Question

You are considering an ARM with the following characteristics: Mortgage amount $100,000 1st year contract rate: 7% 2nd year contract rate: 8% Discount points: 3% Loan maturity with monthly payments: 10 year loan. What is the effective yield if the loan is paid off at the end of year 2?

A. 7.93%

B. 8.23%

C. 8.87%

D. 9.22%

Answer 1

1...First, calculating the monthly pmt. On the mortgage,as if the loan is held for the full 10 years

using the PV of ordinary annuity formula

where, PV of mortgage, incl. discount points= $ 100000

r=7 %/12=0.005833 p.m. for n=10*12= 120 months

ie.100000=Pmt.*(1-1.005833^-120)/0.005833

So, the mthly Pmt.=100000/((1-1.005833^-120)/0.005833)=

1161.06

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

Using the formula

FV of bal. principal =FV of the original single sum of principal-FV of annuity

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-- $ 100000

Pmt.= the equal monthly pmt. On the mortgage---1161.06 (as found above)

r- rate /pmt.--ie. 0.005833 p.m.

n= no.of pmts., ie. 1*12=12

Plugging in all the values, we get the rem. Bal. at end of 12 pmts.(1 yr.) As

FV=(100000*(1+0.005833)^12)-(1161.06*((1+0.005833)^12-1)/0.005833)

92840.07

Now again, calculating as above, for the 2nd year,

ie. calculating the monthly pmt. On the mortgage,as if the loan is held for the balance 9 yrs., ie. 9*12=108 mths.,

using the PV of ordinary annuity formula

where, PV of the mortgage now = $ 92840.07

r=8 %/12=0.006667 p.m. for n=9*12= 108 months

ie.92840.07=Pmt.*(1-1.006667^-108)/0.006667

So, the mthly Pmt.=92840.07/((1-1.006667^-108)/0.006667)=

1208.68

Now,with the above monthly pmt. We find the remaining principal balance at end of 2nd yr. Ie. After 12+12=24 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-- $ 92840.07

Pmt.= the equal monthly pmt. On the mortgage---1208.68(as found above)

r- rate /pmt.--ie. 0.006667 p.m.

n= no.of pmts., ie. 1*12=12

Plugging in all the values, we get the rem. Bal. at end of 2nd yr. As

FV=(92840.07*(1+0.006667)^12)-(1208.68*((1+0.006667)^12-1)/0.006667)=

85498.15

Now, the above is the amt. to be repaid if the loan is paid off at the end of year 2

So, equating cash inflows & outflows of the mortgage,

Loan amt-Pmt. Towards disc.pts.=PV at Yr. 0 of 1st yr. annuity+ PV at Yr. 0 of 2nd yr. annuity+PV of the lumpsum principal repayment-----ie.

100000-3000=(1161.06*(1-(1+r)^-12)/r)+(1208.68*(1-(1+r)^-12)/r/(1+r)^12)+(85498.15/(1+r)^24)

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

0.76860%

converting to annual IRR, ie. effective yield we have,

(1+0.7686%)^12-1

9.62%

Effective Yield / interest rate= 9.62% ---(nearest answer---- D. 9.22%)