88 Financial Management

Chapter 5
Portfolio Theory and
Assets Pricing Models
Learning Objectives
 Discuss the concepts of portfolio risk and return
 Determine the relationship between risk and return of portfolios
 Highlight the difference between systematic and unsystematic risks
 Examine the logic of portfolio theory
 Show the use of capital asset pricing model (CAPM) in the valuation of securities
 Explain the features and modus operandi of the arbitrage pricing theory (APT)

INTRODUCTION pricing theory (APT). In this chapter, we discuss the

portfolio theory and show how CAPM and APT work
A portfolio is a bundle or a combination of individual in valuing assets.
assets or securities. Portfolio theory provides a normative
approach to investors to make decisions to invest their
wealth in assets or securities under risk.1 It is based on PORTFOLIO RETURN: TWO-ASSET CASE
the assumption that investors are risk-averse. This implies
that investors hold well-diversified portfolios instead of The return of a portfolio is equal to the weighted average
investing their entire wealth in a single or a few assets. of the returns of individual assets (or securities) in the
One important conclusion of the portfolio theory, as we portfolio with weights being equal to the proportion of
explain later, is that if the investors hold a well-diversified investment value in each asset. Suppose you have an
portfolio of assets, then their concern should be the opportunity of investing your wealth in either asset X or
expected rate of return and risk of the portfolio rather asset Y. The possible outcomes of two assets in different
than individual assets and the contribution of individual states of economy are given in Table 5.1.
asset to the portfolio risk. The second assumption of the
portfolio theory is that the returns of assets are normally Table 5.1: Possible Outcomes of Two Assets, X and Y
distributed. This means that the mean (the expected
Return (%)
value) and variance (or standard deviation) analysis is
State of Economy Probability X Y
the foundation of the portfolio decisions. Further, we
can extend the portfolio theory to derive a framework A 0.10 –8 14
for valuing risky assets. This framework is referred to as B 0.20 10 –4
the capital asset pricing model (CAPM). An alternative C 0.40 8 6
model for the valuation of risky assets is the arbitrage D 0.20 5 15
E 0.10 –4 20
1. For a simple treatment of return and risk concepts and the capital asset pricing model, see Mullins, D. W., Does the Capital Asset Pricing
Model Work?, Harvard Business Review, Jan–Feb 1982; and Butters, J.K., et. al., Case Problems in Finance, Richard D. Irwin, 1991.
 Portfolio Theory and Assets Pricing Models 89
The expected rate of return of X is the sum of the product Note that w is the proportion of investment in asset
of outcomes and their respective probability. That is: X and (1 – w) is the remaining investment in asset Y.
E( Rx ) = ( − 8 × 0.1) + (10 × 0.2) + (8 × 0.4) + ( 5 × 0.2) Given the expected returns of individual assets, the
portfolio return depends on the weights (investment
+ ( − 4 × 0.1) = 5% proportions) of assets. You may be able to change your
Similarly, the expected rate of return of Y is: expected rate of return on the portfolio by changing
your proportionate investment in each asset. How
E( Ry ) = (14 × 0.1) + ( − 4 × 0.2) + (6 × 0.4) + (15 × 0.2) much would you earn if you invested 20 per cent of
+ ( 20 × 0.1) = 8% your wealth in X and the remaining wealth in Y? The
portfolio rate of return under this changed mix of wealth
We can use the following equation to calculate the in X and Y will be:
expected rate of return of individual asset: E( Rp ) = 0.2 × 5 + (1 − 0.2) × 8 = 7.4%
E( Rx ) = ( R1 × P1 ) + ( R2 × P )2 + You may notice that this return is higher than what
( R3 × P3 ) + ..... + ( Rn × Pn ) you will earn if you invested equal amounts in X and Y.
n The expected return would be 5 per cent if you invested
E( Rx ) = ∑ Ri Pi (1) entire wealth in X (i.e., w = 1.0). On the other hand, the
i =1 expected return would be 8 per cent if the entire wealth
Note that E(Rx) is the expected return on asset X, were invested in Y (i.e., 1 – w = 1, since w = 0). Your
Ri is ith return and Pi is the probability of ith return. expected return will increase as you shift your wealth
Consider an example. from X to Y. Thus, the expected return on portfolio will
Suppose you decide to invest 50 per cent of your depend on the percentage of wealth invested in each
asset in the portfolio.
wealth in X and 50 per cent in Y. What is your expected
rate of return on a portfolio consisting of both X and What is the advantage in investing your wealth in
Y ? This can be done in two steps. First, calculate both assets X and Y when you could expect highest
the combined outcome under each state of economic return of 8 per cent by investing your entire wealth in
condition. Second, multiply each combined outcome by Y? When you invested your wealth equally in assets X
and Y, your expected return is 6.5 per cent. The expected
its probability. Table 5.2 shows the calculations.
return of Y (8 per cent) is higher than the portfolio
There is a direct and simple method of calculating return (6.5 per cent). But investing your entire wealth
the expected rate of return on a portfolio if we know the in Y is more risky. Under the unfavourable economic
expected rates of return on individual assets and their condition, Y may yield a negative return of 4 per cent.
weights. The expected rate of return on a portfolio (or The probability of negative return is eliminated when
simply the portfolio return) is the weighted average of you combine X and Y. Further, the portfolio returns are
the expected rates of return on assets in the portfolio. In expected to fluctuate within a narrow range of 3 to 10
our example, the expected portfolio return is as follows: per cent (see column 3 of Table 5.2). You may also note
E( Rp ) = (0.5 × 5) + (0.5 × 8) = 6.5% that the expected return of X (5 per cent) is not only
less than the portfolio return (6.5 per cent), but it also
In the case of two-asset portfolio, the expected rate shows greater fluctuations. We discuss the concept of
of return is given by the following formula: risk in greater detail in the following sections.
Expected return on portfolio
= weight of security X CHECK YOUR CONCEPTS
× expected return on security X
+ weight of security Y 1. Define the portfolio return.
× expected return on security Y 2. How is the expected return on a portfolio calculated?
E(Rp) = w × E(Rx) + (1 – w) × E(Ry) (2)

Table 5.2: Expected Portfolio Rate of Return

Combined Returns (%)

State of Economy Probability X(50%) & Y (50%) Expected Return (%)
(1) (2) (3) (4) = (2) × (3)
A 0.10 (–8 × 0.5) + (14 × 0.5) = 13.0 0.10 × 3.0 = 0.3
B 0.20 (10 × 0.5) + (–4 × 0.5) = 13.0 0.20 × 3.0 = 0.6
C 0.40 (8 × 0.5) + (6 × 0.5) = 17.0 0.40 × 7.0 = 2.8
D 0.20 (5 × 0.5) + (15 × 0.5) = 10.0 0.20 × 10.0 = 2.0
E 0.10 (–4 × 0.5) + (20 × 0.5) = 18.0 0.10 × 8.0 = 0.8
Expected return on portfolio 6.5
90 Financial Management
PORTFOLIO RISK: TWO-ASSET CASE E( RP ) = 0.5 × 0 + 0.5 × 40 = 20%
We have seen in the previous section that returns on Thus, by investing equal amounts in A and B, rather
individual assets fluctuate more than the portfolio return. than the entire amount only in A or B, the investor is
Thus, individual assets are more risky than the portfolio. able to eliminate the risk altogether. She is assured of
How is the risk of a portfolio measured? As discussed in a return of 20 per cent with a zero standard deviation.
the previous chapter, risk of individual assets is measured It is not always possible to entirely reduce the risk.
by their variance or standard deviation. We can use It may be difficult in practice to find two assets whose
variance or standard deviation to measure the risk of returns move completely in opposite directions like in the
the portfolio of assets as well. Why is a portfolio less
above example of securities A and B. It needs emphasis
risky than individual assets? Let us consider an example.
to state that the risk of portfolio would be less than
Suppose you have two investment opportunities A
the risk of individual securities, and that the risk of
and B as shown in Table 5.3.
a security should be judged by its contribution to the
Table 5.3: Investments in A and B portfolio risk.
Economic Returns (%)
Condition Probability A B Measuring Portfolio Risk for Two Assets
Good 0.5 40 0 Like in the case of individual assets, the risk of a
Bad 0.5 0 40 portfolio could be measured in terms of its variance
Assuming that the investor invests in both the or standard deviation. As stated earlier, the portfolio
assets equally, the expected rate of return, variance and return is the weighted average of returns on individual
standard deviation of A are: assets. Is the portfolio variance or standard deviation a
weighted average of the individual assets’ variances or
E( RA ) = 0.5 × 40 + 0.5 × 0 = 20% standard deviations? It is not. The portfolio variance
or standard deviation depends on the co-movement of
σ A2 = 0.5( 40 − 20)2 + 0.5(0 − 20)2 = 400
returns on two assets.
σ A = 400 = 20%
Covariance When we consider two assets, we are
Here σA represents the standard deviation of assets A, concerned with the co-movement of the assets. Covariance
σA2 is the variance of asset A and E(RA) is the estimated of returns on two assets measures their co-movement.
rate of returns. Note that variance is the square of How is covariance calculated?
standard deviation.
Three steps are involved in the calculation of
Similarly, the expected rate of return, variance and covariance between two assets:
standard deviation of B are:
 Determine the expected returns on assets.
E( RB ) = 0.5 × 0 + 0.5 × 40 = 20%  Determine the deviation of possible returns from
σ B2 = 0.5(0 − 20)2 + 0.5( 40 − 20)2 = 400 the expected return for each asset.
σ B = 400 = 20%  Determine the sum of the product of each
deviation of returns of two assets and respective
Both investments A and B have the same expected probability.
rate of return (20 per cent) and same variance (400) and
Let us consider the data of securities of X and Y
standard deviation (20 per cent). Thus, they are equally
given in Table 5.4. The expected return on security X is:
profitable and equally risky. How does combining
investments A and B help an investor? If a portfolio E( Rx ) = (0.1 × −8) + (0.2 × 10) + (0.4 × 8) + (0.2 × 5)
consisting of equal amount of A and B were constructed, + (0.1 × − 4) = 5%
the portfolio return would be:
Security Y’s expected return is:
E( RP ) = 0.5 × 20 + 0.5 × 20 = 20% E( Ry ) = (0.1 × 14) + (0.2 × −4) + (0.4 × 6) + (0.2 × 15)
This return is the same as the expected return from + (0.1 × 20) = 8%
individual securities, but without any risk. Why? If the
economic conditions are good, then A would yield 40 per If the equal amount is invested in X and Y, the
cent return and B zero and the portfolio return will be: expected return on the portfolio is:

E( RP ) = 0.5 × 40 + 0.5 × 0 = 20%

( )
E Rp = 5 × 0.5 + 8 × 0.5 = 6.5%

Table 5.4 shows the calculations of variations from

When economic conditions are bad, then A’s return the expected return and covariance, which is the product
will be zero and B’s 40 per cent and the portfolio return of deviations of returns of securities X and Y and their
would still remain the same: associated probabilities:
 Portfolio Theory and Assets Pricing Models 91
Table 5.4: Covariance of Returns of Securities X and Y can, however, compute the correlation to measure the
relationship between two returns.
Deviation Product of
State of from Deviation & Correlation How can we find relationship between
Economy Probability Returns Expected Probability two variables? Correlation is a measure of the linear
Returns (Covariance) relationship between two variables (say, returns of two
X Y X Y securities, X and Y in our case). It may be observed from
A 0.1 –8 14 – 13 6 – 7.8 Equation (3) that covariance of returns of securities X and
B 0.2 10 –4 5 – 12 – 12.0 Y is a measure of both variability of returns of securities
C 0.4 8 6 3 –2 – 2.4 and their association. Thus, the formula for covariance
D 0.2 5 15 0 7 0.0 of returns on X and Y can also be expressed as follows:
E 0.1 –4 20 – 9 12 – 10.8
Covariance XY = Standard deviation X
E(RX) E(RY) Covar = – 33.0
× Standard deviation Y
= 5 = 8
× Correlation XY
The covariance of returns of securities X and Y is
Cov xy = σ xσ y Co rxy
–33.0. The formula for calculating covariance of returns (4)
of the two securities X and Y is as follows: Note that σx and σy are standard deviations of
n returns for securities X and Y and Corxy is the correlation
Cov xy = ∑ [Rx − E(Rx )][Ry − E(Ry )] × Pi (3)
between returns of X and Y. From Equation (4), we can
i =1
determine the correlation by dividing covariance by the
Note that Covxy is the covariance of returns on standard deviations of returns on securities X and Y:
securities X and Y, Rx and Ry returns on securities X
and Y respectively, E(Rx) and E(Ry) expected returns of Covariance XY
Correlation X ,Y =
X and Y respectively and Pi probability of occurrence Standard deviation X
of the state of economy i. Using Equation (3), the × Standard deviation Y
covariance between the returns of securities X and Y
Cov xy
can be calculated as shown below: Corxy = (5)
σ xσ y
Cov xy = 0.1( −8 − 5)( −14 − 8) + 0.2(10 − 5)( −4 − 8)
+0.4(8 − 5)(6 − 8) + 0.2( 5 − 5)(15 − 8) The value of correlation, called the correlation
coefficient, could be positive, negative or zero. It depends
+0.1( −4 − 5)( 20 − 8)
on the sign of covariance since standard deviations are
= −7.8 − 12 − 2.4 + 0 − 10.8 = −33.0 always positive numbers. The correlation coefficient
What is the relationship between the returns of always ranges between –1.0 and +1.0. A correlation
securities X and Y? There are following possibilities: coefficient of +1.0 implies a perfectly positive correlation
 Positive covariance X’s and Y’s returns could while a correlation coefficient of –1.0 indicates a perfectly
be above their average returns at the same time. negative correlation. The correlation between the two
Alternatively, X’s and Y’s returns could be below variables will be zero (or not different from zero) if
their average returns at the same time. In either they are not at all related to each other. In a number of
situation, this implies positive relation between situations, returns of any two securities may be weakly
two returns. The covariance would be positive. correlated (negatively or positively).
 Negative covariance X’s returns could be above Let us calculate correlation by using data given in
its average return while Y’s return could be Table 5.4. The covariance is –33.0. We need standard
below its average return and vice versa. This deviations of X and Y to compute the correlation. The
denotes a negative relationship between returns standard deviation of securities X and Y are as follows:
of X and Y. The covariance would be negative. σ x2 = 0.1( −8 − 5)2 + 0.2(10 − 5)2 + 0.4(8 − 5)2
 Zero covariance Returns on X and Y could show +0.2( 5 − 5)2 + 0.1( −4 − 5)2
no pattern; that is, there is no relationship. In this = 16.9 + 5.0 + 3.6 + 0 + 8.1 = 33.6
situation, covariance would be zero. In reality,
covariance may be non-zero due to randomness σ x = 33.6 = 5.80%
and the negative and positive terms may not σ y2 = 0.1(14 − 8)2 + 0.2( −4 − 8)2 + 0.4(6 − 8)2
cancel out each other.
+0.2(15 − 8)2 + 0.1( 20 − 8)2
In our example, covariance between returns on
X and Y is negative, that is, –33.0. This is akin to the = 3.6 + 28.8 + 1.6 + 9.8 + 14.4 = 58.2
second situation above; that is, two returns are negatively σ y = 58.2 = 7.63%
related. What does the number –33.0 imply? As in the
case of variance, covariance also uses squared deviations The correlation of the two securities X and Y is as
and therefore, the number cannot be explained. We follows:
92 Financial Management

−33.0 −33.0 Applying Equation (6), the variance of portfolio of

Corxy = = = −0.746 X and Y will be as follows:
5.80 × 7.63 44.25
Securities X and Y are negatively correlated. The σ p2 = 33.6 (0.5) 2 + 58.2 (0.5) 2
correlation coefficient of –0.746 indicates a high negative + 2 (0.5)(0.5)( 5.80)(7.63)( −0.746)
relationship. If an investor invests her wealth in both = 8.4 + 14.55 − 16.51 = 6.45
instead any one of them, she can reduce the risk. How?
The standard deviation of two-asset portfolio is the
square root of variance:
Variance and Standard Deviation of a
Two-Asset Portfolio σ p = σ x2 w x2 + σ y2 w y2 + 2w x w yσ xσ y Corxy
We know now that the variance of a two-asset portfolio σ p = 6.45 = 2.54% (7)
is not the weighted average of the variances of assets
since they co-vary as well. The variance of two-security What does the portfolio standard deviation of 2.54
portfolio is given by the following equation: per cent mean? The implication is the same as in the
case of the standard deviation of an individual asset
σ p2 = σ x2wx2 + σ y2wy2 + 2wx wy 222xy (security). The expected return on the portfolio is 6.5
per cent, and it could vary between 3.96 per cent [i.e.,
= σ x2wx2 + σ y2wy2 + 2wx wy 2
σ xσ y 222xy 2 (6) 6.5 – 2.54] and 9.04 per cent [i.e., 6.5 + 2.54] within one
It may be noticed from Equation (6) that the variance standard deviation from the mean. There is about 68 per
of a portfolio includes the proportionate variances of the cent probability that the portfolio return would range
individual securities and the covariance of the securities. between 3.96 per cent and 9.04 per cent if we assume
The covariance depends on the correlation between the that the portfolio return is normally distributed.
securities in the portfolio. The risk of the portfolio would
be less than the weighted average risk of the securities
Minimum Variance Portfolio
for low or negative correlation. It is a common practice
to use a tabular approach, as given Table 5.5, to calculate What is the best combination of two securities so that
the variance of a portfolio: the portfolio variance is minimum? The minimum
variance portfolio is also called the optimum portfolio.
Table 5.5: Covariance Calculation Matrix However, investors do not necessarily strive for the
minimum variance portfolio. A risk-averse investor will
I II III have a trade-off between risk and return. Her choice of
a particular portfolio will depend on her risk preference.
σ x2 Cov xy w x2 wx wy σ x2 w x2 wx wy Cov xy
We can use the following general formula for
estimating optimum weights of two securities X and Y
Cov xy σ y2 wx wy w y2 wx wy Cov xy σ y2 w y2 so that the portfolio variance is minimum:

The first two parts of Table 5.5 contain the variance, σ y2 − Cov xy
wx * = (8)
covariance and weights of two securities, X and Y, in the σ x2 + σ y2 − 2 Cov xy
portfolio. The third part gives the cell-by-cell product of
the values in the two part. We can obtain Equation (6) where w* is the optimum proportion of investment or
when we add all values in the third part. weight in security X. Investment in Y will be: 1 – w*.
Using the sequences of Table 5.6, the variance of the In the example above, we find that w* is:
portfolio of securities X and Y is given below: 58.2 − ( −33)
The total of values in the third table: 8.40 – 8.25 – wy * = = 0.578
58.2 + 33.6 − 2( −33)
8.25 + 14.55 = 6.45 is the variance of the portfolio of
securities X and Y.
Table 5.6: Covariance calculation Matrix: Example

I II III

σ x2 Covxy w x2 wxwy σ x2 w x2 wxwyCovxy

33.6 –33.0 (0.5)2 = 0.25 (0.5)(0.5) = 0.25 (33.6)(0.25) = 8.40 (0.25)(–33.0) = –8.25

σ y2 w y2 σ y2 w y2
Covxy wxwy wxwy Covxy
–33.0 58.20 (0.5)(0.5) = 0.25 (0.5)2 = 0.25 (0.25)(–33.0) = –8.25 (58.2)(0.25) = 14.55
 Portfolio Theory and Assets Pricing Models 93
Thus the weight of Y will be: 1 – 0.578 = 0.422. 1.0), then there is no advantage of diversification. The
The portfolio variance (with 57.8 per cent of weighted standard deviation of returns on individual
investment in X and 42.2 per cent in Y) is: securities is equal to the standard deviation of the
σ p2 = 33.6 (0.578) 2 − 58.2 (0.422) 2
portfolio. We may therefore conclude that diversification
always reduces risk provided the correlation coefficient
+0.746)
− 2 (0.578)(0.422)( 5.80)(7.63)(+
= 11.23 − 10.36 + 16.11 = 5.48 is less than 1.
ILLUSTRATION 5.1: Risk of Two-Asset Portfolio
The standard deviation is: σ = 5.48 = 2.34
Securities M and N are equally risky, but they have different
Any other combination of X and Y will yield a expected returns:
higher variance or standard deviation.
M N
Expected return (%) 16.00 24.00
Portfolio Risk Depends on Weight 0.50 0.50
Correlation between Assets Standard deviation (%) 20.00 20.00

We emphasize once again that the portfolio standard What is the portfolio risk (variance) if (a) Cormn = +1.0,
deviation is not the weighted average of the standard (b) Cormn= –1.0, (c) Cormn = 0.0, (d) Cormn = +0.10, and
deviations of the individual securities. In our example (e) Cormn = – 0.10 ?
above, the standard deviation of portfolio of X and Y (a) When correlation is +1.0, Equation (7) will reduce to
is 2.54 per cent. Let us see how much is the weighted
σ p = σ x2w x2 + σ y2w 2y + 2w x w yσ xσ y = σ x w x + σ y w y
standard deviation of the individual securities:
Weighted standard deviation of individual securities The standard deviation of portfolio of M and N is as
= 5.8 × 0.5 + 7.63 × 0.5 = 6.7% follows:
Thus, the standard deviation of portfolio of X and σ p = 20 × 0.5 + 20 × 0.5 = 20.0%
Y is considerably lower than the weighted standard
deviation of these individual securities. This example (b) The portfolio standard deviation is calculated as
shows that investing wealth in more than one security follows:
reduces portfolio risk. This is attributed to diversification σ p = 202 × 0.52 + 202 × 0.52 + 2 × 0.5 × 0.5 × 20 × 20 × −1.0
effect. However, the extent of the benefits of portfolio
diversification depends on the correlation between returns = 100 + 100 − 200 = 0.0%
on securities. In our example, returns on securities X and (c) When the correlation is zero, Equation (7) will
Y are negatively correlated and the correlation coefficient reduce to
is – 0.746. This has caused significant reduction in the
portfolio risk. Would there be diversification benefit (that σ p = σ x2w x2 + σ 22w 2y
is, risk reduction) if the correlation were positive? Let us
For the portfolio of M and N, the standard deviation is:
assume that correlation coefficient in our example is +0.25.
How much is the portfolio standard deviation? (Using σ p = 202 × 0.52 + 202 × 0.52 = 200 = 14.14%
Eq. 7) It is 5.34 as shown below:
(d) The portfolio variance under the weakly positive
σ p2 = 33.6 (0.5)2 + 58.2 (0.5)2 + 2 (0.5)(0.5)( 5.80)(7.63)( +0.25) correlation (+0.10) is given below:
= 8.4 + 14.55 + 5.53 = 28.48 σ p2 = 202 × 0.52 + 202 × 0.52 + 2 × 0.5 × 0.5 × 20 × 20 × 0.10
σ p = 28.48 = 5.34%
= 100 + 100 + 20 = 220 = 14.83%
The portfolio risk (σ = 5.34%) is still lower than (e) The portfolio variance under the weakly negative
the weighted average standard deviation of individual correlated (–0.10) returns of two securities M and N is:
securities (σ = 6.7%). If the returns of securities X and
Y are positively and perfectly correlated (with the σ p2 = 202 × 0.52 + 202 × 0.52 + 2 × 0.5 × 0.5 × 20 × 20 × −0.10
correlation coefficient of 1), then the portfolio standard
= 100 + 100 − 20 = 180 = 13.42%
deviation is as follows:
It may be observed in the above example that a
σ p2 = 33.6 (0.5)2 + 58.2 (0.5)2 + 2 (0.5)(0.5)( 5.80)(7.63)( +1) total reduction of risk is possible if the returns of the
= 8.4 + 14.55 + 22.13 = 45.08 two securities are perfectly negatively correlated, though,
σ p = 45.08 = 6.7% such a perfect negative correlation will not generally
be found in practice. Securities do have a tendency of
When correlation coefficient of the returns on moving together to some extent, and therefore, risk may
individual securities is perfectly positive (i.e., Cor = not be totally eliminated.
94 Financial Management
Suppose two securities, Logrow and Rapidex have
CHECK YOUR CONCEPTS the following characteristics:
1. Define standard deviation and variance. How are Logrow Rapidex
they calculated? Expected Return (%) 12.00 18.00
2. Define covariance. How is it calculated? Variance 256.00 576.00
3. What is meant by correlation? Standard deviation (%) 16.00 24.00
4. What is the relationship between correlation and Further, assume five possible correlations between the
covariance? returns of these securities: perfectly positive correlation
5. How is the standard deviation and variance of a (+1.0); perfectly negative correlation (–1.0); no correlation
two-asset portfolio calculated? (0.0), positive correlation (0.5) and negative correlation (–
6. How do we determine minimum variance portfolio? 0.25). The first three relationships are special situations.
They are not rare, but they may not be very common
7. How the portfolio risk depends on the correlation
in practice. In the real world, returns of securities have
between assets? a tendency to move together in the same direction.
Sometimes they move in opposite direction too. Thus, a
positive or negative correlation is more likely between two
PORTFOLIO RISK-RETURN ANALYSIS:
risky securities. Given the characteristics of Logrow and
TWO-ASSET CASE Rapidex and their correlation, what are the interactions
between risk and return of portfolios that could be formed
Let us recapitulate that the portfolio return depends on
by combining them?
the proportion of wealth invested in two assets, and is in
no way affected by correlation between asset returns. In Logrow is a low return and low risk security as
contrast, the portfolio risk depends on both correlation and compared to Rapidex, which has high return and high
proportions (weights) of the assets forming the portfolio. risk. If you hold 100 per cent investment in Logrow, your
Let us emphasize again that the correlation coefficient expected return is 12 per cent and standard deviation
will always lie between +1.0 and –1.0. Returns on assets 16 per cent. On the contrary, if you invest your entire
or securities vary perfectly together in the same direction wealth in Rapidex you may expect to earn a higher return
when the correlation coefficient is +1.0 and in perfectly of 18 per cent, but the standard deviation, 24 per cent,
opposite directions when it is –1.0. A zero correlation is also higher. How would the expected return and risk
coefficient implies that there is no relationship between change if you form portfolios of Logrow and Rapidex by
the returns of securities. In practice, the correlation combining them in different proportions? In Table 5.7 we
coefficients of returns of securities may vary between show the calculations of the portfolio return and risk for
+1.0 and –1.0. Let us consider an example to understand different combinations (weights) of Logrow and Rapidex
the implications of asset correlation and weights for the under different assumptions regarding the correlation
portfolio risk-return relationship. between them. We use Equation (2) for calculating the
Table 5.7: Portfolio Return and Risk for Different Correlation Coefficients
Portfolio Risk, σp (%)
Portfolio Correlation
Weight Return (%) + 1.00 – 1.00 0.00 0.50 – 0.25
Logrow Rapidex Rp σp σp σp σp σp

1.00 0.00 12.00 16.00 16.00 16.00 16.00 16.00

0.90 0.10 12.60 16.80 12.00 14.60 15.74 13.99
0.80 0.20 13.20 17.60 8.00 13.67 15.76 12.50
0.70 0.30 13.80 18.40 4.00 13.31 16.06 11.70
0.60 0.40 14.40 19.20 0.00 13.58 16.63 11.76
0.50 0.50 15.00 20.00 4.00 14.42 17.44 12.65
0.40 0.60 15.60 20.80 8.00 15.76 18.45 14.22
0.30 0.70 16.20 21.60 12.00 17.47 19.64 16.28
0.20 0.80 16.80 22.40 16.00 19.46 20.98 18.66
0.10 0.90 17.40 23.20 20.00 21.66 22.44 21.26
0.00 1.00 18.00 24.00 24.00 24.00 24.00 24.00
Minimum Variance Portfolio
wL 1.00 0.60 0.692 0.857 0.656
wR 0.00 0.40 0.308 0.143 0.344
σ2 256.00 0.00 177.23 246.86 135.00
σ (%) 16.00 0.00 13.31 15.71 11.62