Interest Rate Differential (IRD)

QUESTION: Two years ago, Bob and Mary borrowed $152,135 amortized for 27 years, and signed a five year term at an interest rate of 11%.

Now, two years later, the three year term interest rates are 7%. Bob and Mary would like to pay 7% interest for the remaining 3 years of monthly payments, instead of 11%. The Lender asked them for a $15,440 interest rate differential (IRD) payment along with their 24th monthly payment if they wanted the interest rate lowered to 7% for the remaining 3 years. How can Bob and Mary make an informed decision about paying the $15,440?

Simple Answer

Bob and Mary would owe the lender $145,803, after 60 monthly payments (12/15/2004), if they do nothing but continue to make their regular payments.

If Bob and Mary decide to pay the $15,440, by adding it to their balance (as a negative prepayment) on the twenty fourth payment, the balance is the same after the 60th payment (12/15/2004) that is, $145,803. By paying the IRD Bob and Mary saved nothing. This should be no surprise because that is what the IRD calculation is meant to do, compensate the Lender for an early release from the 5 year term agreement.

The MORTGAGE2 PRO program handles that calculation automatically as shown below.

Simply highlight the 24th payment line and click on the renewal key icon and the renewal screen appears below;

What could be simpler?

Detailed Answer

Using the MORTGAGE2 PRO software, is easy to verify that $15,440 is the correct amount to be paid to the lender. One merely highlights the 24th payment of the amortization schedule and clicks on the renewal icon and fills in the renewal rate of 7% and the results are below;

If one was only interested in the IRD calculation without using the MORTGAGE2 PRO amortization software then you could use the DISCOUNTING.EXE program to arrive at the same numbers as shown below;


American mortgages have monthly compounding, .. the same example for an American mortgage is also shown below;


For the mathematicians a detailed analysis is shown below. After 24 months the Loan Balance is approximately $150,000 with 36 months left to go until the end of the 5 year Term. If Bob and Mary simply make their payments, with no changes, after the remaining 36 payments they will owe $145,803.85

If Bob and Mary decide to pay the IRD, they could add the $15,440.04 directly onto the balance owing of $150,000 along with the 24th payment and thus the interest rate would be 7% from this point on until 36 payments have been made. From Figure 14 it can be seen that after 36 payments the balance owing is exactly the same (within a few pennies due to rounding off ) as it should be because that is what the IRD calculation is intended to achieve.

For this analysis it is assumed that the IRD money is borrowed. The IRD could be obtained from another investment but that would make the example more complex. Having the IRD money as a windfall (money under the mattress) makes the benefits obvious. If Bob and Mary decide to keep the two loans separate; then, the first loan is $150,000 @ 7% with $1,443.79 per month and the balance owing after 36 payments is $126,824.05

The second loan of $15,440.04 @ 7% with 476.04 per month (except for the last payment of 475.96) and the balance after 36 payments is zero.

An annuity calculation would show that 476.04 deposited every month at a monthly interest of .575% would accumulate to $18,980 after 36 months.

It can also be seen that a negative amortization schedule would give the same balance owing of $18,979.73 (close enough to 18,980). The approximate difference between $145,803 – $126,824 = $18,979. In other words, $15,440 now is worth $18,980 thirty six months in the future (FV) if it is appreciating at .575% per month . Stated differently, $15,440 now is worth $18,980 three years in the future(FV) if it is growing at an effective interest rate of 7.1224415.

Note that most interest rates are not normally quoted to more than two decimal places because of the difficulty in getting agreement in the PV/FV calculations as the equations exponent is very sensitive to the number of decimal places used!

Another way of looking at the numbers is as follows. If one deposits $476.04 per month, every month for 36 months, into an account paying interest at 0.575% per month then after 36 months $18,980 will have accumulated.


Bob and Mary have 36 monthly payments of $1,444. If the Lender uses 11% as the AIR then the balance after 36 months is $145,803. If the Lender uses 7% as the AIR the balance after 36 months will be $126,824. The difference in the two balances, is $18,979 which is money the Lender would lose in the future (36 months from now). The IRD of $15,440 now is worth $18,979 thirty six months into the future. The IRD is a fair method of compensation for both the borrower and the lender.



PV= 15,440.04

FV= 18,979.72

I = 0.575000

N = 36.0000