effective rate of discount

Interest and discount are two ways of describing the same deal seen from opposite ends. Interest looks forward: you lend 100 today and it grows to 105, so you earned interest on what you started with. Discount looks backward and is charged up front: a lender hands you 95 today and you repay 100 in a year. The 5 you gave up is a discount, and it is measured against the future amount you repay, not the present amount you receive.

Formally, the effective rate of discount for a period, written d, is the interest amount divided by the ending balance: d = (amount at end minus amount at start) divided by amount at end. Because the denominator is the larger end value, d is always slightly smaller than the corresponding interest rate i. The two are tied together by clean relations: d = i divided by (1 + i), and equivalently i = d divided by (1 minus d), and 1 minus d equals 1 divided by (1 + i). The factor (1 minus d) is exactly the present value of one dollar due in one period.

Discount is the natural language for instruments sold below face value, such as zero-coupon bonds and short-term Treasury bills, where you pay a discounted price today and collect the full face amount at maturity. Actuaries use d alongside i throughout annuity and insurance formulas, and the relationship A = 1 minus d times an annuity-due is a recurring identity in life-contingent mathematics. A frequent confusion is to expect d and i to be equal; they describe the same growth but are quoted against different bases.