulro LandRover forums? If so howdy from there too, i'm ariddell on those forums.
 
Hello Al,

Glad that you enjoyed the problem,..and yes that is me over at Aulro,...so hello from me over there too... :D

Ron.
 
You must log in or register to reply here.
Back
Top