Present financial position and performance of the …



Net Present Value Analysis

To capture the time value of money when making investment decisions, we can use a capital budgeting technique that accounts for the present value of the entire stream of net cash flows over the life of the project. One such technique is the net present value method. In the case where the discount rate (R) is expected to remain constant over the entire economic life of the investment project (i.e., R1 = R2 = … = RN), the net present value of an investment project (NPV) is given by

1) NPV = NCF1[1/(1+R)] + NCF2[1/(1+R)2] + …. + NCFN[1/(1+R)N] - C

where NCF1 represents the annual net cash flow generated by the project in year 1, NCF2 represents the net cash flow generated by the project in year 2, etc., R is the discount rate and N is the number of years in the life of the project. Finally, C represents the original cash purchase price less any cash discounts (but not the trade-in value of used machinery deducted from the purchase price at the time of the purchase).

The net present value of an investment project can be viewed as the “profit” or dollar measure of the amount saved by making this investment now. Given the assumptions of profit maximization and complete certainty, you should accept those projects whose net present values are positive (i.e., NPV > 0). You will be indifferent between whether or not to invest when the net present value equals zero (i.e., NPV = 0), and you should reject all projects whose net present value is negative (i.e., NPV < 0).

Let us assume that you expect a constant discount rate of 5 percent over the 5-year economic lives of a particular investment project; here project A. The net present value for this project is reported in Table 1 below:

Table 1 - Net Present Value for Project A.

(1) (2) 3)

Net Cash Present Value Present Value

Flow Interest Factors of NFCi

Year (NCFi) PIF0.05,I (1) x (2)

1 $ 2,000 0.952 $ 1,904

2 3,000 0.907 2,721

3 5,000 0.864 4,320

4 2,000 0.823 1,646

5 1,000 0.784 784

$ 13,000 $ 11,375

Less initial cost - 10,000

Net present value $ 1,375

This table shows, for example, that project A has a positive net present value over the 5-year period. Thus we would consider the project feasible from a economic perspective. Keep in mind that we do not know yet whether it fits within out capital budget constraint.

Composition of net cash flows

The net cash flows generated by an investment project represent the net change in the annual net cash flows generated by the firm’s existing resources, or before the investment is made. These annual net cash flows require a forward assessment of all the forces that affect future net cash flows over the economic life of the investment. The following table will help make its measurement clear:

Table 2 – Measuring annual net cash flows.

| |Before new investment|After new |Net change |

|Item: | |investment | |

|1. Cash receipts |$25,000 |$30,000 |$5,000 |

|2. Cash operating expenses |-15,000 |-18,000 |-3,000 |

|3. Depreciation |-3,000 |-4,000 |-1,000 |

|4. Tax deductible expenses (2 + 3) |18,000 |22,000 |4,000 |

|5. Taxable income (1 - 4) |7,000 |8,000 |1,000 |

|6. Income tax payments (5 x rate) |1,750 |2,000 |250 |

|7. Net income after taxes (1 – 4 – 6) |5,250 |6,000 |750 |

|8. Net cash flow (7 + 3) |8,250 |10,000 |1,750 |

The additional cash receipts generated by the investment project are $5,000 annually over the life of the project. The firm’s cash operating expenses (i.e., fuel expenses, hired labor expenses, fertilizer and chemical expenses) are expected to increase $3,000 annually while depreciation expenses are expected to increase by $1,000. This means the firm’s tax deductible expenses will be $4,000 higher than the current level if the investment project is undertaken. Subtracting these tax deductible expenses from cash receipts results in an increase in taxable income of $1,000 annually.

If the tax rate is equation to 25 percent, income tax payments would be $250 higher annually, giving a net income after taxes of $750 annually. Finally, we have to add back in the depreciation expenses used to compute tax deductible expenses (a non-cash flow) to measure the annual net cash flows associated with the project. Table 2 above indicates that this annual net cash flow would be equal to $1,750.

Note this will allow us to use equation (2) when computing the net present value of the project. Why? The annual net cash flows are expected to be identical over the entire economic life of the project.

Economic and service life

The economic life of an investment project represents the length of time the firm intends to hold the assets acquired. It does not necessarily represent the service life (sometimes called the useful life) of the assets, or the amount the amount of time taken before they wear out. For example, suppose that the firm plans to purchase a piece of equipment that normally wears out over a ten-year period but only plans to hold this piece of equipment for three years. Thus, the economic life of the investment is three years while the service life of the equipment is ten years. This distinction is important when accounting for the effects that the terminal value of a project has upon its feasibility.

Original and terminal value

Information is also needed on the original net capital outlay when the assets are acquired as well as their terminal or market value at the end of the economic life of the investment project. The net capital outlay was represented by C in equation (1). This value should be the net cash purchase price after any cash discounts.

The terminal value of assets purchased in an investment project must also be accounted for before the net present value of the project can be determined. This value may be nothing more than the salvage value in the case of depreciable assets if the economic life of the investment project coincides with the service life of these assets.

If the economic life is less than the service life, the terminal value may play a key role in determining the economic feasibility of the project. This value if what you expect to be able to sell these assets at the end of the project’s life. If land is involved, the terminal value may be considerably higher than its original value at the beginning of the project’s economic life.

Let T represent the terminal value of an asset at the end of the economic life of an investment project. Because the firm receives this value at a specific point in the future, we must discount this value back to the present. We can modify equation (1) to include this discounted terminal value as follows:

2) NPV = NCF1[1/(1+R)] + NCF2[1/(1+R)2] + .. + NCFN[1/(1+R)N] – C

+ T[1/(1+R)N]

The selection of an appropriate discount rate when calculating the net present value of an investment project involves finding that rate which reflects the after-tax rate of return the firm requires to cover the opportunity cost of not undertaking its next best alternative of a similar maturity and risk exposure.

Accounting for business risk

Let’s assume a normal triangular probability distribution for the annual net cash flow in the ith year can be expressed mathematically as follows:

(3) E(NCFi) = Pi,1(NCFi,1) + Pi,2(NCFi,2) + Pi,3(NCFi,3)

where:

E(NCFi) Expected additional net cash flow attributable to the project in the ith year

Pi,1 Probability that “optimistic” economic conditions will occur in the ith year

Pi,2 Probability that “most likely” economic conditions will occur in the ith year

Pi,3 Probability that “pessimistic” economic conditions will occur in the ith year

NCFi,1 Net cash flow if “optimistic” economic conditions occur in the ith year

NCFi,2 Net cash flow if “most likely” economic conditions occur in the ith year

NCFi,3 Net cash flow if “pessimistic” economic conditions occur in the ith year

Given the assumption above, the expected value E(NCF1) or mean of this triangular probability distribution is equal to its “most likely” value, or NCFi,2 given in equation (3) above.

There are two traditional measures of business risk, the standard deviation about the mean or expected value, and the coefficient of variation. Using our notation above, the standard deviation associated with the net cash flows generated by the project in the ith year, assuming a normal probability distribution, is given by:

(4) SD(NCFi) = ( 2[Pi,1(NCFi,1 - E(NCFi))2]

While the standard deviation is useful for other reasons, it is not a very good measure of risk is it offers to basis of comparison to the mean of the distribution. We can rectify that by calculating the coefficient of variation as follows:

(5) CV(NCFi) = SD(NCFi) ( E(NCFi)

where CV(NCFi) represents the coefficient of variation for net cash flow in the ith year, or business risk per dollar of expected net cash flow. We will use this annual statistic as our measure of exposure to business risk.

Now that we have a measure of the unique annual exposure to business risk, we need to relate that to the firm’s required rate of return, or the discount rate used in assessing the net present value associated with the investment project. To do this, we must first assess the firm’s aversion to business risk. This can be done in the context of a “hurdle” rate, or the minimum rate of return the firm requires for accepting additional risk.

Firm’s respond to exposure to risk. Few are risk neutral when evaluating investment projects unless they ignore the risk associated with the expected returns from a project.

This suggests that the risk neutral investor will not require any additional return over the risk-free rate of return. The lowly risk-averse investor will require RRRL,i as a hurdle or required rate of return while the highly risk-averse investor will require RRRH,i.

The difference between RF,i and either RRRL,i or RRRH,i represents the business risk premium or additional return for taking additional risks.

Assume you are a consultant discussing an investment project with a client and he has told you that he requires a minimum rate of return of 12% if he is to invest in a project with a risk of 6 cents on the dollar (i.e., a coefficient of variation of 0.06).

This response helps you develop what is know as a risk/return preference function. To see this, let’s use the following general form of the risk/return preference function:

(6) RRRi = RF,i + bi(CVi)

where:

RRRi Required rate of return in the ith year

bi Slope of the firm’s risk/return preference curve ((RRRi /((CVi)

For example, if the risk free rate of return (RF,i) is 5%, then we can solve equation (6) for the slope of the risk/return preference curve bi as follows:

(7) bi = (RRRi - RF,i) ÷ CVi

which in our example above would be equal to:

8) bi = (.12 - .05) ÷ .10

=0.70

Thus the risk/return preference function in this case can be expressed as follows:

(9) RRRi = .05 + 0.70(CVi)

This risk /return preference curve can be displayed graphically as follows:

It is important to note that each year can have a unique required rate of return. Why? There are several reasons: (a) the risk free rate of return (RF,i) can change from one year to the next, (b) the coefficient of variation (CVi) can change from one year to the next, and (c) the slope of the risk/return preference curve can change.

The difference between the required rate of return and the risk free rate of return for an opportunity of equal maturity is known as the business risk premium. This represents the additional rate of return you require over a risk free investment for taking on the business risk involved in the project in the ith year.

These annual values of RRRi represent the discount rates associated with the corresponding annual net cash flows. We can now adjust our net present value model in equation (1) to account for the presence of business risk as follows:

(10) NPV = E(NCF1)((1+RRR1) + E(NCF2) ( [(1+RRR1)(1+RRR2)] + … + E(NCFn) ( [(1+RRR1)(1+RRR2)…(1+RRRn)] + T[(1+RRR1)(1+RRR2)…(1+RRRn)] – C

where:

E(NCF1) Expected additional net cash flow attributable to the project in year 1

1/(1+RRR1) Present value discount factor in year 1 reflecting required rate of return based upon unique risk exposure in year 1

T Expected terminal value of assets acquired

C Initial net outlay for assets acquired

To illustrate, let’s assume the following states of nature facing a firm in year 1 which is considering an investment that will enhance its annual net cash flows:

Table 3 – Elements of Triangular Normal Probability

Distribution.

State of nature: Net cash flow Probability

1. Optimistic $8,382 5.00%

2. Most Likely 7,620 90.00%

3. Pessimistic 6,858 5.00%

We know from our previous discussion that the expected net cash flow in the ith year or E(NCF1) will be $7,620. We can prove that to be true using equation (3) as follows:

11) E(NCF1) = 0.05($8,382) + 0.90($7,620) + 0.05($6,858)

= $419.10 + $6,858.00 + $342.90

= $7,620

Using equation (4), we can calculate the standard deviation associated with the annual net cash flows in year 1 of this project as follows:

12) SD(NCFi) = ( [2.0(0.05($8,382 - $7,620)2 )

= ( 2.0[$29,032.20]

= $240.97

The next step is to calculate the coefficient for the net cash flows expected in year 1 under this investment project. Using the format outlined in equation (5) we see that the coefficient of variation would be:

13) CV(NCFi) = $240.97 ($7,620

= 0.0316

or approximately 3.2 cents per dollar of expected net cash flow in year 1. Given the specification of the risk/return preference function given in equation (9), we see that the required rate of return in year 1 would be:

(14) RRR1 = .0689 + 0.70(0.0316)

= .0689 + .022

= .091

where the risk free rate of return is 6.89%. This process would be completed for each and every year in the economic life of the investment project.

For example, assume the expected value of the net cash flows E(NCFi) over the remaining 3 years of the 4-year economic life of this investment and their corresponding standard deviations are as follows:

Table 4 – Expected Value and Standard Deviation.

Year Expected Standard

Value deviation

1 $ 7,620 $241

2 10,920 488

3 14,220 779

4 14,220 899

The corresponding annual coefficients of variation, business risk premiums and required rates of return using equation (14) would be as shown in the last column of Table 5 below:

Table 5 – Required return and business risk premium.

Year Coefficient Risk-free Business risk Required

of variation rate of return premium rate of return

1 0.0316 6.89% 2.21% 9.10%

2 0.0447 7.16% 3.13% 10.29%

3 0.0548 7.12% 3.83% 10.95%

4 0.0632 7.26% 4.43% 11.69%

In addition to these annual net cash flows, the firm expects to receive a terminal value of $7,810 when it sells the assets acquired under this project at the end of the 4th year. The annual required rates of return in Table 6 above are then included in equation (15) when calculating the net present value for this project costing $45,000 as follows:

(15) NPV = $7,620 ( (1+.0910) + $10,920 ( [(1+.0910)(1+.1029)] + … + $14,220 ( [(1+.0910)(1+.1029)…(1+.1169)] + 7,810[(1+.0910)(1+.1029)…(1+.1169)] – $45,000

We can express this calculation in table form to give you a better idea about the individual components of equation (19) as follows:

Table 6 – Use of Risk Adjusted Discount Rates.

(1) (2) (3)

Net Cash Present Value Present Value

Year Flow Interest Factors of NFCi

(i) (NCFi) (1) x (2)

1 $ 7,620 0.9166 $ 6,984

2 10,920 0.8310 9,075

3 14,220 0.7490 10,651

4 14,220 0.6702 9,530

4 7,810 0.6702 5,234

$ 54,790 $41,474

Less initial cost - 45,000

Net present value $ - 3,526

Thus, we would reject this project after adjusting for risk since the net present value is negative.

Accounting for financial risk

The greater the use of leverage, or greater the debt-to-equity ratio, the greater the potential exposure to loss in equity capital well be. Leverage thus is associated with financial risk.

We can modify the risk/return preference function presented initially in equation (86) to reflect financial risk as follows:

16) RRRi = RF,i + bi(CVi) + ci(Li)

where bi(CVi) represents the business risk premium and ci(Li) represents the financial risk premium. We can visualize the addition of the financial risk premium below:

We earlier showed i

in equation (14) that the required rate of return for the project in the

first year of a project having a risk per dollar of expected net cash flows of 3.16 cents in year 1 would be:

17) RRR1 = .0689 + 0.70(.0316)

= .091

Adding the financial risk premium in year 1 to equation (17), we see that:

18) RRR1 = .091 + c1(L1)

Let’s now assume that the firm said it would require a rate of return equal to 15 percent in year 1 given its exposure to business and financial risk if its leverage ratio was 1.0. Given this information we can compute the coefficient in the financial risk premium for year 1 by transposing terms, or:

19) C1(L1) = .15 - .091

Solving for the coefficient associated with the liquidity variable, we see that

(24) c1 = (.15 - .091) ÷ L1

= .059 ÷ 1.0

= .059

which represents the change in the required rate of return for a given change in the firm’s leverage position, or (RRRi/(Li for the ith year.

With the addition of the financial risk premium, we can now assemble the complete risk/return preference function. This function, which includes both the business risk premium and the financial risk premium as well as the risk-free rate of return on assets of similar maturity in year 1, takes the form:

(25) RRR1 = .0689 + .70(CV1) + .059(L1)

This equation suggests that the higher the coefficient of variation associated with expected annual net cash flows over the life of a project or the higher the firm’s debt relative to equity, the greater “hurdle” or required rate of return a new project will have to “clear” in order to be acceptable to the firm’s decision makers.

-----------------------

Business Risk

premium

RF,i=.05

RRRi=.12

Coefficient of variation

Required

Rate of

return

0.10

}

Risk neutral

Highly risk

averse

Lowly risk averse

CVi

Slope equal to 0.70

RRRL,i

Coefficient of variation

Required

Rate of

return

RF,i

RRRH,i

Business risk premium

+ ci(Li)

Financial risk premium

CVi

RRRi

Coefficient of variation

Required

Rate of

return

RF,i

RRRi

HO #24

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