FUTURE VALUE

DEFINITION

Future value refers to the value of a sum of money at a particular point of time in the future.
Due to the interest-earning capacity of money, a sum of money invested now will accumulate to a greater amount in the future.

Cashflows:

Timing of cashflows:
select timing
Beginning of the time period
End of the time period

Rate of interest: (%)

Present value of cashflows:

Consider a sum of money \(c\) invested into a bank account that pays interest at the rate \(i\). If \(c\) is invested at the beginning of the year, an amount of \(c+ic\) would be accumulated by the end of the year(\(ic\) is the amount of interest paid on \(c\). So, the future value of \(c\) after one year is

\[FV=c+ic=c\left(1+i\right)\]

In the second year, this amount of \(c\left(1+i\right)\) will again accumulate at the rate \(i\). Thus, the future value of this amount by the end of second year would be

\[FV=c\left(1+i\right)\times \ \left(1+i\right)=c{\left(1+i\right)}^2\]

Similarly, for a sum of money \(c\) invested now, the future value at the end of \(n\) years is

\[FV=c{\left(1+i\right)}^n\]

Future value of a series of cashflows:

Consider a series of cash flows c_{t} occurring at times \(t=0,1,2,\dots \dots \dots n.\) The future value of this series of cash flows will be the sum of future values of each cash flow.

\[FV=\ c_o{\left(1+i\right)}^n+c_1{\left(1+i\right)}^{n-1}+c_2{\left(1+i\right)}^{n-2}\dots \dots \dots c_{n-1}{\left(1+i\right)}^1+c_n\]
\[FV=\ \sum^n_{t=1}{c_t}{\left(1+i\right)}^{n-t}\]

Example 1

A cash flow of \(\textrm{₹}\ 100\) occurs today. The future value of this cash flow at \(8\ \%\) at the end of three years from now would be

\[FV=c{\left(1+i\right)}^3\]
\[FV=100{\left(1+0.08\right)}^3=100{\left(1.08\right)}^3\]
\[FV=125.97\]

Example 2

A cash flow of\(\mathrm{\textrm{₹}}\mathrm{\ 100}\) will occur at the end of one year from today. The future value of this cash flow at \(8\ \%\) at the end of three years from now would be

\[FV=c{\left(1+i\right)}^{\left(n-1\right)}\]
\[FV=100{\left(1+0.08\right)}^{\left(3-1\right)}=100{\left(1.08\right)}^2\]
\[FV=116.64\]

Since the cash flow is occurring at the end of the first year, It is accumulated only during the second and third year and hence the accumulation factor is\({\left(1+i\right)}^2\) instead of \({\left(1+i\right)}^3\)

Example 3

The following payments are received at the beginning of every year for the next 6 years:-

\(\mathrm{\$1000,\ \$2000,\ \$3000,\ \$2500,\ \$2000,\ \$1500}\)

The future value of this series of cash flows at the rate of \(10\ \%\) at the end of the sixth year would be

\[FV=1000{\left(1.1\right)}^6+2000{\left(1.1\right)}^5+3000{\left(1.1\right)}^4+2500{\left(1.1\right)}^3+2000{\left(1.1\right)}^2+1500{\left(1.1\right)}^1\]
\[FV=16782.38\]

Example 4

Consider the same series of cash flows as in example 3. But now, these cash flows occur at the end of every year. The future value at \(10\ \%\) at the end of the sixth year would be

\[FV=1000{\left(1.1\right)}^5+2000{\left(1.1\right)}^4+3000{\left(1.1\right)}^3+2500{\left(1.1\right)}^2+2000{\left(1.1\right)}^1+1500\]
\[FV=15256.71\]