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Another way to describe this situation is to say that the future amount 𝑋𝑋𝑇𝑇 is discounted according to the interest rate. The quantity can be called a discount factor, and it is the amount that the future amount needs to be multiplied by in order to get the initial value. The discount factor can be calculated using Equation (6):

(1 + 𝑟𝑟)−𝑇𝑇

Like before, a larger interest rate results in a smaller discount factor, which indicates a greater decrease from the future value. Therefore, we again see the inverse relationship whereby a large interest rate results in a small initial price, and vice versa. Although we will largely ignore default for the remainder of these notes, recall how default risk — if perceived by the market — will reduce the price of a note or bill (via the usual forces of supply and demand); note that this price reduction corresponds to an increase in the interest rate. Remember that interest rates are just a language for expressing how a loan investment increases — the possibility of default is compensated for by a lower price, or, to say this same thing in different language, a higher interest rate. The increase in the interest rate when going from a default-free to a default-risky loan is known as a spread, which we study more specifically in Module 4.

The formula for calculating annual effective rate presented in Equation (2) is just one way — one language — to describe the increase in the loan investment (or, equivalently, the interest charged on the loan). There are other so-called interest-rate (or discounting) convention.

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