HO #24
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:
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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
3
4
5
$ 2,000
3,000
5,000
2,000
1,000
$ 13,000
Less initial cost
Net present value
0.952
0.907
0.864
0.823
0.784
$ 1,904
2,721
4,320
1,646
784
$ 11,375
- 10,000
$ 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.
Item:
1. Cash receipts
2. Cash operating expenses
3. Depreciation
4. Tax deductible expenses (2 + 3)
5. Taxable income (1 - 4)
6. Income tax payments (5 x rate)
7. Net income after taxes (1 – 4 – 6)
8. Net cash flow (7 + 3)
Before new After new
investment investment
$25,000
$30,000
-15,000
-18,000
-3,000
-4,000
18,000
22,000
7,000
8,000
1,750
2,000
5,250
6,000
8,250
10,000
Net
change
$5,000
-3,000
-1,000
4,000
1,000
250
750
1,750
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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
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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:
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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 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.
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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.
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 riskaverse 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).
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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
bi
Required rate of return in the ith year
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:
Required
Rate of
return
Slope equal
to 0.70
RRRi=.12
Risk
} Business
premium
RF,i=.05
0.10
Coefficient of variation
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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:
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Table 3 – Elements of Triangular Normal Probability
Distribution.
State of nature:
1. Optimistic
2. Most Likely
3. Pessimistic
Net cash flow
$8,382
7,620
6,858
Probability
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. 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
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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
Value
Standard
deviation
1
2
3
4
$ 7,620
10,920
14,220
14,220
$241
488
779
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
1
2
3
4
Coefficient
of variation
0.0316
0.0447
0.0548
0.0632
Risk-free
Business risk
rate of return premium
6.89%
7.16%
7.12%
7.26%
2.21%
3.13%
3.83%
4.43%
Required
rate of return
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 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:
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(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)
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
10,920
14,220
14,220
7,810
$ 54,790
Less initial cost
Net present value
0.9166
0.8310
0.7490
0.6702
0.6702
$ 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.
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:
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Required
Rate of
return
RRRi
Financial risk premium
RRRi
+ ci(Li)
We earlier showed
RF,i i
Business risk premium
CVi
Coefficient of variation
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
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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 riskfree 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.