Present financial position and performance of the firm



Handout #5

Agricultural Economics 489/689

Topic #5

Spring Semester 2008

John B. Penson, Jr.

A. Exposure to Business Risk

Expected future net cash flows

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

The mean or expected value of this triangular probability distribution can be expressed mathematically as follows:

(79) 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 of a triangular probability distribution above, the expected value E(NCF1) or mean of this probability distribution is equal to its “most likely” value, or NCFi,2 given in equation (79) above.

Measurement of business risk

There are two traditional measures of business risk, the standard deviation above 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 is given by:

(80) SD(NCFi) = ( [Pi,1(NCFi,1 - E(NCFi))2 + Pi,3(NCFi,3 - E(NCFi))2]

or

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

You will notice several shortcuts taken in equations (80) and (81). First, the deviation between the potential net cash flow associated with the “most likely” scenario and the mean of the probability distribution is absent from equation (80). This term drops out under the normal triangular probability distribution assumed here since these two terms are identical! Second, since both tails of this distribution are identical in absolute terms, we can multiply either one of them by 2.0 and drop the other as illustrated above in equation (81).

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:

(82) 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.

B. Risk/Return Preferences

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 inadvertently 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 19% if he is to invest in a project with a risk of 20 cents on the dollar (i.e., a coefficient of variation of 0.20). 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:

(83) 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 (83) for the slope of the risk/return preference curve bi as follows:

(84) bi = (RRRi - RF,i) ÷ CVi

which in our example above would be equal to:

(85) bi = (.19 - .05) ÷ .20

=0.70

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

(80) 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 the net present value model presented in equation (57) to account for the presence of business risk as follows:

(87) 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, and

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 7 – Elements of Triangular 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. Let’s prove that to be true using equation (79) as follows:

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

(89) SD(NCFi) = ( [0.05($8,382 - $7,620)2 + 0.05($6,858 - $7,620)2]

= ( $29,032.20 + $29,032.20

= $240.97

We could have also used equation (81) to calculate this standard deviation given the normal nature of our triangular probability distribution and achieved the same solution:

(90) 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 (82) we see that the coefficient of variation would be:

(91) 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 (83), we see that the required rate of return in year 1 would be:

(92) RRR1 = .05 + 0.70(0.0316)

= .05 + .022

= .072

or 7.2%. This process is completed for each year in the economic life of the 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 8 – 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 (92) would be:

Table 9 – 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 5.00% 2.21% 8.16%

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 10 above are then included in equation (93) when calculating the net present value for this project costing $45,000 as follows:

(93) NPV = $7,620 ( (1+.0816) + $10,920 ( [(1+.0816)(1+.1029)] + … + $14,220 ( [(1+.0816)(1+.1029)…(1+.1169)] + 7,810[(1+.0816)(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 (93) as follows:

Table 10 – 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 0.9246 $ 7,045

2 10,920 0.8383 9,154

3 14,220 0.7556 10,745

4 14,220 0.6765 9,620

4 7,810 0.6765 5,283

$ 54,790 $41,847

Less initial cost - 45,000

Net present value $ - 3,153

Thus, we would reject this project after adjusting for risk since the net present value is negative. If we discounted the net cash flows above at the risk-free rate of return (RF,i), we would have calculated a net present value of:

Table 11 – Use of Risk Free 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 0.9524 $ 7,257

2 10,920 0.8887 9,705

3 14,220 0.8297 11,798

4 14,220 0.7725 10,985

4 7,810 0.7725 6,033

$ 54,790 $45,778

Less initial cost - 45,000

Net present value $ 778

Using the risk-free discount rate would have lead us to conclude that this was an economically feasible investment opportunity!

Finally, how important was it for us to account for the possibility of increasing risk over time rather than use the interest factor calculated for year 1 in Table 10? This table involves using the 8.16% required rate of return reported for year 1 in Table 9 when calculating the interest factors for the subsequent years. The results of this adjustment are reported in Table 12 below:

Table 12 – Use of Constant Risk 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 0.9246 $ 7,045

2 10,920 0.8548 9,334

3 14,220 0.7903 11,238

4 14,220 0.7307 10,391

4 7,810 0.7307 5,707

$ 54,790 $43,715

Less initial cost - 45,000

Net present value $ - 1,285

Thus we still would have concluded that the business risk involved with this project would have made it an infeasible economic opportunity, although the net present value is less negative than that reported in Table 10.

C. Exposure to Financial Risk

The economic growth model presented in equation (24) helped us see the advantages and disadvantages associated with the use of financial leverage to grow the firm. If the rate of return on assets exceeds the rate of interest on debt capital, leverage will contribute to the growth of the firm’s equity.

However, if the rate of return on assets is less than the rate of interest on debt capital, leverage will detract from the growth of the firm’s equity. Leverage thus is associated with 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.

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

(94) 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 in equations (91) and (92) that the required rate of return for a project in the ith year of a project having a risk per dollar of expected net cash flows of 3.16 cents would be:

(95) RRRi = .05 + 0.70(.0316)

= .05 + 0.022

= .072

Adding the financial risk premium to equation (95), we see that:

(96) RRRi = .072 + ci(Li)

Let’s now assume that the firm said it would require a rate of return equal to 13 percent 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 by transposing terms, or:

(97) ci(Li) = .13 - .072

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

(98) ci = (.13 - .072) ÷ Li

= .058 ÷ 1.0

= .058

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

With the addition of the financial risk premium, we now assemble the entire 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, takes the form:

(99) RRRi = .05 + .70(CVi) + .058(Li)

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.

D. Portfolio Effect

The firm can benefit from diversifying its portfolio of assets and enterprises if certain conditions hold. One of these conditions is that the net cash flows associated with the firm’s existing operations be highly negatively correlated with the net cash flows generated by the new project.

We can illustrate the nature of the path taken for annual net cash flows generated by the firm’s existing assets that are highly negatively correlated returns with the annual net cash flows associated with a new investment project by examining the following figure:

The figure above illustrates the case where the net cash flows generated by the firm’s existing assets are low when the net cash flows from the new project are high, and vice versa. Thus, the peaks of one stream help offset, at least in part, the valleys of the other stream.[1] If the expected net cash flows generated by these two sources are weighted approximately the same, the time path taken by the E(NCFi) will be a relatively flat line parallel to the time axis, reflecting relatively constant net cash flows over time.

When this is the case, the firm’s overall exposure to risk is lowered, allowing us to reduce the required rate of return given by equation (94) due to the investment project’s risk reducing features.

To illustrate how negatively correlated investment projects affect the firm’s exposure to business risk, let’s examine the following situation. Suppose you are considering investment in project C and are concerned about the degree of business risk associated with the project’s expected net cash flows. Let the expected rate of return from project C in the ith year be represented by E(ROAC,i) and the standard deviation of these returns be represented by SD(ROAC,i). Further assume that the expected rate of return from firm’s existing assets in the ith year is represented by E(ROAEX,i) and the standard deviation of these returns be represented by SD(ROAEX,i). Finally, assume that the expected rate of return generated by the new project is highly negatively correlated with the expected rate of returns generated by the firm’s existing assets (see the figure above). The expected rate of return for the entire portfolio in the ith year after the project is completed would be given by:

(100) E(ROAT,i) = WC(E(ROAC,i)) + WEX(E(ROAEX,i))

where WC + WEX = 1.0

The standard deviation in the ith year for the new portfolio of assets would be given by:

(101) SD(ROAT,i) = {WC2(SD(ROAC,i)2 + WEX2(SD(ROAEX,i)2

+ [2(WC)(WEX)(()(SD(ROAC,i)) (SD(ROAEX,i))] }1/2

where ( represents the correlation coefficient between the rate of return generated by the firm’s existing assets and a project it is considering.

▪ If these rates of return are highly negatively correlated, the value of ( will be at or close to –1.0.

▪ If these rates of return are highly positively correlated, the value of ( will be at or close to +1.0.

▪ A value of ( equal to zero means these two annual rates of return are uncorrelated.[2]

The sum of the first two terms in equation (101) represents the weighted average variance for the new portfolio while the entire last term represents the covariance associated with the net cash flows from the new project C and the firm’s existing assets.

Let’s assume that WC = .20 and WEX = .80 in the first year of the project and that the annual rate of return in this year are expected to be E(ROAC,i) = 10% and E(ROAEX,i) = 8%. Using equation (94), the expected rate of return for the entire portfolio after the investment is made is expected in year 1 to be:

(102) E(ROAT,1) = 0.20(0.10) + 0.80(0.08)

= 0.084 or 8.4%

If the values of the corresponding standard deviations are SD(ROAC,i) = 0.02 and SD(ROAEX,i) = 0.03 and the value of the correlation coefficient ( = - 1.0, then the standard deviation for the entire portfolio in year 1 using equation (101) would be:

(103) SD(ROAT,1) = {(0.20)2(0.02)2 + (0.80)2(0.03)2

+ [2(0.20)(0.80)(-1.0)(0.02)(0.03)] } 1/2

= 0.02

which results in a coefficient of variation for the entire portfolio in year 1 after the new investment is made of:

(104) CV(ROAT,1) = SD(ROAT,1) ÷ E(ROAT,1)

= 0.02 ÷ 0.084

= 0.238

or 23.8 cents per dollar of expected net returns.

How can we use the total portfolio information given by equations (102) and (103)?

The coefficient of variation or risk per dollar of expected return for the existing portfolio is equal to:

(105) CV(ROAEX,1) = SD(ROAEX,1) ÷ E(ROAEX,1)

= 0.03 ÷ 0.08

= 0.375

or 37.5 cents per dollar of expected net returns. Therefore, the incorporation of the new project (project C) into the firm’s total portfolio lowers the risk per dollar of expected return from 37.5 cents to 23.8 cents, a 36 percent reduction! This occurs only because the returns from project C are highly negatively correlated with the returns stemming from the firm’s existing assets.

This would suggest that a reduction in the required rate of return for project C when computing its net present value is justified. One approach would be to lower the business risk premium by 36 percent when computing the net present value for project C, leaving the financial risk premium and risk-free rate of return unchanged!!!!

E. Optimal Capital Structure of the Firm

Explicit and implicit costs of capital

Thus far we have focused on the required rate of return, mentioning the cost of debt capital only in comparison to the rate of return on assets (ROA) when discussing the rate of growth in equity capital and when discussing a project’s internal rate of return, or IRR. Even then we only addressed the explicit cost of debt capital, or the externally determined rate specified on the mortgage or note.

There implicit costs of debt capital that cause firms to internally ration their use of debt capital that were more or less implied when we discussed the concept of financial risk and the financial risk premium. As the firm reduces its credit liquidity as it uses up its credit reserves, its implicit cost of debt capital rises, causing the total cost of debt capital to rise as depicted in the graph below. This concept is an important component to analyzing the firm’s weighted average cost of capital and optimal capital structure.

Weighted average cost of capital

The weighted average cost of capital (WACC) employed by the firm is given by the following equation:

WACC = WEQ(rE) + WDT(rD)

where WEQ is the relative importance of equity in the firm’s balance sheet, rE is the cost of equity capital, WDT is the relative importance of debt in the firm’s balance sheet, and rD is the total cost of debt capital. The optimal capital structure of the firm’s balance sheet is given by the least cost combination of debt and equity capital. We can illustrate the point where this occurs is the graph below:

Two features are worth noting in the graph above. The first is the fact that the cost of debt capital is less than the cost of equity capital. How can this be? Think of the cost of debt capital as the minimum opportunity rate of return available to the firm. After all, one of the opportunities available to using the firm’s equity capital is to make loans to others.

The other feature has to do with the shape of the weighted cost of capital curve and the optimal location on that curve. This curve falls sharply at low debt/equity ratios since the cost of equity capital is higher and carries a higher weight. The optimal location on the weighted average cost of capital curve is at its lowest point. At this point, the firm is minimizing its cost of capital. Any other combination of debt and equity capital would reduce the returns from the firm’s portfolio.

Table 14 – Calculation of the Weighted Cost of Capital.

Leverage Source of Unit

ratio capital cost WACC

0.0 Debt 0.04

Equity 0.06 0.060

0.5 Debt 0.04 Equity 0.06 0.053

1.0 Debt 0.04 Equity 0.06 0.050

1.5 Debt 0.05

Equity 0.08 0.062

2.0 Debt 0.06

Equity 0.10 0.074

We see above that the least cost combination of debt and equity capital occurs where the firm achieves a 50-50 balance of debt and equity capital on its balance sheet, or leverage ratio of 1.0. At this point we see that the weighted average cost of capital is 5 percent.

F. Ranking Potential Projects and the Capital Budget

The final topic covered in this handout is the role that the capital budget plays in the selection of economically feasible investments to fund in the current period. Let’s assume the firm is facing the following investment opportunities this year and has a $90,000 capital budget to work with:

Table 13 – Cost and Benefits from Alternative Projects.

(1) (2) (3)

Present Net

Value of Present

Cost of Net Cash Value

Project Project Flows (2) – (1)

A $10,000 $14,500 $4,500

B 24,000 33,120 9,120

C 7,500 8,850 1,350

D 43,000 46,500 3,400

E 5,250 3,360 -1,890

Totaling up the costs of the five projects the firm is considering, we see that this total ($89,750) does not exceed the amount of debt and internal equity capital available this year to the firm ($90,000). What projects would you advise this firm to invest in?

First, we can throw out project E because it has a negative net present value. This leaves us with $84,500 in projects that have a positive net present value. Should the firm invest in all four projects?

If the firm wants to minimize its cost of financing (i.e., use 50 percent retained earnings and 50 percent debt capital given by the minimum point on its weighted average cost curve), the answer is no.[3] For example, if the firm uses all of its retained earnings and sticks by this least-cost decision rule, it would prefer to spend only $77,000 on new investment projects in the current period (i.e., $38,500 in retained earnings and $38,500 in debt capital) and hold $13,000 in reserve ($90,000 - $77,000). The firm would invest in projects A, B, and D, which together cost $77,000 and collectively generate a total net present value to the firm of $17,020 or $4,500 + $9,120 + $3,400. Note the firm selected project D over project C because project C had a lower net present value and fit within the budget constraint.

Another approach is to calculate the profitability index or benefit – cost ration for these investment alternatives. This involves calculating the ratio of the present value of net cash flows to the cost of the project, or column (2) divided by column (1).

The profitability index for the example presented in Table 13 is presented in Table 14 below. The decision rule is that all projects with an index greater than or equal to 1 would be considered economically feasible. Differences between the NPV decision rule and the profitability index can occur if the alternative projects require significantly different project costs. When conflicts arise, the final decision must be made on the basis of other factors. In the absence of an effective constraint on available financing (i.e., no capital

Table 14 – Profitability Index for from Alternative Projects.

(1) (2) (3)

Present Profitability

Value of Index

Cost of Net Cash

Project Project Flows (2) / (1)

A $10,000 $14,500 1.45

B 24,000 33,120 1.38

C 7,500 8,850 1.18

D 43,000 46,500 1.08

E 5,250 3,360 0.64

rationing), the net present value rule is preferred since it will select projects that are expected to generate the largest total dollar increase in the firm’s net worth position. If capital rationing limits available financing, however, the profitability index may be preferred since it indicates which projects maximize the returns per dollar of investment. In the example above, assume the firm is limited to spending $80,000. In this case, the total cost of projects A, B, C and D (omitting project E since the index is less than 1) is equal to $84,500. In this case, the firm may choose projects A, B and D which total $77,000.

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[1] Highly positively correlated returns would have the opposite effect. They would fluctuate in an identical fashion with the flows generated by the firm’s existing operations. Thus, they do not reduce the firm’s overall exposure to risk; they increase it by putting more “eggs into the same basket”.

[2] For a further discussion of this topic when more than two investment projects are being considered, see Van Horne, Financial Management and Policy, Chapter 3.

[3] The least cost the weighted average cost of capital and optimal capital structure of the firm was discussed in the previous section.

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Total cost

Percent

Business Risk

premium

RF,i=.05

RRRi=.19

Annual fluctuations of net cash flows from existing assets

Coefficient of variation

Required

Rate of

return

0.20

}

Annual fluctuations of net cash flows from new project

NCFi

Time

Highly Negatively Correlated Net Cash Flows

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

Reduction in RRR1 due to a reduction in business risk based upon the portfolio effect associated with the new investment.

.238 .375

.150

.132

.100

Optimistic scenario

(Pi,1)

Pessimistic scenario

(P i,3)

Most Likely scenario

(Pi,2)

Ignores changing risk free rates and increasing risk over time

Here we are ignoring risk entirely.

36 percent reduction in business risk premium translated into a .018 or 1.8 percent point reduction in the required rate return used for Project C.

D/E ratio

1.0

Cost of debt capital

Weighted average cost of capital

Cost of equity capital

$/unit

Use of credit capacity

75%

Explicit cost

9%

12%

Implicit cost

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