FRM Fun 15. P2 (mortgage loan term)

[David Harper CFA FRM]

This is a P2 question.

Hi, I noticed a blog entry today at the CFA Institute, Should you pay down your mortgage?

It got me thinking. If a mortgage loan balance is $100,000 and the annual interest rate is 4.80%, then we can compute the monthly mortgage payment (P&I) for a 30-year loan term with:

• n= 360, I/Y = 0.40, PV = -100,000, FV = 0 --> CPT PMT = 524.66

But what if the mortgage holder decided to pay $700.00 per month, rather than the $524.66, with the goal to reduce the loan term? If we assume the same interest rate, then we can easily use the calculator to re-compute the shorter loan term:

• PMT = 700 --> CPT N = 212.25 months
  Cool, an 33% increase in the payment reduces the term from 30 years to ~ 17.7 years.

But this uses the calculator. Isn't there a direct (analytic) function to solve for the new term?

Question: Can we find a relatively simple "analytical" formula (armed only with a P2 FRM formula) for the implied loan term given a higher monthly payment (and loan balance and interest rate)? (given an analytical expression, presumably we can even find an expression for the amount of loan term [time] reduced for each additional monthly dollar paid?)

 

[aadityafrm]

If I understood question correctly, it is quite straightforward –

Monthly payment of a mortgage is given by, E1 = P x r x (1+r)^n1/ (1+r)^n1-1

If we increase the monthly payment to E2 the payment term changes to n2 , i.e.
E2 = P x r x (1+r)^n2/ (1+r)^n2-1

Rearranging the above expression for n2, we get n2 = Log [E2 /(E2 – P x r)]/Log(1+r)

And the amount of loan term reduced for each additional monthly dollar paid (n1-n2) is Log[E1 x (E2 – P x r)]/E2 x (E1 – P x r)]/Log(1+r)


P.S. corrected few mistakes in the above calculations

 

[ShaktiRathore]

MP1=MB0*r/(1-(1+r)^-T1)
1-(1+r)^-T1= MB0*r/ MP1
1-MB0*r/ MP1=(1+r)^-T1
Ln(1-MB0*r/ MP1)=-T1*ln(1+r)
Ln(MP1/MP1-MB0*r)/ln(1+r)=T1
T1/T2= Ln(MP1/MP1-MB0*r)/ Ln(MP2/MP2-MB0*r)
T2= [Ln(MP2/MP2-MB0*r)/ Ln(MP1/MP1-MB0*r)]*T1 …..our analytical formula for calculating time period after change in Monthly Payment from MP1 to MP2
MP1=524.66
MP2=700
MB0*r=.4%*100000=400
T1/T2=ln(524.66/524.66-400)/ln(700/700-400)
T1/T2=ln(524.66/124.66)/ln(7/3)
T1/T2=1.44/.844=1.699
T2=T1/1.699=30/1.699=17.65 yrs