FORMULA AND DERIVATION
Consider a sum of money, \(a\), invested into a bank account that pays interest at the rate \(i\). If \(a\) is invested at the beginning of the year then it will accumulate to \(a\left(1+i\right)\) by the end of the year. Consider this accumulated amount as \(c\). So,
\[a\left(1+i\right)=c\]
The above equation can be explained as follows:-
The future value of \(a\) is \(c\)
or
The present value of \(c\) is \(a\).
The second explanation is of greater interest to us. It tells us that if an amount of \(c\) is required at the end of the year, we need to invest an amount, \(a\), at the beginning of the year. This means that an amount of \(c\), due one year from now, is currently worth \(a\), and thus \(a\) is the present value of \(c\).
\[a\left(1+i\right)=c\] \[\Rightarrow a=\frac{c}{\left(1+i\right)}\]
So, the formula for the present value of a cash flow \(c\), due one year from now, is
\[PV=\frac{c}{1+i}\]
\[PV=cv\]
Where, \(v=\frac{1}{1+i}={\left(1+i\right)}^{-1}\)
Similarly, the formula for the present value of a cash flow \(c\), due \(n\) years from now is
\[PV=\frac{c}{{\left(1+i\right)}^n}=c{\left(1+i\right)}^{-n}={cv}^n\]
Present value of a series of cashflows
Consider a series of cash flows\(c_t\) occurring at times \(t=0,1,2,3,\dots .n\). The present value of this series of cash flows will be the sum of present values of each cash flow.
\[PV=c_0+c_1{\left(1+i\right)}^{-1}+c_2{\left(1+i\right)}^{-2}\dots \dots \dots c_n{\left(1+i\right)}^{-n}\]
\[=c_o+c_1v^1+c_2v^2\dots \dots \dots c_nv^n\]
\[=\sum^n_{t=0}{c_tv^t}\]
\(c_0\) is not multiplied by the discount factor, \(v\), because the present value of a cash flow occurring now, is simply the value of that cash flow.
EXAMPLES
Example 1
A cash flow of \(\textrm{₹}100\) is due at the end of the fifth year from now. A rate of \(10\%\) p.a. is considered to discount the cash flows to their present values.
Here,
\[c=100\]
\[n=5\]
\[i=10\%p.a\]
\[PV={cv}^n\]
\[PV=100v^n=100{(1+0.1)}^{(-5)}\]
\[PV=100{(1.1)}^{(-5)}\]
\[PV=62.09\]
This means that if \(\textrm{₹}62.09\) is invested today, it will be worth \(\textrm{₹}100\) in 5 years’ time, at \(10\%\) p.a.
Example 2
A cash flow of \(\textrm{₹}100\) is due at the beginning of the fifth year from now. The discount rate is \(10\%\) p.a.
\[PV={cv}^{\left(n-1\right)}\]
\[ PV=100v^4=100{(1+0.1)}^{(-4)}\]
\[PV=100{(1.1)}^{(-4)}\]
\[PV=68.3\]
Here, the cash flow is discounted only for 4 years because the cash flow is occurring at the beginning of the fifth year, which is the same as a cash flow of equal amount occurring at the end of the fourth year.
Example 3
\(\textrm{₹}1000\), \(\textrm{₹}2000\), \(\textrm{₹}3000\), \(\textrm{₹}2500\), \(\textrm{₹}2000\), \(\textrm{₹}1500\) are received for the next 6 year at the end of every year. Considering a discount rate of \(10\%\) p.a. the present value of this series of cash flows would be
\[PV=1000v+2000v^2+3000v^3+2500v^4+2000v^5+1500v^6\]
\[PV=8612.02\]
Example 4
Consider the same cash flows as in example 3. But now, the cash flows arise at the beginning of every year. The discount rate is \(10\%\) p.a.
\[PV=1000+2000v+3000v^2+2500v^4+1500v^5\]
\[PV=9473.22\]
The cash flow of \(\textrm{₹}1000\) is not discounted because it is occurring at present time (at the beginning of the year, i.e. now). So, its present value is just \(\textrm{₹}1000\).
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