A loan is provided by a lender to a borrower in return for the payment of interest. The borrower agrees to repay the loan and the interest over the term of the loan with a series of regular periodic payments.

As the regular payments will clear the loan balance over the term, the present value (PV) of the payments must be equal to the value of the loan. This can be demonstrated using the following example.

Suppose a business takes out a loan of 100,000 (PV) for the term of four years at an interest rate of 6%, and agrees to repay the loan in four equal annual installments of 28,859.15 at the end of each year.

Period | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|

Cash flow | Loan ↑ | Pmt ↓ | Pmt ↓ | Pmt ↓ | Pmt ↓ |

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

At the start of the loan, **the present value of the loan installments is equal to the loan amount**.

## Calculating the Outstanding Loan Balance

We have seen above that, the present value of the loan installments is equal to the loan amount, it follows 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 equal to the outstanding loan balance.

We can use this information to calculate the outstanding loan balance at any point in time. For example, after the first installment has been paid, the outstanding loan balance should be the present value of the remaining three installments 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

The same answer as given by the annuity formula applied to the remaining three installments

## Outstanding Loan Balance Example

Suppose a business borrows 150,000 from a lender at an interest rate of 5%. The loan is for a term of 10 years and 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 loan 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

After 68 payments the outstanding loan balance on the original loan amount of 150,000 will be 74,243.84.