Question

Ann is looking for a fully amortizing 30 year Fixed Rate Mortgage with monthly payments for $135,000. Mortgage A has a 5.25% interest rate and requires Ann to pay 1.5 points upfront. Mortgage B has a 6% interest rate and requires Ann to pay zero fees upfront. Assuming Ann makes payments for 30 years, what is Ann’s annualized IRR from mortgage A?

 

Answer 1

 

firstly we will calculate the monthly payments for a mortgage of $135000

in this case Ann has to pay 1.5% of the mortgage value upfront i.e. 1.5% *135000 = $2025

Thus the mortgage amount in this case = 135000-2025 = 132975

Present value of mortgage = 132975

formula for Present value = A*[((1+R)ⁿ -1)/((1+R)ⁿ * R)]

where

A = annual payments for mortgage

R = interest rate on loan = 5.25%

n = 30 years

putting the known values in the above equation

132975 = A*[((1.0525)³⁰ - 1)/((1.0525)³⁰ * 0.0525)] = A* 3.641551/0.243681

132975 = A*14.94392

A = 132975/14.94392 = 8898.267

now the monthly payment = Annual payment/12 = 8898.267/12 = 741.5223

Now we will calculate IRR

The NPV of the mortgage should be equated to zero

for this,3

NPV = sum of present value of annual payments - loan amount

putting NPV = 0

sum of present value of annual payments = loan amount

sum of present value of annual payments = 132975

Left hand sid eof equation = 8898.267*PVIFA(30 Years, IRR%)

PVIFA = present value interest factor

PVIFA =  [((1+IRR)³⁰ -1)/((1+IRR)³⁰ * IRR)]

8898.267*[((1+IRR)³⁰ -1)/((1+IRR)³⁰ * IRR)] = 132975

[((1+IRR)³⁰ -1)/((1+IRR)³⁰ * IRR)] = 132975/8898.267 = 14.9439

You have to calculate IRR by Trial and error , such that by putting in the value in place Of IRR the result is = 14.9439

Using the above equation if you put in 5.25% as IRR

you will get the left hand side = 14.9439

hence annual IRR = 5.25%