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# Annual equivalent method

In the annual equivalent method of comparison, first the annual equivalent cost or the revenue of each alternative will be computed.

In most of the practical decision environments, executives will be forced to select the best alternative from a set of competing alternatives.

Let us assume that an organization has a huge sum of money for potential investment and there are three different projects whose initial outlay and annual revenues during their lives are known. The executive has to select the best alternative among these three competing projects.

There are several bases for comparing the worthiness of the projects. These bases are:

1.   Present worth method

2.   Future worth method

3.   Annual equivalent method

4.   Rate of return method

ANNUAL EQUIVALENT METHOD

ü  In the annual equivalent method of comparison, first the annual equivalent cost or the revenue of each alternative will be computed.

ü  Then the alternative with the maximum annual equivalent revenue in the case of revenue-based comparison or with the minimum annual equivalent cost in the case of cost- based comparison will be selected as the best alternative.

i.Revenue-Dominated Cash Flow Diagram

A generalized revenue-dominated cash flow diagram to demonstrate the annual equivalent method of comparison is presented in Fig.

Fig. Revenue-dominated cash flow diagram.

In Fig. P represents an initial investment, Rj the net revenue at the end of the th year, and S the salvage value at the end of the nth year.

The first step is to find the net present worth of the cash flow diagram using the following expression for a given interest rate, i:

PW(i)  = + R1/(1 + i)1  + R2/(1 + i)2  + ...

Rj/(1 + ij  + ... + Rn/(1 + i)n  + S/(1 + i)n

In the above formula, the expenditure is assigned with a negative sign and the revenues are assigned with a positive sign.

ii.Cost-Dominated Cash Flow Diagram

A generalized cost-dominated cash flow diagram to demonstrate the annual equivalent method of comparison is illustrated in Fig.

In Fig, P represents an initial investment, Cj the net cost of operation and maintenance at the end of the jth year, and S the salvage value at the end of the nth year.

The first step is to find the net present worth of the cash flow diagram using the following relation for a given interest rate, i.

PW(i) = P + C1/(1 + i)1  + C2/(1 + i)2  + ...

Cj/(1 + ij  + ... + Cn/(1 + i)n   S/(1 + i)n

EXAMPLE

A company provides a car to its chief executive. The owner of the company is concerned about the increasing cost of petrol. The cost per litre of petrol for the first year of operation is Rs. 21. He feels that the cost of petrol will be increasing by Re.1 every year. His experience with his company car indicates that it averages 9 km per litre of petrol. The executive expects to drive an average of 20,000 km each year for the next four years. What is the annual equivalent cost of fuel over this period of time?. If he is offered similar service with the same quality on rental basis at Rs. 60,000 per year, should the owner continue to provide company car for his executive or alternatively provide a rental car to his executive? Assume i = 18%. If the rental car is preferred, then the company car will find some other use within the company.

Solution

Average number of km run/year = 20,000 km

Number of km/litre of petrol = 9 km

Therefore,

Petrol consumption/year = 20,000/9 = 2222.2 litre

Cost/litre of petrol for the 1st year = Rs. 21

Cost/litre of petrol for the 2nd year = Rs. 21.00 + Re. 1.00

= Rs. 22.00

Cost/litre of petrol for the 3rd year  = Rs. 22.00 + Re. 1.00 = Rs. 23.00

Cost/litre of petrol for the 4th year  = Rs. 23.00 + Re. 1.00 = Rs. 24.00

Fuel expenditure for 1st year  = 2222.2    21      = Rs. 46,666.20

Fuel expenditure for 2nd year = 2222.2    22      = Rs. 48,888.40

Fuel expenditure for 3rd year = 2222.2     23      = Rs. 51,110.60

Fuel expenditure for 4th year = 2222.2     24      = Rs. 53,332.80

The annual equal increment of the above expenditures is Rs. 2,222.20

(G). The cash flow diagram for this situation is depicted in Fig.

Fig. Uniform gradient series cash flow diagram.

In Fig., A1 = Rs. 46,666.20 and G = Rs. 2,222.20

A   A1 + G(A/G, 18%, 4)

=  46,666.20 + 2222.2(1.2947)

=   Rs. 49,543.28

The proposal of using the company car by spending for petrol by the company will cost an annual equivalent amount of Rs. 49,543.28 for four years. This amount is less than the annual rental value of Rs. 60,000. Therefore, the company should continue to provide its own car to its executive.

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