NATIONAL AND KAPODISTRIAN UNIVERSITY OF ATHENS
Faculty of Economics
Department of Business Economics and Finance
Center of Financial Studies
Laboratory for Investment Applications
Internal Audit Program
Course: Managerial Economics and Financial
Management - Chapter 2
Instructor: Panayotis Alexakis
In cooperation with:
Under the aegis of:
Managerial Economics and
Financial Management
Chapter 2: Decision Making Under Risk and
Uncertainty
Contents of presentation:
Expected present value, decision trees, risk and
uncertainty and the measure of risk
Case 1: Uncertain cash flows
Objective evaluations
Subjective evaluations
Case 2: Uncertainty and time dependence of cash flows
The choice of postponement
Introduction
As we have seen in Chapter 1, decision making takes place under
conditions of uncertainty. Decision makers can only describe the
possible outcomes of their decisions. If their views take an orderly
form then they can be described with the use of a probability
distribution of the outcomes, where the expected value of the
outcomes can be calculated, the variation as well as other
statistical measures.
 However, there is the possibility for the uncertainty to be a
general one with dispersed results, as the quality of information
on future outcomes is very low, or because significant changes are
to happen shortly, independent of the business decision, which can
affect its results, for example the beginning of a war, a big
economic discovery or a disaster.
 All these, the existence of uncertainty not easily measured can
affect business environment, business behaviour, there can be
surprises, unexpected changes affecting values.

3
Introduction


Such uncertainty can not be subject to rules of risk
distribution.
Below we concentrate on the analysis of measurable
uncertainty. The tools that have been described in
Chapter 1 for evaluation and decision making are now
reintroduced through the incorporation of uncertainty
conditions. The discounting of future cash flows and the
NPV estimations are extended and adjusted for the risk
factor. This adjustment does not change the basic concept
or the effectiveness of these tools. NPV remains the right
measure of the contribution of the investment project to
the net wealth position of those who undertake it.
However, the user of this tool must know the difficulties
in measuring investment values under uncertainty and the
way to face such difficulties.
4
Introduction

1.
2.
3.
It is important to note that economic and investment analysis under
uncertainty is based on the collection and processing of information
which forms a fundamental task for every decision maker, as it sets
light of the future outcome of an investment decision, as:
It relates to information on the investment itself, such as the possible
evolution of prices of raw materials, energy, labour or the product.
It relates to the possible evolution of capital markets of all kinds, such
as the real estate values, stock values, interest rates, exchanges rates,
inflation. This information affects the results achieved by a specific
investment, but originates from general phenomena of the economy
which cannot be controlled by the enterprise.
It refers to the developments on the broader legal and regulatory
framework of an investment, such as the change in taxation, new
environmental rules, rules on competition protection, corporate
transparency and governance, protection of labour and employee
protection.
5
Expected value, decisions trees, risk and
uncertainty and the measure of risk
It was mentioned that EPV analysis is a multiperiod analysis
resulting cost and revenue implications both in present and
future periods, while at the same time there is a probability
distribution of outcomes each period.
 «Decision trees» facilitate EPV analysis so that the
consequencecs of a decision can be easily spotted and then one
calculates the probabilities, and the EPV of the decision.
 The following table presents a decision tree which has a
decision problem at its background. The decision maker (a
company) must choose one branch or another (the undertaking
of investment in large machinery equipment or in small
equipment) and in order to decide, the EPV of the profits
promised by each alternative must be evaluated, using a 10%
opportunity discount rate, having also assigned probabilities to
the possible demand situations presented for each year. This is
done in the following tables and the EPV criterion suggests that
the large equipment should be utilized.

6
Expected value, decisions trees, risk and
uncertainty and the measure of risk
Machine size
Year 1
Demand Profits (€)
Heavy 20,000
Large
Small
Medium 8,000
Light
-2,000
Heavy
14,000
Medium 10,000
Light
2,000
Year 2
Demand
Heavy
Medium
Light
Heavy
Medium
Light
Heavy
Medium
Light
Heavy
Medium
Light
Heavy
Medium
Light
Heavy
Medium
Light
Profits (€)
25,000 25,000
10,000 10,000
2,000
25,000
10,000
10,000 2,000
25,000
10,000
2,000
16,000
12,000
4,000
16,000
12,000
4,000
16,000
12,000
4,000
7
Year 1
Year 2
Calculation of EPV
Machine
Expected
value,PVdecisions
trees, Profits
risk and
Demand
Profits [3]
[4]
Demand
[6]
uncertainty
and
measure
of
PV [7]
Total
PV the
Joint
Prob. Weighted
cost [1]
(prob)risk
[2]
(DF=0.826)
[8]
[9]
NPV [10]
25,000
20,661
34,842
0,08
2,787.36
10,000
8,264
22,445
0,08
1,795.60
2,000
1,652
15,833
0,04
633.32
25,000
20,661
23,933
0,12
2,871.96
10,000
8,264
11,536
0,12
1,384.32
2,000
1,652
4,924
0,06
295.44
25,000
20,661
14,843
0,20
2,968.60
10,000
8,264
2,446
0,20
489.20
2,000
1,652
-4,166
0,10
-416.60
(DF=0.909)
(prob) [5]
Heavy
(P=0.4)
Heavy
20,000
18,181
(P=0.2)
Medium
(P=0.4)
Light
(P=0.2)
Heavy
(P=0.4)
Large
Medium
(€4,000)
(P=0.3)
8,000
7,272
Medium
(P=0.4)
Light
(P=0.2)
Heavy
(P=0.4)
Light
(P=0.5)
-2,000
-1,818
Medium
(P=0.4)
Light
(P=0.2)
Expected Present Value
8
12,809.20
Year 1
Machine
Demand
cost [1]
(prob) [2]
Year 2
Profits [3]
PV [4]
Demand
(DF=0.909)
(prob) [5]
Heavy
Profits [6]
Calculation of EPV
PV [7]
Total PV
Joint Prob.
Weighted
(DF=0.826)
[8]
[9]
NPV [10]
16,000
13,223
22,550
0,08
1,804.00
12,000
9,917
19,244
0,08
1,539.52
4,000
3,305
12,632
0,04
505.28
16,000
13,223
18,913
0,12
2,269.56
12,000
9,917
15,607
0,12
1,879.84
4,000
3,305
8,995
0,06
539.70
16,000
13,223
11,641
0,20
2,328.20
12,000
9,917
8,335
0,20
1,667.00
4,000
3,305
1,723
0,10
172.30
(P=0.4)
Heavy
14,000
12,727
(P=0.2)
Medium
(P=0.4)
Light
(P=0.2)
Heavy
(P=0.4)
Small
Medium
(€3,400)
(P=0.3)
10,000
9,090
Medium
(P=0.4)
Light
(P=0.2)
Heavy
(P=0.4)
16,000
13,223
22,550
Medium
(P=0.4)
Light
(P=0.2)
Expected Present Value
12,698.40
9
Expected value, decisions trees, risk and
uncertainty and the measure of risk
Risk



Coming now to risk and uncertainty, the appropriate measure of risk
allows decision makers to quantify the risk of an investment and also
compare risks of competing decision alternatives.
The degree of risk of a particular decision is related to the
dispersion of outcomes, the range of outcomes within which the
actual outcome may fall after the decision is made. It is defined as the
dispersion of the probability distribution of possible outcomes around
the expected value of these outcomes. Some decision alternatives may
have greater distribution of outcomes, therefore having greater degree
of risks, being more risky.
The appropriate measure of dispersion around the expected value is
the standard deviation of the probability distribution. It shows
the weighted average absolute deviation of all possible outcomes from
the expected value of that probability distribution. The deviation of
each possible outcome from the expected value is weighted by the
probability of each outcome occurring, to find the weighted-average
deviation.
10
Expected value, decisions trees, risk and
uncertainty and the measure of risk

n
2
(
X

EPV
)
Pi
 i
i 1

Where σ denotes standard deviation, Σ presents the sum
of the series of squared and weighted deviations from i =
1,2,3,..,n, Xi represents the ith possible outcome, Pi is the
probability of that outcome and EPV is the expected value
of the probability distribution.
Example:
Calculation of σ for the large machinery equipment and the
small machinery equipment. It is derived that the degree
of risk of large machine (10,326.70) is much greater to
that of small machine (5,870.10).
11
Expected value, decisions trees, risk and
uncertainty and the measure of risk
Calculation of the Standard Deviation for the Large Equipment Decision
Large Equipment
Xi – EPV
(Xi – EPV)2
Pi
(Xi – EPV)2 Pi
Xi (€000)
EPV (€000)
(€000)
(€000)
(€000)
34.830
12.809
22.0186
484.819
0.08
38.7855
22.445
12.809
9.6286
92.710
0.08
7.4168
15.833
12.809
3.0206
9.1240
0.04
0.3649
23.933
12.809
11.1806
125.006
0.12
5.0002
11.536
12.809
-1.2794
1.6368
0.12
0.1964
4.924
12.809
-7.8874
62.2110
0.06
3.7327
14.843
12.809
2.0206
4.0829
0.20
0.8166
2.446
12.809
-10.3694
107.525
0.20
21.5050
-4.166
12.809
-16.9774
288.232
0.10
28.8232
1.00
Variance =106.6413
Variance
12
Expected value, decisions trees, risk and
uncertainty and the measure of risk
Small Equipment
Xi (€000)
EPV (€000)
Xi – EPV
(€000)
(Xi – EPV)2
(€000)
Pi
(Xi – EPV)2 Pi
(€000)
22.550
12.698
9.8489
97.001
0.08
7.7601
19.244
12.698
6.5449
42.836
0.08
3.427
12.632
12.698
-0.0631
0.004
0.04
0.000
18.915
12.698
6.2129
38.600
0.12
4.632
15.607
12.698
2.9089
8.462
0.12
1.015
8.995
12.698
-3.6991
13.683
0.06
0.821
11.641
12.698
-1.0591
1.122
0.20
0.224
8.335
12.698
-4.3631
19.037
0.20
3.807
1.723
12.698
-10.9711
120.365
0.10
12.036
1.00
Variance = 33.7221
Standard Deviation =
Variance
13
= 5.8071 or €5,807.10
Expected value, decisions trees, risk and
uncertainty and the measure of risk
Risk Aversion
 Coming now to risk aversion it is defined as the
feeling of disutility caused by uncertainty. In general,
business decision makers are risk averse, that is they
do not like risk and are only prepared to undertake
risky situations if they are adequately compensated for
bearing the risk involved, gaining in this way sufficient
utility from the return associated with the proposed
investment project.
 Hence, the greater the risk perceived, the greater the
return the investor requires to offset that risk. The
risk (σ) - return (EPV) trade off is the characteristic of
a risk averter.
 We can depict all the above using indifference
curves.
14
Expected value, decisions trees, risk and
uncertainty and the measure of risk
Indifference Curves for a Risk Averter in Risk-Return
Space

Different people have different degrees of risk aversion, reflected
on flatter or steeper indifference curves in the risk-return space.
15
Expected value, decisions trees, risk and
uncertainty and the measure of risk
Comparison between decision alternatives on a risk-adjusted basis
 The risk averse decision maker can compare decision alternatives and select
the one that best serves the company objectives.
Methods:
1. Coefficient of variation.
2. The expected value criterion using different discount rates.
3. The uncertainty – equivalent criterion.
Coefficient of variation:
 Is defined as the ratio of the standard deviation to the expected
present value. So, the coefficient of variation for a probability
distribution indicates the amount of standard deviation or risk, per
euro of expected present value, or return. This criterion involves
choosing the decision alternative with the lowest positive coefficient
of variation.
16
Expected value, decisions trees, risk and
uncertainty and the measure of risk

In the previous example for the large machinery equipment σ =
€10,326.70 and the EPV = 12,809.10, while for the small machine the
respective numbers were €5,807.10 and 12,698.40, rendering CVs
equal to 0,8060 and 0,4575, respectively, opting finally for the small
machine decision alternative, which practically produces the larger
risk-adjusted return (reciprocal of CV).
Expected value criterion
 We now introduce the expected value criterion which refers to the
adjustment of the expected value with the use of higher discount
rates for more risky decision alternatives. As it was mentioned in
Chapter1 , the opportunity discount rate (ODR) is the best interest
rate that could be earned elsewhere, for the same degree of risk.
Therefore, the expected value criterion, adjusted for different risks
of decision alternatives, involves selecting the alternative with the
greater expected value (EPV), after such alternative has been
discounted using the ODR that is specifically appropriate for that
particular decision alternative.
17
Expected value, decisions trees, risk and
uncertainty and the measure of risk

In the previous example, when considering the
significantly different dispersion of the two
machines, suppose that 10% is the correct ODR
for the small machinery and 12% is the correct
ODR for the large machine equipment. It can be
seen in the following table that the large machine
EPV is at 12,301.30€, now less than that of the
small machine (12,698.40€), favouring now the
purchase of the small machine, just as the
previous method (CV) did.
18
YEAR 1
YEAR 2
Initial Cost
Demand
[1]
(prob) [2]
Profits [3]
PV [4]
Demand
(DF=0.8929)
(prob) [5]
Heavy
CALCULATION OF EPV
Profits [6]
PV [7]
Total PV [8]
(DF=0.7972)
Joint Prob.
Weighted
[9]
PV [10]
25,000
19,930
33,788
0,08
2,703.04
10,000
7,972
21,830
0,08
1,746.40
2,000
1,595
15,453
0,04
618.12
25,000
19,930
23,073
0,12
2,768.76
10,000
7,972
11,115
0,12
1,333.80
2,000
1,595
4,738
0,06
284.28
25,000
19,930
14,144
0,20
2,828.80
10,000
7,972
2,186
0,20
437.20
2,000
1,595
-4,191
0,10
-419.10
(P=0.4)
Heavy
20,000
17,858
(P=0.2)
Medium
(P=0.4)
Light
(P=0.2)
Heavy
(P=0.4)
-4,000
Medium
8,000
7,143
(P=0.3)
Medium
(P=0.4)
Light
(P=0.2)
Heavy
(P=0.4)
Light
(P=0.5)
-2,000
-1,786
Medium
(P=0.4)
Light
(P=0.2)
Expected value
12,301.30
19
Expected value, decisions trees, risk and
uncertainty and the measure of risk
Note:
 The two methods developed so far are basically similar and
should rank decision alternatives in the same manner.
However, in terms of data required, the CV criterion can be
used with much lower search costs. It is also implemented
using the firm’s cost of capital as the discount, rates which is
known to the firm.
Certainty equivalent criterion (CE)
 The CE criterion, the certainty equivalent for a decision
alternative with more than one possible outcomes, is the
sum of money, available with certainty, that would cause the
decision maker to be indifferent between that decision
alternative and accepting the certain sum of money. This
criterion also involves the concepts of utility and
indifference.
20
Expected value, decisions trees, risk and
uncertainty and the measure of risk
Certainly Equivalents of the Large-Machine/Small Machine Decision
So, the CE criterion, involves selecting the decision alternative which
has the highest certainty equivalent.
Example:
 Suppose that one is offered the certain sum of €0.50 or the toss for
€1.00. If he comes to be different between the two alternatives, then
his CE for that case is €0.50. CE is the sum of money that will
compensate you to give up the risky situation.
21

Expected value, decisions trees, risk and
uncertainty and the measure of risk

Further, one way to find the CE of an uncertain
venture without involving indifference curves is
to decide what fraction of the EPV would make
you indifferent between the EPV of the uncertain
venture and this fraction of the EPV, if it were
available with certainty. That is, to decide «how
many cents to a euro» you would consider to be
equivalent if these were available with certainty.
This fraction, say 0.65, is known as the certainty
equivalent factor (CEF) and multiplied with
EPV provides the CE. Riskier ventures will have
smaller CEFs.
22
Expected value, decisions trees, risk and
uncertainty and the measure of risk
A «critique» of the three methods
The three methods should be used with caution and reservations. The,
choice among them depends upon three major factors:
1. The frequency with which one is confronted with a particular
decision.
2. The magnitude of the worst possible outcomes.
3. The decision maker’s attitude towards risk and uncertainty, that
is the degree of risk preference or aversion and therefore the
willingness to accept risk and its consequences.
 Furthermore, non-monetary factors could be involved in
decision making, such as environment protection, philanthropic
considerations relating to laying off people in periods of low
demand, avoidance of highly competitive markets and highstress management situations. These may drive managers to
forgo alternatives that serve the purely monetary objectives of
the firm and its managers.
23
Expected value, decisions trees, risk and
uncertainty and the measure of risk
The importance of information for the evaluation of decisions
 It is worth referring to information for evaluation decisions.
 In an environment of uncertainty the firm lacks information about
actual outcomes of its decision. When this lack of information is
valuable to the firm it may engage in search activity and undertake
costs in order to retrieve it. The value of information is the
difference between what you can earn with the information
already held and what you could earn if you were to know with
certainty the outcome prior to making the decision. Search
costs are defined as the costs of obtaining information in the
form needed by the decision maker and within the time
constraints required by the decision maker. Now, in the case that
search costs are greater than the value of information, the
decision maker should proceed on the basis of information
already held, as its cost outweighs the benefits derived.
24
Expected value, decisions trees, risk and
uncertainty and the measure of risk
Evaluating decisions
 Coming now to the question of how to evaluate decisions
and when a decision is good, as a decision cannot be bad if
only the actual outcome comes to be bad, the quality of
the decision depends on three main issues:
1. Whether the information search was undertaken to the
point where it would not be marginally profitable to
continue the search procedure.
2. Whether the information obtained was used in the
appropriate form
3. Third whether the appropriate decision criterion was
used.
25
Expected value, decisions trees, risk and
uncertainty and the measure of risk
Useful note:
 Since information tends to grow over time,
decisions should not be made before they
must be made, and the decision maker
should aim at quantifying the advantages of
waiting and actually make the decision
when it is apparent that more would be
lost than gained by further waiting.
26
Expected value, decisions trees, risk and
uncertainty and the measure of risk
Sensitivity analysis:
 Finally, the decision maker should consider the
sensitivity of the decision on the assumptions on
which it is based.
 One could refer to assumptions such as that the data
is accurate, that the probability distributions will be
validated by future events, that the discount rate
used is appropriate, and that fixed cost categories do
not vary with output levels, among others.
 Sensitivity analysis, is defined as the examination of a
decision to find the degree of inaccuracy in the
underlying assumptions that can be tolerated
without causing the decision to be inappropriate.
27
The choice of postponement
A project can be undertaken now or
postponed for one or more periods. The
choice of the time of undertaking an
investment is always present in investment
analysis. It is rational for each manager who
decides to undertake an investment during the
present period, to be able to justify why, for
example, he does not choose to implement it
during the next period.
 The decision to undertake an investment in
the present or to delay its undertaking for the
future, forms mutually exclusive decisions. It is
not possible for both to be accepted.

28
The choice of postponement




Suppose that an investment undertaken today
produces positive NPV. Say,
NPV(0) = Y > 0
Suppose now that this investment can be
postponed for 1 year, while all its features
remain unchanged. In this case, the investment
1 year from today will offer the same NPV and
therefore the current value (NPV*) of the
future investment is expressed as:
NPV*(0) = NPV(1) / (1+i) = Y / (1+i) < Y
29
The choice of postponement
Under these conditions, it is always
preferable
to
undertake
profitable
investment earlier rather than later.
 A postponement becomes of interest when
the investment undertaken in the next year
offers:
 NPV(1) = X > Y(1+i)
Then,
 NPV*(0) = NPV(1) / (1+i) = X / (1+i) > Y

30
The choice of postponement
Which factors would bring forth such a result?
1. When the investment cost is reduced in the
course of time, say, due to technological
advancements, or positive developments in raw
material prices.
2. If the prediction is for future interest rates to be
reduced, improving the discounted value of longterm future cash flows.
3. It could be that an important uncertainty
appearing today related to the investment is to be
reduced along time, for example, if the investment
is to be under conditions of high or low demand
in the future, as in our example above with the
evolving «trees».
31
The choice of postponement

Finally, more advanced applications of the
decision analysis for the investment
postponement are taking place with the
use of models of valuation of financial
options. Those models are applicable as a
postponed investment entails the right to
undertake it in the future.
32
Questions
In decision making under uncertainty, the
valuation tools remain useful and effective
provided that these incorporate the
uncertainty conditions. Discuss.
2. How
could one measure risk and
uncertainty?
3. Define risk aversion. Does a risk averter
denies any risk taking? Explain.
4. How could one express the uncertain cash
flows related to the sales of a product?
5. In which way can a manager incorporate risk
in his decision making process?
1.
33
Questions
Describe the main methods by which a risk
averse decision maker can compare decision
alternatives on a risk adjusted basis.
7. In the case of subjective evaluation, how can
he end up with the certainty equivalent of a
cash flow.
8. Describe the usefulness and operation of the
«decision tree» technique for the valuation of
an investment?
9. Can various risk criteria rank differently
decision alternatives? Discuss
6.
34
Exercises
1.
PEGASOS SA purchases a production line
of fish feed at the price of 250,000 euros.
However, the cash flow that is to occur
from the three year production depends
on the demand conditions of the fish
farming sector. More specifically, three
outcomes are possible, namely, high,
regular and low demand, depicting equal
probabilities for each outcome. Demand is
expressed as follows (in euros).
35
Exercises
Year 1
2
3
Demand
High
330.000
370.000
425.000
Regular
300.000
350.000
400.000
Low
260.000
315.000
370.000

How could this investment project be valued, if the
machinery is to have a residual value of 40.000 euros
at the end of year 3, the market interest rate on such
activities is at 7% and the risk free rate is at 4%?
36
Exercises
2.
A supermarket store is considering
opening a new store in a town suburb,
and two store sizes have been suggested.
The regular size of 6,000 sq. metres and
the large size of 10,000 sq. metres. The
initial costs, expected demand situation,
profits and probabilities are as follows:
37
Exercises
(Thousand euros)
Store
Initial
Demand
size
cost
situation
Large
2000
Year 2
Profits Prob
Profits
Prob
600
0,2
1000
0,2
Medium 1000
0,5
1600
0,4
High
1600
0,3
2000
0,4
Low
600
0,6
1200
0,4
Medium 1200
0,3
1800
0,3
High
0,1
2400
0,3
Low
Regular 1700
Year 1
2000
38
Exercises

If the appropriate discount rate is at 10%,
which store size promises the larger
expected net present value? If you were
the manager of the supermarket store,
which would be your decision and why?
39
Exercises
3.
A firm is considering the introduction of a
new product and its management is
required to set the price. There are three
price strategies under consideration,
namely, the high (6 €), the medium (4 €)
and the low (2.5 €). Market research
indicates that the probability distribution
of sales at these prices are as follows:
40
Exercises
High price
Medium price
Low price
Sales
Prob
Sales
Prob
Sales
Prob
7,000
0,1
10,000
0,2
20,000
0,4
5,000
0,3
8,000
0,5
15,000
0,3
3,000
0,6
6,000
0,3
10,000
0,3
Second 10,000
Year
0,2
16,000
0,3
24,000
0,3
8,000
0,3
13,000
0,4
18,000
0,5
6,000
0,5
10,000
0,3
15,000
0,2
First
Year
41
Exercises
4.
A company producing batteries has found
that its leading product has suffered
market share losses, due to the
competition from other producers. It is
considering to either somehow lift the
existing product or introduce a totally
new product. The result of these
alternatives depends on the state of the
economy:
42
Exercises
Decision alternatives (net
profit)
State of the
Minor lift
New product
Downturn
20,000
-40,000
Constant
60,000
40,000
Upturn
160,000
300,000
economy
43
Exercises
The probabilities of a downturn, constant
and upturn are 30%, 50% and 20%,
respectively.
a) Calculate the EV, SD and CV for each
decision alternative.
b) Which decision alternative is indicated
under the various criteria?
c) Explain, which alternative will have the
highest CE.

44
Exercises
5. You plan to operate either a hot-dog
stand or an ice-cream stand at every
home game of your local football team.
The actual outcomes of each decision
alternative depend largely on weather.
The following table presents the
expected value of profits for each
football game, and for each decision
alternative, under the three different
weather conditions.
45
Exercises
State of
weather
Rain
Cloud
Sun
Decision alternatives
Hot-dog
Ice-cream
€900
€225
€750
€450
€300
€1,200
46
Exercises

The information data from the meteorological
office indicate that over the past ten football
seasons it was raining for 15% of the games, it
was cloudy for 55% of the games and it was
sunny for 30% of the games. You have to design
which product to choose for the entire season,
either hot-dog or ice-cream. Calculate the
expected value and coefficient of variation of
each decision alternative. Which alternative
would you choose? Explain your answer and
also include a statement about your certainly
equivalent for each of the two alternatives.
47
Exercises
6. A company examines an investment
project of a time horizon of 10 years. The
Net Cash Flows (NCF) of this project are
as follows:
Probability
2/3
1/3
NCF
€120,000
€300,000
48
Exercises
The above probability distribution of NCFs is
the same for each year for the next 10 years.
The risk free rate is 8% and the cost of the
investment project is at €800,000. The Net
Present Value (NVP) of this project, according
to the evaluation undertaken, is at €70,000.
 Estimate, in the context of the objective
evaluation of this investment, the risk premium
that is implied for the above information, as well
as the NPV provided by the company
management.

49
Exercises
7. DANAOS S.A. evaluates an investment
proposal with a life span of 2 years. The
investment expenditure is at 4,000 euros.
The probability distribution of the cash
flows stemming from the investment
depends on the market conditions during
the year, presented in the table below:
50
Exercises
Cash flows stemming from the investment
Market
condition
1
2
3
4
Year 1
Probabilities
NCF
0.20
0.40
0.20
0.20
2,000
3,000
4,000
5,000
Year 2
Probabilities
NCF
0.20
0.40
0.20
0.20
1,200
3,600
5,000
6,000
51
Exercises
The company faces a risk free cost of
capital of 10%.
a) Estimate the expected NCF for each
year.
b) Estimate σ and CV of the NCFs per
year.
c) On the basis of return and risk, in which
of the two years the company faces the
biggest problem?

52
Solutions to Exercises
Exercise 1

Expected Cash flow: [Note Pi=100/3=0.333]

Year 1= 0.333 x (330,000 + 300,000 + 260,000) = 296,666.6
Year 2 = 0.333 x (370,000 + 350,000 + 315,000) = 345,000
Year 3 = 0.333 x (425,000 + 400,000 + 370,000)= 398,333.3



For the valuation of this specific investment, we need to
estimate the NPV.

NPV= 296,666.6xPVIF (1,7%) + 345,000 x PVIF (2,7%) +
398,333.3 x PVIF (3,7%) + 40,000xPVIF (3,4%) – 250,000 =
689,313.3 > 0

Therefore, the project is beneficial.
54
Exercise 2

Regular

Year 1 (0.2 x 600) + (0.5 x 1,000) + (0.3 x
1,600) = 120 +500+480 = 1,100.0

Year 2 (0.2 x 1,000) + (0.4 x 1,600) + 0.4 x
2,000) = 200 +640 + 800 = 1,640

NPV = 1,100.0 X 0.909 + 1,640 X 0.8254 –
1,700 = 1,000.01 + 1,353.656 – 1,700 =
=653.666
55
Exercise 2

Large

Year 1 (0.6 x 600 + 1,200 x 0.3 + 0.1 x 2,000) =
360 +360+200 = 920

Year 2 (0.4 x 1,200 + 0.3 x 1,800 + 0.3 x 2,400) =
480 +540 +720 = 1,740

NPV= 920 x 0.9091 + 1,740 x 0.8254 – 2,000 =
836.372 + 1,436.196 – 2,000 = 272.568
56
Exercise

3
High Price
Year 1 (0.1 x 7,000 + 0.3 x 5,000 + 0.6 x 3,000) = 700
+ 1,500 + 1,800 = 4,000
 Year 2 (0.2 x 10,000 + 0.3 x 8,000 + 0.5 x 6,000) =
2,000 + 2,400 + 3,000 = 7,400


NFC 1= 4,000 X (6-1) = 4,000 X 5 = 20,000

NFC 2 = 7,400 X (6-1) = 7,400 X 5 = 37,000

NPV = 20,000 X 0.8929 + 37,000 X 0.7972 – 44,000=
= 17,858 + 29,496.4 – 44,000 = 26,354.4 = 3,354.4
57
Exercise
3

Medium Price

Year 1 (0.2 x 10,000 + 0.5 x 8,000 + 0.3 x 6,000) =
7,800

Year 2 (0.3 x 16,000 + 0.4 x 13,000 + 0.3 x 10,000) =
13,000

NCF 1 = 7,800 (4-1) = 7,800 x 3 = 23,400

NCF 2 = 13,000 (4-1) = 13,000 x 3 = 39,000

NPV = 23,400 x 0.8929 + 39,000 x 0.7972 – 44,000 =
20,893.86 + 31,090.8 – 44,000 = 7,984.66
58
Exercise
3

Low Price

Year 1 (0.4 x 20,000 + 0.3 x 15,000 + 0.3 x 10,000) = 15,500

Year 2 (0.3 x 24,000 + 0.5 x 18,000 + 0.2 x 15,000) = 19,200

NCF 1 = 15,500 (1.5)= 23,250

NCF 2 = 19,500 (1.5) = 29,250

NPV = 23,250 x 0.8929 + 29,250 x 0.7972 – 44,000 =
20,759.9 + 23,318 – 44,000 = -102.1
a.
b.
Medium
No, EPV of funds is zero.
59
Exercise 4

EV 1= (0.30 x 20 + 0.50 x 60 + 0.20 x 160) = 6
+ 30 + 32 = 68

EV 2 = (0.30 X (-40) + 0.50 X 40 + 0.20 X
300) -12 + 20 + 60 = 68

SD1=[0.30x(20-68)2+0.50x(60-68)2+0.20x(16068)2]1/2=[0.30x(-48)2+0.5x(8)2+0.20x922]1/2=[0.30x2,304+0.5x64+0.2x8,46
4]1/2=[691.2+32+1,692.8]1/2]=[2,416]1/2=49.160
60
Exercise 4

SD2=[0.30x(-40-68)2+0.50x(40-68)2+0.20x(30068)2]1/2=[0.30x(-108)2+0.50x(28)2+0.20x2322]1/2=[0.30x11,664+0.50x784+0.20x53,824]
1/2=[3,499.2+392+10,764.8]1/2=1,465.61/2≈121

CV1=

CV2=
a.
b.

49.160

 0.7229
ECF
68

121
ECF

68
 1.7794
EV criterion ranks them equally. CV favours Minor lift.
Minor lift.
61
Exercise 5
EVH-D = 900 x 0.15+750 x 0.55+300 x
0.30 = 637.5
 EVI-C = 225 x 0.15+450 x 0.55+1,200 x
0.30 = 641.25
 σH-D ≈ 226.8670
CVH-D ≈ 0.35587
 σI-C ≈ 373.8545
CVI-C ≈ 0.583

62
Exercise 5

Under the EV approach one would
choose to operate the ice-cream unit.
However, with the use of CV criterion the
hot-dog operation looks more favorable.
Finally, CVH-D < CVI-C. Hence, the hot-dog
stand is preferable to the ice-cream stand.
63
Exercise 6
The expected NCF of the investment
project is:
 (2/3x120,000)+(1/3x300,000)= €180,000
 In order to estimate the risk premium or
risk adjusted interest rate (k) (k= risk
free rate plus risk premium), we firstly
derive k used for the NPV estimation.

64
Exercise 6
From the relation:
 ENCFxPVIFA-Ko=NPV
 By substituting:
 180,000xPVIFA-800,000=70,000
 PVIFA=870,000/180,000=4.8333

65
Exercise 6
From the relevant table it is shown that
the interest rate reflecting this PVIFA for
10 years is 16%.
 Therefore, k=i+ risk premium
 where:
 k=16%
 i=8%
 and therefore the risk premium equals
16%-8%=8%.

66
Exercise 7
a) Expected NCFs for each year:
 x̄1=€3,400, x̄2=€3,880
 b) Estimation of the standard deviation of
NCFs for each year:
 σ1=1,019.80 , σ2=1,617.89
 CV1=(1,019.80/3,400)=0.299 or 29.99%
 CV2=(1,617.89/3,880)=0.416 or 41.69%

67
Exercise 7

We recapitulate as follows:
Expected NCF
NCF1
NCF2
3,400
3,880
Standard
deviation
1,019.80
1,617.89
Coefficient of
variation
29.99%
41.69%
68
Exercise 7

c) It is observed that the company faces a
bigger problem in year 2, as both σ and
CV are higher, implying that risk has
increased in relation to the return of the
expected NCFs.
69