# calulating interest and principal components

#### missl

##### New member
Hi, I need some help. I am looking at this problem on the text book. Where the questions says:

Given:
9500 personal loan at 4.5% compounded monthly is to be repaid over a four year term by equal monthly payment .

{9500*0.625} / {1-1/+0.625)^48} =229.70 PMT

and they want us to find: How much interest will be paid in the second year of the loan?

Answers on the text book:

total interest paid in year 2=12 (PMT)- Total principal paid in year 2
total prinical paid in year 2=Balance after year 1 - balance after year 2
=(Balance after 12 payment)-(Balance after 24 payment)

okay here I am having trouble with it why did they use 12 and 24 can someone please explain that to me ??

#### Subhotosh Khan

##### Super Moderator
Staff member
missl said:
Hi, I need some help. I am looking at this problem on the text book. Where the questions says:

Given:
9500 personal loan at 4.5% compounded monthly is to be repaid over a four year term by equal monthly payment .

{9500*0.625} / {1-1/+0.625)^48} =229.70 PMT

and they want us to find: How much interest will be paid in the second year of the loan?

Answers on the text book:

total interest paid in year 2=12 (PMT)- Total principal paid in year 2
total prinical paid in year 2=Balance after year 1 - balance after year 2
=(Balance after 12 payment)-(Balance after 24 payment)

okay here I am having trouble with it why did they use 12 (months in year 1) and 24 (months in 2 years) can someone please explain that to me ??

#### missl

##### New member
Oh I see.. But what happen if they ask How much interest will be paid in the third year of the loan?

so then I use 12 (months in year 1) and 24 (months in 2 years) and 36 (months in 3 years ??? is that correct ??

#### Denis

##### Senior Member
Code:
MONTH    PAYMENT  INTEREST   BALANCE
00                           9500.00
01
02
....
12
13
....
24                               $X 25 .... 36$Y
To find interest cost during 3rd year means you must use the balances owing
at end of year 2 (AFTER 24TH PAYMENT) and at end of year 3 (AFTER 36TH PAYMENT).
That's what the "24 months" and "36 months" mean: NOTHING ELSE.

So in above, $X is owing (after 24 payments), then$Y is owing (after 36 payments).

To get the interest, X and Y need to be calculated.
Then using those, you calculate the interest using your book's formula.

#### missl

##### New member
Hi Denis,,

I did the way you told me and I also follow the text book as follow but for some reason the number seem to be very big..

Q.How much interest will be paid in the third year of the loan

PMT= 37,000 / 12.050167 = 3070.50

= 37,000 ( 1.054) ^24 – 3,070.50 [ (1.054)^24 -1/0.054 ]
= 13,313.21

= 37,000 (1.054)^36 – 3070.50 [ (1.054)^36 -1 / 0.054]
= 75,045.19 this number seem to be very big

Therefore,
Total principal paid in year 3 13,313.21 – 75 045.19 = 61,731.98
And
Total interest paid in year 3 = 36 (3070.50) – 61,731.98 = 48,806.02

#### tkhunny

##### Moderator
Staff member
What on Earth are you doing? The loan started at $9500. How do you get things greater that$9500? Don't EVER go with things that make absolutely no sense.

Where did you get 0.625? 0.045/12 = 0.00375
What does the notation "/+" mean?

Try again on the regular payment. Once you get that...

What is the outstanding balance at the end of each year?

Beginning: $9500 End of year 1, right after payment:$7125
End of year 2, right after payment: $4750 End of year 3, right after payment:$2375
End of year 4, right after payment: $0 How much did the outstanding balance change during the interesting year? How much was paid in total during the interesting year? Note: I REALLY don't like this question. There are MANY ways to charge interest. Without a very clear specification and disclosure, one has no basis to assume anything. Look up the "Rule of 78's", sometime. This sort of arrangement is rather common for short-term, small-value loans. It gives VERY different answers. #### TchrWill ##### Full Member Given: 9500 personal loan at 4.5% compounded monthly is to be repaid over a four year term by equal monthly payment . {9500*0.625} / {1-1/+0.625)^48} =229.70 PMT and they want us to find: How much interest will be paid in the second year of the loan? Answers on the text book: total interest paid in year 2=12 (PMT)- Total principal paid in year 2 total prinical paid in year 2=Balance after year 1 - balance after year 2 =(Balance after 12 payment)-(Balance after 24 payment) okay here I am having trouble with it why did they use 12 and 24 can someone please explain that to me ? I do not know where you are getting some of the numbers you use. Also, you are not using the correct numbers in the formula. <-----------------------------------------------------------------> What is the periodic payment required to retire a debt of P dollars in n periods (months or years) if payments start at the end of the first period and bear I% interest compounded periodically? For this typical loan payment calculation, ..............R = Pi/[1 - (1 +i)^(-n)] where R = the rent (periodic payment), P = the amount borrowed, n = the number of payment periods, and i = I/100n. Example: What is the annual payment required to retire a loan of$10,000 over a period of 5 years at an annual interest rate of 8%? Here, P = 10,000, n = 5, and i = .08 resulting in

......R = 10000(.08)/[1 - (1.08)^-5] = $2504.56 per year Had the question been what is the monthly payment required to retire a loan of$10,000 over a period of 5 years at an annual interest rate of 8%, compounded monthly? Here, P = 10,000, n = 60, and i = .006666 resulting in

..........R = 10000(.006666)/[1 - (1.006666)^-5] = $202.76 per month or$71.44 less than making yearly payments.

In your case, Given:
$9500 personal loan at 4.5% compounded monthly is to be repaid over a four year term by equal monthly payment . Here, P = 9500, n = 4(12) = 48 and i = 4.5/100(48) = .00375. Therefore, R = 9500(.00375)/[1 - (1+.00375)^(-48)] ...............9500(.00375)/[1 - 1/1.19681)] ...............9500(.00375)/.1644485 =$216.63 per month.

As Denis suggested, make a table of the monthly outlays and balances from which you can determine the annual outlays and balances. For instance:
End.of..Starting...Month End...Monthly..Toward....Final
Month...Balance....Interest....Payment...Debt....Balance
.1.......9500.......35.62......216.63....181.....9318.99
.2.....9318.99......34.94......216.63...181.68...9137.30
.3.....9137.30......34.26......216.63...182.36...8954.93
.4.....8954.93......33.58......216.63...183.00...8771.88
.5.....8771.88......32.89......216.63...183.73...8588.14
.6.....8588.14......32.70......216.63...184.42...8403.72
.7.....8403.72......31.511.....216.63...185.11...8218.60
.8.....8218.60......30.82......216.63...185.81...8033.39
.9.....8033.39......30.12......216.63...186.50...7846.89
10.....7846.89......29.42......216.63...187.20...7659.68
11.....7659.68......28.72......216.63...187.91...7471.77
12.....7471.77......28.02......216.63...188.61...7283.16

1 represents month 1
The starting balance is $9500 The end of month interest due is .00375(9500) =$35.62
The monthly payment is $216.63$216.63 - $35.62 =$181.1 which goes toward reducing the balance.
The end of month 1 balance is therefore $9318.99. 2 represents month 2 The starting balance is$9318.99
The end of month interest due is .00375(9318.99) = $34.94 #216.63 -$34.94 = $181.68 The end of month 2 balance is therefore$9137.30

As you can see from the table, at the end of month 12, only 23.33% of the loan has been repaid.

You can continue to fill in the chart to find the sum of yearly interest for any year you wish. I know of no formula that willl give you the individual yearly sums of interest, debt reduction, etc. If I run across some, I will let you know.

#### Denis

##### Senior Member
Formula for interest during a given period:
Total payments - (opening balance - closing balance).

In TchrWill's example:

216.63 * 12 - (9500.00 - 7283.16) = 382.72

#### TchrWill

##### Full Member
Denis said:
Formula for interest during a given period:
Total payments - (opening balance - closing balance).

In TchrWill's example:

216.63 * 12 - (9500.00 - 7283.16) = 382.72
Thanks Denis. But this only works if you have the nth (24th, 36th, 48th) month balance which requires you to fill out the entire chart. What would be great is to have a formula that gives the nth month balance without completing the chart. I thought I had it but alas, it was not to be.

#### Denis

##### Senior Member
Yes, but the balances owing are easily obtainable by formula:
A(1 + i)^n - P[(1 + i)^n - 1] / i

So as example: a $3000 loan at rate 12% cpd monthly is being repaid at$100 per month;
how much interest is paid between payment number x and payment number y (y > x) ?

100(y - x) - {3000(1 + .01)^x - 100[(1 + .01)^x - 1] / .01 - 3000(1 + .01)^y - 100[(1 + .01)^y - 1] / .01}

Could be simplified some.

#### TchrWill

##### Full Member
Denis said:
Yes, but the balances owing are easily obtainable by formula:
A(1 + i)^n - P[(1 + i)^n - 1] / i

So as example: a $3000 loan at rate 12% cpd monthly is being repaid at$100 per month;
how much interest is paid between payment number x and payment number y (y > x) ?

100(y - x) - {3000(1 + .01)^x - 100[(1 + .01)^x - 1] / .01 - 3000(1 + .01)^y - 100[(1 + .01)^y - 1] / .01}

Could be simplified some.
9500(1.00375)^12 - 216.63(1.00375^12 - 1)/.00375
9936.43 - 2653.85 = $7282.57 vs$7283.16 Close!)

Nice job! Where were you earlier?

#### missl

##### New member
OMG I think I confuse everyone here.. I mixed up with another question .. Sorry

I was trying to refer to this new question here.

$37,000 loan at 10.8% compounded semi-annually is to be repaid by equal semi-annual payments over 10 years. How much interest will be paid in the third year of the loan? Answer that I got. Just wondering is this the right way of doing it to find the interest that was paid on the third year of the loan... 37,000 ( 1.054) ^24 – 3,070.50 [ (1.054)^24 -1 / 0.054 ] = 13,313.21 = 37,000 (1.054)^36 – 3070.50 [ (1.054)^36 -1 / 0.054 ] = 75,045.19 Therefore, Total principal paid in year 3 13,313.21 – 75 045.19 = 61,731.98 And Total interest paid in year 3 = 36 (3070.50) – 61,731.98 = 48,806.02 #### Denis ##### Senior Member TchrWill said: 9500(1.00375)^12 - 216.63(1.00375^12 - 1)/.00375 9936.43 - 2653.85 =$7282.57 vs $7283.16 Close!) T'would be exactly same if you used payment (formula generated) of 216.633117..... #### Denis ##### Senior Member missl said:$37,000 loan at 10.8% compounded semi-annually is to be repaid by equal semi-annual payments over 10 years. How much interest will be paid in the third year of the loan?
Total interest paid in year 3 = 36 (3070.50) – 61,731.98 = 48,806.02
Are you willing to pay me $48,806.02 in interest for only 1 year if I lend you$37,000 ?????