Value and return in terms of present value

How compounding technique can be used for adjusting for the time value of money.It increases an investor analytical power to compare cash flows that are separated more than one period,given her interest rate pr period.With the compounding technique ,the amount of present cash can be converted into their present value.Present value of a future cash flows (inflow or outflow) is the amount of current cash that is of equivalent value to the decision maker.Discounting is the process of determining present values of a series of future cash flows .The compound interest rate used for discounting cash flows is also called the discount rate.

Present value of a single cash flow
Now we assume Rs 1\times 1.10^{2} =Rs.1.21 after two years,or Re 1\times 1.10^{3}  = Rs.1.33 after years.How much would the investor give up now to get an amount of Rs.1 at the end of one year.Assuming a 10 percent interest rate ,we know that an amount sacrificed in the beginning of year will grow to 110 percent or 110 percent or 1.10 after a year.Thus the amount to be sacrificed today would be 1/1.10=Rs.0.909.In other words,at a 10 percent rate ,Re 1 to be received after a year is 110 percent of Re.0.909 sacrificed now,Stated differently ,Rs.0.909 deposited now at 10 percent rate of interest will grow to Rs.1 after one year.If 1 is received after two years,then the amount needed to be sacrificed today would be :1/1.10^{2}
How can we express the present value calculation formally?Let i represent the interest rate per period,n the number of periods,F the future value (or cash flow) and P the present value (cash flow).We know the future value after one year ,F_{1}(present value (principal) plus interest),will be.
F_{1}=P(1+i)
The present value,P, will be equal to
p=\frac{F_{1}}{(1+i)^{1}}

The future value after two years is
F_{2}=P(1+i)^{^{2}}

The present value ,P, will be
p=\frac{F_{2}}{(1+i)^{2}}

The present value can be worked out for any combination number of years and interest rate .The following general formula can be employed to calculate the present value of a lump sum to be received after some future periods:

 p=\frac{F_{n}}{(1+i)^{n}} =  F_{n}[(1+i)^{^{-n}}]

 p=F_{n}[\frac{1}{(1+i)^{n}}]

The term is parentheses is the discount factor or present value factor(PVF), and it is always less than 1.0 for positive i, indicating that a future amount has a smaller present value .
Present value= Future value*Present value factor of Re1
 PV=F_{n_{}}\times PVF_{n,i}

 PVF_{n,i} is the present value factor for n periods at i rate of interest.

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