Cash Flow: Bases for Comparison of Alternatives

CASH FLOW

Introduction

In this method of comparison, the cash flows of each alternative will be reduced to time zero by assuming an interest rate i. Then, depending on the type of decision, the best alternative will be selected by comparing the present worth amounts of the alternatives.

The sign of various amounts at different points in time in a cash flow diagram is to be decided based on the type of the decision problem.

In a cost dominated cash flow diagram, the costs (outflows) will be assigned with positive sign and the profit, revenue, salvage value (all inflows), etc. will be assigned with negative sign.

In a revenue/profit-dominated cash flow diagram, the profit, revenue, salvage value (all inflows to an organization) will be assigned with positive sign. The costs (outflows) will be assigned with negative sign.

In case the decision is to select the alternative with the minimum cost, then the alternative with the least present worth amount will be selected. On the other hand, if the decision is to select the alternative with the maximum profit, then the alternative with the maximum present worth will be selected.

BASES FOR COMPARISON OF ALTERNATIVES

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

1.PRESENT WORTH METHOD

ü  In this method of comparison, the cash flows of each alternative will be reduced to time zero by assuming an interest rate i.

ü  Then, depending on the type of decision, the best alternative will be selected by comparing the present worth amounts of the alternatives.

In a cost dominated cash flow diagram, the costs (outflows) will be assigned with positive sign and the profit, revenue, salvage value (all inflows), etc. will be assigned with negative sign.

ü  In a revenue/profit-dominated cash flow diagram, the profit, revenue, salvage value (all inflows to an organization) will be assigned with positive sign. The costs (outflows) will be assigned with negative sign.

a.Revenue-Dominated Cash Flow Diagram

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

To find the present worth of the above cash flow diagram for a given interest rate, the formula is

PW(i) =  – P + R1[1/(1 + i)¹] + R2[1/(1 + i)²] + ...

+  Rj [1/(1 + i) ^j] + Rn[1/(1 + i)ⁿ] + S[1/(1 + i)ⁿ]

b.Cost-Dominated Cash Flow Diagram

A generalized cost-dominated cash flow diagram to demonstrate the present worth method of comparison is presented in Fig.

To compute the present worth amount of the above cash flow diagram for a given interest rate i, we have the formula

PW(i) = P + C1[1/(1 + i)¹] + C2[1/(1 + i)²] + ... + Cj[1/(1 + i) ^j]

+ Cn[1/(1 + i)ⁿ] – S[1/(1 + i)ⁿ]

EXAMPLE

Alpha Industry is planning to expand its production operation. It has identified three different technologies for meeting the goal. The initial outlay and annual revenues with respect to each of the technologies are summarized in Table 1. Suggest the best technology which is to be implemented based on the present worth method of comparison assuming 20% interest rate, compounded annually.

Solution

In all the technologies, the initial outlay is assigned a negative sign and the annual revenues are assigned a positive sign.

TECHNOLOGY 1

Initial outlay, P = Rs. 12,00,000

Annual revenue, A = Rs. 4,00,000

Interest rate, i = 20%, compounded annually

Life of this technology, n = 10 years

The cash flow diagram of this technology is as shown in Fig. 4.3.

Fig. Cash flow diagram for technology 1.

The present worth expression for this technology is

PW(20%)1  = –12,00,000 + 4,00,000 (P/A, 20%, 10)

= –12,00,000 + 4,00,000 (4.1925)

= –12,00,000 + 16,77,000

= Rs. 4,77,000

TECHNOLOGY 2

Initial outlay, P = Rs. 20,00,000

Annual revenue, A = Rs. 6,00,000

Interest rate, i = 20%, compounded annually

Life of this technology, n = 10 years

The cash flow diagram of this technology is shown in Fig. 4.4.

Fig.    Cash flow diagram for technology 2.

The present worth expression for this technology is

PW(20%)2  = – 20,00,000      + 6,00,000  (P/A, 20%, 10)

= – 20,00,000       + 6,00,000  (4.1925)

=       – 20,00,000 + 25,15,500

=       Rs. 5,15,500

TECHNOLOGY 3

Initial outlay, P = Rs. 18,00,000

Annual revenue, A = Rs. 5,00,000

Interest rate, i = 20%, compounded annually

Life of this technology, n = 10 years

The cash flow diagram of this technology is shown in Fig. 4.5.

Fig.    Cash flow diagram for technology 3.

The present worth expression for this technology is

PW(20%)3  = –18,00,000       + 5,00,000  (P/A, 20%, 10)

= –18,00,000        + 5,00,000  (4.1925)

=       –18,00,000 + 20,96,250

=       Rs. 2,96,250

From the above calculations, it is clear that the present worth of technology 2 is the highest among all the technologies. Therefore, technology 2 is suggested for implementation to expand the production.

2.  FUTURE WORTH METHOD

ü  In the future worth method of comparison of alternatives, the future worth of various alternatives will be computed.

ü  Then, the alternative with the maximum future worth of net revenue or with the minimum future worth of net cost will be selected as the best alternative for implementation.

i.Revenue-Dominated Cash Flow Diagram

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

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

The formula for the future worth of the above cash flow diagram for a given interest rate, i is

FW(i) = –P(1 + i)ⁿ  + R1(1 + i)^n–1  + R2(1 + i)^n–2  + ...

+ R j(1 + i)^n–j  + ... + Rn + S

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

ii.Cost-Dominated Cash Flow Diagram       

A generalized  cost-dominated  cash      flow  diagram       to  demonstrate  the  future worth method

of comparison is given in Fig.

In Fig. 5.2, P represents an initial investment, Cj the net cost of operation and maintenance at the end of the j th year, and S the salvage value at the end of the nth year.

The formula for the future worth of the above cash flow diagram for a given interest rate, i is

FW(i) = P(1 + i)ⁿ + C1(1 + i )^n–1 + C2(1 + i)^n–2 + ...  + Cj(1 + i)^n–j  + ... + Cn – S

EXAMPLE

Consider the following two mutually exclusive alternatives:

At i = 18%, select the best alternative based on future worth method of comparison.

Solution Alternative A

Initial investment, P = Rs. 50,00,000

Annual equivalent revenue, A = Rs. 20,00,000

Interest rate, i = 18%, compounded annually

Life of alternative A = 4 years

The cash flow diagram of alternative A is shown in Fig.

The future worth amount of alternative B is computed as

FW_A(18%)  = –50,00,000(F/P, 18%, 4) + 20,00,000(F/A, 18%, 4)

=  –50,00,000(1.939) + 20,00,000(5.215)

 = Rs. 7,35,000

Alternative B

Initial investment, P = Rs. 45,00,000

Annual equivalent revenue, A = Rs. 18,00,000

Interest rate, i = 18%, compounded annually

Life of alternative B = 4 years

The cash flow diagram of alternative B is illustrated in Fig..

The future worth amount of alternative B is computed as

FW_B(18%) = – 45,00,000(F/P, 18%, 4) + 18,00,000 (F/A, 18%, 4)

=  – 45,00,000(1.939) + 18,00,000(5.215)

=  Rs. 6,61,500

3.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 j 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)  = –P + R1/(1 + i)¹  + R2/(1 + i)²  + ...

+ Rj/(1 + i) ^j  + ... + 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 + i) ^j  + ... + 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.

4.RATE OF RETURN METHOD

ü The rate of return of a cash flow pattern is the interest rate at which the present worth of that cash flow pattern reduces to zero.

ü In this method of comparison, the rate of return for each alternative is computed. Then the alternative which has the highest rate of return is selected as the best alternative.

ü A generalized cash flow diagram to demonstrate the rate of return method of comparison is presented in Fig

In the above cash flow diagram, P represents an initial investment, Rj the net revenue 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 expression at a given interest rate, i.

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

+ Rj/(1 + i) ^j  + ... + Rn/(1 + i)ⁿ  + S/(1 + i)ⁿ

EXAMPLE

A person is planning a new business. The initial outlay and cash flow pattern for the new business are as listed below. The expected life of the business is five years. Find the rate of return for the new business.

Solution

Initial investment = Rs. 1,00,000 Annual equal revenue = Rs. 30,000 Life = 5 years

The cash flow diagram for this situation is illustrated in Fig.

Fig.  Cash flow diagram.

The present worth function for the business is

PW(i) = –1,00,000 + 30,000(P/A, i, 5)

When  i  =  10%,

PW(10%)  = –1,00,000 + 30,000(P/A, 10%, 5)

=  –1,00,000 + 30,000(3.7908)

=      Rs. 13,724.

When  i  =  15%,

PW(15%) =  –1,00,000  +  30,000(P/A,  15%,  5)

=    –1,00,000  +  30,000(3.3522)

=    Rs.  566.

When i = 18%,

PW(18%)  = –1,00,000 + 30,000(P/A, 18%, 5)

=    –1,00,000  +  30,000(3.1272)

=    Rs. – 6,184

i=  15% + 0.252% = 15.252%

Therefore, the rate of return for the new business is 15.252%.