# 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 $$X_{t+T}$$.