The document discusses various techniques for risk analysis in project finance, including sensitivity analysis, scenario analysis, break-even analysis, and simulation analysis. It defines key risk analysis terms and provides examples of calculating expected net present value and standard deviation of NPV using the Hiller model under both uncorrelated and perfectly correlated cash flows. Simulation analysis involves modeling the relationship between variable factors and NPV, specifying probability distributions, running simulations to obtain multiple NPV outcomes, and analyzing the results. Project selection under risk may involve judgmental evaluation, payback period requirements, or risk-adjusted discount rates.

1. Project: Risk Analysis
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2. Risk Analysis
▪ Risk is inherent in almost every business decision
▪ Risk refers to variability
▪ Capital budgeting decision involves cost and benefits over a long period of time
▪ Financial analysis has two phases
▪ feasibility analysis
▪ Risk analysis
▪ Sources of Risk
▪ Project Specific Risk: factors specific to project like quality, production
▪ Corporate Risk: action of competitors
▪ Industry specific Risk: technological developments and regulatory charges
▪ Market risk: Changes in microeconomic factors have impact project
▪ International risk: in case of foreign projects or political risk
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3. Techniques of Risk Analysis
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Techniques for risk
analysis
Analysis of Stand
alone Project
Analysis of
Contextual Risk
Sensitivity Analysis Scenario Analysis
Break Even Analysis Hillier Model
Simulation Analysis Decision tree analysis
Corporate Risk
Analysis
Market Risk Analysis

4. Measures of Risk
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Range : Range of variance is difference of maximum and minimum value
Mean or average ҧ𝑥 = 𝐸 𝑥 = σ𝑖 𝑝𝑖 𝑥𝑖
Standard deviation : standard deviation (𝜎 ) is defined as 𝜎 = 𝐸 𝑥 − ҧ𝑥 2 = σ𝑖 𝑝𝑖 𝑥 − ҧ𝑥 2
Coefficient of Variation: standard deviation (𝜎 ) is not adjusted for scale. Coefficient of
variation is adjusted for scale and is defined as 𝐶𝑉 =
𝜎
𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑀𝑒𝑎𝑛 𝑣𝑎𝑙𝑢𝑒
Semi Variance: since investors are concerned with negative variations only so computing
variance with only negative errors (outcome less than mean) gives semi variance
standard deviation (𝜎 ) is defined as
𝑠𝑒𝑚𝑖 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 = 𝐸 𝑥 − ҧ𝑥 2 𝑓𝑜𝑟 𝑜𝑛𝑙𝑦 𝑣𝑎𝑙𝑢𝑒𝑠 𝑜𝑓 𝑥 < ҧ𝑥

5. Measures of Risk
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NPY ( thousands) x Probability p px x- ҧ𝑥 𝑥 − ҧ𝑥 2 p * sq. error
200 0.3 60 -340 115600 34680
600 0.5 300 60 3600 1800
900 0.2 180 360 129600 25920
mean NPV ( ҧ𝑥) 540 Sq. variance 62400
std deviation 249.8
For the example, Range = 900-200 = 700 K
𝑀𝑒𝑎𝑛 𝑁𝑃𝑉 = 0.3 𝑥 200 + 0.5 𝑥 600 + 0.2 𝑥900 = 540
𝜎 = 𝐸 𝑥 − ҧ𝑥 2 = 0.3 𝑥 (−340)2+0.5 𝑥 602 + 0.2 𝑥 3602 = 249.8 𝐾
Coefficient of Variation, 𝐶𝑉 =
249.8
540
= 0.46
Semi standard deviation= 0.3 𝑥 (−340)2= 186.2 𝐾

6. Measures of Risk
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• Standard deviation is most commonly employed as measure of risk in finance.
• For computing mean and dispersion variables, probability distribution is required,
• If sufficient records for similar ventures are available, probability distribution is quite
‘objective,
• If sufficient records are not available, probability distribution is quite ‘subjective’,
Prospective on risk
There are three perspectives of the risk
• Stand alone risk: risk of the project when viewed in isolation
• Firm risk or Corporate risk: contribution of a project to the risk of the firm
• Systematic risk or market risk: risk of a project from view point of diversified investor

7. Sensitivity analysis
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Sensitivity analysis is ‘what if’ analysis
NPV = -20000+4000 x PVIFA
= -20000+4000 x 5.65
= 2600 K
Cash flows depends on various factors and can
vary widely.
So optimistic and pessimistic estimates for
variables defined and NPV calculated
Cash Flow of ABC LTD project
(in thousands) Year 0 year 1-10
1 Investment 20000
2 Sales 18000
3 Variable cost ( 2/3 of sales) 12000
4 Fixed cost 1000
5 Depreciation 2000
6 Pre tax profit 3000
7 Taxes (@ 33.3 %) 1000
8 PAT 2000
9
cash flow from operation
(PAT+ depreciation) 4000
10 Net cash flow 2000 4000
Cost of capital 12%
Accumulated PV indexing factor (PVIFA) @12% 5.65

8. Sensitivity analysis
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• NPV calculated by varying one variable at a time
• NPV is more sensitive to sales and least sensitive to fixed cost. For more sensitive
variable, it may be explored how variability of the factor can be contained.
• In real situation, many variable may change at a time so interpretation of results is
subjective
Sensitivity of NPV to variations in the value of key variables in ABC LTD project
Range (in thousands) NPV
Pessimistic Expected Optimistic Pessimistic Expected Optimistic Variation
1 Investment 24000 20000 18000 -1400 2600 4600 6000
2 Sales 15000 18000 21000 -1147 2600 6366 7513
3
Variable cost
as percentage
70 66.67 65 340 2600 3730 3390
4 Fixed cost 1300 1000 800 1470 2600 3353 1883

9. Scenario analysis
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• Scenario analysis is beneficial when
various scenarios are well defined. It
considers several variable at a time
• More variable are required to be
estimated
• Normally scenarios are not discretely
defined
Sensitivity of NPV to variations in the value of key
variables in ABC LTD project
Range (in thousands)
Pessimistic Expected Optimistic
1 Investment 24000 20000 18000
2 Sales 15000 18000 21000
3
Variable cost
as percentage
70 66.67 65
4 Fixed cost 1300 1000 800
NPV -8180 2600 10438

10. Break-Even analysis
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• It tells what is minimum value of key variables/
revenues so that project does not ‘lose money’
• Variable cost to sales ratio= 12/18= 0.667
• Contribution to margin ratio =0.333
• Accounting break even=
𝐹𝑖𝑥𝑒𝑑 𝑐𝑜𝑠𝑡+𝑑𝑒𝑝𝑟𝑒𝑐𝑖𝑎𝑡𝑖𝑜𝑛
𝑐𝑜𝑛𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 𝑡𝑜 𝑚𝑎𝑟𝑔𝑖𝑛 𝑟𝑎𝑡𝑖𝑜
=
1000+2000
0.333
= 9000 𝐾
• For sales 0f 9000 K , PBT, PAT will be zero
• At accounting break even project gives zero %
return
Cash Flow of ABC LTD project
(in thousands) Year 0 year 1-10
1 Investment 20000
2 Sales 18000
3 Variable cost ( 2/3 of sales) 12000
4 Fixed cost 1000
5 Depreciation 2000
6 Pre tax profit 3000
7 Taxes (@ 33.3 %) 1000
8 PAT 2000
9
cash flow from operation
(PAT+ depreciation) 4000
10 Net cash flow 2000 4000
Cost of capital 12%
Accumulated PV indexing factor (PVIFA) @ 12 % 5.65

11. Financial Break-Even analysis
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• Focus is on NPV and not on accounting profit
Variable cost =0.667 x sales
Contribution = 0.333 x sales
Fixed cost = 1000
Depreciation =2000
PBT= 0.333 x sales-3000
Tax= 0.333 x (0.333 x sales-3000)=0.111 x sales-1000
PAT= 0.667 x (0.333 x sales-3000)=0.222 x sales-2000
Cash flows= PAT+ depreciation= 0.222 x sales
PV of cash flows= Cash flows x PVIFA= 1.255 x sales
Breakeven NPV = 20000- 1.255 x sales= 0
so Breakeven sales= 15936 K
Cash Flow of ABC LTD project
(in thousands) Year 0 year 1-10
1 Investment 20000
2 Sales 18000
3 Variable cost (2/3 of sales) 12000
4 Fixed cost 1000
5 Depreciation 2000
6 Pre tax profit 3000
7 Taxes (@ 33.3 %) 1000
8 PAT 2000
9
cash flow from operation
(PAT+ depreciation) 4000
10 Net cash flow 2000 4000
Cost of capital 12%
Accumulated PV indexing factor (PVIFA) @ 12% 5.65

12. Hiller Model : Un correlated cash flows
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interest rate = 12 %
Expected NPV, 𝑁𝑃𝑉 = σ 𝑡=1
3 𝐴 𝑡
(1+𝑟) 𝑡 =
5000
1.12
+
4000
(1.12)2 +
5000
(1.12)3
= 4464.3 + 3188. 8 + 3558.9 = 11211.9
Standard deviation 𝜎 𝑁𝑃𝑉 = σ 𝑡=1
3 𝜎𝑡
2
(1+𝑟)2𝑡 =
2400000
1.122 +
1600000
1.124 +
2400000
1.126 = 2036.18
Analytical derivation to find expected NPV and standard deviation of NPV
Year 1 Year 2 Year 3
Net Cash flow (Rs) Probability Net Cash flow (Rs) Probability Net Cash flow (Rs) Probability
3000 0.3 2000 0.2 3000 0.3
5000 0.4 4000 0.6 5000 0.4
7000 0.3 6000 0.2 7000 0.3
Mean 5000 4000 5000
Standard variance 2400000 Standard variance 1600000 Standard variance 2400000

13. Hiller Model : Perfectly correlated cash flows
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interest rate = 12 %
Expected NPV, 𝑁𝑃𝑉 = σ 𝑡=1
4 𝐴 𝑡
(1+𝑟) 𝑡 − 1 =
5000
1.12
+
3000
(1.12)2 +
4000
(1.12)3 +
3000
(1.12)3 = 11609.54
Standard deviation 𝜎 𝑁𝑃𝑉 = σ 𝑡=1
4 𝜎𝑡
(1+𝑟) 𝑡 =
1500
1.12
+
1000
1.122 +
2000
1.123 +
1200
1.124 = 4322.66
Analytical derivation to find expected NPV and standard deviation of NPV for perfectly
correlated cash flows
A investment project involves a current outlay of Rs 10000. mean and standard deviation of cash
flows , which are perfectly correlated are as follows
year 𝐴 𝑡 𝜎 𝑁𝑃𝑉
1 5000 1500
2 3000 1000
3 4000 2000
4 3000 1200

14. Simulation Analysis
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• Sensitivity of criterion of merit (NPV IRR, ROI etc.) may not be adequate for decision making
• Likelihood of occurrence of circumstances (probability profile) can be obtained by
simulation
• The steps in simulation analysis are (criterion of merit is NPV)
1. Model the project : how criterion of merit is related to variables
• Parameter variables : input variables given by decision maker held constant in
simulation
• Exogenous variable: which are random in nature and can not be controlled
2. Specify the value of parameters and probability distribution of exogenous variables
3. Select a value randomly from probability distribution of exogenous variables.
4. Determine NPV
5. Repeat steps 2 to 4 a number of times to get large number of simulated NPV’s
6. Plot the frequency distribution of NPV

15. Project selection under risk
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• Judgemental Evaluation: Accept or reject the project based on the risk and return
characteristics without any formal method for incorporating risk in decision making
• Payback period requirement: payback period requirement is applied to control risk.
• Risk adjusted discount method:
• If the risk of the project is equal to risk of existing investment, discount rate is
average cost of capital
• If the risk of the project is greater than risk of existing investment, discount rate is
higher than average cost of capital

16. Thank You
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Contact
Email: naimkidwai@gmail.com
https://nrkidwai.wordpress.com/