The debt constant sometimes referred to as the loan constant or mortgage constant is the ratio of the constant periodic payment on a loan to the original loan amount.

The debt constant is only relevant to loans that have a fixed interest rate over the period of the loan, and is used to make quick calculations of the amount needed to repay a loan over its term, and the balance outstanding at any point in time.

## Debt Constant Formula

The periodic payment on a loan is based on the present value of an annuity formula given as follows:

Pmt = PV x i / (1 - 1 / (1 + i)^{n})

The debt constant for a period is then given by the ratio of the payment (Pmt) to the loan amount (PV)):

## Mortgage Constant Example

Suppose a mortgage is for a term of 30 years at a rate of 5% with monthly repayments at the end of each month.

The mortgage constant is calculated as follows:

Mortgage loan constant = i / (1 - 1 / (1 + i)^{n}) i = 5%/12 per month n = 30 x 12 = 360 months Mortgage loan constant = (5%/12) / (1 - 1 / (1 + (5%/12))^{360}) Mortgage loan constant = 0.537% per month

This calculation shows that monthly payments amounting to 0.537% of the mortgage would clear the mortgage after 30 years providing the rate is constant at 5%.

The debt constant is independent of the amount of the mortgage. In the above example, if the mortgage was for 100,000, then monthly repayments of 0.537% x 100,000 = 537 would need to be made to clear the mortgage after 30 years at an interest rate of 5%.

In order that comparisons can be made, mortgage constants are often quoted for a year. Again using the numbers in the example above, a monthly mortgage constant of 0.537%, gives an annual mortgage constant of 0.537% X 12 = 6.442%.

For our mortgage, the monthly payments in a year would total to 6.442% of the mortgage = 6.442% x 100,000 = 6,442 per year (537 x 12 allowing for rounding).

## Link Between the Debt Constant and the Annuity Factor

The annuity factor given by the annuity tables and the debt constant are derived from the same present value of an annuity formula.

Pmt = PV x i / (1 - 1 / (1 + i)^{n})

The debt constant is equal to Pmt/PV whereas the annuity factor is given by PV/Pmt. The relationship between the debt constant and the annuity factor is therefore given by the formula.

Debt constant = 1 / Annuity factor

Consider another example of a loan for 25 years at a rate of 6% with annual payments at the end of each year.

The debt constant or loan constant is calculated using the formula as follows:

Debt constant = i / (1 - 1 / (1 + i)^{n}) i = 6% n = 25 Debt constant = 6% / (1 - 1 / (1 + 6%)^{25}) Debt constant = 7.8227% per year

The alternative to using the debt constant formula is to use the present value of an annuity tables.

From the annuity tables, the annuity factor for 25 years and 6% is given as 12.7834, and from this the debt constant is calculated as.

Debt constant = 1 / Annuity factor Debt constant = 1 / 12.7834 Debt constant = 7.8227% (as above)

## Outstanding Loan Balance and Debt Constant

The debt constant can be used to calculate the outstanding or unpaid balance on a loan.

The outstanding balance on a loan is the present value of the outstanding annuity payments at that point in time, this is given by the present value of an annuity formula.

PV = Pmt x (1 - 1 / (1 + i)^{m}) / i

**Variables used in the formula**

PV = Present Value at the date the outstanding balance is required for.

Pmt = Periodic payment

i = Discount rate

m = Number of periods out standing on the loan

But we know from earlier, that the payment (Pmt) is the debt constant for the loan multiplied by the loan amount, and the remaining function is the annuity factor for the outstanding period of the loan.

Outstanding loan = Pmt x (1 - 1 / (1 + i)^{m}) / i Outstanding loan = Loan x Debt constant (n) x Annuity factor(m) But the Debt constant = 1 / Annuity factor Outstanding loan / Loan = Debt constant (n) / Debt constant (m) or expressed as a percentage of the loan amount Outstanding loan % = Debt constant (n) / Debt constant (m)

## Outstanding Loan Balance Example

If a loan for 250,000 has a 30 year term and a rate of 7%, what is the outstanding balance after 21 years?

The debt service constant for the loan is given as follows:

Debt constant (30)= i / (1 - 1 / (1 + i)^{n}) Debt constant (30) = 7% / (1 - 1 / (1 + 7%)^{30}) Debt constant (30) = 8.0586%

After 21 years there is 9 years left to pay. The debt constant for the remaining term is given by:

Annuity factor of remaining payments = (1 - 1 / (1 + i)^{m}) / i Debt constant (9) = 1 / Annuity factor Debt constant (9) = 7% / (1 - 1 / (1 + 7%)^{9}) Debt constant (9) = 15.349%

The outstanding loan balance is then calculated using the debt constants

Outstanding loan % = Debt constant (n) / Debt constant (m) Outstanding loan % = Debt constant (30) / Debt constant (9) Outstanding loan % = 8.0586% / 15.349% Outstanding loan % = 52.504%

At the end of 21 years 52.504% of the loan balance would be outstanding, on the 250,000 loan, this amounts to 250,000 x 52.504% = 131,260.

## About the Author

Chartered accountant Michael Brown is the founder and CEO of Double Entry Bookkeeping. He has worked as an accountant and consultant for more than 25 years and has built financial models for all types of industries. He has been the CFO or controller of both small and medium sized companies and has run small businesses of his own. He has been a manager and an auditor with Deloitte, a big 4 accountancy firm, and holds a degree from Loughborough University.