1  Simple Interest and Discount

1.1 Simple Interest Rates

Definition: Simple interest rates can be used to compute the interest for a very simple transaction. If the simple interest rate is \(i\) per unit of time, then:

\[ \text{interest per unit of time} = \text{principal} \times i \]

If the loan lasts for \(t\) units of time then: \[ \text{interest} = \text{principal} \times i \times t \]

We often use one year as our unit of time, in which case \(i\) is an annual interest rate.

1.1.1 Accumulation Function

The ratio of the accumulated value to the present value is known as the accumulation function, denoted by a(t). Under simple interest, the accumulation function is:

\[ a(t) = \frac{\text{Accumulated Value}}{\text{Present Value}} = 1 + it \]

Rearranging the terms, we also have

\[ AV_t =PV_0(1 + it) \qquad \]

1.2 Simple Discount Rates

A simple discount rate multiplied with the accumulated value can obtain the interest per unit of time, which in this context is often called the discount per unit of time. If the simple discount rate is \(d\) per unit time, then the discount per unit of time is:

\[ \text{discount per unit of time} =\text{(accumulated value)} \times d \]

Using simple discount, the amount of discount is calculated by multiplying the simple discount rate times the accumulated value times the amount of time that the loan is in effect. If the loan lasts for \(t\) units of time then:

\[ \text{discount} = \text{interest} = \text{(accumulated value)} \times d \times t \]

We have the following identity:

\[ PV_0 = AV_t(1 - dt) \]

1.3 Equivalent Simple Interest and Discount Rates

If a loan is to be repaid with a single payment at time \(t\), then the equivalent simple interest and discount rates satisfy:

\[ 1 + it = \frac{1}{1-dt} \]