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RISK ANALYSIS IN CAPITAL BUDGETING
Discuss the concept of risk in investment decisions. Understand some commonly used techniques, i.e., payback,
certainty equivalent and risk-adjusted discount rate, of risk analysis in capital budgeting.
Focus on the need and mechanics of sensitivity analysis and scenario analysis.
Highlight the utility and methodology simulation analysis. Explain the decision tree approach in sequential investment
decisions. Focus on the relationship between utility theory and capital
budgeting decisions.
1
Nature of Risk
Risk exists because of the inability of the decision-maker to make perfect forecasts.
In formal terms, the risk associated with an investment may be defined as the variability that is likely to occur in the future returns from the investment.
Three broad categories of the events influencing the investment forecasts: General economic conditions Industry factors Company factors
2
TECHNIQUES FOR RISK ANALYSIS
Techniques of riskanalysis
Analysis of stand-alone risk
Analysis of contextual risk
Sensitivityanalysis
Break-evenanalysis
Simulationanalysis
Scenarioanalysis
Corporate risk analysis
Market riskanalysis
Hilliermodel
Decision treeanalysis
Techniques for Risk Analysis
Statistical Techniques for Risk Analysis Probability Variance or Standard Deviation Coefficient of Variation
Conventional Techniques of Risk Analysis Payback Risk-adjusted discount rate Certainty equivalent
4
Probability
A typical forecast is single figure for a period. This is referred to as “best estimate” or “most likely” forecast: Firstly, we do not know the chances of this figure actually occurring, i.e.,
the uncertainty surrounding this figure. Secondly, the meaning of best estimates or most likely is not very clear. It
is not known whether it is mean, median or mode.
For these reasons, a forecaster should not give just one estimate, but a range of associated probability–a probability distribution.
Probability may be described as a measure of someone’s opinion about the likelihood that an event will occur.
5
Assigning Probability
The probability estimate, which is based on a very large number of observations, is known as an objective probability.
Such probability assignments that reflect the state of belief of a person rather than the objective evidence of a large number of trials are called personal or subjective probabilities.
6
Risk and Uncertainty
Risk is referred to a situation where the probability distribution of the cash flow of an investment proposal is known.
If no information is available to formulate a probability distribution of the cash flows the situation is known as uncertainty.
7
Expected Net Present Value Once the probability assignments have been made to
the future cash flows the next step is to find out the expected net present value.
Expected net present value = Sum of present values of expected net cash flows.
8
= 0
ENPV = (1 )
n
tt
ENCF
k
ENCF = NCF × t jt jtP
Example
Suppose an investment project has a life of three years, and it would involve an initial cost of Rs 10,000.
If the discount rate is 15 per cent, calculate the expected NPV.
9
Expected Cash Flow
Example10
Variance or Standard Deviation Variance measures the deviation about expected cash
flow of each of the possible cash flows. Standard deviation is the square root of variance.
Absolute Measure of Risk.
11
2 2
=1
(NCF) = (NCF – ENCF)n
j jj
P
Coefficient of Variation
Coefficient of variation is relative Measure of Risk.
It is defined as the standard deviation of the probability distribution divided by its expected value:
Coefficient of variation = standard deviation / expected value
12
Coefficient of Variation
The coefficient of variation is a useful measure of risk when we are comparing the projects which have
same standard deviations but different expected values, or different standard deviations but same expected values, or different standard deviations and different expected values.
13
Exercise14
The following information is available regarding the expected cash flows generated, and their probability for company X. What is the expected return on the project? Assuming 10 per cent as the discount rate, find out the present values of the expected monetary values.
Year 1 Year 2 Year 3
Cash flows Probability Cash flows Probability Cash flows Probability
Rs 3,0006,0008,000
0.250.500.25
Rs 3,0006,0008,000
0.500.250.25
Rs 3,0006,0008,000
0.250.250.50
Solution15
TABLE 3 (i) Calculation of Expected Monetary Values
Year 1 Year 2 Year 3
Cash flows
Probability
Monetary values
Cash flows
Probability
values
Monetary Cash flows
Probability
values
Monetary
Rs 3,0006,0008,000Total
0.250.500.25
Rs 7503,0002,0005,750
Rs 3,0006,0008,000
0.500.250.25
Rs 1,5001,5002,0005,000
Rs 3,0006,0008,000
0.250.250.50
Rs 750
1,5004,0006,250
(ii) Calculation of Present Values
Year 1 Rs 5,750 × 0.909 = Rs 5,226.75
Year 2 5,000 × 0.826 4,130.00
Year 3 6,250 × 0.751 4,693.75
Total 14,050.50
HILLIER MODEL
Uncorrelated Cash Flows
n Ct
NPV = – I t = 1 (1 + i)t
n t2 ½
(NPV) = t = 1 (1 + i)2t
Perfectly Correlated Cash Flows
n Ct
NPV = – I t = 1 (1 + i) t
n t
(NPV) = t = 1 (1 + i)t
HILLIER MODELIndependent Cash Flows Over Time The mathematical formulation to determine the expected values of the probability distribution of NPV for any project is:
The above calculations of the standard deviation and the NPV will produce significant volume of information for evaluating the risk of the investment proposal. The calculations are illustrated in Example 6.
(8)m
1j jt.P
2
tCFjt
CFtσ
:follows as calculated be would
tα t, periodfor flows cash expected of ondistributi yprobabilit the of deviation standard the is tα where
(7)n
1t 2ti1
σ2tNPVσ
to equal isNPV of ondistributi yprobabilit the of deviation standard Theinterest. of rate riskless the is i andt period in CFATnet of value expected the is CF where
(6)n
1tCOti1
tCFNPV
Example 6
Suppose there is a project which involves initial cost of Rs 20,000 (cost at t =
0). It is expected to generate net cash flows during the first 3 years with the
probability as shown in Table 7.
TABLE 7 Expected Cash Flows
Year 1 Year 2 Year 3
Probability Net cash
flows
Probability Net cash
flows
Probability Net cash
flows
0.10 Rs 6,000 0.10 Rs 4,000 0.10 Rs 2,000
0.25 8,000 0.25 6,000 0.25 4,000
0.30 10,000 0.30 8,000 0.30 6,000
0.25 12,000 0.25 10,000 0.25 8,000
0.10 14,000 0.10 12,000 0.10 10,000
Solution
1 . Expected Values: For the calculation of standard deviation for different periods, the expected values are to be calculated first. These are calculated in Table 8.
(3) NPV= Rs 10,000 (0.909) + Rs 8,000 (0.826) + Rs 6,000 (0.751) – Rs 20,000 = Rs 204.
2,280. Rs to out workalso )σ and (σ 3 and 2 periods for deviations standard the lines similar on calculated When
Rs2,280]10,00014,0000.1010,00012,0000.25
10,00010,0000.3010,0008,0000.2510,0006,000[0.10σ
:is 1 period for deviation standard the Thus,
.PCFCFσ
: isflow cash net posible of deviation standard The (ii)
32
22
222
1
m
1jjt
2
tjtt
Calculation of Expected Values of Each Period
Time period
Probability(1)
Net cash flow(2)
Expected value (1 × 2)(3)
Year 1
Year 2
Year 3
0.100.250.300.250.10
0.100.250.300.250.10
0.100.250.300.250.10
Rs 6,0008,000
10,00012,00014,000
4,0006,0008,000
10,00012,000
2,0004,0006,0008,000
10,000
Rs 6002,0003,0003,0001,400
= 10,000400
1,5002,4002,5001,200
= 8,000200
1,0001,8002,0001,000
= 6,000
1CF
2CF
3CF
3,283Rs1.10
2,280Rs
1.10
2,280Rs
1.10
2,280Rs
i1σ
:time over flows cash of ceindependen of assumption the under deviation standard (iv)The
6
2
4
2
2
2n
1t2t
2
tσ
Normal Probability Distribution
We can make use of the normal probability distribution to further analyze the element of risk in capital budgeting. The use of the normal probability distribution will enable the decision maker to have an idea of the probability of different expected values of NPV, that is, the probability of NPV having the value of zero or less; greater than zero and within the range of two values, say, Rs 1,000 and Rs 1,500 and so on.
The normal probability distribution as shown in Figure has a number of useful properties.
The area under the normal curve, representing the normal probability distribution, is equal to 1 (0.5 on either side of the mean). The curve has its maximum height at its expected value (mean). The distribution (curve) theoretically runs from minus infinity to plus infinity. The probability of occurrence beyond 3σs is very near zero (0.26 per cent).
-3σ -2σ -1σ X +1σ +2σ +3σ
99.74%
95.46%
Normal Curve
68.26%
Example
Assume that a project has a mean of Rs 40 and standard deviation of Rs 20. The management wants to determine the probability of the NPV under the following ranges: (i) Zero or less, (ii) Greater than zero, (iii) Between the range of Rs 25 and Rs 45, (iv) Between the range of Rs 15 and Rs 30.
Solution
(i) Zero or less: The first step is to determine the difference between the expected outcome X and the expected net present value. The second step is to standardize the difference (as obtained in the first step) by the standard deviation of the possible net present values. Then, the resultant quotient is to be seen in statistical tables of the area under the normal curve. Such a table (Table Z) is given at the end of the book. The table contains values for various standard normal distribution functions. Z is the value which we obtain through the first two steps, that is:
This is also illustrated in Fig. 3.
The figure of –2 indicates that a NPV of 0 lies 2 standard deviation to the left of the expected value of the probability distribution of possible NPV. Table Z indicates that the probability of the value within the range of 0 to 40 is 0.4772. Since the area of the left-hand side of the normal curve is equal to 0.5, the probability of NPV being zero or less would be 0.0228, that is, 0.5 – 0.4772. It means that there is 2.28 per cent probability that the NPV of the project will be zero or less.
0.220Rs
40Rs0Z
-20
Figure 3
Expected Outcomes (X Values)
0 +20 +40 +60 +80 +100
Co
nti
nu
ou
s P
rob
ab
ilit
y d
istr
ibu
tio
n
• •
••
Exercise
The Cautious Ltd is considering a proposal for the purchase of a new machine requiring an outlay of Rs 1,500 lakh. Its estimate of the cash flow distribution for the three-year life of the machine is given below (amount in Rs lakh):
Period 1 Period 2 Period 3
Cash flows Probability Cash flows Probability Cash flows Probability
Rs 800 0.1 Rs 800 0.1 Rs 1,200 0.2
600 0.2 700 0.3 900 0.5
400 0.4 600 0.4 600 0.2
200 0.3 500 0.2 300 0.1
The probability distribution is assumed to be independent. Risk-free rate of interest is 5 per cent. From the above information, determine the following: (i) the expected NPV of the project; (ii) the standard deviation of the probability distribution of NPV; (iii) the probability that the NPV will be (a) zero or less (assuming that the distribution is normal); (b) greater than zero; and (c) at least equal to the mean; (iv) the profitability index of the expected value; and (v) the probability that the profitability index will be less than 1.
Table 9 Determination of Expected NPV (Rs lakh)
Period 1 Period 2 Period 3
CF Pj Cash flow CF Pj Cash flow CF Pj Cash flow
(CF × Pj) (CF × Pj) (CF × Pj)
800 0.1 80 800 0.1 80 1,200 0.2 240
600 0.2 120 700 0.3 210 900 0.5 450
400 0.4 160 600 0.4 240 600 0.2 120
200 0.3 60 500 0.2 100 300 0.1 30
Mean ( ) 420 Mean ( ) 630 Mean ( ) 840
NPV = Rs 420 (0.952) + Rs 630 (0.907) + Rs 840 (0.864) – Rs 1,500 = Rs 197 lakh.
1CF 2CF 3CF
12 - 26
18835,6001
σ
35,600 j1
P2)1
CF- j1
Σ(CF
14,520 0.3 x 48,400160 0.4 x 400
6,480 0.2 x 32,40014,440 Rs 0.1 x 1,44,400 Rs
j1P2)
1CF-
j1(CF
j1(x)P 2)
1CF
j1(CF
:t Period, forflow Cash Expected of Deviation Standard (ii)
1 Period
908,1002
σ
8,100 j2
P2)2
CF-j2
Σ(CF
3,380 0.2 x 16,900 x 9001,470 0.3 x 4,900
2,890 Rs 0.1 x 28,900 Rsj2
P2)2
CF- j2
(CF j2
(x)P 2)2
CF-j2
(CF
2 Period
262684003
σ
68400 j3
P2)3
CF-j3
Σ(CF
29,160 0.1 x 2,91,60011,520 0.2 x 57,6001,800 0.5 x 3,600
25,920 Rs 0.2 x 1,29,600 Rsj3
P2)3
CF- j3
(CF j3
(x)P 2)3
CF-j3
(CF
3 Period
300Rs1.34068,400 Rs
1.216Rs81,00
1.10235,520 Rs
60.05)(1
2262Rs40.05)(1
290Rs20.05)(1
2188Rsn
1t 2ti1
σ2tσ(NPV)
:NPV about deviation standard of nCalculatio
(iii) (a) Calculation of Probability of the NPV Being Zero or Less: Z = [(0-197)/300]=-.6567
According to Table Z, the probability of the NPV being zero is = 0.2454, that is, 24.54 per cent. Therefore, the probability of the NPV being zero or less would be 0.5 – 0.2454 = 0.2546 or 25.46 per cent.
(b)The probability of the NPV being greater than zero would be 1 – 0.2546 = 0.7454 or 74.54 per cent
(c)At least equal to mean: Z = [(197-197)/300] = 0
Reading from the normal distribution table, we get the probability corresponding to 0 as 0. Therefore, the probability of having NPV at least equal to mean would be equivalent to the area to the right of the curve, that is, 0.5 = 50 per cent.
(iv) Profitability Index: (PV of cash inflows/PV of cash outflows) = [(Rs 197 + Rs 1,500) / Rs 1,500] = 1.13
(v) The probability of the index being less than 1: For the index to be 1 or less, the NPV would have to be zero or negative. Thus, the probability would be equal to 25.46 per cent as calculated in part (iii) (a) of the answer.
Decision-tree Approach
Decision tree is a pictorial representation in tree from which indicates the magnitude, probability and inter-relationships of all possible outcomes.
Example: Suppose a firm has an investment proposal, requiring an outlay of Rs 2,00,000 at present (t = 0). The investment proposal is expected to have 2 years’ economic life with no salvage value. In year 1, there is a 0.3 probability (30 per cent chance) that CFAT will be Rs 80,000; a 0.4 probability (40 per cent chance) that CFAT will be Rs 1,10,000 and a 0.3 probability (30 per cent chance) that CFAT will be Rs 1,50,000. In year 2, the CFAT possibilities depend on the CFAT that occurs in year 1. That is, the CFAT for the year 2 are conditional on CFAT for the year 1. Accordingly, the probabilities assigned with the CFAT of the year 2 are conditional probabilities. The estimated conditional CFAT and their associated conditional probabilities are as follows:
If CFAT1 = Rs 80,000 If CFAT1 = Rs 1,10,000 If CFAT1 = Rs 1,50,000
CFAT2 Probability CFAT2 Probability CFAT2 Probability
Rs 40,000 0.2 Rs 1,30,000 0.3 Rs 1,60,000 0.1
1,00,000 0.6 1,50,000 0.4 2,00,000 0.8
1,50,000 0.2 1,60,000 0.3 2,40,000 0.1
12 - 30
Solution The estimated values have been portrayed in Fig. 4.
Time: 0
Path Expected NPVat 8% rate of
discount
JointProbability
(Pj)**
Expected NPVNPV (×) Pj
Year 1 Year 2
Probabilities
CFAT Probabilities
CFAT
Cash Outlays
Rs 2,00,000
0.3
0.4
0.3
Rs 80,000
1,10,000
1,50,000
0.2
0.6
0.2
0.3
0.4
0.3
0.1
0.8
0.1
Rs 40,000
1,00,000
1,50,000
1,30,000
1,50,000
1,60,000
1,60,000
2,00,000
2,40,000
1
2
3
4
5
6
7
8
9
Rs (– 91,640)
(–40,220)
(–2,630)
13,270
30,410
38,980
76,020
1,10,300
1,44,580
0.06
0.18
0.06
0.12
0.16
0.12
0.03
0.24
0.031.00
Rs (– 5,498.4)
(–7,239.6)
(–157.8)
1,592.4
4,865.6
4,677.6
2,280.6
26,472.0
4,337.431,329.8
* PV factors for years 1 and 2 at 8% discount rate as per Table A-3 are 0.926 and 0.857 respectively. Multiply CFAT1 by 0.926 and CFAT2 by 0.857; summing up, we get total PV for individual possible CFAT; substracting Rs. 2,00,000 (CO), we get the NPV.
** Product of probabilities of CFAT for years 1 and 2.
Figure 4: Decision Tree
The DT shows 9 distinct possibilities, the project could assume if accepted. For example, one possibility is that the CFAT for the year one may amount to Rs 80,000 and for the year 2 Rs 40,000. A close perusal of Fig. 4 would also indicate that this is the worst event that could happen. Assuming a 8 per cent risk free/discount rate for the project, the NPV would be negative. Likewise, the best outcome that could occur is CFAT1 = Rs 1,50,000 and CFAT2 = Rs 2,40,000. The NPV would be the highest among all the 9 possible combinations. Figure 4 shows the NPV at 8 per cent discount rate of each of the estimated CFATs.
The expected NPV of the project is given by the following mathematical formulation:
wherePj =The probability of the jth path occurring which is equal to the joint probability along the path;
NPVj = NPV of the j th path occurring.
In our example, the joint probability, Pj for the worst path is 0.06 (0.3 × 0.2) and for the best path is 0.03 (0.3 × 0.1). The sum of all these joint probabilities must be equal to 1. The last column shows the expected NPV, which is obtained by summing up the product of NPV of jth path and the corresponding probability of jth path (EPj × NPVj). The sum of these weighted NPVs is positive and, therefore, the project should be accepted.This approach has the advantage of exhibiting a bird’s eye view of all the possibilities associated with the proposed project.
m
1jjj )9(NPVPNPV
CONVENTIONAL TECHNIQUES OF RISK ANALYSIS
Payback Risk-adjusted discount rate Certainty equivalent
32
Risk Analysis in Practice
Most companies in India account for risk while evaluating their capital expenditure decisions.
The following factors are considered to influence the riskiness of investment projects: price of raw material and other inputs price of product product demand government policies technological changes project life inflation
33
Risk Analysis in Practice
Four factors thought to be contributing most to the project riskiness are: selling price product demand technical changes government policies
Methods of risk analysis in practice are: sensitivity analysis conservative forecasts
34
Sensitivity Analysis & Conservative Forecasts
Sensitivity analysis allows to see the impact of the change in the behaviour of critical variables on the project profitability.
Conservative forecasts include using short payback or higher discount rate for discounting cash flows.
Except a very few companies most companies do not use the statistical and other sophisticated techniques for analysing risk in investment decisions.
35
Payback
This method, as applied in practice, is more an attempt to allow for risk in capital budgeting decision rather than a method to measure profitability.
The merit of payback Its simplicity. Focusing attention on the near term future and thereby emphasising
the liquidity of the firm through recovery of capital. Favouring short term projects over what may be riskier, longer term
projects.
Even as a method for allowing risks of time nature, it ignores the time value of cash flows.
36
Risk-Adjusted Discount Rate Risk-adjusted discount rate, will allow for both time
preference and risk preference and will be a sum of the risk-free rate and the risk-premium rate reflecting the investor’s attitude towards risk.
Under CAPM, the risk-premium is the difference between the market rate of return and the risk-free rate multiplied by the beta of the project.
37
= 0
NCFNPV =
(1 )
nt
tt k
f rk = k + k
We shall be using the following equation for the purpose of determining NPV under the RAD method.
where CFATt = expected CFAT in year t, Kr = risk-adjusted discount rate, CO = cash outflows.Thus, projects are evaluated on the basis of future cash flow projections and an appropriate discount rate. Example 5 clarifies how the Kr can be used to evaluate capital budgeting projects.
n
1tt
r
t (3)COk1
CFATNPV
Example
Cash outlays (Rs 1,00,000)50,00060,00040,000
CFAT Year 1Year 2Year 3
Riskless rate of return = 6 per cent
Risk-adjusted rate of return for the current project = 20 per cent
Solution
6,410 Rs (0.579)] 40,000 [Rs (0.694)] 60,000 [Rs (0.833)] 50,000 [Rs
1,00,000 Rs.201
40,000Rs
.201
60,000Rs
.201
50,000 Rs 1,00,000) (Rs NPV 32
Risk-adjusted Discount Rate: Merits
It is simple and can be easily understood. It has a great deal of intuitive appeal for risk-averse
businessman. It incorporates an attitude (risk-aversion) towards
uncertainty.
39
Risk-adjusted Discount Rate: LimitationsThere is no easy way of deriving a risk-adjusted discount rate.
CAPM provides a basis of calculating the risk-adjusted discount rate.
It does not make any risk adjustment in the numerator for the cash flows that are forecast over the future years.
It is based on the assumption that investors are risk-averse. Though it is generally true, yet there exists a category of risk seekers who do not demand premium for assuming risks; they are willing to pay a premium to take risks.
40
Example41
Example42
Certainty-Equivalent
Reduce the forecasts of cash flows to some conservative levels.The certainty-equivalent coefficient assumes a value between 0 and 1, and varies inversely with risk. Decision-maker subjectively or objectively establishes the coefficients.
The certainty—equivalent coefficient can be determined as a relationship between the certain cash flows and the risky cash flows.
43
= 0
NCFNPV =
(1 )f
nt t
tt k
*NCF Certain net cash flow =
NCF Risky net cash flowt
tt
We illustrate below the certainty-equivalent approach to adjust risk to capital budgeting analysis on the basis of Previous Example.
Year Coefficient
1
2
3
0.90
0.70
0.60
The certainty-equivalent cash inflows would be as follows:
Year 1 = Rs 45,000 (coefficient 0.9 × Rs 50,000, the expected cash inflows)
Year 2 = Rs 42,000 (0.70 × Rs 60,000)
Year 3 = Rs 24,000 (0.60 × Rs 40,000)
This would be discounted by the riskless rate of return, which is, 6 per cent. Substituting the value in Equation (5),
Since the NPV is negative, the project should be rejected. This decision is in conflict with the decision using the risk-adjusted discount rate where K = 20 per cent. Thus, both these methods may not yield identical results.
25Rs1,00,000Rs0.84024,000Rs0.89042,000 Rs(0.943) 45,000 Rs
1,00,000Rs0.061
24,000Rs
0.061
42,000Rs
0.061
45,000RsNPV 321
Certainty-Equivalent: Evaluation
First, the forecaster, expecting the reduction that will be made in his forecasts, may inflate them in anticipation.
Second, if forecasts have to pass through several layers of management, the effect may be to greatly exaggerate the original forecast or to make it ultra-conservative.
Third, by focusing explicit attention only on the gloomy outcomes, chances are increased for passing by some good investments.
45
Example46
Risk-adjusted Discount Rate Vs. Certainty-Equivalent The certainty-equivalent approach recognises risk in capital
budgeting analysis by adjusting estimated cash flows and employs risk-free rate to discount the adjusted cash flows.
On the other hand, the risk-adjusted discount rate adjusts for risk by adjusting the discount rate. It has been suggested that the certainty-equivalent approach is theoretically a superior technique.
The risk-adjusted discount rate approach will yield the same result as the certainty-equivalent approach if the risk-free rate is constant and the risk-adjusted discount rate is the same for all future periods.
47
SENSITIVITY ANALYSIS
Sensitivity analysis is a way of analysing change in the project’s NPV (or IRR) for a given change in one of the variables.
The decision maker, while performing sensitivity analysis, computes the project’s NPV (or IRR) for each forecast under three assumptions: pessimistic, expected, and optimistic.
48
SENSITIVITY ANALYSIS
The following three steps are involved in the use of sensitivity analysis:
1. Identification of all those variables, which have an influence on the project’s NPV (or IRR).
2. Definition of the underlying (mathematical) relationship between the variables.
3. Analysis of the impact of the change in each of the variables on the project’s NPV.
49
SENSITIVITY ANALYSIS(‘000)
YEAR 0 YEAR 1 - 101. INVESTMENT (20,000)2. SALES 18,0003. VARIABLE COSTS (66 2/3 % OF SALES) 12,0004. FIXED COSTS 1,0005. DEPRECIATION 2,0006. PRE-TAX PROFIT 3,0007. TAXES 1,0008. PROFIT AFTER TAXES 2,0009. CASH FLOW FROM OPERATION 4,00010. NET CASH FLOW 4,000
NPV = -20,000,000 + 4,000,000 (5.650) = 2,600,000 ( discount rate = 12 % )
RS. IN MILLIONRANGE NPV
KEY VARIABLE PESSIMISTIC EXPECTED OPTIMISTIC PESSIMISTIC EXPECTED OPTIMISTIC INVESTMENT (RS. IN MILLION) 24 20 18 -0.65 2.60 4.22 SALES (RS. IN MILLION) 15 18 21 -1.17 2.60 6.40 VARIABLE COSTS AS A 70 66.66 65 0.34 2.60
3.73 PERCENT OF SALES FIXED COSTS 1.3 1.0 0.8 1.47 2.60 3.33
DCF Break-even Analysis
Sensitivity analysis is a variation of the break-even analysis.
DCF break-even point is different from the accounting break-even point. The accounting break-even point is estimated as fixed costs divided by the contribution ratio. It does not account for the opportunity cost of capital, and fixed costs include both cash plus non-cash costs (such as depreciation).
51
BREAK-EVEN ANALYSIS
• ACCOUNTING BREAK-EVEN ANALYSISFIXED COSTS + DEPRECIATION 1 + 2
= = RS. 9 MILLION
CONTRIBUTION MARGIN RATIO 0.333
CASH FLOW FORECAST FOR NAVEEN’S FLOUR MILL PROJECT
(‘000)
YEAR 0 YEAR 1 - 10
1. INVESTMENT (20,000)
2. SALES 18,000
3. VARIABLE COSTS (66 2/3% OF SALES) 12,000
4. FIXED COSTS 1,000
5. DEPRECIATION 2,000
6. PRE-TAX PROFIT 3,000
7. TAXES 1,000
8. PROFIT AFTER TAXES 2,000
9. CASH FLOW FROM OPERATION 4,000
10. NET CASH FLOW (20,000) 4,000
• CASH BREAK-EVEN ANALYSIS
Sensitivity Analysis: Pros and Cons It compels the decision-maker to identify the variables, which
affect the cash flow forecasts. This helps him in understanding the investment project in totality.
It indicates the critical variables for which additional information may be obtained. The decision-maker can consider actions, which may help in strengthening the ‘weak spots’ in the project.
It helps to expose inappropriate forecasts, and thus guides the decision-maker to concentrate on relevant variables.
53
Sensitivity Analysis: Pros and ConsIt does not provide clear-cut results. The terms
‘optimistic’ and ‘pessimistic’ could mean different things to different persons in an organisation. Thus, the range of values suggested may be inconsistent.
It fails to focus on the interrelationship between variables. For example, sale volume may be related to price and cost. A price cut may lead to high sales and low operating cost.
54
SCENARIO ANALYSIS
One way to examine the risk of investment is to analyse the impact of alternative combinations of variables, called scenarios, on the project’s NPV (or IRR).
The decision-maker can develop some plausible scenarios for this purpose. For instance, we can consider three scenarios: pessimistic, optimistic and expected.
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SCENARIO ANALYSISPESSIMISTIC, NORMAL AND OPTIMISTIC
Pessimistic Scenario
Expected Scenario
Optimistic Scenario
1. Investment 24 20 18
2. Sales 15 18 21
3. Variable costs 10.5 (70%) 12 (66.7%) 13.65 (65%)
4. Fixed costs 1.3 1.0 0.8
5. Depreciation 2.4 2.0 1.8
6. Pre-tax profit 0.8 3.0 4.75
7. Tax 0.27 1.0 1.58
8. Profit after tax 0.53 2.0 3.17
9. Annual cash flow from operations 2.93 4.0 4.97
10. Net present value (9) x PVIFA (12%, 10 yrs) – (1)
(7.45) 2.60 10.06
SIMULATION ANALYSIS
The Monte Carlo simulation or simply the simulation analysis considers the interactions among variables and probabilities of the change in variables. It computes the probability distribution of NPV.
The simulation analysis involves the following steps: First, you should identify variables that influence cash inflows and
outflows. Second, specify the formulae that relate variables. Third, indicate the probability distribution for each variable. Fourth, develop a computer programme that randomly selects one
value from the probability distribution of each variable and uses these values to calculate the project’s NPV.
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Simulation Analysis: Shortcomings The model becomes quite complex to use. It does not indicate whether or not the project
should be accepted. Simulation analysis, like sensitivity or scenario
analysis, considers the risk of any project in isolation of other projects.
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Decision Trees for Sequential Investment DecisionsInvestment expenditures are not an isolated
period commitments, but as links in a chain of present and future commitments.
An analytical technique to handle the sequential decisions is to employ decision trees.
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Steps in Decision Tree ApproachDefine investment Identify decision alternatives Draw a decision tree
decision points chance events
Analyse data
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Usefulness of Decision Tree Approach
Clarity: It clearly brings out the implicit assumptions and calculations for all to see, question and revise.
Graphic visualization: It allows a decision maker to visualise assumptions and alternatives in graphic form, which is usually much easier to understand than the more abstract, analytical form.
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Decision Tree Approach: Limitations The decision tree diagrams can become more and
more complicated as the decision maker decides to include more alternatives and more variables and to look farther and farther in time.
It is complicated even further if the analysis is extended to include interdependent alternatives and variables that are dependent upon one another.
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SOLVED PROBLEMS
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Consumer demand for a new toy
Probability of occurrence
Estimated sales in year (Rs in lakh)
1 2 3
Above average 0.30 12 25 6
Average 0.60 7 17 4
Below average 0.10 2 9 1.5
SOLVED PROBLEM 1
Toy Enterprises Ltd designs and manufactures toys. Past experience indicates that the product life of a toy is 3 years. Promotional advertising produces an increase in sales in the early years, but there is a substantial sales decline in the final year of a toy’s life.
Consumer demand for new toys placed on the market tends to fall into three classes. About 30 per cent of the new toys sell well above expectations, 60 per cent sell as anticipated, and 10 per cent have poor consumer acceptance.
A new toy has been developed. The following sales projections were made by carefully evaluating the consumer demand.
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Variable costs are estimated at 30 per cent of the selling price. Special machinery must be purchased at a cost of Rs 8,60,000 which will be installed in an unused portion of the factory. The company has been trying unsuccessfully for several years to rent out the vacant portion at Rs 50,000 per year. Fixed expenses (excluding depreciation) are estimated at Rs 50,000 per year. The new machinery will be depreciated by the written down value method @ 25 per cent with an estimated value of Rs 1,10,000 at the end of the third year. Assume this is the only asset in the block. Advertising and promotional expenses will be incurred uniformly, and will total Rs 1,00,000 in the first year, Rs 1,50,000 in the second year, and Rs 50,000 in the third year.
The company is subject to a corporate tax rate of 35 per cent. Its cost of capital is 10 per cent.
(i)Prepare a schedule computing the probable sales of this new toy in each of the three years. Also, determine the NPV of the proposal.
(ii)Assuming that cash flows occur uniformly throughout each year, determine the NPV of the proposal. The present value of Re 1 earned uniformly throughout the year discounted at 10 per cent is as follows:
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Solution
(i) Schedule showing probable sales of the new toy, years 1–3(Rs in lakh)
Consumer demand
for new toy
Probability of occurrence
(Pj )
Years (estimated sales)
Probable sales per year
1 2 3 1 2 3
Above average 0.30 12 25 6 3.6 7.5 1.80
Average 0.60 7 17 4 4.2 10.2 2.40
Below average 0.10 2 9 1.5 0.2 0.9 0.15
8.0 18.6 4.35
Year Discount factor
1 0.95
2 0.86
3 0.78
(iii) Give your recommendations in both the situations.
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Determination of CFAT
Particulars Years
1 2 3
Probable sales revenue Rs 8,00,000
Rs 18,60,000
Rs 4,35,000
Less: Variable costs (0.30) 2,40,000 5,58,000 1,30,500
Less: Depreciation 2,15,000 1,61,250 Nil*
Cash fixed costs 50,000 50,000 50,000
Advertising expenses 1,00,000 1,50,000 50,000
EBT 1,95,000 9,40,750 2,04,500
Less: Taxes (0.35) 68,250 3,29,263 71,575
EAT 1,26,750 6,11,487 1,32,925
CFAT (EAT + Depreciation) 3,41,750 7,72,737 1,32,925
Add: Salvage value — — 1,10,000
Add: Tax savings on short-term capital loss**
— — 1,30,812
3,41,750 7,72,737 3,73,737
* No depreciation in terminal year.** (Rs 3,73,750 × 0.35)
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(ii) Determination of NPV assuming CFAT occurs uniformly throughout the year
Year CFAT PV factor (0.10)
Total PV
1 Rs 3,41,750 0.95 Rs 3,24,662
2 7,27,737 0.86 6,25,854
3 1,32,925 0.78 1,03,681
3 1,10,000 (salvage value) 0.751 82,610
3 1,32,812 (tax savings on short-term capital loss) 0.751 98,240
Total present value 12,35,047
Less: Cash outflows 8,60,000
NPV 3,75,047
(iii) Recommendation The project should be accepted in both the situations.
Determination of NPV
Year CFAT PV factor (0.10) Total PV
1 Rs 3,41,750 0.909 Rs 3,10,651
2 7,27,737 0.826 6,01,111
3 3,73,737 0.751 2,80,676
Total present value 11,92,438
Less: Cash outflows 8,60,000
NPV 3,32,438
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SOLVED PROBLEM 2
A company has the following estimates of the present values of the future
cash flows after taxes associated with the investment proposal, concerned
with expanding the plant capacity. It intends to use a decision-tree approach
to get a clear picture of the possible outcomes of this investment. The plant
expansion is expected to cost Rs 3,00,000. The respective PVs of future CFAT
and probabilities are as follows:
PV of future CFAT
With expansion Without expansion Probabilities
Rs 3,00,000 Rs 2,00,000 0.2
5,00,000 2,00,000 0.4
9,00,000 3,50,000 0.4
Advise the company regarding the financial feasibility of the project.
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Solution
The relevant computation are depicted below
Decision Tree
Time:0 Year 1
Probabilities (Pi)
PV of CFAT Expected PV
(CFAT) x (Pi)
0.2 Rs 3,00,000 Rs 60,000
0.4 5,00,000 2,00,000
0.4 9,00,000 3,60,000
6,20,000
Less: Cash Outflows - 3,00,000
NPV 3,20,000
0.2 2,00,000 40,000
0.4 2,00,000 80,000
0.4 3,50,000 1,40,000
2,60,000
Less: Cash outflows Nil
NPV 2,60,000
The expected NPV with plant expansion and without expansion is Rs 3,20,000 and Rs 2,60,000 respectively. Therefore, the company is advised to expand the plant capacity
Cash outlays
Nil
Cash outlays
Rs 3,00,000
Do not expand plant
Expand plant
Decision tree
