FINANCE & PRICING
STRATEGY
Project evaluation
Mushegh Harutyunyan
m.harutyunyan@imperial.ac.uk
Net Present Value (NPV)
• Converts future cash flow into today’s money
• Allows for more consistent comparison between projects
• More efficient decisions
• Example
Project X cash flow information
t=0 t=1 t=2 t=3 t=4
-15,000 6,500 6,500 4,500 3,000
• How much is £6,500 in one year worth today?
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Net Present Value (NPV)
• $1 tomorrow is worth less than $1 today
• The difference is determined by the opportunity cost
• $1 today can be used to earn $1 + $X tomorrow
• Hence, tomorrow’s $1 must be discounted to find its worth
today
• The discount rate is influenced by inflation rate, risk level, etc.
• Risk-free discount rate
• Gov. bonds, savings account return
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Net Present Value (NPV)
• How much is £6,500 in one year worth today?
• Let’s put it differently: how much do I need to invest today to
receive £6,500 in one year?
• Suppose the project is risk free => use risk-free discount rate
• Let r = 2% be the risk-free discount rate
1/1.02 is the 1
discount factor £6,500 = £6,373
(1 + 0.02)
• => if I invest £6,373, then I will receive £6,500 in one year
• Therefore, £6,373 is the present value of £6,500 in one year
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Net Present Value (NPV)
• Next, how much is £6,500 in two years worth today?
1
£6,500 = £6,248
(1 + 0.02)2
• If I invest £6,248, then I will receive £6,500 in two years
• Therefore, £6,500 in two years is worth £6,248 today
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Net Present Value (NPV)
• NPV of Project X is as follows
Project X cash flow information
t=0 t=1 t=2 t=3 t=4
(15,000) 6,500 6,500 4,500 3,000
Disc factor
PV
NPV
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Net Present Value (NPV)
• NPV of Project X is as follows
Project X cash flow information
t=0 t=1 t=2 t=3 t=4
(15,000) 6,500 6,500 4,500 3,000
Disc factor 1 0.980 0.961 0.942 0.924
PV
NPV
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Net Present Value (NPV)
• NPV of Project X is as follows
Project X cash flow information
t=0 t=1 t=2 t=3 t=4
(15,000) 6,500 6,500 4,500 3,000
Disc factor 1 0.980 0.961 0.942 0.924
PV (15,000) 6,373 6,248 4,240 2,772
NPV
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Net Present Value (NPV)
• NPV of Project X is as follows
Project X cash flow information
t=0 t=1 t=2 t=3 t=4
(15,000) 6,500 6,500 4,500 3,000
Disc factor 1 0.980 0.961 0.942 0.924
PV (15,000) 6,373 6,248 4,240 2,772
NPV 4,632
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Net Present Value (NPV)
• More generally, consider a project with a cash flow CF0, CF1,
… CFT
• Annual interest rate is r
• Then, the NPV of the project is as follows:
$
CF!
NPV = " !
1+r
!"#
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Planning horizon
• What is T?
• Varies from business to business
• Having more resources allows a longer planning horizon
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Internal Rate of Return (IRR)
• IRR is the discount rate that ensures NPV = 0
$
CF!
" ! =0
1 + 𝐈𝐑𝐑
!"#
• After finding IRR, compare it to the actual discount rate (r)
• What if IRR > discount rate? accept
• What if IRR < discount rate? Reject
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Example
• Consider project X
Project X cash flow information
t=0 t=1 t=2 t=3 t=4
-15,000 6,500 6,500 4,500 3,000
• Let’s find its IRR
• WolframAlpha is a convenient tool for solving equations
• Copy-paste this into WolframAlpha: -15000 + 6500/(1+x) +
6500/(1+x)^2 + 4500/(1+x)^3 + 3000/(1+x)^4 = 0
• Click Enter!
• Can use Excel too
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Example
• Consider project X
Project X cash flow information
t=0 t=1 t=2 t=3 t=4
-15,000 6,500 6,500 4,500 3,000
• IRR is about 16%
• Compare with discount rate of 2%
• => very attractive opportunity
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Practice
• ABC Inc. owns a small plot of land
• The land can be used for one of the following projects
Project A
0 1 2 3 4
(10,000) 3,500 3,500 3,500 3,500
Project B
0 1 2 3 4
(100,000) 30,000 30,000 30,000 30,000
• What are the project IRR’s? What are the NPV’s?
• For NPV, assume r = 4%
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IRR vs NPV
• ABC Inc. example illustrates a drawback of IRR
Project A Project B
IRR 15% 8%
NPV £2,705 £8,897
• IRR ignores absolute numbers, focuses on percentages
• ABC Inc. is clearly better off by choosing Project B, although
its IRR is smaller
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RISK
RISK
Imperial College Business School
RISK
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Risk
• What is risk?
• Do you have risk in your life? How do you manage it?
• How is risk managed at work?
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Risk versus uncertainty
• Risk can be “quantified”
• For example, we can estimate probabilities for events
• With uncertainty, we may not identify events or assign
probabilities
• Scenario analysis
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Risk
• Objective probabilities
• Can be established mathematically
• Or based on historical data
• Subjective probabilities
• Based on experience, intuition
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Analyzing risk
i. Risk-adjusted discount rate
ii. Sensitivity analysis
iii. Scenario analysis
iv. Probability analysis
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Risk-adjusted discount rate
• Example with three levels of risk: low, medium, high
Risk-free rate Risk premium Risk-adjusted rate
Low risk 2% 3% 5%
Medium risk 2% 7% 9%
High risk 2% 15% 17%
• Let’s evaluate project A using risk-adjusted discount rates
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Risk-adjusted discount rate
Project A
0 1 2 3 4
(10,000) 3,500 3,500 3,500 3,500
Low risk Medium risk High risk
NPV £2,411 £1,399 -£399
Decision Accept Accept Reject
• What are the drawbacks of using risk-adjusted discounting?
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Sensitivity analysis
• So far, NPV analysis has been very static
• All variables are given, nothing moves anywhere
• How sensitive is our NPV estimate to the assumptions we
make?
• Sensitivity analysis seeks an answer
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Example
• Gerry & Co. is considering launching a new product
targeting Gen Z
• The following estimates are available
Estimate
Price £200
Per-unit production cost (e.g. labour, materials) £145
Initial investment in design and production £3,750,000
First-year overhead (e.g. ads, management salaries) £1,500,000
Overhead after the first year £450,000
Projected Sales 60,000
Risk level Medium (2%+7%)
Tax rate 20%
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Example
Time
0 1 2 3 4
Revenue £12,000,000 £12,000,000 £12,000,000 £12,000,000
Cost of Sales (£8,700,000) (£8,700,000) (£8,700,000) (£8,700,000)
Gross Profit £3,300,000 £3,300,000 £3,300,000 £3,300,000
Overhead (£1,500,000) (£450,000) (£450,000) (£450,000)
R&D (£3,750,000)
Operating
Income £1,800,000 £2,850,000 £2,850,000 £2,850,000
Net Income £1,440,000 £2,280,000 £2,280,000 £2,280,000
Discount factor 1.0000 0.9174 0.8417 0.7722 0.7084
PV (£3,750,000) £1,321,101 £1,919,030 £1,760,578 £1,615,209
NPV £2,865,919
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Example
• What if the market turns out to be too competitive?
• Reduce price by £10 (5%)
Time
0 1 2 3 4
Revenue £11,400,000 £11,400,000 £11,400,000 £11,400,000
Cost of Sales (£8,700,000) (£8,700,000) (£8,700,000) (£8,700,000)
Gross Profit £2,700,000 £2,700,000 £2,700,000 £2,700,000
Overhead (£1,500,000) (£450,000) (£450,000) (£450,000)
R&D (£3,750,000)
Operating
Income £1,200,000 £2,250,000 £2,250,000 £2,250,000
Net Income £960,000 £1,800,000 £1,800,000 £1,800,000
Discount factor 1.0000 0.9174 0.8417 0.7722 0.7084
PV (£3,750,000) £880,734 £1,515,024 £1,389,930 £1,275,165
NPV £1,310,854
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Example
• What if the product turns less popular?
• Sales are lower by 3,000 (5%)
Time
0 1 2 3 4
Revenue £11,400,000 £11,400,000 £11,400,000 £11,400,000
Cost of Sales (£8,265,000) (£8,265,000) (£8,265,000) (£8,265,000)
Gross Profit £3,135,000 £3,135,000 £3,135,000 £3,135,000
Overhead (£1,500,000) (£450,000) (£450,000) (£450,000)
R&D (£3,750,000)
Operating
Income £1,635,000 £2,685,000 £2,685,000 £2,685,000
Net Income £1,308,000 £2,148,000 £2,148,000 £2,148,000
Discount factor 1.0000 0.9174 0.8417 0.7722 0.7084
PV (£3,750,000) £1,200,000 £1,807,929 £1,658,650 £1,521,697
NPV £2,438,276
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Example
• What if problems with product development?
• R&D costs higher by £250,000 (7%)
Time
0 1 2 3 4
Revenue £12,000,000 £12,000,000 £12,000,000 £12,000,000
Cost of Sales (£8,700,000) (£8,700,000) (£8,700,000) (£8,700,000)
Gross Profit £3,300,000 £3,300,000 £3,300,000 £3,300,000
Overhead (£1,500,000) (£450,000) (£450,000) (£450,000)
R&D (£4,000,000)
Operating
Income £1,800,000 £2,850,000 £2,850,000 £2,850,000
Net Income £1,440,000 £2,280,000 £2,280,000 £2,280,000
Discount factor 1.0000 0.9174 0.8417 0.7722 0.7084
PV (£4,000,000) £1,321,101 £1,919,030 £1,760,578 £1,615,209
NPV £2,615,919
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Sensitivity graph
NPV
3.5m
3.0m
2.5m
2.0m
1.5m
% change in
-10 -5 0 5 10 variable
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Sensitivity graph
NPV
3.5m Price
3.0m
2.5m
2.0m
1.5m
% change in
-10 -5 0 5 10 variable
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Sensitivity graph
NPV
3.5m Price
Sales
3.0m
R&D
2.5m
2.0m What do you
see?
1.5m
% change in
-10 -5 0 5 10 variable
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Sensitivity analysis
• What drawbacks does sensitivity analysis have?
• Some events are more likely than others but SA does not assign
probabilities
• All variables are assumed to be constant except one
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Scenario analysis
• Scenario analysis allows us to move more than one variable
at a time
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Example
• Let’s return to Gerry & Co. example
• Suppose that these estimates represent the neutral scenario
Estimate
Price £200
Per-unit production cost (e.g. labour, materials) £145
Initial investment in design and production £3,750,000
First-year overhead (e.g. ads, management salaries) £1,500,000
Overhead after the first year £450,000
Projected Sales 60,000
Risk level Medium (2%+7%)
Tax rate 20%
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Example
• Worst-case scenario
Estimate
Price £180
Per-unit production cost (e.g. labour, materials) £150
Initial investment in design and production £4,250,000
First-year overhead (e.g. ads, management salaries) £1,500,000
Overhead after the first year £450,000
Projected Sales 52,000
Risk level Medium (2%+7%)
Tax rate 20%
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Example
• Best-case scenario
Estimate
Price £200
Per-unit production cost (e.g. labour, materials) £130
Initial investment in design and production £3,750,000
First-year overhead (e.g. ads, management salaries) £1,500,000
Overhead after the first year £450,000
Projected Sales 72,000
Risk level Medium (2%+7%)
Tax rate 20%
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Scenario analysis
• Evaluate NPV’s in each scenario
• How bad is the worst-case scenario?
• How good is the best-case scenario?
NPV
Best case £7,375,609
Neutral case £2,865,919
Worst case -£2,143,771
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Scenario analysis
• Evaluate NPV’s in each scenario
• How bad is the worst-case scenario?
• How good is the best-case scenario?
• For a sound evaluation, we still need to estimate
probabilities
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Probability analysis
• Let’s start with some basics
• What is the probability of getting tails?
• Lottery ticket price £25
Probability Outcome
Heads 0.5 £100
Tails 0.5 -£80
• Will you buy the ticket?
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Probability analysis
• Expected return from the lottery:
0.5*(£100 –£25) + 0.5*(–£80 –£25) = –£15
• This lottery is a bad deal and you should reject!
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Probability analysis
• More generally, suppose there are N possible
outcomes/events
• Outcome #i gives payoff Xi
• Probability of outcome i is pi
• Note that (p1+p2+…+pN) = 1
'
• Then,
Expected return = " X % p%
%"&
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Practice
• Recall the three scenarios from Gerry & Co
• Here are the NPV’s corresponding to each
NPV Event Probability
Best case £7,375,609 Boom 0.1
Neutral case £2,865,919 Stagnation 0.6
Worst case -£2,143,771 Recession 0.3
• What is the expected return of Gerry & Co?
• £1,813,981
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Probability analysis
• Consider this example: 4 projects need to be evaluated
Payoffs
Project A Project B Project C Project D
Boom (0.1) 90 150 150 200
Stagnation (0.6) 90 100 100 0
Recession (0.3) 90 60 -50 -90
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Probability analysis
• Let’s find Expected Return first
Payoffs
Project A Project B Project C Project D
Expected Return 90 93 60 -7
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Probability analysis
• Standard deviation shows how much payoffs are dispersed
from the expected return
Std Deviation = Variance
'
Variance = " X % − Expected Value (p%
%"&
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Probability analysis
Payoffs
Project A Project B Project C Project D
Expected Return 90 93 60 -7
Variance 0 681 5400 6381
Std. Deviation 0 26 73 80
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Probability analysis
Project A Project C
Project B Project D
100
Mean-variance rule
Expected Return
Projects with higher
expected return and 50
lower standard deviation
are more preferable.
0 40 80
Standard Deviation
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Probability analysis
Project A Project C
Project B Project D
• Who might prefer Project D? 100
Expected Return
50
0 40 80
Standard Deviation
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Risk analysis in practice
• Percentage of firms that use a specific method
Firm Size
Small Medium Large
Risk-adjusted discount rate 42% 71% 50%
Sensitivity/scenario analysis 82% 83% 89%
Probability analysis 27% 21% 42%
• Note: Some firms use more than one method
Survey data from Arnold and Hatzopoulos (2000)
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Main takeaways
• Time affects the value of money
• Discount cash flows to compare apples to apples
• NPV and IRR help evaluate projects based on discounted
cash flows
• Risk assessment and incorporation into project evaluation
• 4 different methods
• Conduct sensitivity analysis to measure how sensitive your
qualitative conclusions (accept vs reject) are with respect to
quant. Assumptions
• Probability analysis for more sophisticated decision making
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