Question

Randi can get: a fully amortizing 30 year fixed rate mortgage with monthly payments for $500,000 at an annual interest rate of 9%, compounded monthly. Or a fully amortizing 15 year fixed rate mortgage with monthly payments for $500,000 at an annual interest rate of r, compounded monthly. What does the interest rate on the 15 year fixed rate mortgage need to be so that Randi s payment on the 15 year fixed rate mortgage is the same as the 30 year fixed rate mortgage?

Betti gets a fully amortizing 30 year fixed rate mortgage with monthly payments for $1,000,000. If the annual interest rate is 3.25%, compounded monthly, and Betti must pay 1.75 points in closing costs, what will Ann s true APR be

Answer 1

Note: The question regarding only Randi's mortgage is answered below as per guidelines

We need to first calculate the total monthly which comes with a $500,000 loan @9% interest rate compounded monthly.

This can be given with the excel formula =PMT, which needs the following syntax

=PMT(Rate,NPer,PV,FV,Type).

In this formula, the inputs are Rate=9%/12 (since we need to input rate per month), nper=30*12 and PV=-$500,000; we now calculate the PMT which is the repayment amount per month.

=PMT(9%/12,30*12,-500000)

= $4,023.11

Note that FV is 0 (since loan is value will be amortuzed fully in 30 years) and type is 0 (end repayments), and therefore not input here. Also, negative value of PV will give a positive value of PMT and vice versa.

Now, Randi is to make the same monthly payments of $4023.11 and the amount of loan is also same $500,000, but for only 15 years. It can be logically infered that this can only happen if the rate of interest (asked as r in the question) paid is lesser than 9%per annum.

With these inputs of PV=$500,000 and nper=15*12 and PMT=4023.11, we now need to calculate rate of interest, with rate function which is Rate(NPer,PMT,PV,FV,Type)

=RATE(15*12,-4023.11,500000)

=0.4387%

The negative value of PMT denotes outflow and positive value of PV denotes inflow, as required by the function. The formula will generate an error if two opposing signs are not put here.

The result is the per month rate which multiplied by 12 will give yearly rate.

Thus yearly rate = 0.4387*12=5.26%.

The interest rate that should result in same payments for Randi is thus 5.26%