Equivalent worth method and rate of return method

Equivalent worth Methods:

Present worth Method:

The Net Present worth (NPW) or Present worth or Net Present Value (NPV) of a given series of cash flow is the equivalent values of the cash flows at the end of year zero i.e. the beginning of year 1. In other words, how much money we have to set aside to provide for future cash flows.

Net Present worth = Equivalent present worth of future cash flow – Initial investment

Under NPW method, PW of all cash inflows are compared with PW of all cash outflows. The difference between these PV’s is called Net Present Value (NPW).

The basic procedure to apply the NPW criterion is stated below.

1. Determine the interest rate that the firm wishes to earn on its investments which are often referred to as either a required rate of return or a minimum attractive rate of return (MARR).
2. Estimate the service life of a project.
3. Estimate the cash inflow and cash outflow for each period over the service life.
4. Determine the net cash flow for each period.

Net cash flow = Cash inflow – Cash outflow

5. Find the present worth of each net cash flow at the MARR and add up the present worth figures hence their sum is defined as the project’s NPW which is given by the expression below.

Where, A_n = net cash flow at the end of period n

i = MARR

N = Service life of the project

Here, An will be positive if the corresponding period has a net cash inflow and negative if there is a net cash outflow.

In a single project evaluation, the decision rule is:

If PW(i) > 0, accept the investment

If PW(i) = 0, remain indifferent

If PW(i) < 0, reject the investment

In comparing mutually exclusive alternatives, select the one with the largest PW(i). When comparing with the same revenues, they are compared on a cost only basis. In this situation, you should agree on the project that results in the smallest or least negative NPW (because you are minimizing cost rather than maximizing profit).

Future worth Method:

NPW measures the surplus in an investment project at time ‘0’. But Net future worth (NFW) measures the surplus or equivalent worth or value of a project at the end of the investment period.

The net future worth (NFW) expression at the end of period ‘N’ is given by,

The decision rule is same as that of NPW criterion. For a single project evaluation,

If FW(i) > 0, accept the investment

If FW(i) = 0, remain indifferent

If FW(i) < 0, reject the investment

Annual worth method:

The annual equivalent worth criterion provides a foundation for measuring the value of an investment by determining equal payments on an annual basis. Knowing that any lump sum cash amount can be transformed into a series of equal payments, we may first find the net present worth (NPW) of the original cash series and then multiply this amount by the capital recovery factor:

AW (i) = PW(i).(A/P, i, N)

Also, AW of a project is its annual equivalent receipts (R) minus annual equivalent expenses (E) minus the annual equivalent capital recovery (CR) at the certain specific MARR.

i.e. AW(i) = R – E – CR

The decision rule from this method is also same as that of NPW and NFW.

If AW(i) > 0, accept the investment

If AW(i) = 0, remain indifferent

If AW(i) < 0, reject the investment

Rate of Return Methods:

The rate of Return (ROR) is the interest rate earned on the unpaid balance of an amortized (installment) loan. Suppose that bank lends Rs.10,000 and is repaid Rs.4,021 at the end of each year for three years. As we learned in the earlier chapter, we would set up the equivalence equation as 10,000 = 4,021(P/A, i, 3) and solving we get, i = 10%.

The rate of return is the break-even interest rate i* that equates the present worth of a project’s cash inflows to the present worth of its cash outflows.

Mathematically, PW(i*) = PW_{cash inflows} – PW_{cash ouflows} = 0 and solve for i*.

Internal Rate of Return (IRR):

A project’s return is referred to as the internal rate of return (IRR) promised by an investment project over its useful life. The internal rate of Return (IRR) is the interest earned on the unrecovered project balance of investment such that when the project terminates, the unrecovered project balance will be zero. In more familiar terms, the IRR of an investment is the interest rate at which the costs of the investment lead to the benefits of the investment.

NPV = A₀/ (1 + i*)⁰ + A₁/ (1 + i*)¹ + A₂/ (1 + i*)² + …… + A_n/ (1 + i*)ⁿ = 0

Solve for i* which is required IRR.

Decision rule:

If IRR > MARR, accept the project

If IRR = MARR, remain indifferent

If IRR < MARR, reject the project

We classify the investment into two types.

1. Simple Investment: It is defined as that investment when the sign change in the project cash flow occurs only once.
2. Non-simple Investment: It is that investment where the sign change in the project cash flow occurs more than once.

Methods of determining IRR:

1. Direct Solution Method: For a project with only a two flow transaction or a project with a service life of return in two years, we can find the direct mathematical solution for determining the rate of return.
2. Trial and error method: At first, we guess the interest rate to compute the present worth of net cash flows and notice whether it is positive, negative or zero. Here we are aiming to make PW = 0. If PW is negative, we must raise the PW for which the interest rate is to be lowered and repeat the process. If the PW is positive, we raise the interest rate to lower PW. Whenever we reach the point where PW is bounded by one negative and one positive value, we use linear interpolation to approximate i*. This process somewhat tedious and inefficient.
3. Graphical Method: Here we create the NPW profile calculating the value of NPW at different interest rates. Points are plotted and curves are sketched where the horizontal axis indicates the interest rate and vertical indicates the NPW. The point observed at which the curve crosses the horizontal axis closely approximates i*. The graphical approach works for both simple and non-simple investments.

External Rate of Return (ERR)/ Modified IRR:

The drawbacks of the IRR method (reinvestment assumption) may not be valid in the engineering economy. For example, if a firm’s MARR is 25 % per year and the IRR for a project is 45.4 %, it may not be possible for the firm to reinvest net cash proceeds from the project at much more than 25%. This situation, coupled with the computational demands and possible multiple interest rates associated with the IRR method, has given rise to another rate of return methods than can remedy this weakness which is referred to as External Rate of Return or Modified IRR.

The external rate of return (i’) is the unique rate of return for a project that assumes that net positive cash flows, which represent money not immediately needed by the project, are reinvested at the reinvestment rate ε%. The reinvestment rate depends upon the market rate available for investments.

Steps for ERR calculation:

1. All cash outflows are discounted to period zero (present) at ε% per compounding period.
2. All cash inflows are compounded to period N at ε%
3. This net cash outflows at present are converted to future values at the rate of i’ and equated to future net cash inflows found earlier and solved for ERR (i’).

Present Net cash Inflows × (F/P, i’, N) = Future Net Cash Outflows

Decision Rule:

If ERR > MARR, accept the project

If ERR = MARR, remain indifferent

If ERR < MARR, reject the project

The advantages of ERR over IRR are as follows:

1. It does not need trial and error approach for determination of i’%
2. There is no possibility of multiple rates of return

BIBLIOGRAPHY:

Chan S.Park, Contemporary Engineering Economics, Prentice Hall, Inc.
E. Paul De Garmo, William G.Sullivan and James A. Bonta delli, Engineering
Economy, MC Milan Publishing Company.
James L. Riggs, David D. Bedworth and Sabah U. Randhawa,Engineering
Economics, Tata MCGraw Hill Education Private Limited.