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Mortgage Math. Fixed-Rate Mortgage Calculations Payments Balances Yields Annual Nominal Rate APR True Annual Nominal Rate Effective Yields. Fixed-Rate Mortgages. Let’s look at this mortgage in terms of “time value of money” principles 1. The borrower’s payments are a simple annuity.

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Mortgage Math

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Mortgage Math

  • Fixed-Rate Mortgage Calculations

    • Payments

    • Balances

  • Yields

    • Annual Nominal Rate

    • APR

    • True Annual Nominal Rate

    • Effective Yields

Fixed-Rate Mortgages

  • Let’s look at this mortgage in terms of “time value of money” principles

  • 1. The borrower’s payments are a simple annuity.

    • The present value of an annuity is given by the formula:

      • P=804.62

      • i=.09/12 or

      • n=30*12

      • fv=0

Fixed-Rate Mortgages

  • Typically, we use a financial calculator to calculate the payment of the annuity described in the note as a function of the other terms:

    • N=360

    • i=9%/12

    • PV=-$100,000

    • FV= 0

    • Solve for PMT=804.62

Fixed-Rate Mortgages

  • The present value of the annuity described in the note is the principal amount of the loan.

  • Just as with all time value calculations, if we know all but one of the key factors of the mortgage, we can calculate the last one.

    • The key factors for a mortgage are:

    • 1) Principal (PV)

    • 2) Contractual Interest rate (i%)

    • 3) Payment (PMT)

    • 4) Maturity (N)

    • Note: By definition, the FV for an amortizing mortgage=0.

Calculate the required payment if maturity is 30 years, interest rate is 8% and principal is $150,000

N=360 months




Calculate the maximum loan size you can borrow if you can pay $1000 per month, the interest rate is 7.5%, and the maturity is 25 years.





Fixed-Rate Mortgages

Calculate the maturity of a loan if the monthly payment is $843.86, the principal balance is $100,000 and the interest rate is 6%




N=??? (180 months)

Calculate the interest rate of a loan if the principal amount of the loan is $175,000, the monthly payment is $1,367.36 and the maturity of the loan is 30 years.




i=??? (.72292% /month)


Fixed-Rate Mortgages

Fixed-Rate Mortgages

  • Now lets look at fixed rate mortgages from a different perspective

Amortization Schedule for 8% 30 Year FRM

Paydown Pattern of 8% Fixed Rate Mortgage

Components of Payment

Formulas For Calculating Payment in EXCEL

Column I

Column H

Note that EXCEL wants the interest rate in decimal form.

Formulas for Amortizing Loan

Column F

Column B

Column C

Column D

Column E





Fixed Rate Mortgages

  • Shortly after origination, the borrower’s payment is almost all interest

  • Near maturity, the borrower’s payment is almost all principal.

  • Result: Loans with higher coupons amortize more slowly.

    • Intuition?

Calculating the Outstanding Balance

  • In real estate, we frequently need to know what the remaining unpaid principal balance (UPB) of the loan will be at some future date.

    • Calculating payoff balance

      • borrower wants to move or refinance

      • adjustable rate mortgages

      • default

      • PMI

Calculating the Outstanding Balance

  • There are basically two equivalent ways to calculate a remaining balance outstanding after K payments are made.

    • Amortize the loan forward from the beginning and take the EOP balance after k payments.

      • Right hand column from the spreadsheet

    • Calculate the PV of all the remaining payments using the loan rate as the discount rate.

Calculating the Outstanding Balance

  • Calculate the remaining balance after 12 payments for a mortgage loan with original balance of $100,000, loan rate of 8% and original term of 30 years.

    • Method 1: Return to the spreadsheet we calculated earlier.

      • $99,164.64

Amortization Schedule for 8% 30 Year FRM

Calculating the Outstanding Balance

  • Method 2:

    • N=360-12=348 =# of payments remaining

    • I=8/12=.666667= contractual periodic rate of interest

    • PMT=733.76457 (based on original loan terms)

    • PV=??? 99,164.64

  • The concept that the unpaid principal balance on a loan is equal to the present value of all remaining payments (when discounted using the contractual interest rate) is important to understand.


True Annual Nominal Rates, Yields, APRs and Other Mysteries

  • Lenders frequently charge fees for originating a new loan. These cover the cost of underwriting the loan, appraising the property and the lender’s time and effort in making the loan.

    • How do these fees affect the borrower’s true cost of borrowing?

    • Why do lenders charge these fees? Why not just increase the interest rate?

Loan Fees and Borrowing Costs

  • A lender offers a $100,000 loan to a borrower with the following terms:

    • Maturity : 30 yrs

    • Interest rate: 8%

    • Total Loan Fees: $3000

  • What is the true cost of the loan to the borrower?

    • While we will soon adopt the conventional shorthand of calling this cost a “yield”, for a while, I will be more precise and call it the “trueannual nominal mortgage rate”.

      • I call the interest rate in the note; the contractual interest rate.

Loan Fees and Borrowing Costs

  • Step 1: Calculate the payments required by the note itself:

    • N=360

    • i=8/12

    • PV=-100,000

    • PMT=733.76

  • The note requires that the borrower make 360 monthly payments equal to $733.76

Loan Fees and Borrowing Costs

  • Step 2: Determine the net amount of cash the borrower receives:

    • Loan Amount: 100,000

    • Less Fees: (3,000)

    • Net Amount: 97,000

  • The borrower will actually receive only $97,000 at time zero and must make 360 payments of 733.76 to the lender.


  • Step 3: We can use this formula to calculate a value of i; given that Pi=733.76 (for i=1…360), PV=97,000. Using a financial calculator:

    • N=360

    • PMT=733.76

    • PV=-97,000

    • i=???.69364

  • Because we are accustomed to thinking in terms of annual rates, the accepted custom is to multiply this monthly rate by 12. or .69364*12=8.32%

  • We call this annualized rate the trueannualnominal (mortgage) rate.


  • In order to make the present value of the future payments equal to a smaller number ($97,000), we must increase the discount rate.

    • The calculator tells us that the rate must be increased to 8.32%

      • Check: Calculate the PV of 360 monthly payments discounted at 8.32% annual rate

Yields & APR

  • In order to facilitate borrower comparisons of loan costs when both the coupon rate and fees can change, all lenders are required to compute and disclose the Annual Percentage Rate (APR) being charged on the loan.

    • The Federal Reserve (through Req Z) specifies how the APR is calculated


  • To calculate the APR for a loan, one follows the steps just applied (and can then round the result to the nearest 1/8 th %).

    • The steps are:

      • Use the loan rate, maturity and principal balance to calculate the monthly payment.

      • Assume the borrower makes that monthly payment for the full loan maturity

      • Calculate the net cash received by the borrower by deducting points and other loan fees.

      • Calculate the periodic rate that equates the present value of the future monthly payments to the net cash received.

      • Annualize the periodic rate by multiplying it by 12.

Loan Fees and Borrowing Costs

  • In calculating the APR of 8.32% for the loan with $3,000 of fees, we assumed that the borrower paid the loan over 30 years.

    • Is this normally what happens?

    • How many people do you know who have stayed in one home for thirty years and never refinanced their original loan?

Loan Fees and Borrowing Costs-- Impact of Early Payoff

  • What if Jane Doe takes the loan described above, pays the $3,000 in fees, but moves after five years? What is Jane’s real borrowing cost?

  • To calculate the real cost of Jane’s loan, we need to put on the RHS the payments Jane actually makes.

    • Jane will make 60 payments and then payoff the remaining balance of her loan.

Loan Fees and Borrowing Costs-- Impact of Early Payoff

  • Before we had 360 periodic payments and no balance due at the end of the loan(FV=0)

    • Now we need to determine the outstanding balance after 60 payments have been made in order to be able to solve for the i=r/12 that equates the RHS with the LHS.

    • Note: I use r for true annual nominal rate.

Loan Fees and Borrowing Costs-- Impact of Early Payoff

  • Calculate the remaining balance:

    • N=360-60=300

    • i=8/12 (use the loan contract rate)

    • PMT = 733.76

    • PV=95,069.26

Loan Fees and Borrowing Costs-- Impact of Early Payoff

  • Now we can plug this in and solve for the periodic discount rate i.

    • N=60

    • PMT=733.76

    • FV=95,069.26

    • PV=-97,000

    • i=???(.7299)

  • Calculate r, the true annual nominal rate, by multiplying by 12.

    • r=.7299*12=8.76%

Loan Fees and Borrowing Costs-- Impact of Early Payoff

  • The example shows that if the borrower pays the loan off before maturity, the impact of the fees on the borrower’s cost can be much greater than that implied by the APR.

Charging Fees to Achieve Yield

  • What if a lender needs to earn a return (yield) of 9% on its new loans, but a borrower can afford to pay, at most, $733.76/month (the payment on a 8% loan)?

    • How much in fees would the lender need to charge the borrower to achieve a 9% yield?

Charging Fees to Achieve Yield

  • Here we know the required periodic discount rate, the loan maturity and the monthly payments.

    • If the lender assumes the borrower will never prepay the loan:

      • N=360

      • PMT=733.76

      • i=9/12

      • Solve for PV=91,193

    • The lender would need to charge $8,807 in fees so that its net outflow at origination is only 91,193.

    • By custom, fees are often expressed as a % of the loan or 8.8 “points”

Charging Fees to Achieve Yield

  • If the lender charges $8,807 in fees, the present value of the payments the borrower is expected to make, discounted at 9%, is exactly equal to the amount the lender advances.

    • The lender earns an IRR of 9% on its money.

      • Would the lender rather own a mortgage with an true annual nominal rate of 9% or a treasury bond with a coupon rate of 9.125%?

        • When comparing securities with different compounding frequencies-- you need to use effective annual rates

Effective Yields

  • To Compare instruments with different compounding rates, we annualize the rate in a different way.

  • i=the periodic compounding rate

    • i=r/n when r is the true annual nominal rate.

    • n=the number of compounding periods in a year

Effective Yields

  • When comparing similar instruments (like mortgages to mortgages), annual nominal rates will give the right ordering by cost.

  • When comparing two instruments with same annual nominal rate but different compounding, the one with more frequent compounding will have higher effective yield.

  • When comparing instruments with different true annual nominal rates and different compounding, the lower rate instrument might have higher annual effective rate.

Charging Fees to Achieve Yield

  • What if the lender believes that the borrower will likely prepay the loan in a shorter period of time?

    • Will the lender need to charge a larger fee or a smaller fee?

Assume Prepays in 5 years:






The lender needs to charge 3,932 in fees

Assume Prepays in 10 years






The lender needs to charge 6,290 in fees

Charging Fees to Achieve Yield

Pricing Loans

  • Finally, assume that Lender A (who has just originated an 8% 30 year mortgage) approaches Lender B and offers to sell Lender B the new loan.

    • If Lender B’s required rate of return is 9%, what will Lender B pay for the loan?

Pricing Loans

  • A Lender who is purchasing loans should follow the same steps:

    • 1. Determine the expected cash flows the borrower will make

      • contractual cash flows are based on loan amount, maturity and contractual interest rate.

      • Actual cash flows will depend on prepayment assumption

    • 2. Discount the expected cash flows at the lender’s required yield

      • Caution: Never confuse the contractual loan rate that determines the borrower’s periodic loan payments and the lender’s required yield.


Measures of Maturity

  • Final Maturity

    • A Constant Payment Mortgage with its last payment scheduled for 10/2030 has 30 years until final maturity.

    • The Constant Amortization Loan with final maturity of 10/2030 is a 30 year mortgage.

      • Compare the payments and paydown patterns of these two mortgages.

        • Do they seem to be equally as long-lived an investment to you?

    • A thirty year bond pays interest every six months and returns all principal at the end of 30 years.

      • Is this effectively” shorter or longer than the mortgages described above?

Measures of Maturity

  • Weighted average life

    • One solution to the weakness of using final maturity is to use the weighted average time that your principal comes back.

Measures of Maturity

  • Duration

    • An even better solution is to calculates the weighted average time of receipt of present value.


  • Duration is the Best Measure of Effective Maturity

    • It considers all cash flows-- not just the last and not just principal

    • It considers the present value of the cash flows not just their dollar amount

    • It provides a measure of price sensitivity to interest rates.

Using Duration

  • The Key Relationships:

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