Future value without additional contribution

Exponential and logarithmic growth: Future value

Future value without additional contribution

We assume a start capital #S_0# (also called initial value) that is deposited in a bank over a number #n# periods (e.g., years) on the basis of compound interest at a certain rate per period. We are interested in the future value (also known as the terminal value) after the duration #n# if no additional money is deposited in the interim. The interest rate determines the growth rate. We describe the results in terms of growth rate.

Formula for future value without additional contribution

The future value #S(n)# of an initial capital #S_0# after #n# periods with growth rate #i# equals

\[ S(n)=S_0\cdot\left(1+i\right)^n\]

As the capital is multiplied each period by the same factor, we are dealing with exponential growth. Thus, the formula for #S(n)# is the model for growth with growth factor #1+i# with initial value #S_0#.

If the percentage is #p\%#, then the growth rate is #i=\frac{p}{100}#. The growth factor (the factor by which the capital at the beginning of the period must be multiplied in order to obtain the capital at the end of that period) is #1+i=1+\frac{p}{100}#.

The growth rate is also called growth perunage or interest rate.

In Financial Arithmetic, the factor #\left(1+i\right)^n# is often denoted by #S_{\left.n\right \rceil i}#.

Since the growth rate is equal to #i#, the growth factor is equal to #1+i# (see Exponential growth model). By multiplying the initial value by #1+i#, the value after 1 year, #S(1)#, arises.

The value after 2 years is obtained by multiplying #S(1)# by #1+i#. This gives \[S(2)=S(1) \cdot (1+i) = S_0 \cdot (1+i) \cdot (1+i) = S_0 \cdot (1+i)^2\]

We can use this fact to derive \[S(3)=S(2) \cdot (1+i) = S_0 \cdot (1+i)^2 \cdot (1+i) =S_0 \cdot (1+i)^3\]

More generally, if we know that formula is satisfied for #n-1# instead of #n# (that is: if we know that #S(n-1) = S_0\cdot (1+i)^{n-1}#), then we can deduce the formula itself:

\[S(n) = S(n-1)\cdot (1+i)=S_0\cdot (1+i)^{n-1}\cdot(1+i) = S_0 \cdot (1+i)^n\]

We find \[S(n)=S_0 \cdot (1+i)^n\]

(This way of reasoning is known as complete induction.)

Notation

The future value is also abbreviated as #FV#.

The example below shows what this means in practice.

A capital of #\euro \, 10000# is deposited for #8# years against #1.4\%# interest per year on the basis of compound interest.

Calculate the future value rounded to two decimal places.

Future value: \(\euro \, 11176.44\)

Here, the growth rate is #i = \frac{1.4}{100}=0.014#. The number of periods is #n=8#. According to the formula, the future value is
\[ S_0\cdot\left(1+i\right)^n = 10000 \cdot(1.014)^{8} =10000 \cdot{1.11764}\cdots\approx 11176.44 \]

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