Question

a. What is the effective annual interest rate (EAR) of the APR of 10.5% given that it is compounded quarterly? Monthly? Annually? Daily?

b. If you purchase a new home for $250,000 today, what is your monthly payment if you have to pay 4.25% annual interest compounded monthly? Assume a 30‐year fixed mortgage (360 months) and 25% down on the home (this is of the purchase price above).

Answer 1

1.) EAR is the actual rate paid (or received) after accounting for compounding that occurs during the year

Effective annual interest rate (EAR) Formula

  i = (1 + r/m)^m – 1

where, r = the nominal interest rate per year

i = is the effective annual interest rate

m = is the number of interest periods per year

a) Daily Rate

i=(1+0.105/365)³⁶⁵−1

i=0.110694

I=i×100=11.0694%

b) Quarterly

i=(1+0.105/4)⁴−1

i=0.109207

I=i×100=10.9207%

c) Annually

i=(1+0.105/1)¹−1

i=0.105

I=i×100=10.5%

d)Monthly

i=(1+0.105/12)¹²−1

i=0.110203

I=i×100=11.0203%

2) PMT=PVi(1+i)ⁿ

            (1+i)ⁿ−1

where n = is the term in number of months,

PMT = monthly payment,

i = monthly interest rate as a decimal (interest rate per year divided by 100 divided by 12)

PV = mortgage amount

Applying values in formula

Purchase price =$250,000

Down payment= 25% of purchase price

= 25% of 250,000= $62,500

PV = $250,000-$62,500= $187500

i= 4.25%/12 =0.354166 %

       in decimals= 0.00354166

n= 360 months

PMT= $187500*0.00354166 (1+0.00354166)³⁶⁰

            (1+0.00354166)³⁶⁰ -1

        =$187500*0.00354166*3.570649

            3.570649 - 1

      =     $2,371.129638

               2.570649

= $922.3856

$922.3856 is the monthly payment for this mortgage