Present value

Exponential and logarithmic growth: Future value

Present value

We have already seen how to calculate the future value of a capital after several years in a bank based on a compound interest. It is also interesting to consider the amount we have to put in the bank in order to have a certain amount at our disposal, after it has been sitting in the bank for a selected number of years and has increased in value on the basis of compound interest.

Present value

The present value or start value of a capital #S# is the initial capital #S(0)# needed on the basis of compound interest in order to obtain #S_n# as the future value after a certain duration of #n# periods.

The question: "After #15# years Benjamin wants to have #\euro \, 100000# in the bank on the basis of compound interest of #1\%#. How much should he deposit in the bank?" is a question regarding present value.

The notation employed here deviates a little from the notation of the formula for future value: here we use #S_n# instead of #S(n)# and #S(0)# instead of #S_0#. The expressions #S_0# and #S_n# indicate values that are being given when the problem is posed and #S# is the function of #n#, so #S(n)# is the value at #n# of the function #S#. For the future value we have #S(0) = S_0# and here, for the present value, #S(n)=S_n#.

Notation

The present value is abbreviated as #PV#.

For calculating the present value we can use the same formula as for calculating the future value. Below we give a rewritten form of this formula.

Calculating present value

The present value #S(0)# of a capital #S_n# at duration of #n# periods and growth rate #i# is:

\[S(0)=\frac{S_n}{(1+i)^n}\]

The formula for the value is:

\[S_n=S(0) \cdot (1+i)^n\]

Dividing the left and right hand sides by #(1+i)^n# we find:

\[S(0)=\frac{S_n}{(1+i)^n}\]

This is the formula for calculating the present value (or start value).

The factor #\frac{1}{(1+i)^n}#, also written as #(1+i)^{-n}#, is sometimes also denoted by #A_{\left .n\right \rceil i}#.

In general, we have: #A_{\left .n\right \rceil i}=\frac{1}{S_{\left .n\right \rceil i}}#. This relation can also be written as #A_{\left .n\right \rceil i} \cdot S_{\left .n\right \rceil i}=1#.

In the examples below, we see how this works in practice.

Calculate the present value of capital #\euro \, 40000# at a compound interest of #2.0\%# per annum and duration of #10# years.

The present value is: #\euro# #32813.93#

In order to calculate the present value we use the formula #S(0)=\frac{S_n}{(1+i)^n}#, where #S(0)# is the present value, #S_n# is the terminal value, #i# is the growth rate, and #n# is the duration.
In this case, the final value equals #S_n =40000#, the growth rate #i=\frac{2.0}{100}=0.020#, and the duration is #n=10#. This gives:
\[\begin{array}{rcl}
S(0)&=& \dfrac{S_n}{(1+i)^n}\\
&& \phantom{xxxxx}\color{blue}{\text{the formula}}\\
S(0)&=& \dfrac{40000}{(1+0.020)^{10}}\\
&& \phantom{xxxxx}\color{blue}{\text{values substituted}}\\
S(0)&=&\displaystyle 32813.93 \\
&& \phantom{xxxxx}\color{blue}{\text{calculated}}\\
\end{array}\]
Thus, the present value is #\euro \, 32813.93#.

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