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This problem has four stages.

Stage 1: find the monthly mortgage payment ignoring points.

Stage 2: calculate your net cash flow with the bank at closing when you take the points into account. Your net cash flow equals mortgage loan minus points (expressed in dollars).

Stage 3: using this net cash flow as present value and the mortgage payment calculated in Stage 1 as the annuity payment, apply the financial function keys to calculate the actual interest rate per month.

Stage 4: calculate the EAR (effective annual rate), given the monthly rate from Stage 3.

Example 3.

You take out a mortgage loan from Bank A with the following characteristics:

compounding period is monthly

loan is for $100,000

APR = 12%; hence r = 12%/12 = 1%

life of loan is 30 years; hence T = 12 x 30 = 360 months

this mortgage loan has no points

This mortgage loan is the same as in Example 1, so the monthly payment is $1,028.61.

Calculate the effective annual interest rate:

EAR = (1 + 0.12/12)12 - 1 = 0.126825 = 12.68% (rounded to two decimal places)

From the lenders perspective, this rate is the true total return on their investment in the borrowers mortgage. From the borrowers perspective (on the other side of the table), the EAR is the true cost of the loan.

Now suppose that Bank B offers a mortgage loan with the same features as the loan offered by Bank A, except that Bank B offers one point. In dollar terms:

one point = 0.01 x $100,000 = $1,000

Bank B calculates the monthly mortgage payment in exactly the same way as Bank A. In particular, the initial loan principal is $100,000, hence you pay $1,028.61 each month, beginning one month from after the closing. The difference lies in the cash flows today. Ignoring all other fees and closing costs, your cash flows today with respect to the bank are:

cash inflow: $100,000 (the initial principal of the mortgage loan)

cash outflow: $1,000 (the point that you pay immediately to Bank B)

your net cash inflow: $99,000 (of course, this cash immediately becomes part of your total payment to the seller)

Lets look closely at the annuity in this situation.

PVA = $99,000 (your net cash inflow today)

C = $1,028.61 (the future monthly annuity payment)

T = 360 months (the life of the mortgage loan)

Calculate the corresponding effective annual rate. First, find the monthly rate using the financial function keys.

Press 2nd, then CLR TVM.

Type 360, then press N.

Type 99000, then press PV.

Type 1028.61, press +/-, then press PMT.

Press CPT, then press I/Y.

The BA II Plus displays I/Y= 1.011232679. This value is r, the monthly rate, in percent.

Recall that the formula for the effective annual rate is

EAR = (1 + APR/m)m - 1

where r = APR/m (in decimal form). Thus,

EAR = (1 + r)m - 1

In our current example,

r = 0.01011232679

m = 12

Thus, EAR = (1 + 0.01011232679)12 - 1 = 0.128329784 (the monthly rate in decimal form)

Convert to percent (by multiplying by 100) and then round to two decimal places: EAR = 12.83%.

Summary:

Bank A offers a 30-year, conventional fixed-rate mortgage loan at an APR of 12% with no points. Hence, its EAR = 12.68%.

Bank B offers a 30-year, conventional fixed-rate mortgage loan at an APR of 12% with one point. Hence, its EAR = 12.83%.

Actually, we did not need to calculate the EARs to know that Bank B offered the worse deal. The mortgage loans are otherwise identical exceptthat that borrower pays a $1,000 fee up front to Bank B.

However, that is too obvious. Here is what you are more likely to see advertised.

Bank A offers a 30-year, conventional fixed-rate mortgage loan at an APR of 12.13% with no points.

Bank B offers two types of loans:

a 30-year, conventional fixed-rate mortgage loan at an APR of 12.13% with no points (same as Bank A); or

a 30-year, conventional fixed-rate mortgage loan at an APR of 12.00% with one point.

Moreover, the marketing division at Bank B promotes their second loan as an opportunity to buy down the rate from an APR of 12.13% to an APR of 12.00%, thus implying that you get a lower rate and hence a better deal.

We showed above that Bank Bs loan with one point has an EAR = 12.83%. What is the EAR on their no point loan (or, equivalently, the loan offered by their competitor, Bank A)?

EAR = (1 + 0.1213/12)12 - 1 = 0.1283 = 12.83%

In other words, both of Bank Bs mortgage loans cost the borrower the same effective annual interest rate of 12.83%.

In short, points are a marketing ploy. Nonetheless, it is worth your while to calculate the effective annual rate for the various combinations of APR and points in order to determine which one really is the best deal.

Here is the actual Problem!

You take out a mortgage loan from Bank A with the following characteristics:

compounding period is monthly

loan is for $250,000

APR = 5%

life of loan is 30 years

this mortgage loan has 2 points

What is your true cost of borrowing? That is, what is the effective annual rate on your mortgage loan?

Do not round at intermediate steps in your calculation. Report the rate in percent to three decimal places.

4.813%

5.116%

4.950%

5.303%