Question

- What is the effective rate of a 4.25%, $250,000 30 year fixed rate mortgage with 1.5 discount points, if the mortgage is held for 10 years?

- What is the effective rate of a 4.25%, $250,000 30 year fixed rate mortgage with 1.5 discount points, if the mortgage is held for 1 year?

Answer 1

1...First, calculating the monthly pmt. On the mortgage,

using the PV of ordinary annuity formula

where, PV of mortgage, incl. discount points= $ 250000

r=4.25%/12=0.003542 p.m. for n=30*12=360 months

ie.250000=Pmt.*(1-1.003542^-360)/0.003542

So, the mthly Pmt.=250000/((1-1.003542^-360)/0.003542)=

1229.91

Now,with this monthly pmt. We find the remaining principal balance at end of 10 yrs. Ie. 10*12=120 months

Using the formula

FV=PV*(1+r)^n-(Pmt.*((1+r)^n-1)/r)

where

FV= the future value , ie. Remaining principal balance ----??

PV=PV or original loan balance-- $ 250000

Pmt.= the equal monthly pmt. On the mortgage---1229.91 (as found above)

r- rate /pmt.--ie. 0.003542 p.m.

n= no.of pmts., ie. 10*12=120

Plugging in all the values, we get the rem. Bal. at end of 120 pmts. As

FV=(250000*(1+0.003542)^120)-(1229.91*((1+0.003542)^120-1)/0.003542)

198610.44

Now, equating the cash flow on the mortgage,

250000+3750=(1229.91*(1-(1+r)^-120)/r)+(198610.44/(1+r)^120)

& solving for r, we get the IRR (monthly) as

0.33192%

converting to annual IRR, we have,

(1+0.33192%)^12-1

4.06%

Effective interest rate= 4.06%

2...Now,with the above monthly pmt. We find the remaining principal balance at end of 1 yr. Ie.12 months

Using the formula

FV=PV*(1+r)^n-(Pmt.*((1+r)^n-1)/r)

where

FV= the future value , ie. Remaining principal balance ----??

PV=PV or original loan balance-- $ 250000

Pmt.= the equal monthly pmt. On the mortgage---1229.91 (as found above)

r- rate /pmt.--ie. 0.003542 p.m.

n= no.of pmts., ie. = 12

Plugging in all the values, we get the rem. Bal. at end of 12 pmts. As

FV=(250000*(1+0.003542)^12)-(1229.91*((1+0.003542)^12-1)/0.003542)

245785.61

Now, equating the cash flow on the mortgage,

250000+3750=(1229.91*(1-(1+r)^-12)/r)+(245785.61/(1+r)^12)

& solving for r, we get the IRR (monthly) as

0.19428%

converting to annual IRR, we have,

(1+0.19428%)^12-1=

2.36%

Effective interest rate= 2.36%