TOPIC FIVE
INVESTMENT APPRAISAL TECHNIQUES
5.1.0 Introduction
This lecture is concerned with strategic investment decisions, a forward-looking process through
which the future of the firm and that of its shareholder wealth is determined. Consequently, it is
imperative that such decisions are properly appraised and evaluated before being implemented.
In this lecture, we shall review both the investment process and the investment analysis
techniques commonly used by managers to evaluate such alternatives. Precisely, the concept of
net present value will be introduced and the calculation of the NPV of a simple project will be
demonstrated. The lecture will also look at other measures of an investment‘s attractiveness
(internal rate of return, payback period, the average return of return and profitability index
techniques). It shall assume that both the cash flows and the cost of capital are known. The
former will be left for a later course but the latter will be dealt with in a later topic in this course.
It shall also leave out the more complex investment proposals such as those involving, machine
replacement decision, decisions of when to invest, investment under capital rationing, and
incorporation of risk in project analysis for a later course.
5.2.0 Assessment Criteria and Learning Objectives
S/N
Sub Enabling
Outcomes
Related Tasks
Assessment Criteria
6.1.7
Apply investment
appraisal techniques
in evaluating
transport projects
(a) Discuss the need for appraising
investment projects
(b) Evaluate investment projects
using non-discounted cash flows
methods
(c) Evaluate investment projects
using discounted cash flows
methods
Investment appraisal
techniques in
evaluating transport
projects are applied
By the end of this lecture, you should be able to:

Develop familiarity with both the investment process and the most commonly used
investment appraisal techniques
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
Distinguish between the traditional and the discounted cash flow techniques and
understand why the discounted cash flow techniques are preferred to the traditional
techniques

Understand why NPV is considered superior to IRR and how the two techniques
can be reconciled in situations where they offer conflicting advice.
5.3.0 The Investment Setting
5.3.1 Investment Defined
Investment means forgoing present consumption of resources in order to increase the total
amount of resources, which can be consumed in the future. It involves making an outlay of cash
now in the expectation of extra cash inflow in the future. The objective of investment is to
acquire an asset (real or financial) for less than its value in order to add value.
5.3.2 Investment Appraisal (Capital Budgeting) Process
An investment appraisal or capital budgeting is a strategic decision-making process for
determining how a firm‘s management should allocate limited capital resources to long-term
investment opportunities. At the outset, the investment problem is that it is often difficult to
determine which assets or projects will be wealth enhancing and which ones will be wealth
reducing.
5.3.3 Usefulness of Investment Appraisal
Investment appraisal is not restricted to private sector business enterprises only as it may sound
to many people. Many public sector, and Not-for Profit Organisations (NPOs), such as hospitals,
trusts, charities, etc, must also invest in fixed assets, usually in the context of having very limited
resources and time frame. For such organisations, investing in fixed assets also presents a
strategic investment decision, as these are the key assets necessary for delivering effective and
efficient services. Organisations make investment for various reasons, which may include;
competitiveness, expansion and growth, replacement, renewal, refitting/refurbishment, mergers
and acquisitions, foreign direct investments, research and developments, explorations, social
investments, etc.
5.3.4 Characteristics of Investment Projects
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Investments have certain characteristics, which lead to the need for careful considerations before
a decision is taken. For example, investment projects;
(i)
Commit resources into the future;
(ii)
Involve substantial amounts of cash;
(iii)
Are difficult and costly to reverse once taken;
(iv)
Usually include intangible costs and benefits which are difficult to evaluate;
(v)
Require approval by higher organs within the organisation.
5.3.5 The Appraisal Process
Investment appraisal may involve six interrelated stages:
(i)
Proposal generation
(ii)
Proposal review and evaluation
(iii)
Decision-making
(iv)
Implementation
(v)
Follow-up and control
(vi)
Post implementation audit
In this lecture, we shall focus on the second and third stage but assuming that both the cash flows
and the cost of capital for the projects are known. However, at this point it suffices to point out
what kinds of cash flows are relevant.
5.3.6 Project Cash flows
(a)
Relevant Cash Flows
To perform an investment analysis you need to establish the cash flow consequences of the
proposal and take a decision based on the value of such cash flows at the time of decision against
a predetermined criterion or rule. It involves an assessment of the projects costs and benefits.
However, not all the cash flows of the project will be relevant in taking investment decisions.
The process demands for the identification, and the evaluation, of all the relevant or incremental
costs and benefits that are likely to flow from the decision to invest.
Relevant cash flow may be defined as the future differential costs and benefits, i.e. those costs
and benefits, which will change in the future as a direct result of undertaking the investment. In
other words, a relevant cash flow is the difference between cash flow items with, and without,
the proposed project.
(b)
Types of Incremental Cash flow
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Cash flow from any investment may be classified into capital and revenue items:
(i)
Capital items

Initial outlay

Costs necessary to make the equipment operational

Any incremental working capital for example, increases in inventory, increases in
debtors, necessary for the project to become operational
(ii)
Revenue items
The day-to-day recurring costs and benefits

Running costs and benefits

Incidental costs and benefits

Opportunity costs
5.3.7 Types of projects
Projects may be classified into independent and mutually exclusive project.
This kind of
classification is crucial as it affects the way decision rules under each investment analysis
techniques work.
(a)
Independent projects
Projects are independent when the acceptance of one does not prevent acceptance of other
projects. This implies that the projects are not in any way related and funds permitting,
management may decide to undertake all value enhancing projects.
(b)
Mutually Exclusive Projects
Projects are mutually exclusive when the acceptance of one eliminates the chance to
accept the rest. In this case, the decision becomes an ―either-or‖ choice of projects. The
projects are in fact substitute for each other. Projects may be mutually exclusive because
they fulfil the same function or provide the same solution or may be because they
compete for limited funds.
5.4.0 Investment Appraisal Techniques
Appraisal techniques will enable management to select projects that will advance organisation‘s
objectives and plans and add to shareholders wealth. Investment techniques are decision-making
aids. In this section, we shall look at the most commonly used method, which are also classified
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into too major categories; traditional or non-discounted cash flow (NDCF) and discounted cash
flow (DCF) techniques.
Traditional techniques include mainly the Payback Period (PBP) and the Accounting Rate of
Return (ARR) or Return on Investment (ROI). On the other hand, the DCFs include the Net
Present Value (NPV), Internal Rate of Return (IRR), and the Profitability Index (PI). There are
also modified techniques from some of the commonly used ones but these are considered here to
be outside the scope of the course, although you can read about them from most of the standard
corporate finance text books.1
5.4.1 Non-discounted Cash flow Techniques
5.4.1.1 The Pay Back Period
PBP is the ratio of initial outlay on the project to the annual net cash inflows from the project.
PBP 
initial cash outlay
annualnet cash inf lows
This is the most popular and easily understood technique. It is essentially an expression of ―how
long‖ it will take to recover the initial cash outlay on an investment from the investment‘s cash
flow. Many companies set a maximum PBP for a project given its characteristics and use this as
their decision rule. The decision rule is:
―If the actual PBP is less than the pre-determined maximum period, the project is acceptable,
otherwise the project is rejected‖.
Example 1
Mazuri, a sporting equipment servicing company is considering an investment in one of the two
machine tools projects, A and B. Project A requires an investment of 100m/= and expected to
yield cash inflows of 40m/= per annum for 3 years. Project B requires 110m/= and expected to
yield 50m/= per annum for three years
The pay back period for the two projects A, and B, respectively are:
1
Give an example of such books and pages.
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PBPA 
100,000,000
 2.5 years
40,000,000
PBPB 
110,000,000
 2.2 years
50,000,000
Using the PBP technique, Project B will be preferred because it returns the initial cash outlay
sooner than project A does. However, this assumes the management accepts PBP higher than 2.5
years for projects of this kind; otherwise, it would be difficult to judge the two results.
In addition, looking closely at the solution you will realise that the model works for projects with
level cash flows only. Cases with non-level cash flows will require cumulative summation of the
cash flows starting from period one to the period when the initial cash lows are recovered.
Example 2
If we assume that the same two projects A and B have the following net cash inflows
after tax.
Period
Project A
(‗000)
Project B
Cumulative
sum(‗000)
Cumulative
(‗000)
sum(‗000)
0
-100,000
-110,000
1
60,000
60,000
20,000
20,000
2
40,000
100,000
40,000
60,000
3
20,000
90,000
150,000
Project A will take 2 years whereas project B will now take 2.6 years [i.e. 2 years and (110,00060,000)/90,000)], making project A preferred.
Example 3
What if Mazuri Company also considers the following two-other projects; X and Y
which have the following cash flows?
Period
X
Y
0
-25,000
-25,000
1
4,000
15,000
2
5,000
6,000
3
16,000
4,000
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Both projects have identical PBP and identical cash out flows. How then can one make a choice
between these two projects? Definitely, the only criterion left is the timing of the cash flows. A
rational investor would prefer getting more cash sooner than later, making project Y preferable.
This brings us to the limitation of PBP technique, which may be summarised as follow:
(i)
It does not take account of all cash flows (i.e. it ignores the cash inflows occurring after
the PBP cut-off point);
(ii)
All cash flows occurring within the PBP are given equal weighting;
(iii)
It does not measure profitability or returns;
(iv)
It does not provide a rational decision making rule i.e. there is no generally acceptable
method of determining an appropriate PBP; and
(v)
It ignores the time value of money by not discounting the cash flows to find their present
values. This is the most serious problem.
Despite the problems above, the PBP is very useful because:
(i)
It deals with cash flows rather than accounting profits;
(ii)
It is used as a risk screening device; the longer it takes to recover the initial cash flow the
greater the chances of something going wrong;
(iii)
It is frequently used to supplement more sophisticated investment appraisal techniques;
and
(iv)
It is useful in circumstances where the company is facing liquidity difficulties. In such
circumstances, the company picks projects that return cash flows quicker, and thus
reducing the risk of insolvency.
5.4.1.2 Accounting Rate of Return
The accounting rate of return, also known as the return on investment (ROI) or return on capital
employed (ROCE) is defined as the ratio of accounting profit to investment in the project
expressed as percentage.
ARR 
Pr oject' s averageaccounting profits
 100%
 initial outlay  residualvalue 


2


where the residual value is the expected realisable value of the asset. A variant of the model use
initial outlay only in the denominator, which in turn results into a lower rate of return.
These are:
ARR = (Profit for the year/Asset book value at start of the year)x100%
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ARR = (Average annual profit/initial capital employed)x100%
ARR = (Average annual profit/average capital employed)x100%
Many firms will set a minimum acceptable target ARR for projects against which projects are
measured. The decision rule is:
―If the actual ARR is higher than, equal to, the minimum required ARR (a hurdle
rate), the project is acceptable, otherwise the project is rejected.‖
Example 3:
The following are cash flows from a proposed machine purchase for Mazuri
Company.
Period
Cash flow
Depreciation
Profit/loss Average
ARR
Profit/loss
0
(100,000)
1
60,000
30,000
30,000
2
40,000
30,000
10,000
3
20,000
30,000
(10,000)
120,000
90,000
30,000
ARR 
10,000
18.2%
10,000
 100%  18.1818%
 100,000  10,000 


2


Where depreciation is given as (100,000-10,000)/3 and residual is given as 100,000 – 90,000 =
10,000.
Note that depreciation would have been excluded from the calculation under PBP because it is
not a cash flow item. The decision whether or not the proposed project is acceptable would
depend on the ARR acceptable by management. (See the rule above).
Like the PBP, ARR is not without its own limitations:
(i)
It ignores cash flows and uses accounting profits instead;
(ii)
It ignores the timing of return by taking a simple average, which gives equal weighting to
each year‘s returns; and
(iii)
It ignores the time value of money
It is rather interesting to know that despite all these limitation, the ARR is one of the most used
techniques, probably because it does suggest something about a project‘s profitability. Managers
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are also familiar with it because it is an ancient measure of profitability. Even divisional
performances are measures in such manners i.e. profits as a ratio of capital employed, hence its
familiarity amongst managers. This is lacking in PBP technique. In addition, it is simple to use
and understand. McMenamin (1999: 362) shades some light on why it is so preferred over other
techniques.
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5.4.2 Discounted Cash flow Methods
5.4.2.1 Net Present Value (NPV)
Net present value is the difference between the present value of future cash inflows and the
present value of the initial outlay, discounted at the firm‘s cost of capital. If you have a project
with n-period cash flows, the NPV model is stated as follows:
NPV  C 0  C1
1
1
1
 C2
 ...  C n
2
1  r 
1  r 
1  r n
N
NPV  C0   Ct
t 1
1
1  r t
The discount rate used to determine the present value of the investment cash flows is the
minimum accepted rate of return, and should reflect the rate of return available on similar risk
investments. It reflects the opportunity cost of capital and is often referred to as the company‘s
cost of capital.
The decision rule:
―If the actual present value of the expected benefits exceeds costs (i.e. NPV  0) ,
accept the project, otherwise reject.
Back to our example on Mazuri‘s Projects A and B, we calculate NPV for each project as
follows;
Period
PVIF(r,n)
CA,t
PVA
CB,t
PVB
0
1.000
-100,000
-100,000
-110,000
-110,000
1
0.909
60,000
54,540
20,000
18,180
2
0.826
40,000
33,040
40,000
33,040
3
0.751
20,000
15,020
90,000
67,590
NPVA =
2,600
NPVB =
8,810
What does NPV tell us? If NPV > 0 means that, the project earns more than it costs, implying
that the project earns more than the rate of return. If NPV = 0 means that the project earns just
sufficient returns to compensate investors; and finally, if NPV < 0, the project does not earn
adequate return.
Since both projects yield positive present values of cash flows, the NPV rule suggests that we
should accept both projects. However, such a suggestion depends on whether or not funding is
72
not a problem and whether the two projects are independent or mutually exclusive. For example,
if independent and funding is not a problem, both will be worth undertaking. Otherwise, for
mutually exclusive projects, project B will be preferred because it yields more NPV than project
A.
5.4.2.2 Internal Rate of Return
The IRR is defined as the rate of return, which equates the present value of future cash flows to
the initial outlay. That is, the IRR is the rate of return such that the outlay equals future cash
flows discounted at rate r. IRR is the yield of the project.
C 0  C1
1
1
1
 C2
 ...  C n
2
1  r 
1  r 
1  r n
Or
n
C 0   Ct
t 1
1
1  r t
IRR is the discounted rate at which the NPV is zero, i.e. r such that
NPV  C 0  C1
1
1
1
 C2
 ...  C n
0
2
1  r 
1  r 
1  r n
Or
n
NPV  C 0   Ct
t 1
1
0
1  r  t
IRR is the unknown r in the equations above. It is the highest rate of interest that could be paid
on a loan used to finance the investment and still allow the investment to breakeven. However,
the little trouble is how to determine the IRR. The following are some of the available options.
(i)
use trial and error approach in which you try a number of rates, guided by the relationship
between interest rate of the corresponding NPV.
(ii)
Financial calculators or computers software programmes, own designed or spread sheet
functions such as in Microsoft Excel could be used to easy the burden.
(iii)
Alternatively, you can use a combination of trial and error and the linear interpolation
technique. Hereunder we shall see an example (i) and (iii) techniques combined.
Example:
A project costs 16,000/= and is expected to generate cash inflows of 8,000/=, 7,000/=,
and 6,000/= at the end of each year for 3 years. What is the project‘s IRR?
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By trial and error
At k=20%
NPV  16,000 
8,000 7,000 6,000


 1,000
1.2
1.2 2
1.2 3
We are looking for a rate that will make the NPV zero.
From the negative
relationship between the discount rate and the NPV, the true rate of return must then
be lower than 20%.
At k=15%,
NPV  16,000 
8,000 7,000 6,000


 194.60
1.15 1.152 1.153
We have shown that the true rate is in between 15% and 20%. To find the exact rate
we must bring in the linear interpolation technique. We have seen that if k=20% NPV
is -1,000/= and if k=15%, NPV is 194/60. We need a rate r at which NPV is 0. So;
k  20 
0  0  1,000
 15  20  4.1855  4.19
194.60  (1,000)
k  4.19  20  15.8%
Decision Rule.
If k=IRR, and r = opportunity cost of capital, then the decision rule is, reject the
project if k<r. That is, the investor is better served by not going ahead with the project
and applies the money to the best alternative use. If k>=r then accept the project, i.e.
the project under consideration produces the same or higher yield than investment
elsewhere for the same level of risk
Problems with IRR
It must be used with great care especially when we have (i) different cash flow profiles (The
earlier he cash flows the more attractive the project is likely to be), (ii) differing size and scale of
projects, (iii)multiple rates of returns. In the latter condition, projects may have NIL or multiple
IRRs under certain conditions; for example, when a project has a non-conventional cash flow.
What are conventional cash flows? One large outflow followed by a series of positive cash
inflows. On the other hand, unconventional cash flow refers to the periodic cash outflows during
the life of the project; the number depends on the sign reversals.
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5.4.2.3 Profitability Index
Profitability index (PI) also referred to as the benefit cost ratio is defined as follows:
PI 
PV of benefits
PV of initial outlay
The main difference between NPV and PI is that NPV is an absolute measure of a project‘s
acceptability while PI is a relative measure of benefits relative to initial outlay.
Back to the example;
PIA = 102,600/100,000 = 1.03
PIB = 118,810/110,000 = 1.08
The rule is as follows:
Accept the project is PI >=1.0, otherwise reject.
You should have notice that whereas NPV would have preferred Project B, PI would prefer
project A. Is that rational? In case of mutually exclusive projects, the PI would not provide a
ranking system, which the firm would wish to follow.
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TOPIC SIX
RISK AND RETURN
6.0
INTRODUCTION
This lecture introduces you to the concepts of risk and returns. You will agree from this lecture
that the aim of both companies and individuals is to either, minimize the risk they face given the
return that they expect to receive or to maximize the expected return given the risks they are
willing to take. This represents a trade-off between risk tolerable and the return expected. It is
important therefore for managers first to understand why risk exists in the first place and how to
quantify it so that they can manage or control it. Thus, this lecture introduces you to how risk
and returns are related, how they are measures both for an individual asset and for asset held in a
portfolio. This lecture will form an important input to the further lectures on portfolio theory as
well as on capital budgeting decisions under uncertainty.
Investments and financing decisions and future oriented decisions. One takes a decision today
based on his or her expectation about its impact in the future days. But since it is generally
impossible to forecast with complete accuracy what the future will bring, most investment and
financing decisions are characterized by risk and uncertainty. Exposure to risk is considered to
be unwelcome and will only be accepted by investors if they are offered an inducement in the
form of higher expected rate of return. An investment‘s expected return is the investment‘s most
likely return, and it is measured in terms of future cash flows the investment is expected to
generate.
Investors can however limit their exposure to risk by diversifying and investing in portfolios of
assets. They achieve this by balancing the risk level they consider affordable for a given level of
expected rate of return. This represents a risk-return trade off which a wealth maximizing
investor must strike and it provides a benchmark for decision taking by management attempting
to maximize the welfare of their shareholders. That is, no financing and investment proposal
should be accepted unless it can offer a return comparable to that available to shareholders in the
financial market on similar risk investment.
The key issues that this lecture will deal with include; how are these risk and expected returns
defined, what is a portfolio investment and how they are constructed and how the widespread of
their use has changed the way we view risk for decision making and so on. This topic will
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highlight the relationship that exists between risk and return. In so doing, the topic will look at
the risk in and return on individual assets as well as on portfolio basis following the fact that
investors normally hold assets is portfolios rather than individual assets. Further the scope for
risk reduction or diversification through portfolio investments will be explored.
6.1 Assessment Criteria and Lecture Objectives
S/N
Sub Enabling
Outcomes
Related Tasks
Assessment Criteria
6.1.4
Determine risk and
rates of return in
transport investment
projects
(a) Define expected returns and risk
(b) Compute expected returns and
risk of an individual asset
(c) Compute expected returns and
risk of a portfolio of assets
(d) Discuss systematic and
unsystematic risks
Risk and rates of
return are explained
6.1.1 Learning Objectives
At the end of this topic you should be able to:
At the end of this lecture you will be able to:
(i)
Define and measure the risk and expected rate of return of an individual asset
(ii)
Define and measure the risk and expected rate of return of a portfolio of assets
(iii)
Differentiate between systematic and unsystematic risk and identify the reasons why
investors may not be concerned with both
6.1.2 Content

Expected Returns and Risk Defined

Calculating expected return

Calculating risk (Variance and Standard deviation)

Portfolios and portfolio weights

Portfolio expected returns

Portfolio variance and standard deviation

Systematic and unsystematic risk
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Risk defined
There are two ways in which risk is defined.
The first is risk as a hazard, a peril, or an exposure to loss or injury. This means that its focuses
on the possibility of an unfavourable outcome or loss occurring. This is the view is the most
popular one and it is normally applicable in insurance and related fields.
The second is risk that refers to the situation in which an outcome of an investment can not be
specified with complete confidence. This later view is the most commonly view that is used in
the finance literature and it corresponds more to the most popular notion of uncertainty, which
reflects the idea that it is difficult to foretell how things are going to turn out in the future.
In this lecture we shall focus on the latter version of risk, leaving the former to another module in
risk management and insurance.
Expected return defined
Expected rate of return is the average rate of return that could be anticipated if the proposed
investment could be repeated on a large number of occasions.
Measuring risk and return
Risk in finance is measured in terms of the dispersion of possible outcomes. The wider the range
of possibilities the greater the risk, and the less desirable an investment is considered to be. This
dispersion of possible outcomes is normally measured by standard deviation or variance of the
distribution of possible returns around the expected return. This expected return is defined as the
weighted average of the possible returns, where the weights are provided by the probabilities of
these returns occurring. It is important to note that the tighter the distribution of possible returns
around the expected return the lower the standard deviation and the perception of an
investment‘s risk.
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Since the risk of an investment in finance is assessed inn the context of the dispersion of its
possible outcomes, there are a number of ways in which it can be measured and a number of
statistical concepts that can be used to accomplish this.
The range of outcomes: this is computed as the difference between the highest and the lowest
possible returns and it gives an indication of the best and the worst case outcomes. However, by
focusing on the extreme outcomes it gives no indication of the more likely outcome. It is
therefore more appropriate and more informative to consider all the deviations from the expected
return rather than simply the extreme possibilities
The mean absolute deviation: this is a measure that takes the average deviations. However
taking the average would be of little assistance because the positive and the negative values
would cancel out leaving a value of zero. So ignoring the signs of the deviations and measuring
the mean absolute deviation overcomes the difficulty. But despite its simplicity it is not a widely
used measure, most probably perhaps, it is due to its failure to provide a simple basis for
measuring the risk of a portfolio in terms of the risk of the individual assets contained in the
portfolio and some measure of the interdependence of their returns.
Semi variance: this considers the weighted squares of the deviations of possible outcomes below
the expected value. Although focusing on the adverse outcome may be intuitively appealing, it
also poses difficulties once more than one investment is being considered.
The variance and its square root (standard deviation): this is the most commonly used measure of
risk. The variance is calculated as the expected value of the square of the deviations from the
expected outcomes. This measure takes into account all deviations and it also provides the most
convenient basis for bridging the gap between the risk of a single asset and that of a portfolio.
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Implication for investment decision
In this lecture we shall discuss risk in finance but as it is applied on financial investments i.e. on
securities. However the analysis is not limited to that; it also has implication for investments in
real assets such as plant and machinery. The trade-off between return and risk as established in
the financial markets provides a benchmark for the appraisal of proposed investments in real
assets. For example:

an investor who act in the best interest of shareholders should not invest in a risky real
asset unless it promises an expected return comparable to that available on similar risk
investments in the financial market.

It is considerably easier to measure the risk in the financial than real assets and data on
average return earned by different types of financial investments, on the dispersion of
these returns and on the relationship between return and their risk

The outcomes of the more highly differentiated investments in real assets tend to be far
more difficult to specify, measure and evaluate.

S a result of the pros and cons identified against each of the statistical based measures
detailed above, this lecture shall focus on the variance and of its square root – the
standard deviation.
Return defined
The rate of return (actual or expected) on any investment over a given period of time is defined
as in the following equation.
R
R
Income
 100 , or
Investment
ending value  begining value  income
begining value
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By considering what makes an income and what makes an investment, these formulae can be
rearranged depending on the type of investments under consideration. For example, in a share
investment we have
R
P1  P0   D1  100
P0
where, D1 is the end of the period‘s dividend, P0 and P1 are respectively
the initial price and the price at which the current investor will sell the share to the next investor
at the end of the period.
And also in bond valuation,
R
P1  P0   C1  100
P0
where C1 is the end of the period coupon (interest) payment, P0 and P1
are respectively the initial price and the price at which the current investor will sell the bond to
the next investor at the end of the period.
Expected Rate of Return calculated
Expected rate of return is the average rate of return that could be anticipated if the proposed
investment could be repeated on a large number of occasions. Thus in general terms we need to
consider a range of outcomes when computing. We will now consider a simple numerical
example to illustrate the calculation of expected return. We will assume that the returns on an
investment depend simply on how well the economy performs in the future. We consider that
there are only three possible future states of the economy, each of which is expected to occur
with some levels of probabilities (stated in decimal form, sum of which must add up to unity. For
example let us assume that the returns from an investment are expected with the follow
probability distribution.
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Table 1
Anticipated returns on investment conditional to the state of the economy
State of the economy
Probability of state
Percentage return on investment Ri 
Pi 
Recession (1)
0.25
18
Slow growth (2)
0.50
16
Boom (3)
0.25
10
The expected return is computed as a weighted average of possible outcomes (returns), where the
weights are the respective probabilities of each possible outcome (return) occurring.
E R   P1 R1  P2 R2  ...  Pn Rn
N
E R    Pi Ri
i 1
Plugging the data with n = 3, we have
ER  .2518  .516  .2510  15%
n
Note that
P
i 1
i
 1.0 and that the probabilities weightings will have been determined
subjectively by the firm management.
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Activity 1.
The financial manager of ABC ltd wishes to determine the E(R) from a proposed investment
projects. The expected returns from the project are related to the future performance of the
economy over the period as presented in the table below. Calculate the expected return for the
financial manager.
Economic Scenario
Probability of the occurrence
Percentage return if the state
of the economic scenario
occurs
Strong growth
0.25
15
Moderate growth
0.50
12
Low growth
0.25
8
Required Rate of Return defined
The required rate of return, unlike the expected return is the minimum rate of return an investor
requires an investment to earn, given its risk characteristics for the investment to be considered
worthwhile. This is estimated to be the rate of return given by a risk free/safe investment plus a
risk premium, where the risk premium is necessary to compensate the investor for under taking a
risky investment. The risk free or safe investment is proxied by the rate of return promised by an
investment in Government‘s treasury bills.
Required rate of return = Risk free rate + Risk Premium
Note that an investment‘s expected return may or may not be the same as the investor‘s required
rate of return
If
E(R) > required rate of return the investment is worthwhile
E(R) < required rate of return the investment is not worthwhile
E(R) = required rate of return the investment yields just enough to cover its cost.
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1.3 Risk Defined
Risk is broadly defined as the chance that the actual return will differ from the expected return.
There is a chance that the actual return will be greater, less or equal to the expected return. Thus,
it is this potential variability of return that we call risk in finance
In order to judge whether a given risk is tolerable or not, it is important we identify various
attitudes to risk. These are risk averse, risk taking, and risk neutral. These attitudes represent the
investor‘s risk propensity and it affects his/her choice to take or avoid risk.
Altitudes to Risk
(a)
Risk–averse – this represents a low risk propensity (it involves preference for some risk;
not complete avoidance)
(b)
Risk-taker (or risk seekers, risk lovers, risk mongers). This represents high propensity or
a positive desire to risk.
(c)
Risk – neutral (indifference to risk). This represents the attitude of an investor where for
an increase in risk they do not necessarily require an in increase in return.
Note that under normal circumstances both share holders and managers are generally considered
to be risk–averse. That is for an increase in risk they require commensurate increase in return.
Risk measured
(a)
Probability weightings
The potential variability or distribution of returns around an expected value is an indication of
the degree of risk.
(b)
Std deviation and variance
We can also measure risk by considering two measures – variance and standard deviation
From example 2 we can calculate variance as the sum of the squired deviations of observed
returns from the expected return each weighed appropriately by its probability of occurrence.
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From Example 2 the variance is calculated as:
3
Var   2   Pi Ri  E R   6.19%
2
i 1
Economic Scenario
Pi
Ri
Ri-E(R)
[Ri-E(R)]2
Pi[Ri-E(R)]2
Strong growth
0.25
15
3.25
10.5625
2.641
Moderate growth
0.50
12
0.25
0.0625
0.03125
Low growth
0.25
8
-3.75
14.0625
3.515625
Expected Return
E(R)
11.75
2 
6.187875
And the standard deviation will be the square root of the variance calculated above.
S tan dard Deviation     2  6.187875  2.487543969%  2.49%
But what does this figure 2.49 mean to an investor? We need a way to interpret it. Is this a high
risk or low risk investment? To interpret this figure we shall need to compare the risk–return
characteristics of this investment with those of other available investment opportunities.
However, in the final analysis, the investment decisions will be influenced by the investor‘s
attitude to risk, i.e. whether the investor is risk averse, risk neutral or risk taker. The following
two-case discussion will shade some light on how such comparison can be made.
Two Investments:
In this section we shall consider three scenarios. The first is for two investments with the same
expected returns but different risk measure (standard deviation).
The second is for two
investments with the same risk measures (standard deviations) but the same expected returns.
And the last one is for two investments with different expected returns as well as different
measures of risk.
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(a)
Investments with same E R  but diff 
A rational risk–averse investor would select the investment with lower standard deviation
(total risk). In the example below, investment A will be preferred.
(b)
A
B
E R 
10%
10%
 
5%
7%
Investment with same risk   but diff ER s
A rational risk–averse investor would choose investment with higher return. In the
example below investment B will be preferred.
(c)
A
B
E R 
10%
12%
 
5%
5%
Investments with different risk measures and different expected rates of return.
If both assets have different ER and  , the decision is not that simple. We shall need another
measure to be able to select one of the two projects. This is the coefficient of variation (CV).
Example
A
B
E R 
10%
20%
 
5%
8%
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Note that the ER and the  are absolute measure. The coefficient of variation is a relative
measure. To make a valid comparison between these two investments we shall use the relative
measures of risk and return rather than the absolute measures.
Coefficient of var iation  CV 

E R 
And the rule is that the investment with the highest CV has more risk and should be avoided by a
risk averse investor, unless a commensurate compensation in a form of return is offered.
In the example above
CV 

E R 
=
A
B
5
 0 .5 0
10
8
 0.4
20
Thus, although Assets B had a higher absolute risk measure  it has a lower CV . This means
that B has a lower risk per unit of return.
The portfolio approach
Since investors tend to hold a portfolio of assets rather than one asset, an alternative approach to
evaluate return and risk would be the portfolio return and portfolio risk.
The expected return on a portfolio investment would be the weighted average of the expected
returns from each investment, where the weights are the proportion of each amount invested in
each investment project. Note that the probabilities would have been used to compute the
expected returns for each project. For a two asset portfolio, for example,
Portfolio return
ER p   WA ERA   WB ERB 
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And the portfolio risk will be as follows
 p2  WA 2 A2  W B B2  2W AW B COV R A R B 
2
Correlation Coefficient
From COV R A R B    A, B  A  B
Where  A, B is correlation coefficient we can have the correlation coefficient defined as
 A, B 
COV R A RB 
 A B
The correlation coefficient takes the values -1 to +1 i.e.
 1   A , B  1
Uncorrelated returns
If returns from the two investment projects are uncorrelated,
 A,B  0, then cov R A R B   0 and the term 2WAWB CovRA RB   0
Thus;  P2  W A2 A2  WB2 B2
Positively correlated Returns
If the returns from the two projects are correlated, it means that
P A, B  0 and when P A, B  1 we have perfect positively correlated returns. The portfolio risk is
 P2  W A2  A2  WB2  B2  2W A W B  A, B  A  B
if  A, B  1
 P2  W A2  A2  WB2 B2  2W AW B  A  B 
 P2  W A  B  W B  B 2
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The standard deviation is
   P2 
WA A  WB B 2
 W A A  WB B
The standard deviation becomes the weighted average of the individual asset‘s standard
deviation. This implies that each investment will contribute its full risk to the risk of the
portfolio. Here there is no scope whatsoever of reducing risk by holding a portfolio of the two
investments. As one asset‘s return falls the return on the other also falls and vice versa.
Negatively Correlated returns
If  A, B   1 were perfectly correlated returns and the portfolio variance becomes
 P2  W A2 A2  WB2 B2  2  A, B W AW B  A B
This presents an opportunity for risk reduction because the contribution of the individual asset‘s
risk is reduced by the last term because of the negative correlation. If the correlation coefficient
is equal to -1, there is scope for complete elimination of risk. However, ability of the investor to
select assets such that he or she achieves perfectly negative correlation may not exist perhaps
because assets which can be combined in a manner that you can achieve such a perfection may
not be available and if they are it may not be possible for an investor to identify them.
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SYSTEMATIC AND UNSYSTEMATIC RISKS
Introduction
The unanticipated part of the return, that portion resulting from surprises, is the true risk of any
investment. After all, if we always receive exactly what we expect, then the investment is
perfectly predictable and, by definition, risk-free. In other words, the risk of owning an asset
comes from surprises—unanticipated events.
Systematic risk (market risk)
A systematic risk is one that influences a large number of assets, each to a greater or lesser
extent. Because systematic risks have market wide effects, they are sometimes called market
risks. Examples of systematic risk: Uncertainties about general economic conditions, such as
GDP, interest rates, or inflation, are examples of systematic risks. These conditions affect nearly
all companies to some degree. An unanticipated increase, or surprise, in inflation, for example,
affects wages and the costs of the supplies that companies buy; it affects the value of the assets
that companies own; and it affects the prices at which companies sell their products. Forces such
as these, to which all companies are susceptible, are the essence of systematic risk.
Unsystematic risk (unique or asset specific risk)
An unsystematic risk is one that affects a single asset or a small group of assets. Because these
risks are unique to individual companies or assets, they are sometimes called unique or assetspecific risks. Examples unsystematic risks: The announcement of an oil strike by a company
will primarily affect that company and, perhaps, a few others (such as primary competitors and
suppliers). It is unlikely to have much of an effect on the world oil market, however, or on the
affairs of companies not in the oil business, so this is an unsystematic event.
Question for Practice
a) What are the two basic types of risk?
b) What is the distinction between the two types of risk?
NB:
Portfolio: A group of assets such as stocks and bonds held by an investor.
Portfolio weight: A percentage of a portfolio‘s total value that is in a particular asset.
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GROUP ASSIGNMENT
Examine the following new projects‘ information
State of Economy
Probability of State of Economy
Rate of return if state occurs
Project A
Project B
Recession
0.20
15%
30%
Normal
0.50
20%
25%
Boom
0.30
25%
20%
Project A requires initial investment of TZS.20 million and Project B requires initial investment of
TZS.30 million. Assume returns of the two projects are perfectly negative correlated.
Required:
a) calculate the expected return of your portfolio
b) calculate the risk of your portfolio
c) If the two projects are mutually exclusive, which project will be preferred by a risk-averse
investor?
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