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 $X_{t+T}$.