A loan is provided by a lender to a borrower in return for the payment of interest. The borrower repays the loan and interest over the term of the loan with a series of regular periodic payments. Consequently an outstanding loan balance is the remaining amount that a borrower owes on a loan after making one or more of those payments.
As can be seen the regular payments will clear the loan balance over the term. It follows therefore that the present value (PV) of these future payments must be equal to the outstanding loan balance. This can be demonstrated using the following example.
To illustrate suppose a business takes out a loan of 100,000 (PV) for the term of four years at 6%. The business agrees to repay the loan in four equal annual installments of 28,859.15 at the end of each year.
The present value of the annuity payments is given by the present value of an annuity formula as follows:
Pmt = Periodic loan payment = 28,859.15 i = Loan interest rate per period = 6% per year n = Number of loan payments required = 4 PV = Pmt x (1 - 1 / (1 + i)n) / i PV = 28,859.15 x (1 - 1 / (1 + 6%)4) / 6% PV = 100,000
As can be seen, at the start of the loan the present value of the installments equals the loan amount.
Outstanding Balance Examples
Example 1
We have seen above that, the present value of the loan installments is equal to the loan amount. It follows therefore that as loan installments are paid, the present value of the remaining loan installments must be equal to the loan amount at that particular point. That is to say the PV of the remaining installments is equal to the outstanding loan balance.
Outstanding loan balance = PV of remaining loan installments
We can use this information to calculate the outstanding loan balance at any point in time. For example after the first installment is paid, the outstanding balance should be the PV of the remaining three installments. This is calculated as follows.
Pmt = Periodic loan payment = 28,859.15 i = Loan interest rate per period = 6% per year n = Number of loan payments remaining = 3 PV = Pmt x (1 - 1 / (1 + i)n) / i PV = 28,859.15 x (1 - 1 / (1 + 6%)3) / 6% PV = 77,140.85 Outstanding loan balance = 77,140.85
As a check, we can show this to be the case by calculating the outstanding loan balance without using the annuity formula. As each payment is made the balance on the loan falls. So for example after the first repayment, the outstanding loan balance will be the original loan, plus the interest for a year, less the first installment, as follows:
Loan balance = Loan amount + Interest - Installment Loan balance = 100,000 + 100,000 x 6% - 28,859.15 Outstanding loan balance = 77,140.85
As can be seen this is the same answer as given by the annuity formula applied to the remaining three installments
Example 2
Suppose a business borrows 150,000 from a lender at an interest rate of 5% for 10 years. The loan is repaid by monthly installments at the end of each month. Calculate the outstanding loan balance after 68 months.
The first step is to calculate the loan installments using the annuity payment formula PV as follows:
PV = Loan amount = 150,000 i = Loan interest rate per period = 5%/12 a month n = Number of loan payments required = 10 x 12 = 120 Pmt = PV x i / (1 - 1 / (1 + i)n) Pmt = 150,000 x 5%/12 / (1 - 1 / (1 + 5%/12)120) Pmt = 1,590.9827
The next step is to calculate the outstanding balance after 68 months by calculating the present value of the remaining installments, using the present value of an annuity formula.
Pmt = Periodic mortgage payment = 1,590.9827 i = Mortgage interest rate per period = 5%/12 a month n = Number of loan payments remaining = 120 - 68 = 52 PV = Pmt x (1 - 1 / (1 + i)n) / i PV = 1,590.9827 x (1 - 1 / (1 + 5%/12)52) / (5%/12) PV = 74,243.84 Outstanding loan balance = 74,243.84
As can be seen after 68 payments the outstanding loan balance on the original loan amount of 150,000 will be 74,243.84.
In summary, an outstanding loan balance is the remaining amount that a borrower owes on a loan after making one or more payments. The outstanding balance can be calculated using a simple time value of money formula. The formula that takes into account the principal amount borrowed, the interest rate, and the number of payments made.
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.