As an illustration, suppose a mason acquires a machine at a price of Rs 4,000 for cutting and polishing granite sheets. The lifespan of the machine is three years. The machine yields an income of Rs 1,100 at the end of the first year, Rs 2,420 at the end of the second year and Rs 1,331 at the end of the third year.

Thereafter, it reduces to scrap. Given that the market rate of interest is 10% per annum and it remains unchanged at least over the lifespan of the machine is the mason’s decision to invest. Rs 4000 in the machine wise?

Before we set ourselves to seek an answer through the NPV of the income stream in the illustration let us first understand the concept of present value and net present value in their true connotations.

The present value of an income expected after ‘n’ years, assuming the rate of interest to remain unchanged at ‘r’ per cent per annum throughout the period, may be treated as the principal ‘P’ that is lent out now at ‘r’ per cent per annum compounded annually for ‘n’ years.

Amount accruing to the lender, in consequence, may be treated as the income expected by an investor after ‘n’ years of purchase of a capital asset equal in value to the principal lent and generating a profit at the rate equal to the rate of interest.

For instance, suppose that a money lender lends Rs 10,000 for a period of 3 years at 10% compounded annually. The amount ‘A’ that accrues to him in consequence is given by the compound interest formula,

A = P(1 + r)n

Substituting P = Rs 10,000; r = 10% or 0.10; and n = 3 years in equation 5.2, we have

A = 10,000 (1 + 0.10)3 = Rs 13,310

We can repose the problem in a slightly different way. Suppose the money lender wants to know as to how much to lend out now so as to receive a sum of Rs 13,310 after 3 years at 10% per annum compounded annually. The answer is an obvious one— Rs 10,000! Rewriting equation 5.2 as

That is, an amount of Rs 10,000 today is the same as that of Rs 13,310 three years hence so-long-as the rate of interest is stable at 10% per annum. The former (Rs 10,000) is known as the present value of the latter (Rs 13,130). If the amount of 13,310 is due after 4 years and the rate of interest continues to be 10%, its present value is given as

Note that the present value decreases due to an increase in the time period, other things remaining same. In like manner, if the rate of interest is 12% instead of 10%, the present value of Rs 13,310 expected three years hence is

That is, the present value decreases also due to an increase in the rate of interest, other things remaining the same. The present value, thus, varies inversely with the rate of interest and the time-period. It can also be shown that the present value varies directly with the amount expected, other things remaining the same.

Now, suppose that an investor spends the same amount on purchase and use of a machine and receives Rs 15,000 after 3 years. If the rate of interest is 10%, the present value of this amount is

The present value net of the investment = Rs (11,270 – 10,000) = Rs 1,270.

This is the Net Present Value (NPV) of the investment. NPV represents the excess of the sum of the present values of the future incomes over the cost of investment. In our example, NPV is Rs 1,270. Had the income from the capital investment been only Rs 13,310 instead of Rs 15,000, the present value would have been Rs 10,000 and the NPV would have been zero.

A capital investment that generates annual incomes A1 A2, …, An respectively after 1, 2, …, n years and has a scrap value of ‘S’ at the end of its lifespan of ‘n’ years, given the rate of interest as stable at ‘r’ per cent per annum, the expression for the sum of the present values of the entire income stream can be laid down as

Where, Av A2 A3… An = the incomes expected after 1, 2, 3, …, ‘n’ years respectively

PV = the present value of income expected in future.

‘r’ = the rate of interest compounded annually.

S = the scrap value of the capital asset at the end of its lifespan of ‘n’ years.

Returning back to the illustration in the beginning of this section, the present value of the income stream comprising,

A1 = 1100, the amount expected after 1 year from now,

A2 = 2420, the amount expected after 2 years from now, and

A3 = 1331, the amount expected after 3 years from now.

When the rate of interest is 10%, the sum of the present values may be calculated as

Hence mason’s investment in the machine leads to no profit, no loss. If not wise, it is not even unwise.

If the scrap is disposed off for Rs 500 to a scrap vendor at the end of the lifespan of the machine, it would lead to a further addition to the PV by [500 / (1 + 0.10) 3].

The total PV would then work out at Rs 4375.66 [4,000 + 375.66], The NPV in that case would be 375.66 and the mason’s decision, certainly a wise one.

Investment in a capital asset is not viable when NPV < 0. When NPV > 0, it is certainly rewarding but when NPV = 0 investment decision involves indifference on the part of the investor.

In case the expected annual amounts of income, also called as the annuities, be constant, i.e., A1 = A2 = A3 = … = An = A; and the lifespan of the machine be infinite (i.e., n -?) with no scrap value (i.e., S = 0), expression in 5.4 reduces to

?PV= (A/r)*

Expression in equation 5.5 represents the present value of a perpetual income of ‘A’ per annum when the rate of interest is ‘r’ per cent.

Expression in equation 5.6 below follows from equation 5.4. When the rate of interest changes from year to year.

Equations 5.4, 5.5 and 5.6 provide expressions for the sum of the present values of income streams accruing to an investor under different situations. The value of NPV can be determined by subtracting the cost of investment from PV.

Investment decisions of a firm depend on this value of NPV. If NPV > 0, investment is viable. If more than one investment options are available, one with highest NPV is chosen by the investor.