Which formula is used to calculate the annual cost of an early settlement discount?

Which formula is used to calculate the annual cost of an early settlement discount?

• A. (1 + (discount left to pay / amount left to pay))^number of periods - 1
• B. (discount left to pay / amount left to pay) x number of periods
• C. 365 / number of periods
• D. (amount left to pay / discount left to pay) ^ number of periods - 1

Answer: (A)

The idea being tested is turning a per-period saving from an early settlement discount into an annual rate by compounding. The per-period rate you effectively gain is the discount amount divided by the amount payable after taking the discount: i = discount left to pay / amount left to pay. If you could repeat this same period in a year, you annualize it by compounding: the yearly cost equals (1 + i) raised to the power of the number of discount periods in a year, minus 1. This is the true effective annual cost of not taking the discount. For example, if the discount is 20 on a 980 payment, i ≈ 0.0204. If you could apply that period about 36.5 times in a year, the annual cost would be (1 + 0.0204)^36.5 − 1, which reflects the compounding effect over the year. The other forms don’t capture this compounding: they either use simple interest, invert the ratio, or divide by periods without exponentiating, which underestimates the true annual cost.

The idea being tested is turning a per-period saving from an early settlement discount into an annual rate by compounding. The per-period rate you effectively gain is the discount amount divided by the amount payable after taking the discount: i = discount left to pay / amount left to pay. If you could repeat this same period in a year, you annualize it by compounding: the yearly cost equals (1 + i) raised to the power of the number of discount periods in a year, minus 1. This is the true effective annual cost of not taking the discount.

For example, if the discount is 20 on a 980 payment, i ≈ 0.0204. If you could apply that period about 36.5 times in a year, the annual cost would be (1 + 0.0204)^36.5 − 1, which reflects the compounding effect over the year. The other forms don’t capture this compounding: they either use simple interest, invert the ratio, or divide by periods without exponentiating, which underestimates the true annual cost.