Commercial Discount: Formula, Calculation and Use in Credit Instruments

Discount is the operation of receiving a future value early by giving up part of that value in exchange for immediate liquidity. If you hold an invoice for $1,000 due in 3 months and need cash now, a bank can advance that amount and retain a percentage as compensation for the service.

The future value stated on the instrument is called the face value (N) or nominal value. The amount received today, after the discount, is called the present value (A). The difference between them is the discount (D): D = N − A.

There are two main discount regimes: commercial discount (also called bank discount or discount on the face value) and rational discount (also called mathematical discount or discount on the present value). They differ in the base to which the rate is applied.

In commercial discount, the rate is applied to the face value of the instrument — the future value. The formula is D = N × r × t, where N is the face value, r is the discount rate per period (as a decimal), and t is the time until maturity (in the same unit as the rate).

The present value is A = N − D = N − N × r × t = N × (1 − r × t). Example: an invoice for $2,000 maturing in 4 months is discounted at a commercial rate of 3% per month. The discount is D = 2,000 × 0.03 × 4 = $240. The amount received today is A = 2,000 − 240 = $1,760.

Note that the discount base is always N (the face value), not A. This means the effective rate from the recipient's perspective is higher than the nominal contracted rate — because you receive only $1,760 but pay $240 in discount on the $2,000. Calculating the true effective rate matters when comparing different early-collection options.

In rational (or mathematical) discount, the rate is applied to the present value — the amount you will actually receive. The formulas are: D = A × r × t and A = N ÷ (1 + r × t). Using the same example: A = 2,000 ÷ (1 + 0.03 × 4) = 2,000 ÷ 1.12 ≈ $1,785.71. The rational discount is D = 2,000 − 1,785.71 ≈ $214.29.

Comparing the two regimes: under commercial discount, the deduction was $240 and the amount received was $1,760. Under rational discount, the deduction was only $214.29 and the amount received was $1,785.71. For the same nominal rate, commercial discount always removes more money from the seller than rational discount.

Rational discount is mathematically consistent with simple interest: it is as if you were calculating the present value of N using the simple interest formula. Commercial discount, on the other hand, is a market convention that favours the party discounting the instrument (the bank), because the calculation base is larger.

In many markets, invoice discounting is one of the main sources of working capital for small and medium-sized businesses. A company that sold on credit can accelerate the receipt of those sales by discounting the instruments at the bank before maturity.

In this operation, the bank assesses the default risk of the debtor, applies a discount rate, and advances the present value to the company. The rate varies according to the term, the volume of instruments, and the credit profile.

To evaluate whether early collection is worthwhile, compare the discount rate with the cost of other working-capital sources (overdraft, direct working-capital loans). Also calculate the effective rate of the commercial discount using r_eff = r ÷ (1 − r × t), which will always be higher than the contracted nominal rate.

For the same face value, term and nominal rate, commercial discount always results in a larger deduction (less money received) than rational discount. This is because the calculation base for commercial discount is larger (N versus A).

An important caution: in exam problems, the question always specifies which type of discount to use. Do not assume the regime without checking. Terms such as 'commercial discount rate', 'bank discount', or 'discount on the face value' indicate D = N × r × t. Terms such as 'rational discount', 'mathematical discount', or 'discount on the present value' indicate A = N ÷ (1 + r × t).

Another frequent error is confusing the discount rate with the equivalent interest rate. A commercial discount rate of 3% per month is not the same as an interest rate of 3% per month: the effective interest rate embedded in the commercial discount is r_i = r_d ÷ (1 − r_d × t), which always exceeds the nominal discount rate.