Question

Over Spring Break you shopped for a mortgage for your first house. You got a quote from a bank with an effective annual rate of 4.0% on a 30-year mortgage, that will be paid back in monthly installments. The loan amount is $100,000. However, in order to receive this relatively low interest rate, the bank required that you pay 2-points during the origination of your loan (a point is 1% of your loan). With you paying this one-time fee at the start of your loan, what is the actual effect annual rate that you will be paying on your loan?

Answer 1

As effective annual rate is given, we can do our calculation with anuual payment(annuity).

Calculation of Value of Annual payment if loan amount is$100000.

PV of annuity= pv factor o annuity@4% 30years*Annuity amount

so Annuity amount=100000/17.2920333

=5783.010

As bank is charging 2% charge i.e.2000 the pv of loan shall be $102000

Present value of this annuity amout can be put equal to the initial loan that is 102000.

Now

We know that if we discount 5783.01 annual receivable till 30 years for getting present value equal to 100000, the rate shall be 4%.

So if we want to get present value equal to 102000 the rate should be lower than 4%

So we shall use interpolation method to find the rate which shall lie between 3% to 4%.

NPV at Higher rate(4%) =[(5783.01*17.2920333)-102000]

=-2000

NPV at Lower rate(3%) =[(5783.01*19.6004413)-102000]

=113349.55-102000

=11349.55

=3+(11349.55/13349.55)*1

=3.85% (approx).

If we solve this using irr formula (as shown below) then we get 3.8377% which is very much similar to our answer calculated using interpolation method.

this has been calculated using formula =irr(N1:N31)

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Good Luck!