How Much House Can You Afford? Mortgage Payment Problem

What You Will Learn

• How to apply the present value of annuity formula to mortgage calculations
• Converting between annual and monthly interest rates in financial problems
• Understanding the relationship between payment capacity and borrowing power
• Working with compound interest in reverse—from payments to principal
• Interpreting mortgage mathematics that underlies every home purchase

Visualizing the Problem

The loan amount today equals the present value of all future payments discounted at the interest rate.

Solution: Method 1 — Present Value of Annuity Formula

This is a classic mortgage calculation where we need to find the present value of a stream of future payments. The key insight is that a loan is simply the present value of all the payments the borrower will make.

Step 1 — Convert the annual interest rate to monthly

Since mortgage payments are made monthly, we need the monthly interest rate:

Step 2 — Calculate the total number of payments

For a 30-year loan with monthly payments:

Step 3 — Apply the present value of annuity formula

The present value formula for an ordinary annuity is:

Where PV = present value (loan amount), PMT = monthly payment ($1,450), r = monthly rate (0.006667), n = number of payments (360)

Step 4 — Calculate the bracket term

Let's compute the complex part step by step:

Step 5 — Calculate the maximum house price

Now we can find the present value:

Rounding to the nearest cent: $197,242.05

Solution: Method 2 — Geometric Series Approach

We can also think of this as a geometric series problem. Each payment has a different present value depending on when it's made.

Step 1 — Set up the series

The present value of all payments is:

Step 2 — Factor out the payment amount

This becomes:

Step 3 — Recognize the geometric series

The bracket is a geometric series with first term a = 1/1.006667 and common ratio r = 1/1.006667:

Step 4 — Simplify the expression

After algebraic manipulation, this reduces to the same formula:

Step 5 — Complete the calculation

Therefore:

Verification

Let's verify by checking that $197,242.05 borrowed at 8% annual interest results in exactly $1,450 monthly payments over 30 years.

Forward calculation check

Using the standard mortgage payment formula:

Total payments check

The borrower will pay $1,450 × 360 = $522,000 total over 30 years. The interest paid is $522,000 - $197,242.05 = $324,757.95, which represents about 165% of the original loan amount—typical for a 30-year mortgage at 8%.

Does This Seem Reasonable?

Let's put this answer in perspective. At $1,450 per month, this borrower will pay $17,400 per year. A house price of about $197,000 means the annual payments are roughly 8.8% of the home's value.

Compare this to some boundary cases:

• If interest were 0%: The maximum loan would simply be $1,450 × 360 = $522,000
• If the loan term were infinite: The maximum would approach $1,450 ÷ 0.006667 = $217,500 (just enough for interest-only payments)
• Our answer ($197,242): Falls reasonably between these extremes, closer to the interest-only limit because 8% is relatively high

The fact that the borrower ends up paying $522,000 total for a $197,242 house illustrates why mortgage interest is such a significant factor in home affordability.

Common Pitfalls

The Broader Principle

This problem demonstrates the fundamental relationship in all loan calculations:

This formula works for any annuity: mortgages, car loans, lottery payouts, lease payments, or retirement planning. The present value factor depends only on the interest rate and number of payments—not on the specific dollar amounts.

For reference, here are common PV factors for 8% annual interest:

• 15 years monthly: Factor = 104.64
• 30 years monthly: Factor = 136.03 (our problem)
• 30 years annually: Factor = 11.26

Notice how the 30-year monthly factor is much larger than the 30-year annual factor—that's the power of more frequent compounding working in the borrower's favor.

What If?