[savio13]

The economic evaluation of investment proposals
The time value of money

Recall that the interaction of lenders with borrowers sets an equilibrium rate of interest. Borrowing is only worthwhile if the return on the loan exceeds the cost of the borrowed funds. Lending is only worthwhile if the return is at least equal to that which can be obtained from alternative opportunities in the same risk class.

The interest rate received by the lender is made up of:

i) The time value of money: the receipt of money is preferred sooner rather than later. Money can be used to earn more money. The earlier the money is received, the greater the potential for increasing wealth. Thus, to forego the use of money, you must get some compensation.
ii) The risk of the capital sum not being repaid. This uncertainty requires a premium as a hedge against the risk; hence the return must be commensurate with the risk being undertaken.
iii) Inflation: money may lose its purchasing power over time. The lender must be compensated for the declining spending/purchasing power of money. If the lender receives no compensation, he/she will be worse off when the loan is repaid than at the time of lending the money.

a) Future values/compound interest
Future value (FV) is the value in Rs. at some point in the future of one or more investments.

FV consists of:

i) the original sum of money invested, and
ii) the return in the form of interest.
The general formula for computing Future Value is as follows:
FVn = Vo (l + r)n
Where
Vo is the initial sum invested
r is the interest rate
n is the number of periods for which the investment is to receive interest.
Thus we can compute the future value of what Vo will accumulate to in n years when it is compounded annually at the same rate of r by using the above formula.

Future values/compound interest

i) What is the future value of Rs.10 invested at 10% at the end of 1 year?
ii) What is the future value of Rs.10 invested at 10% at the end of 5 years?
We can derive the Present Value (PV) by using the formula:
FVn = Vo (I + r)n
By denoting Vo by PV we obtain:
FVn = PV (I + r)n
by dividing both sides of the formula by (I + r)n we derive:

Rationale for the formula:

As you will see from the following exercise, given the alternative of earning 10% on his money, an individual (or firm) should never offer (invest) more than Rs.10.00 to obtain Rs.11.00 with certainty at the end of the year.

b) Net present value (NPV)
The NPV method is used for evaluating the desirability of investments or projects.


where:
Ct = the net cash receipt at the end of year t
Io = the initial investment outlay
r = the discount rate/the required minimum rate of return on investment
n = the project/investment's duration in years.
The discount factor r can be calculated using:

Examples:

Decision rule:

If NPV is positive (+): accept the project
If NPV is negative(-): reject the project

Net present value Example

A firm intends to invest Rs.1,000 in a project that generated net receipts of Rs.800, Rs.900 and Rs.600 in the first, second and third years respectively. Should the firm go ahead with the project?

c) Annuities

A set of cash flows that are equal in each and every period is called an annuity.
Example:
Year Cash Flow (Rs.)
0 -800
1 400
2 400
3 400
PV = Rs.400(0.9091) + Rs.400(0.8264) + Rs.400(0.7513)
= Rs.363.64 + Rs.330.56 + Rs.300.52
= Rs.994.72
NPV = Rs.994.72 - Rs.800.00
= Rs.194.72
Alternatively,
PV of an annuity = Rs.400 (PVFAt.i) (3,0,10)
= Rs.400 (0.9091 + 0.8264 + 0.7513)
= Rs.400 x 2.4868
= Rs.994.72
NPV = Rs.994.72 - Rs.800.00
= Rs.194.72

d) Perpetuities
Perpetuity is an annuity with an infinite life. It is an equal sum of money to be paid in each period forever.

where:
C is the sum to be received per period
r is the discount rate or interest rate

Example:

You are promised a perpetuity of Rs.700 per year at a rate of interest of 15% per annum. What price (PV) should you be willing to pay for this income?
PV = 700/0.15
= Rs.4,666.67

A perpetuity with growth:

Suppose that the Rs.700 annual income most recently received is expected to grow by a rate G of 5% per year (compounded) forever. How much would this income be worth when discounted at 15%?
Solution:
Subtract the growth rate from the discount rate and treat the first period's cash flow as a perpetuity.

= 700(1.05)/(0.15-0.05)
= Rs.735/0.10