2004_05_ACCA_F9_P4_sa_lynch_portfolio
The risk-return relationship is explained in two separate back-to-back articles in this month’s issue. This approach has been taken as the risk-return stor y is included in two separate but interconnected parts of the syllabus. We need to understand the principles that underpin por tfolio theory, before we can appreciate the creation of the Capital Asset Pricing Model (CAPM).
In this ar ticle on portfolio theory we will review the reason why investors should establish por tfolios. This is neatly captured in the old saying ‘don’t put all your eggs in one basket’. The logic is that an investor who puts all of their funds into one investment risks ever ything on the performance of that individual investment. A wiser policy would be to spread the funds over several investments (establish a por tfolio) so that the unexpected losses from one investment may be offset to some extent by the unexpected gains from another. Thus the key motivation in establishing a por tfolio is the reduction of risk. W e shall see that it is possible to maintain returns (the good) while reducing risk (the bad).
LEARNING OBJECTIVES
By the end of this article you should be
able to:
understand an NPV calculation from an
investor’s perspective
calculate the expected return and
standard deviation of an individual
investment and for two asset por tfolios
understand the significance of correlation
in risk reduction
prepare a summar y table
understand and explain the nature of risk
as por tfolios become larger
understand and be able to explain why
the market only gives a return for
systematic risk.UNDERSTANDING AN NPV CALCULATION
FROM AN INVESTOR’S PERSPECTIVE
Joe currently has his savings safely deposited
in his local bank. He is considering buying
some shares in A plc. He is trying to
determine if the shares are going to be a
viable investment. He asks the following
questions: ‘What is the future expected return
from the shares? What extra return would I
require to compensate for undertaking a risky
investment?’ Let us tr y and find the answers
to Joe’s questions. First we turn our attention
to the concept of expected return.
EXPECTED RETURN
Investors receive their returns from shares in
the form of dividends and capital gains/
losses. The formula for calculating the annual
return on a share is:
Annual return = D
1
+ (P
1
- P
)
P
where:
D
1
= dividend per share
P
1
= share price at the end of a year
P
= share price at the start of a year.
Suppose that a dividend of 5p per share was
paid during the year on a share whose value
was 100p at the start of the year and 117p
at the end of the year:
Annual return =
5 + (117 - 100) × 100 = 22%
100
The total return is made up of a 5% dividend
yield and a 17% capital gain. We have just
calculated a historical return, on the basis
that the dividend income and the price at the
end of year one is known. However,
calculating the future expected return is a lot
more difficult because we will need to
estimate both next year’s dividend and the
share price in one year’s time. Analysts
normally consider the different possible
returns in alternate market conditions and try
and assign a probability to each. The table in
Example 1 shows the calculation of the
expected return for A plc. The current share
price of A plc is 100p and the estimated
returns for next year are shown. The
investment in A plc is risky. Risk refers to the
possibility of the actual return var ying from
the expected return, ie the actual return may
be 30% or 10% as opposed to the expected
return of 20%.
REQUIRED RETURN
The required return consists of two elements,
which are:
Required return =
Risk-free return + Risk premium
Risk-free return
The risk-free return is the return required by
investors to compensate them for investing
in a risk-free investment. The risk-free return
compensates investors for inflation and
consumption preference, ie the fact that they
are deprived from using their funds while
tied up in the investment. The return on
treasur y bills is often used as a surrogate for
the risk-free rate.
Risk premium
Risk simply means that the future actual
return may vary from the expected return. If
an investor undertakes a risky investment he
needs to receive a return greater than the
risk-free rate in order to compensate him. The
more risky the investment the greater the
compensation required. This is not surprising
and it is what we would expect from risk-
averse investors.
54 student accountant May 2004
he would have to receive an extra 5% of return to compensate for the market risk. Thus 5% is the historical average risk premium in the UK.
Suppose that Joe believes that the shares in A plc are twice as risky as the market and that the use of long-term averages are valid. The required return may be calculated as follows:
Required=Risk free+Risk return of A plc return premium 16%=6%+(5% × 2)
Thus 16% is the return that Joe requires to compensate for the perceived level of risk in A plc, ie it is the discount rate that he will use to appraise an investment in A plc.
THE NPV CALCULATION
Suppose that Joe is considering investing ￡100 in A plc with the intention of selling the shares at the end of the first year. Assume that the expected return will be 20% at the end of the first year. Given that Joe requires a return of 16% should he invest?Decision criteria: accept if the NPV is zero or
positive. The NPV is positive, thus Joe should
invest. A positive NPV opportunity is where
the expected return more than compensates
the investor for the perceived level of risk, ie
the expected return of 20% is greater than
the required return of 16%. An NPV
calculation compares the expected and
required returns in absolute terms.
Calculation of the risk premium
Calculating the risk premium is the essential
component of the discount rate. This in turn
mak es the NPV calculation possible. T o
calculate the risk premium, we need to be
able to define and measure risk.
THE STUDY OF RISK
The definition of risk that is often used in
finance literature is based on the variability
of the actual return from the expected return.
Statistical measures of variability are the
variance and the standard devi ation (the
square root of the variance). R eturning to the
example of A plc, we will now calculate
variability. Thus the variance represents ‘rates
of return squared’. As the standard deviation is
the square root of the variance, its units are in
rates of return. As it is easier to discuss risk as
a percentage rate of return, the standard
deviation is more commonly used to measure
risk. In the exam it is unlikely that you will be
ask ed to undertake these basic calculations.
The exam questions normally provide you with
the expected returns and standard deviations
of the returns.
Shares in Z plc have the following returns
and associated probabilities:
Probability Return %
0.135
0.820
0.15
Let us then assume that there is a choice of
investing in either A plc or Z plc, which one
should we choose? T o compare A plc and Z
plc, the expected return and the standard
deviation of the returns for Z plc will have to
be calculated.
May 2004 student accountant 55
56 student accountant May 2004
The expected return is: (0.1) (35%) + (0.8)(20%) + (0.1) (5%) = 20%
The variance is: = σ2z = (0.1) (35% - 20%)2+ (0.8) (20% - 20%)2 + (0.1) (5% - 20%)2= 45%
The standard deviation is:= σz
= 45 = 6.71%Summary table Investment Expected
Standard
return
deviation A plc 20% 4.47%Z plc 20%
6.71%
Given that the expected return is the same for both companies, investors will opt for the one that has the lowest risk, ie A plc. The
decision is equally clear where an investment gives the highest expected return for a given level of risk. However, these only relate to specific instances where the investments
being compared either have the same expected return or the same standard deviation. Where investments have increasing levels of return accompanied by increasing levels of standard deviation, then the choice between
investments will be a subjective decision based on the investor ’s attitude to risk.
RISK AND RETURN ON TWO -ASSET PORTFOLIOS
So far we have confined our choice to a single investment. Let us now assume investments can be combined into a two -asset portfolio.The risk-return relationship will now be
measured in terms of the por tfolio’s expected return and the portfolio’s standard deviation.
The following table gives information about four investments: A plc, B plc, C plc,and D plc. Assume that our investor , Joe has decided to construct a two-asset portfolio and that he has already decided to invest 50% of the funds in A plc. He is currently tr ying to decide which one of the other three
investments into which he will invest the remaining 50% of his funds. See Example 2.The expected return of a two-asset portfolio The expected return of a portfolio (Rpor t ) is simply a weighted average of the expected returns of the individual investments.Rpor t =x.R A + (1 - x).R B
x
=
the proportion of funds invested in A
(1 - x) =the proportion of funds invested in B
R A + B =0.5 × 20 + 0.5 × 20 = 20R A + C =0.5 × 20 + 0.5 × 20 = 20R A + D =
0.5 × 20 + 0.5 × 20 = 20
Given that the expected return is the same for all the por tfolios, Joe will opt for the portfolio that has the lowest risk as measured by the por tfolio’s standard deviation .
The standard deviation of a two-asset por tfolio We can see that the standard deviation of all the individual investments is 4.47%.
Intuitively , we probably feel that it does not matter which por tfolio Joe chooses, as the standard deviation of the portfolios should be the same (because the standard deviations of the individual investments are all the same).
However, the above analysis is flawed, as the standard deviation of a por tfolio is not simply the weighted average of the standard deviation of returns of the individual
investments but is generally less than the weighted average . So what causes this
reduction of risk? What is the missing factor?The missing factor is how the returns of the two investments co-relate or co -var y , ie move up or down together. There are two ways to measure
Return on investments (%)
Market conditions Probability A plc B plc C plc D plc Boom 0.130301010Normal 0.820202022.5Recession
0.1
10103010Expected return 20202020Standard deviation
4.47
4.47
4.47
4.47
covariability . The first method is called the
covariance and the second method is called the correlation coefficient. Before we perform these calculations let us review the basic logic behind the idea that risk may be reduced depending on how the returns on two investments co-vary.Portfolio A+B – per fect positive correlation The returns of A and B move in perfect lock step, (when the return on A goes up to 30%,the return on B also goes up to 30%, when the return on A goes down to 10%, the return on B also goes down to 10%), ie they move in the same direction and by the same degree. See Example 3.
This is the most basic possible example of perfect positive correlation , where the
forecast of the actual returns are the same in all market conditions for both investments and thus for the portfolio (as the portfolio return is simply a weighted average). Hence there is no reduction of risk. The por tfolio’s standard deviation under this theoretical extreme of perfect positive correlation is a simple weighted average of the standard deviations of the individual investments:σpor t (A,B) = 4.47 × 0.5 + 4.47 × 0.5= 4.47
Portfolio A+C – per fect negative correlation The returns of A and C move in equal but opposite ways (when the return on A goes up
EXAMPLE 2Return on investments (%)
Mark et Conditions A plc B plc Portfolio A + B Boom 303030Normal 202020Recession
10
10
10
EXAMPLE 3
the expected return under normal market conditions and almost the same under boom mark et conditions (20 v 21.25). Therefore, we can say that the forecast actual and expected returns are almost the same in two out of the three conditions. This compares with only one condition when there is perfect positive correlation (no reduction of risk) and all three conditions when there is perfect However, this approach is not required in the
exam, as the exam questions will generally
contain the covariances when required.
a b
The correlation coefficient as a relative
measure of covariability expresses the strength
of the relationship between the returns on two
investments. It is strictly limited to a range
from -1 to +1. See Example 6.
May 2004 student accountant 57
= 4.47
The second version of the formula is the one that is nearly always used in exams and it is the one that is given on the formula sheet.This can be proved quite easily, as a portfolio’s
expected return is equal to the weighted
average of the expected returns on the
individual investments, whereas a portfolio’s
risk is less than the weighted average of the risk
or lesser degree because of common macro-
economic factors affecting all investments. The
risk contributed by the covariance is often
called the ‘market or systematic risk’. This risk
cannot be diversified away.
58 student accountant May 2004
The risk reduction is quite dramatic. We find that two thirds of an investment’s total risk can be diversified away, while the remaining one third of risk cannot be diversified away. A
well-diversified portfolio is very easy to obtain, all we have to do is buy a portion of a larger fund that is already well-diversified, like buying into a unit trust or a track er fund.
Remember that the real joy of diversification is the reduction of risk without any consequential reduction in return. If we assume that investors are rational and risk averse, their portfolios should be
well-diversified, ie only suffer the type of risk that they cannot diversify away (systematic risk).
An investor who has a well-diversified portfolio only requires compensation for the risk suffered by their portfolio (systematic risk). Therefore we need to re-define our understanding of the required return: Required return = Risk free return + Systematic risk premium
Investors who have well-diversified portfolios dominate the mark et. They only require a return for systematic risk. Thus we can now appreciate the statement ‘that the market only gives a return for systematic risk’.
The next question will be how do we measure an investment’s systematic risk? The answer to this question will be given in the following article on the Capital Asset P ricing Model (CAPM).
10 KEY POINTS TO REMEMBER
1The expected return on a share consists of a dividend yield and a capital gain/loss
in percentage terms.
2The required return on a risky investment consists of the risk-free rate (which
includes inflation) and a risk premium.
3Total risk is normally measured by the standard deviation of returns (σ).
4Portfolio theor y demonstrates that it is possible to reduce risk without having a
consequential reduction in return, ie the
portfolio’s expected return is equal to the
weighted average of the expected returns
on the individual investments, while the
portfolio risk is normally less than the
weighted average of the risk of the
individual investments.
5The extent of the risk reduction is
influenced by the way the returns on the
investments co-vary. Covariability is
normally measured in the exams by the
correlation coefficient.
6In reality, the correlation coefficient between returns on investments tend to
lie between 0 and +1. Thus total risk can only be partially reduced, not eliminated.
Maximum
Partial No
reduction reduction reduction
7 A por tfolio’s total risk consists of
unsystematic and systematic risk.
However, a well-diversified portfolio only
suffers from systematic risk, as the
unsystematic risk has been diversified
away.
8An investor who holds a well-diversified
portfolio will only require a return for
systematic risk. Thus their required return
consists of the risk-free rate plus a
systematic risk premium.
9Investors who have well-diversified
por tfolios dominate the market. Thus
the market only gives a return for
systematic risk.
10 The preparation of a summary table and
the identification of the most efficient
portfolio (if possible) is an essential exam
skill.
Patrick L ynch is a lecturer at FTC London Unsystematic Systematic
risk risk
Company General
specific factors economic factors
Can be eliminated Cannot be eliminated
SYSTEMATIC AND UNSYSTEMA TIC RISK
The total risk of a portfolio (as measured by
the standard deviation of returns) consists of
two types of risk: unsystematic risk and
systematic risk. If we have a large enough
portfolio it is possible to eliminate the
unsystematic risk. However, the systematic
risk will remain. See Example 7.
Unsystematic/Specific risk: refers to the
impact on a company’s cash flows of largely
random events like industrial relations
problems, equipment failure, R&D
achievements, changes in the senior
management team etc. In a portfolio, such
random factors tend to cancel as the number
of investments in the portfolio increase.
Systematic/Market risk: general
economic factors are those macro-economic
factors that affect the cash flows of all
companies in the stock mark et in a consistent
manner, eg a country’s rate of economic
growth, corporate tax rates, unemployment
levels, and interest rates. Since these factors
cause returns to move in the same direction
they cannot cancel out. Therefore, systematic/
mark et risk remains present in all portfolios.
WHAT IS THE IDEAL NUMBER OF
INVESTMENTS IN A PORTFOLIO?
Ideally, the investor should be fully-diversified,
ie invest in every company quoted in the stock
mark et. They should hold the ‘Mark et portfolio’
in order to gain the maximum risk reduction
effect. The good news is that we can construct
a well-diversified por tfolio, ie a portfolio that will
benefit from most of the risk reduction effects of
diversification by investing in just 15 different
companies in different sectors of the market.
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