# D Simple interest rate calculation

For the simple interest on an amount \(X_t\) at the interest rate \(r\):

\[X_{t+1} = X_{t}(1+r)\]

For example, if we pay interest on 100€ at an annual rate of 10%, we get:

\[100 € + 0.10 \cdot 100 € = 100 € \cdot(1 +0.10) = 110 €\]

If we add interest to the new value \(X_{t+1}\) again (compound interest), we get:

\[X_{t+2} = X_{t+1}(1+r)\]

By substituting the first into the second equation, we get:

\[X_{t+2} = X_{t}(1+r)(1+r) = X_{t}(1+r)^2\]

For \(T\) interest periods, we obtain:

\[X_{t+T} = X_{t}(1+r)^T\]

We can rearrange this equation according to \(X_t\) to determine the initial amount \(X_t\) that would have to be invested at interest \(r\) in order to obtain an amount \(X_{t+T}\) at time \(t+T\):

\[X_{t} = \frac{X_{t+T}}{(1+r)^T}\]

In this form, \(X_t\) is sometimes referred to as the **cash value** (present value) of the future value \(X_{t+T}\).