ACCA P4 Advanced Financial Management - 2010 - Study text

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Chapter 5: Investing: portfolio theory and the CAPM

When a company invests in a new project, there will be an investment risk. This is

the risk that actual returns from the investment will not be the same as the expected

returns but could be higher or lower. This investment risk for companies is similar

to the investment risk facing equity investors.

1.3 Measuring risk as a variance or standard deviation of expected returns

The risk of variations in annual profits and dividends can be measured. When

annual returns can differ by a large amount from normal or expected (average)

returns, and there are considerable differences in returns from one year to the next,

returns are volatile.

High volatility in returns is associated with high investment risk.

Risk can be measured statistically, either from an analysis of historical returns

achieved in the past, or from probability estimates of returns in the future. The

measurement of risk is a measurement of the volatility of possible returns.

This volatility can be measured as either the variance or standard deviation of

expected returns over a given period of time, such as a variance or standard

deviation of expected annual returns.

For an investment, we can measure both:

 an expected or average annual return for the time period, and

 a variance or standard deviation of the returns.

Calculating the variance or standard deviation of expected returns

The variance of expected returns is calculated as ∑

p(r - r )

where:

p is the probability of a given return on the investment

r is the amount of that return

is the average expected return on the investment. This is the expected value (EV)

of the return.

The

standard deviation of expected returns is the square root of the variance.

 The symbol for variance is either s

or σ

 The symbol for standard deviation is either s or σ.

If you are not familiar with the calculation of a variance, study the following

example carefully.

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Example

From historical analysis, it has been established that the probable returns on equity

investment A are:

Probability

Return

Expected value

(EV) of return

% %

0.2 2

0.4

0.5 6

3.0

0.3 8

2.4

5.8

The variance of the returns is calculated as follows:

Probability

Return

r – r

(r – r

)

p (r – r

)

% %

0.2 2 -

3.8

14.44

2.888

0.5 6 + 0.2

0.04

0.020

0.3 8 + 2.2

4.84

1.452

Variance s

4.360

The variance of the returns is 4.36% and the standard deviation (the square root of

the variance, s) = 2.09%.

1.4 Assessing the investment risk: coefficient of variation

The expected return from the investment is therefore 5.8% with a standard

deviation of 2.09%. The statistical significance of the standard deviation depends on

whether the variability in possible returns has a recognisable statistical distribution,

such as a standard normal distribution.

However, even when the standard deviation of returns is not normally distributed,

the risk in an investment can be assessed by comparing the size of the risk (standard

deviation of returns) with the expected return.

The coefficient of variation is the ratio of the standard deviation to the expected

return.

Coefficient of variation = Standard deviation of expected returns

Expected (value of) returns

A high coefficient of variation indicates high investment risk, because the risk is

large in relation to the expected returns.

Chapter 5: Investing: portfolio theory and the CAPM

Example

From historical analysis, it has been established that the probable returns on equity

investment B are:

Probability

Return

Expected value

(EV) of return

% %

0.4 (2)

(0.8)

0.3 7

2.1

0.3 15

4.5

5.8

The variance of the returns is calculated as follows:

Probability

Return

r – r

p (r – r

)

% %

0.4 (2) -

7.8

24.336

0.3 7 + 1.2

0.432

0.3 15 + 9.2

25.392

Variance s

50.160

The variance of the returns is 50.16% and the standard deviation (the square root of

the variance, s) = 7.08%.

The coefficient of variation for Investment B is 7.08/5.8 = 1.22. This is higher than

the coefficient of variation for Investment A in the previous example, which has the

same expected return of 5.8% but a coefficient of variation of 2.09/5.8 = 0.36.

Clearly, the investment risk with Investment B is higher than the risk with

Investment A.

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Portfolio theory

 Introduction to portfolio theory

 Diversification to reduce risk: building a portfolio

 Two-asset portfolios

 Expected return and standard deviation of return for a two-asset portfolio

2 Portfolio theory

2.1 Introduction to portfolio theory

Portfolio theory is concerned with how investors should build a portfolio of

investments that gives them a suitable balance between return and investment risk.

The theory may seem complex, but it is not often examined. However, portfolio

theory provides a theoretical basis for the capital asset pricing model, which is an

important model in financial management.

2.2 Diversification to reduce risk: building a portfolio

To a certain extent, an investor can reduce the investment risk – in other words,

reduce the volatility of expected returns – by diversifying his investments, and

holding a portfolio of different investments.

Creating a portfolio of different investments can reduce the variance and standard

deviation of returns from the total portfolio, because if some investments provide a

lower-than-expected return, others will provide a higher-than-expected return.

Extremely high or low returns are therefore less likely to occur.

(Similarly, a company could reduce the investment risk in its business by

diversifying, and building a portfolio of different investments. However, it can be

argued that there is no reason for a company to diversify its investments, because an

investor can achieve all the diversification he requires by selecting a diversified

portfolio of equity investments.)

Diversification to reduce risk: the correlation of investment returns

The extent to which investment risk can be reduced by building a portfolio of

different investments depends on the

correlation of the returns from the different

investments in the portfolio

 When returns from different investments in a portfolio are positively correlated,

this means that when the return from one of the investments is higher than

expected, the returns from the other investments will also be higher than

expected. Similarly, when returns from one investment are lower than expected,

the returns from all the investments in the portfolio will be lower than expected.

 When returns from two different investments in a portfolio are negatively

correlated,

this means that when the returns from one of the investments is

Chapter 5: Investing: portfolio theory and the CAPM

higher than expected, the returns from the other investment will be lower than

expected.

 When returns from two investments are neither positively nor negatively

correlated, this means that when the returns from one investment are higher

than expected, the returns from the other investment might be either higher or

lower than expected.

Investment risk is reduced by building up a portfolio of investments whose returns

are negatively correlated, or where correlation is low.

Correlation coefficient of investment returns

The correlation of the returns from different investments can be measured

statistically by a

correlation coefficient, from an analysis of historical data about

returns (or from probability estimates of future returns).

A correlation coefficient can range in value from +1 (perfect positive correlation) to

– 1 (perfect negative correlation).

A correlation coefficient close to zero indicates very little correlation between

investment returns.

Investment risk is reduced most effectively by having investments in a portfolio

whose returns are negatively correlated, or where there is not much correlation.

You might be required to calculate the correlation coefficient of the returns from

two investments in a portfolio (a two-investment portfolio). You will not be

required to make any calculations of correlation for a portfolio of more than two

investments, but the same principles apply to larger portfolios.

2.3 Two-asset portfolios

One way of calculating the correlation coefficient of returns between two

investments in a two-asset portfolio is to begin by calculating the

covariance of the

returns

from the two investments.

Covariance of returns from investment A and investment B (cov

A,B

)

∑ p (r

–

) (r

–

)

Example

There are two investments in a portfolio, investment A and investment B. The

probabilities of annual returns from the investments are as follows:

Probability

Return from

investment A

Return from

investment B

% %

0.4 5.0 12.0

0.4 7.0 10.0

0.2 11.0 1.0

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The expected value of the return from investment A is 7% and the expected value of

the return from investment B is 9%.

(For investment A, this is calculated as (0.4 × 5%) + (0.4 × 7%) + (0.2 × 11%). For

investment B, the expected return is calculated as (0.4 × 12%) + (0.4 × 10%) + (0.2 ×

1%).)

The covariance of returns is calculated as follows:

Probability (r

– r

) (r

– r

) p (r

– r

) (r

– r

)

p % %

0.4 - 2.0 + 3.0

2.4

0.4 0 + 1.0

0.0

0.2 + 4.0 -

8.0

6.4

8.8

Correlation coefficient of investment returns

The correlation coefficient for the returns of two investments (A and B) in a two

asset portfolio is calculated as:

Correlation coefficient, ρ

A,B

Covariance

A,B

This is the covariance of returns from investments A and B, divided by [the

standard deviation of returns from investment A multiplied by the standard

deviation of returns from investment B].

Example

Continuing the previous example, the standard deviation of returns for investment

A is 2.19% and the standard deviation of returns for investment B is 4.10%. (The

workings are not shown here, but you should be able to calculate them, using the

formula described earlier.)

The correlation coefficient for the returns from the two investments, is therefore:

()()

0.98

4.102.19

8.8

t,coefficiennCorrelatio

BA,

−=

−

=ρ

Investment returns that are not correlated

When the returns from two investments are not correlated, the value of the

correlation coefficient should be 0.

Chapter 5: Investing: portfolio theory and the CAPM

2.4 Expected return and standard deviation of return for a two-asset

portfolio

Expected return from a two-asset portfolio

The expected return from a two-asset portfolio is the weighted average of the

returns from the two investments.

= R

x + R

(1 – x)

where:

is the expected return from the two-asset portfolio

is the expected return from investment A

is the expected return from investment B

x is the proportion of investment A in the portfolio (for example, 40% = 0.40)

(1 – x) is the proportion of investment A in the portfolio

Standard deviation of return for a two-asset portfolio

The overall standard deviation of the return from a two-asset portfolio can be

calculated using the following formula, which is given to you in the formula sheet

for the examination.

() ()

BAB,A

x12xx1x sssssp

−+−+=

where:

x is the proportion of the portfolio consisting of investment A

(1 – x) is the proportion of the portfolio consisting of investment B

and

are the standard deviation of returns for investment A and investment B

A,B

is the covariance of returns for investment A and investment B

This formula is included in the formula sheet for your examination, with x and (1 –

x) represented by w

and w

respectively.

Example

Returning to the previous example, suppose that the portfolio of an investor consists

of 70% of investment A and 30% of investment B. The expected return is 7% for

investment A and 9% for investment B.

The expected return from the portfolio is therefore:

(7% × 70%) + (9% × 30%) = 7.6%.

The standard deviation of the returns from this portfolio would be:

()()()() ()()( )()()

0.41%0.16764.102.190.980.300.7020.304.100.702.19

2222

==−++

Paper P4: Advanced Financial Management

The portfolio therefore has an expected return of 7.6% with a standard deviation of

0.41%.

The risk in this portfolio is significantly less than the risk from the two individual

investments (whose standard deviations are 2.19% and 4.10%). This is largely due to

the fact that the returns from the two investments have strong negative correlation.

Conclusion: diversification to reduce investment risk

The general principle, even when correlation is not negative, is that:

 diversification within a portfolio reduces risk

 this rule applies to portfolios with many different investments, not just to two-

investment portfolios.

Example

A multinational company is considering new investments in three countries. An

estimate of the expected returns and the investment risk has been made as follows:

Country Expected

investment

return

Investment

risk (= s)

A 28% 36%

B 24% 29%

C 20% 23%

The company has decided that it will invest equal amounts in two of the countries,

but will not invest in the third. The returns from the investments in each country are

not correlated.

Required

(a) Calculate the expected return, risk and coefficient of variation for each of the

possible investment combinations.

(b)

Suggest how this analysis can help the company to decide which investment

portfolio to select.

Answer

The returns from the investments in the different countries would not be correlated,

therefore the correlation coefficient is 0. This means that the final term in the

formula for the standard deviation of a two-asset portfolio must also be 0.

Two-asset portfolio Expected return from

the portfolio

Country A + Country B (0.5 × 28%) + (0.5 × 24%) 26.0%

Country A + Country C (0.5 × 28%) + (0.5 × 20%) 24.0%

Country B + Country C (0.5 × 24%) + (0.5 × 20%) 22.0%

Chapter 5: Investing: portfolio theory and the CAPM

The standard deviation of the returns from each two-asset portfolio is calculated as

follows (allowing for the fact that the coefficient of correlation is 0 for each

portfolio):

Two-asset portfolio Standard deviation of

expected return

Country A + Country B [(0.5

× 36

) + (0.5

× 29

) ]

1/2

23.1%

Country A + Country C [(0.5

× 36

) + (0.5

× 23

) ]

1/2

21.4%

Country B + Country C [(0.5

× 29

) + (0.5

× 23

) ]

1/2

18.5%

Two-asset portfolio Coefficient of

variation

Country A + Country B 23.1/26.0 0.89

Country A + Country C 21.4/24.0 0.89

Country B + Country C 18.5/22.0 0.84

This information might be used by the company’s management to make the

investment decision and select the preferred two-asset portfolio.

 Investing in Country A and Country B would offer the highest expected return.

The investment risk, measured simply as the coefficient of variation, indicates

that it would have similar risk to a portfolio of Country A and Country C, for

which the expected return is lower.

 Investing in Country B and Country C offers the lowest expected return,

although the investment risk (measured by the coefficient of variation) appears

to be slightly lower.

On the basis of the available information, it seems probable that the company will

choose to invest in either Country A and Country B or in Country B and Country C.

The choice of one portfolio in preference to the other will depend on the ‘risk

appetite’ of the company, and whether it is willing to take on the higher investment

risk in the hope of obtaining a higher investment return.

Paper P4: Advanced Financial Management

Choosing an investment portfolio

 Investor preferences

 Efficient portfolios and the efficient frontier

 The market portfolio

 Risk-free investments and choosing an investment portfolio

 The market premium

 The beta factor of a portfolio

 Systematic and unsystematic risk

3 Choosing an investment portfolio

3.1 Investor preferences

Investors will usually avoid risk, in the sense that if they invest in a risky portfolio,

they will expect a higher return than from investing in a less risky portfolio. A

higher return is required to compensate the investor for the higher risk.

Example

Suppose that an investor has a choice of selecting any of the following portfolios:

Portfolio Expected return Risk (=

)

% %

A 6.0

4.0

B 7.0

3.0

C 9.0

5.5

D 8.0

6.0

 A normal (risk-averse) investor will not select Portfolio A, because Portfolio B

offers a higher return than Portfolio A for less risk

 Similarly, a risk-averse investor will not select Portfolio D, because Portfolio C

offers a higher return than Portfolio D for less risk

 It is not clear whether a risk-averse investor would select Portfolio B or Portfolio

C, because Portfolio C offers a higher return for greater risk, and Portfolio B

offers a lower return but less risk.

3.2 Efficient portfolios and the efficient frontier

Investors can assemble a wide variety of different portfolios, each consisting of

different investments and in differing proportions. However, normal investors will

always select a portfolio that offers a higher expected return or a lower risk than

other portfolios that could be assembled.

ACCA P4 Advanced Financial Management - 2010 - Study text - Emile Woolf Publishing

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