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# Principal and Interest payments

#### Angelinelyt

##### New Member
Hi David, i have come across certain practice questions which require us to calculate how much principal is being paid in nth month given that the balance loan amount is X dollars.

please can you explain how do we do this without using the calculator? i can use the calculator to calculate how much is being paid monthly (interest and principal) but i do not know the portion of each part.

many thanks!

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @Angelinelyt Can you post the question(s) to make it easier to illustrate? The BA II+ actually has a function that displays the principal paid as part of an amortization schedule (but it's not needed for the FRM). Are you referring to an amortizing mortgage? If yes, the payments are the same each month and you can subtract the interest paid to determine the principal. For example, if the yield is 6.0%, then the next interest payment, IPMT(i) = 0.060/12*(outstanding balance) and the principal portion is given by the payment minus IPMT(i). It's easier with an actual question ....

#### Angelinelyt

##### New Member
thanks David. Lets say this question:
A 30-year mortgage has an original balance of \$160,000 and a fixed rate of 5.0% per annum with typical monthly payments. Which of the following is nearest to the principal reduced by the first month's mortgage payment?

this is simple if we have to calculate the first month. how do the principal reduced after the 26th interest payment? and how do we calculate the portion of principal that is being paid in the 27th interest payment?
i assume that such questions are testable in the frm part 1.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Thank you @ShaktiRathore, awesome! @Angelinelyt If I were forced to use the calculator, my BA II+ does have amortization keys (apparently) for this, but I don't actually know how to use them directly, so for this I would use logic with the standard TVM keys. The cool thing about the TVM keys is that you can change a single assumption and re-compute, so this is easier than you might think:
1. First, I'd solve for the monthly payment for the original 30-year mortgage loan: N = 30*12, I/Y = 5/12, PV = -160000, FV = 0 and CPT PMT = 858.91
2. Second, I'd use this payment as the input and solve for the principal (FV) at the end of the 26th month, so we only need to re-key a new N which is cool: 26 N, then CPT FV = 154,732
3. The next payment's interest will by 5%/12*154,732, and the principal portion will be the rest (ie, of the 858.91). I hope that helps!

#### ShaktiRathore

##### Well-Known Member
Subscriber
Hi,
To make it a little more easier you can use alternative formula for principal paid in the month n=MP/(1+r/12)^(N-n+1) where N=Total months =30*12,r=5%
Put values to get principal in the month (n=27)=858.91/(1+5%/12)^(360-27+1)=214.20
thanks

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@ShaktiRathore oh wow, that's dead simple. It seems to work! I must be dense, I'm not sure I see how it works ... i'd be interested in any derivation or intuition? Thank you!!

#### ShaktiRathore

##### Well-Known Member
Subscriber
Hi,
Its derivation is also simple: We need to find the Outstanding Mortgage Balances at start of month n and the end of month n and take the difference.
Outstanding Mortgage Balances at start of month n(remaining months=N-(n-1)) =MB1=(MP/r)*(1-(1+r)^-(N-n+1)) ...Here r is the monthly rate.
Outstanding Mortgage Balances at end of month n(remaining months=N-n) =MB2=(MP/r)*(1-(1+r)^-(N-n))
Take the difference to get the principal paid in nth month
=MB1-MB2
=(MP/r)*(1-(1+r)^-(N-n+1))-(MP/r)*(1-(1+r)^-(N-n))
=(MP/r)*(1-(1+r)^-(N-n+1)-(1-(1+r)^-(N-n)))
=(MP/r)*(1-(1+r)^-(N-n+1)-1+(1+r)^-(N-n))
=(MP/r)*(-(1+r)^-(N-n+1)+(1+r)^-(N-n))
=(MP/r)*((1+r)^-(N-n))*(-(1/(1+r))+1)
=(MP/r)*((1+r)^-(N-n))*(r/(1+r))
=(MP)*((1+r)^-(N-n+1))
=MP/((1+r)^(N-n+1))
thanks

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Super cool, I would not have found this, thank you!