Handout #5
Agricultural Economics 432
Part I - Financial Analysis
Topic #5
Spring Semester 2007
John B. Penson, Jr.
48
Agricultural Economics 432
Part I - Financial Analysis
Outline
Topic 1: Review of Financial Analysis Concepts
A. Introduction to Terminology
B. Key Financial Indicators to Track
C. Other Variables to Track
D. Financial Strength and Performance of the Firm
Topic 2: Growth of the Firm
A. Economic Climate for Growth
B. An Economic Growth Model
Topic 3: Valuing Investment Opportunities
A. Time Value of Money
B. Capital Budgeting Methods
C. Overview of Capital Budgeting Information Needs
D. Specific Applications of Net Present Value Method
Topic 4: Valuation of Externalities
A. Historical Assessments
B. Projecting Future Values
Topic 5: Incorporation of Risk
A. Exposure to Business Risk (p. 49)
B. Risk/Return Preferences (p. 51)
C. Exposure to Financial Risk (p. 57)
D. Portfolio Effect (p. 59)
E. Optimal Capital Structure of the firm (p. 62)
F. Ranking Potential Projects and the Capital Budget (p. 65)
49
Topic 5. Incorporation of Risk
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:
Pessimistic
scenario
(P i,3)
Most Likely
scenario
(Pi,2)
Optimistic
scenario
(Pi,1)
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)
Pi,1
Pi,2
Pi,3
NCFi,1
NCFi,2
NCFi,3
Expected additional net cash flow attributable to the project in the ith year
Probability that “optimistic” economic conditions will occur in the ith year
Probability that “most likely” economic conditions will occur in the ith year
Probability that “pessimistic” economic conditions will occur in the ith year
Net cash flow if “optimistic” economic conditions occur in the ith year
Net cash flow if “most likely” economic conditions occur in the ith year
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.
50
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.
51
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.
Required
Rate of
return
Highly risk
averse
RRRH,i
Lowly risk
averse
RRRL,i
RF,i
Risk neutral
CVi
Coefficient of variation
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:
(83)
RRRi = RF,i + bi(CVi)
where:
RRRi
bi
Required rate of return in the ith year
Slope of the firm’s risk/return preference curve (RRRi /(CVi)
52
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 = (.12 - .05) ÷ .10
=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:
Required
Rate of
return
Slope equal
to 0.70
RRRi=.12
Risk
} Business
premium
RF,i=.05
0.10
Coefficient of variation
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:
53
(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)
1/(1+RRR1)
T
C
Expected additional net cash flow attributable to the project in year 1
Present value discount factor in year 1 reflecting required rate of return
based upon unique risk exposure in year 1
Expected terminal value of assets acquired
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:
1. Optimistic
2. Most Likely
3. Pessimistic
Net cash flow
Probability
$8,382
7,620
6,858
5.00%
90.00%
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
54
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
1
2
3
4
Expected
Value
$ 7,620
10,920
14,220
14,220
Standard
deviation
$241
488
779
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
1
2
3
4
Coefficient
of variation
0.0316
0.0447
0.0548
0.0632
Risk-free
rate of return
6.89%
7.16%
7.12%
7.26%
Business risk Required
premium rate of return
2.21%
3.13%
3.83%
4.43%
9.10%
10.29%
10.95%
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
55
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+.0910) + $10,920  [(1+.0917)(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 (93) as follows:
Table 10 – Use of Risk Adjusted Discount Rates.
(1)
Net Cash
Year
Flow
(i)
(NCFi)
1
2
3
4
4
(2)
(3)
Present Value Present Value
Interest Factors
of NFCi
(1) x (2)
$ 7,620
0.9166
10,920
0.8310
14,220
0.7490
14,220
0.6702
7,810
0.6702
$ 54,790
Less initial cost
Net present value
$ 6,984
9,075
10,651
9,530
5,234
$41,474
- 45,000
$ - 3,526
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)
Net Cash
Year
Flow
(i)
(NCFi)
1
2
3
4
4
(2)
(3)
Present Value Present Value
Interest Factors
of NFCi
(1) x (2)
$ 7,620
0.9355
10,920
0.8753
14,220
0.8167
14,220
0.7625
7,810
0.7625
$ 54,790
Less initial cost
Net present value
$ 7,129
9,558
11,613
10,843
7,518
$46,661
- 45,000
$ 1,662
56
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 9.1% 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)
Net Cash
Year
Flow
(i)
(NCFi)
1
2
3
4
4
(2)
(3)
Present Value Present Value
Interest Factors
of NFCi
(1) x (2)
$ 7,620
0.9166
10,920
0.8401
14,220
0.7701
14,220
0.7058
7,810
0.7058
$ 54,790
Less initial cost
Net present value
$ 6,984
9,174
10,951
10,036
5,512
$42,657
- 45,000
$ - 2,343
Ignores changing
risk free rates and
increasing risk
over time
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.
57
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:
Required
Rate of
return
RRRi
Financial risk premium
RRRi
+ ci(Li)
Business risk premium
RF,i
CVi
Coefficient of variation
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)
= .072
Adding the financial risk premium to equation (95), we see that:
(96)
RRRi = .072 + ci(Li)
58
Let’s now assume that the firm said it would require a rate of return equal to 10 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) = .10 - .072
Solving for the coefficient associated with the liquidity variable, we see that
(98)
ci = (.10 - .072) ÷ Li
= .028 ÷ 1.0
= .028
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) + .028(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.
59
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:
Highly Negatively Correlated Net Cash Flows
NCFi
Annual fluctuations of net cash
flows from existing assets
Annual fluctuations of net cash
flows from new project
Time
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
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”.
60
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:
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.
61
(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!!!!
Reduction in RRR1 due
to a reduction in
business risk based
upon the portfolio
effect associated with
the new investment.
.150
.132
.100
.238
.375
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.
62
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
Percent
Total cost
12%
Implicit cost
9%
Explicit cost
75%
Use of credit capacity
to rise as depicted in the graph above. 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
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given by the least cost combination of debt and equity capital. We can illustrate the point
where this occurs is the graph below:
$/unit
Cost of equity capital
Weighted average cost of capital
Cost of debt capital
1.0
D/E ratio
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
ratio
0.0
0.5
1.0
1.5
2.0
Source of
capital
Unit
cost
Debt
Equity
Debt
Equity
Debt
Equity
Debt
Equity
Debt
Equity
0.04
0.06
0.04
0.06
0.04
0.06
0.05
0.08
0.06
0.10
WACC
0.060
0.053
0.050
0.062
0.074
64
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.
65
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
B
C
D
E
$10,000
24,000
7,500
43,000
5,250
$14,500
33,120
8,850
46,500
3,360
$4,500
9,120
1,350
3,400
-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.
3
The least cost the weighted average cost of capital and optimal capital structure of the firm
was discussed in the previous section.