Portfolio Management and Risk-Return Analysis

Unit 5 – Portfolio
Management
Based on Fisher Jordan and Pradhan

Introduction
1. Most investors invest to earn a return on their money. However,
selecting stocks exclusively on the basis of maximization of return is
not enough.
2. The fact that most investors do not place available funds into 1,2 or
3 stocks promising the greatest returns suggest that other factors
must be considered besides return in the selection process.
3. Investors not only like returns, they dislike risk.
4. To facilitate our job of analysing securities and portfolios within a
return-risk context, we must begin with a clear understanding of
what risk and return are, what cerates them and how they should
be measured.

Introduction 2
5. The ultimate decisions to be made in investments are (1) what
securities should be held (2) how many dollars or rupees should be
allocated to each.
6. These decisions are made in 2 steps – (1) estimates are made of the
returns and risk associated with available securities over a forward
holding period. This step is known as security analysis. (2) return-risk
estimates must be compared in order to decide how to allocate
available funds among these securities on a continuing basis. This step
comprises portfolio analysis, selection and management.
7. The primary purpose of this section is to focus on return and risk and
how they are measured.

Security Returns
1. Investors want to maximize expected returns subject to their
tolerance for risk.
2. Return is the key method available to investors in comparing
alternative investments.
3. Measuring historical returns allows investors to assess how well
they have done, and it plays a part in the estimation of future
unknown returns.
4. We need to distinguish between realized return and expected
return.

Security Returns 2
5. Realized return is after the fact return – return that was earned (or
could have been earned). Realized return is history.
6. Expected return is the return from an asset that investors anticipate
they will earn over some future period. It is a predicted return. It may
or may not occur.

Elements in return
1. Return on a typical investment consists of two components.
2. The basic component is the periodic cash receipts (or income ) on
the investment, either in the form of interest or dividends.
3. The second component is the change in the price of the asset ,
commonly called the capital gain or loss. This element of return is
the difference between the purchase price and the price at which
the asset can be or is sold; therefore, it can be a gain or a loss.
4. Total return = Income + Price change (+/-)

Return Measurement
𝑇𝑜𝑡𝑎𝑙 𝑅𝑒𝑡𝑢𝑟𝑛 =
𝐶𝑎𝑠ℎ 𝑝𝑎𝑦𝑚𝑒𝑛𝑡𝑠 𝑟𝑒𝑐𝑒𝑖𝑣𝑒𝑑 + 𝑃𝑟𝑖𝑐𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑜𝑣𝑒𝑟 𝑡ℎ𝑒 𝑝𝑒𝑟𝑖𝑜𝑑
𝑃𝑢𝑟𝑐ℎ𝑎𝑠𝑒 𝑝𝑟𝑖𝑐𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑠𝑠𝑒𝑡
The total return is an acceptable measure of return for a specified
period of time. But we also need statistics to describe a series of
returns.

Arithmetic & Geometric Returns
1. The arithmetic average return is appropriate as a measure of the
central tendency of a number of returns calculated for a particular
time, such as a year. However, when percentage changes in value
over time are involved, the arithmetic means of these changes can
be misleading.
2. For eg. Price moves like this : 50 , 100 , 50
3. AM = (100% - 50%)/2 = 25%
4. Actual return = 0%
5. A different average is needed to describe accurately the true rate of
return over multiple periods.

Arithmetic & Geometric returns 2
6. The geometric average return measures compound, cumulative
returns over time. It is used in investments to reflect the realized
change in wealth over multiple periods.
7. For eg. Price : 50 , 100 , 50
𝐺𝑀 = [
100
50
∗
50
100
]
1
2 − 1
𝐺𝑀 = [2.00 ∗ 0.50]
1
2 – 1
𝐺𝑀 = 1 − 1 = 0%

Arithmetic & Geometric Returns
𝐺 = [ 1 + 𝑅1 1 + 𝑅2 … (1 + 𝑅𝑛)]
1
𝑛 − 1
Compute the geometric returns for the following series of returns, Y1 =
20%, Y2 = 25%, Y3 = -12.25% , Y4 = -14.63%, Y5 = 32%
𝐺 = [ 1.20 1.25 0.8775 0.8537 1.32 ]
1
𝑛 − 1
𝐺 = [1.5001]
1
5 − 1 = 1.0845 − 1 = 0.0845 = 8.45%

Stating Predictions ‘Scientifically’
1. Security analysts cannot be expected to predict with certainty
whether a stock’s price will increase or decrease or by how much.
2. This existence of uncertainty does not mean that security analysis is
valueless.
3. It does mean that analysts must strive to provide not only careful
and reasonable estimates of return but also some measure of the
degree of uncertainty associated with these estimates of return.
4. Most important, the analyst must be prepared to quantify the risk
that a given stock will fail to realize its expected return.

5. The quantification of risk is necessary to ensure uniform
interpretation and comparison. Verbal definitions simply do not lend
themselves to analysis.
6. A more precise measurement of uncertainty would be gauge the
extent to which the actual return is likely to differ from the predicted
return – that is the dispersion around the expected return.

7. Suppose that stock A, in the opinion of the analyst, could provide
returns as this:
Return (%) Likelihood
7 1 chance in 20
8 2 chances in 20
9 4 chances in 20
10 6 chances in 20
11 4 chances in 20
12 2 chances in 20
13 1 chance in 20

8. The likelihood of outcome could be expressed in fractional or
decimal terms. Such a figure is referred to as a probability. Let us recast
our ‘likelihoods’ into ‘probabilities’.
Return (%) Probability
7 0.05
8 0.10
9 0.20
10 0.30
11 0.20
12 0.10
13 0.05
Total 1.00

9. Based upon his analysis of economic, industry & company factors ,
the analyst assigns probabilities subjectively.
10. Security analysts use the probability distribution of return to specify
expected return as well as risk.
11. The expected return is the weighted average of the returns.
12. The expected return lies at the center of the distribution.
13. The spread of possible returns about the expected return can be
used to give a proxy of risk.
14. Two stocks can have identical expected returns but quite different
spreads and thus different risks.

(1)
Return (%)
(2)
Probability (1) * (2)
7 0.05 0.35
8 0.1 0.8
9 0.2 1.8
10 0.3 3
11 0.2 2.2
12 0.1 1.2
13 0.05 0.65
1 10

(1)
Return (%)
(2)
9 0.3 2.7
10 0.4 4
11 0.3 3.3
1 10
15. Consider stock B:
16. Stocks A & B have identical expected average returns of 10%,
but the spreads for stocks A & B are not the same.

17. The spread or the dispersion of the probability distribution can also
be measured by the degree of variation around the expected return.
𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 ∗ (𝑂𝑢𝑡𝑐𝑜𝑚𝑒 −𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑅𝑒𝑡𝑢𝑟𝑛 )2

(1)
Return (%)
(2)
Probability
(1) * (2)
(3)
Return -
Expected
Return
(4)
Difference
Squared
(4) * (2)
7 0.05 0.35 -3 9 0.45
8 0.10 0.80 -2 4 0.40
9 0.20 1.80 -1 1 0.20
10 0.30 3.00 0 0 0.00
11 0.20 2.20 1 1 0.20
12 0.10 1.20 2 4 0.40
13 0.05 0.65 3 9 0.45
1 10 Total 2.10
Variance 2.10
SD 1.45
Calculation of Variance for Stock A
(1)
Return (%)
(2)
(3)
Return -
Expected
Return
(4)
Difference
Squared
(4) * (2)
9 0.3 2.7 -1 1 0.3
10 0.4 4 0 0 0
11 0.3 3.3 1 1 0.3
1 10 0.6
Variance 0.6
SD 0.77
Variance for Stock B

𝑅 = 𝑃𝑖
𝑛
𝑖=1
𝑂𝑖
𝜎2
= 𝑃𝑖 (𝑂𝑖 − 𝑅)2
𝑛
𝑖=1
𝜎 = 𝜎2

Normal Distribution
1. How do we assume that the geometric mean of returns and the
standard deviation of returns give us a measure of expected returns
and risk?
2. Here we are using the properties of what is statistically called the
normal distribution.
3. Normal distribution appears naturally in many situations where
individual events are independent of each other.
4. We assume that the returns from stocks are very nearly normally
distributed over a short time period. This assumption is of great
help in our security analysis.

Normal Distribution
5. Another property of normal distribution is that it is characterized as
stable, i.e. when two or more normally distributed returns are
combined to form a portfolio, the returns of the combined portfolios
are also normally distributed.
6. How far do the real returns conform to a normal distribution?
7. In normal distribution, the outcomes shall spread to infinity on both
sides, which obviously is not true of stock returns.
8. However, normal distribution can still be useful in predicting future
returns from past data and for comparing stocks with different
expected returns and risks.

Year NIFTY-JUNE Y-O-Y RETURN Geometric Return
1995 961.23
1996 1122 0.1673 1.1673
1997 1192.4 0.0627 1.0627
1998 941.65 -0.2103 0.7897
1999 1187.7 0.2613 1.2613
2000 1471.45 0.2389 1.2389
2001 1107.9 -0.2471 0.7529
2002 1057.8 -0.0452 0.9548
2003 1134.15 0.0722 1.0722
2004 1488.5 0.3124 1.3124
2005 2194.35 0.4742 1.4742
2006 2997.7 0.3661 1.3661
2007 4318.3 0.4405 1.4405
2008 4136.65 -0.0421 0.9579
2009 4375.5 0.0577 1.0577
2010 5269.05 0.2042 1.2042
2011 5471.75 0.0385 1.0385
Geomteric Mean 11.48%
Arithmetic Mean 13.45%
Variance 452.910
SD 21.28
Skewness -0.1601
Nifty values return geometric mean variance SD & skew
Skewness is a pure number. No
units
Annual returns are
negatively skewed
slightly.

Capital Allocation
1. We have now defined risk & return. In this scenario, how shall an
investor decide on an investment in risky or risk free assets?
2. Investors will invest in risky stocks only in the expectation of
commensurate returns. But commensurate has to specified
mathematically.
3. We have defined risk as standard deviation which is a surrogate for
risk.
4. The history of rates of returns on government securities, corporate
bonds and common stocks has shown that stocks have on the
whole outperformed the other two categories of financial assets.

Capital Allocation
5. At the same time the variance of their returns has been higher than
either.
6. Thus riskier assets have to command a premium over risk free assets
due to their increased risks measured by their higher SD, otherwise
investors may not find them attractive as investments.
7. A risk averse investor will judge the returns from a riskier asset by
putting a negative value for the risk - a risk premium which will
compensate him for the risk. The greater the risk the greater shall be
the risk premium. But here a problem arises in choosing from among
risky assets.

Capital Allocation
8. Suppose the investor can get a return of 6% on a risk free asset and
he is considering 3 risky portfolios with the following risk return
characteristics:
Asset Portfolio Return % Risk Premium % Risk SD % Ratio
X 8 2 5 0.4
Y 10 4 10 0.4
Z 14 8 20 0.4
Now the risk premium to the standard deviation ratio is the same for
all three portfolios. This ratio is called the Sharpe ratio.

Capital Allocation
9. The choice among portfolios is clearer if S is not the same; the
investor would like to invest in the feasible portfolio with the highest
value of S. However, we have to derive some other criterion to
distinguish between portfolios with the same values of S. For furthering
our analysis, we have quantify risk aversion by assigning a utility score
to each portfolio.
𝑈 = 𝐸(𝑟) −
1
2
𝐴𝜎2
U = utility ; E(r) = expected return ; A = Coefficient of Risk Aversion [ higher values of A indicate higher risk
aversion]

Capital Allocation
10. With these criteria for utility, we can calculate the utility values of
all the 3 portfolios for different values of A. Let us take 3 different
investors with risk aversion of low (A =2 ), medium (A = 3) and high (A =
5 ).
A Values Port X Port Y Port Z
2 0.0775 0.0900 0.1000
3 0.0763 0.0850 0.0800
5 0.0675 0.0750 0.0400

Capital Allocation
11. In the table below we have selected an investor with A =3. Now let
us see how we can plot indifference curves in the risk return plane.
𝑈 = 𝐸(𝑟) − 0.5 ∗ 𝐴𝜎2
𝐸(𝑟) = 𝑈 + 0.5 ∗ 𝐴𝜎2

Capital Allocation
A SD Var U E(r)
3 0.0000 0.0000 0.07 0.0700
3 0.0500 0.0025 0.07 0.0738
3 0.1000 0.0100 0.07 0.0850
3 0.1500 0.0225 0.07 0.1038
3 0.2000 0.0400 0.07 0.1300
3 0.2500 0.0625 0.07 0.1638
3 0.3000 0.0900 0.07 0.2050
3 0.3500 0.1225 0.07 0.2538
3 0.4000 0.1600 0.07 0.3100
3 0.4500 0.2025 0.07 0.3738
3 0.5000 0.2500 0.07 0.4450
3 0.5500 0.3025 0.07 0.5238
0.0000
0.1000
0.2000
0.3000
0.4000
0.5000
0.6000
0.0000 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000
E(r)
Similar indifference curves can be plotted for different utility levels to get an indifference map
for a given level of A.

Capital Allocation
A SD Var U E(r) A SD Var U E(r)
3 0.0000 0.0000 0.07 0.0700 3 0.0000 0.0000 0.1 0.1000
3 0.0500 0.0025 0.07 0.0738 3 0.0500 0.0025 0.1 0.1038
3 0.1000 0.0100 0.07 0.0850 3 0.1000 0.0100 0.1 0.1150
3 0.1500 0.0225 0.07 0.1038 3 0.1500 0.0225 0.1 0.1338
3 0.2000 0.0400 0.07 0.1300 3 0.2000 0.0400 0.1 0.1600
3 0.2500 0.0625 0.07 0.1638 3 0.2500 0.0625 0.1 0.1938
3 0.3000 0.0900 0.07 0.2050 3 0.3000 0.0900 0.1 0.2350
3 0.3500 0.1225 0.07 0.2538 3 0.3500 0.1225 0.1 0.2838
3 0.4000 0.1600 0.07 0.3100 3 0.4000 0.1600 0.1 0.3400
3 0.4500 0.2025 0.07 0.3738 3 0.4500 0.2025 0.1 0.4038
3 0.5000 0.2500 0.07 0.4450 3 0.5000 0.2500 0.1 0.4750
3 0.5500 0.3025 0.07 0.5238 3 0.5500 0.3025 0.1 0.5538
A SD Var U E(r) A SD Var U E(r)
3 0.0000 0.0000 0.13 0.1300 3 0.0000 0.0000 0.16 0.1600
3 0.0500 0.0025 0.13 0.1338 3 0.0500 0.0025 0.16 0.1638
3 0.1000 0.0100 0.13 0.1450 3 0.1000 0.0100 0.16 0.1750
3 0.1500 0.0225 0.13 0.1638 3 0.1500 0.0225 0.16 0.1938
3 0.2000 0.0400 0.13 0.1900 3 0.2000 0.0400 0.16 0.2200
3 0.2500 0.0625 0.13 0.2238 3 0.2500 0.0625 0.16 0.2538
3 0.3000 0.0900 0.13 0.2650 3 0.3000 0.0900 0.16 0.2950
3 0.3500 0.1225 0.13 0.3138 3 0.3500 0.1225 0.16 0.3438
3 0.4000 0.1600 0.13 0.3700 3 0.4000 0.1600 0.16 0.4000
3 0.4500 0.2025 0.13 0.4338 3 0.4500 0.2025 0.16 0.4638
3 0.5000 0.2500 0.13 0.5050 3 0.5000 0.2500 0.16 0.5350
3 0.5500 0.3025 0.13 0.5838 3 0.5500 0.3025 0.16 0.6138

Capital Allocation
SD E(r)U=0.07 E(r)U = 0.10 E(r)U = 0.13 E(r)U = 0.16
0.0000 0.0700 0.1000 0.1300 0.1600
0.0500 0.0738 0.1038 0.1338 0.1638
0.1000 0.0850 0.1150 0.1450 0.1750
0.1500 0.1038 0.1338 0.1638 0.1938
0.2000 0.1300 0.1600 0.1900 0.2200
0.2500 0.1638 0.1938 0.2238 0.2538
0.3000 0.2050 0.2350 0.2650 0.2950
0.3500 0.2538 0.2838 0.3138 0.3438
0.4000 0.3100 0.3400 0.3700 0.4000
0.4500 0.3738 0.4038 0.4338 0.4638
0.5000 0.4450 0.4750 0.5050 0.5350
0.5500 0.5238 0.5538 0.5838 0.6138
0.0000
0.1000
0.2000
0.3000
0.4000
0.5000
0.6000
0.7000
0.00000.05000.10000.15000.20000.25000.30000.35000.40000.45000.50000.5500
Indifference Map
E(r) U=0.07 E(r) U = 0.10 E(r) U = 0.13 E(r) U = 0.16
*More on this later

Why Portfolios?
1. The fact that securities carry differing degrees of expected risk leads
most investors to the notion of holding more than one security at a
time, in an attempt to spread risks by not putting all their eggs into
one basket.
2. Most investors hope that if they hold several assets, even if one
goes bad, the others will provide some protection from an extreme
loss.

Diversification
1. Efforts to spread and minimize risk take the form of diversification.
2. Harry M Markowitz is said to be the father of modern portfolio
theory.
3. The key assumption in Markowitz’s theory is that investors’
attitudes towards portfolio selection depend exclusively upon (1)
quantification of risk (2) risk return optimization
4. In this situation risk refers to the statistical notion of variance or SD
5. These notions of risk return optimization are the foundation of
modern portfolio theory and are widely used in practice, but they
have recently been challenged by fields such as behavioral finance.

Portfolio Risk
1. The risk involved in individual securities can be measured by
standard deviation or variance.
2. When two securities are combined we need to consider their
interactive risk or covariance.
3. If the rates of returns of two securities move together, we say their
interactive risk or covariance is positive.
4. If rates of return are independent, covariance is 0.
5. Inverse movement results in covariance that is negative.
𝐶𝑜𝑣𝑥𝑦 =
1
𝑁
[𝑅𝑥 − 𝑅𝑏𝑎𝑟
𝑥 ] 𝑅𝑦 − 𝑅𝑏𝑎𝑟
𝑦
𝑁
1

Portfolio Risk
6. The coefficient of correlation is another measure designed to
indicate the similarity or dissimilarity in the behavior of two variables.
𝑟𝑥𝑦 =
𝐶𝑜𝑣𝑥𝑦
𝜎𝑥𝜎𝑦

Portfolio Effect in two security case
1. Markowitz’s efficient diversification involves combining securities
with less than positive correlation in order to reduce risk in the
portfolio without sacrificing any of the portfolio’s returns.
𝜎𝑝 = (𝑊
𝑥
2
𝜎𝑥
2
+ 𝑊
𝑦
2
𝜎𝑦
2
+ 2𝑊
𝑥 𝑊
𝑦 𝑟𝑥𝑦 𝜎𝑥𝜎𝑦)
1
2

X Y
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
SD 10 25
W 0 1 25.00 25.00 25.00 25.00 25.00 25.00 25.00 25.00 25.00 25.00 25.00
0.05 0.95 23.76 23.81 23.86 23.90 23.95 24.00 24.05 24.10 24.15 24.20 24.25
0.1 0.9 22.52 22.62 22.72 22.82 22.92 23.02 23.11 23.21 23.31 23.40 23.50
0.15 0.85 21.30 21.45 21.60 21.75 21.89 22.04 22.18 22.33 22.47 22.61 22.75
0.2 0.8 20.10 20.30 20.49 20.69 20.88 21.07 21.26 21.45 21.63 21.82 22.00
0.25 0.75 18.92 19.16 19.41 19.65 19.88 20.12 20.35 20.58 20.80 21.03 21.25
0.3 0.7 17.76 18.05 18.34 18.62 18.90 19.18 19.45 19.72 19.98 20.24 20.50
0.35 0.65 16.62 16.96 17.29 17.62 17.94 18.25 18.56 18.87 19.17 19.46 19.75
0.4 0.6 15.52 15.91 16.28 16.64 17.00 17.35 17.69 18.03 18.36 18.68 19.00
0.45 0.55 14.47 14.89 15.30 15.70 16.09 16.47 16.84 17.20 17.56 17.91 18.25
0.5 0.5 13.46 13.92 14.36 14.79 15.21 15.61 16.01 16.39 16.77 17.14 17.50
0.55 0.45 12.52 13.01 13.47 13.93 14.36 14.79 15.20 15.60 15.99 16.38 16.75
0.6 0.4 11.66 12.17 12.65 13.11 13.56 14.00 14.42 14.83 15.23 15.62 16.00
0.65 0.35 10.90 11.41 11.90 12.37 12.82 13.25 13.68 14.09 14.48 14.87 15.25
0.7 0.3 10.26 10.76 11.24 11.69 12.13 12.56 12.97 13.37 13.76 14.13 14.50
0.75 0.25 9.76 10.23 10.68 11.11 11.52 11.92 12.31 12.69 13.05 13.40 13.75
0.8 0.2 9.43 9.85 10.25 10.63 11.00 11.36 11.70 12.04 12.37 12.69 13.00
0.85 0.15 9.29 9.63 9.95 10.27 10.57 10.87 11.16 11.44 11.72 11.99 12.25
0.9 0.1 9.34 9.58 9.81 10.04 10.26 10.48 10.69 10.90 11.10 11.30 11.50
0.95 0.05 9.58 9.71 9.83 9.95 10.07 10.18 10.30 10.41 10.53 10.64 10.75
1 0 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00
rxy

0.00
5.00
10.00
15.00
20.00
25.00
30.00
0 0.2 0.4 0.6 0.8 1 1.2
rxy = 0
Varying the proportions of securities when rxy = 0

0.00
5.00
10.00
15.00
20.00
25.00
30.00
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
Risk For all rxy
Series1 Series2 Series3 Series4 Series5 Series6 Series7 Series8 Series9 Series10 Series11

X Y
0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1
SD 10 25
W 0 1 25.00 25.00 25.00 25.00 25.00 25.00 25.00 25.00 25.00 25.00 25.00
0.05 0.95 23.76 23.71 23.66 23.60 23.55 23.50 23.45 23.40 23.35 23.30 23.25
0.1 0.9 22.52 22.42 22.32 22.22 22.12 22.02 21.91 21.81 21.71 21.60 21.50
0.15 0.85 21.30 21.15 21.00 20.85 20.70 20.54 20.39 20.23 20.07 19.91 19.75
0.2 0.8 20.10 19.90 19.70 19.49 19.29 19.08 18.87 18.65 18.44 18.22 18.00
0.25 0.75 18.92 18.67 18.41 18.16 17.90 17.63 17.37 17.09 16.82 16.54 16.25
0.3 0.7 17.76 17.46 17.15 16.84 16.53 16.21 15.88 15.55 15.21 14.86 14.50
0.35 0.65 16.62 16.28 15.92 15.56 15.19 14.81 14.42 14.02 13.61 13.19 12.75
0.4 0.6 15.52 15.13 14.73 14.32 13.89 13.45 13.00 12.53 12.04 11.53 11.00
0.45 0.55 14.47 14.03 13.59 13.12 12.64 12.14 11.62 11.08 10.50 9.90 9.25
0.5 0.5 13.46 12.99 12.50 11.99 11.46 10.90 10.31 9.68 9.01 8.29 7.50
0.55 0.45 12.52 12.02 11.49 10.94 10.36 9.74 9.09 8.38 7.60 6.74 5.75
0.6 0.4 11.66 11.14 10.58 10.00 9.38 8.72 8.00 7.21 6.32 5.29 4.00
0.65 0.35 10.90 10.37 9.80 9.20 8.56 7.87 7.11 6.26 5.27 4.05 2.25
0.7 0.3 10.26 9.73 9.18 8.59 7.95 7.26 6.50 5.63 4.61 3.28 0.50
0.75 0.25 9.76 9.27 8.75 8.20 7.60 6.96 6.25 5.45 4.51 3.31 1.25
0.8 0.2 9.43 9.00 8.54 8.06 7.55 7.00 6.40 5.74 5.00 4.12 3.00
0.85 0.15 9.29 8.94 8.58 8.20 7.80 7.38 6.93 6.46 5.94 5.38 4.75
0.9 0.1 9.34 9.10 8.85 8.59 8.32 8.05 7.76 7.47 7.16 6.84 6.50
0.95 0.05 9.58 9.46 9.33 9.20 9.07 8.94 8.81 8.67 8.53 8.39 8.25
1 0 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00
rxy

0.00
5.00
10.00
15.00
20.00
25.00
30.00
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
Risk for all -rxy
Series1 Series2 Series3 Series4 Series5 Series6 Series7 Series8 Series9 Series10 Series11

Scatter of risk & return with different
portfolio weights & correlations
Wa Wb SDp0 Rp SDp1 Rp SDp-1 Rp
0 1 25 18 25 18 25 18
0.05 0.95 23.75526 18.35 24.25 18.35 23.25 18.35
0.1 0.9 22.52221 18.7 23.5 18.7 21.5 18.7
0.15 0.85 21.30288 19.05 22.75 19.05 19.75 19.05
0.2 0.8 20.09975 19.4 22 19.4 18 19.4
0.25 0.75 18.91593 19.75 21.25 19.75 16.25 19.75
0.3 0.7 17.75528 20.1 20.5 20.1 14.5 20.1
0.35 0.65 16.62265 20.45 19.75 20.45 12.75 20.45
0.4 0.6 15.52417 20.8 19 20.8 11 20.8
0.45 0.55 14.46764 21.15 18.25 21.15 9.25 21.15
0.5 0.5 13.46291 21.5 17.5 21.5 7.5 21.5
0.55 0.45 12.52248 21.85 16.75 21.85 5.75 21.85
0.6 0.4 11.6619 22.2 16 22.2 4 22.2
0.65 0.35 10.90011 22.55 15.25 22.55 2.25 22.55
0.7 0.3 10.25914 22.9 14.5 22.9 0.5 22.9
0.75 0.25 9.762812 23.25 13.75 23.25 1.25 23.25
0.8 0.2 9.433981 23.6 13 23.6 3 23.6
0.85 0.15 9.290452 23.95 12.25 23.95 4.75 23.95
0.9 0.1 9.340771 24.3 11.5 24.3 6.5 24.3
0.95 0.05 9.581884 24.65 10.75 24.65 8.25 24.65
1 0 10 25 10 25 10 25
rxy = 0 rxy = 1 rxy = -1

Portfolios of two securities with differing
correlations of returns

Portfolio Risk Formula
𝐺𝑒𝑛𝑒𝑟𝑎𝑙 𝐹𝑜𝑟𝑚𝑢𝑙𝑎 𝑜𝑓 𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑅𝑖𝑠𝑘 𝜎𝑝
2
= 𝑋𝑖𝑋𝑗 𝑟𝑖𝑗 𝜎𝑖𝜎𝑗
𝑛
𝑗=1
𝑛
𝑖=1
𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑅𝑖𝑠𝑘 2 𝑠𝑒𝑐𝑢𝑟𝑖𝑡𝑦 𝑐𝑎𝑠𝑒 𝜎𝑝
2
= 𝑋1
2
𝜎1
2
+ 𝑋2
2
𝜎2
2
+ 2𝑋1𝑋2𝑟12𝜎1𝜎2
𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑅𝑖𝑠𝑘 3 𝑠𝑒𝑐𝑢𝑟𝑖𝑡𝑦 𝑐𝑎𝑠𝑒 𝜎𝑝
2
= 𝑋1
2
𝜎1
2
+ 𝑋2
2
𝜎2
2
+ 𝑋3
2
𝜎3
2
+
2𝑋1𝑋2𝑟12𝜎1𝜎2 + 2 𝑋1𝑋3𝑟13𝜎1𝜎3 + 2 𝑋2𝑋3𝑟23𝜎2𝜎3

3 security case example
1 2 3
X 0.25 0.35 0.40
SD 10 14 18
r12 0.65
r13 0.55
r23 0.45
Xi sqr * SD sqr 6.25 24.01 51.84
2*X1*X2*r12*SD1*SD2 15.9
2*X1*X3*r13*SD1*SD3 19.8
2*X2*X3*r23*SD2*SD3 31.8
Variance 150
SDp 12.2

Tracing out the efficiency locus

The N security case
The inputs to the portfolio analysis of a set of N securities are:
(1) N expected returns
(2) N variances of returns
(3) (N sqr. – N)/2 covariances
Thus Markowitz calculation requires a total of [N(N+3)/2] separate
pieces of information before efficient portfolios can be calculated and
identified.

The N Security case
Number of
securities
Bits of
information
10 65
50 1325
100 5150
1000 501500
This gave rise to the single index model! Will be discussed later, Meanwhile lets
look at Minimum Variance Portfolio & optimal risky portfolio in a 2 security case!

Minimum Variance Portfolio (2 sec case)
𝑋𝐴 =
[(𝜎𝐵)2
− 𝑟𝐴𝐵 𝜎𝐴 𝜎𝐵]
[𝜎𝐴
2
+ 𝜎𝐵
2
−2𝑟𝐴𝐵𝜎𝐴𝜎𝐵]
; 𝑋𝐵 = 1− 𝑋𝐴

Index Models
1. The Markowitz portfolio selection model has two major
shortcomings:
a. When dealing with a large number of securities, the list of
inputs becomes too large.
b. It provides no guidance on predicting the risk premium of a
security, i.e. the return above the risk free rate of interest, which
would justify investment in the particular security.

Sharpe’s Paper (Abstract)
1. Describes the advantages of using a particular model of the
relationships among securities for practical applications of the
Markowitz Portfolio Analysis technique.
2. Preliminary evidence suggests that the relatively few parameters
used by the model can lead to very nearly the same results
obtained with much larger sets of relationships among securities.

Sharpe’s paper (Introduction)
1. The preliminary sections state the problems in its general form and
describe Markowitz’ solution technique.
2. The remainder of the paper presents a simplified model of the
relationship among the securities, indicates the manner in which it
allows the portfolio analysis problem to be simplified and provide
evidence on the costs as well as the desirability of using the model
for practical applications of the Markowitz technique.

Sharpe’s Paper (PA Problem)
1. A security analyst has provided the following predictions concerning
the future returns from each of N securities:
𝐸𝑖 = 𝑇ℎ𝑒 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑅𝑖 𝑡ℎ𝑒 𝑟𝑒𝑡𝑢𝑟𝑛 𝑓𝑟𝑜𝑚 𝑠𝑒𝑐𝑢𝑟𝑖𝑡𝑦 𝑖
𝐶𝑖1 𝑡ℎ𝑟𝑜𝑢𝑔ℎ 𝐶𝑖𝑛 ; 𝐶𝑖𝑗 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑠 𝑡ℎ𝑒 𝑐𝑜𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑅𝑖 𝑎𝑛𝑑 𝑅𝑗
𝑎𝑠 𝑢𝑠𝑢𝑎𝑙, 𝑤ℎ𝑒𝑛 𝑖 = 𝑗 𝑡ℎ𝑒 𝑓𝑖𝑔𝑢𝑟𝑒 𝑖𝑠 𝑡ℎ𝑒 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑜𝑓 𝑅𝑖

Sharpe’s Paper (PA Problem)
2. The portfolio analysis problem is as follows. Given such a set of
predictions determine the set of efficient portfolios.
3. 𝐿𝑒𝑡 𝑋𝑖 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡 𝑡ℎ𝑒 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛 𝑜𝑓 𝑎 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑖𝑛𝑣𝑒𝑠𝑡𝑒𝑑 in
security i

Sharpe’s Paper (Diagonal Model)
1. The diagonal model is a set of assumptions which reduces the
computational task involved in making security-wise comparisons.
2. This model has 2 virtues:
1. It is one of the simplest which can be constructed without assuming away
the existence of interrelationships among securities and
2. there is considerable evidence that it can capture a large part of such
interrelationships.

3. The major characteristic of the diagonal model is the assumption that the
returns of various securities are related only through common relationships
with some basic underlying factor.
4. The return from any security is determined solely by random factors and
this single outside element; more explicitly:
𝑅𝑖 = 𝐴𝑖 + 𝐵𝑖 𝐼 + 𝐶𝑖
Where
𝐴𝑖 𝑎𝑛𝑑 𝐵𝑖 𝑎𝑟𝑒 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟𝑠 𝑎𝑛𝑑
𝐶𝑖 is a random variable with an expected value of 0 and variance 𝑄𝑖, and
I is the level of some index. Qi may be treated as unsystematic risk.

5. The future level of I is determined in part by random factors:
𝐼 = 𝐴𝑛+1 + 𝐶𝑛+1
Where
𝐴𝑛+1 𝑖𝑠 𝑎 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑎𝑛𝑑
𝐶𝑛+1 𝑖𝑠 𝑎 𝑟𝑎𝑛𝑑𝑜𝑚 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝑤𝑖𝑡ℎ 𝑎𝑛 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒 of 0 and a variance of
𝑄𝑛+1
𝐴𝑛+1 𝑖𝑛𝑑𝑖𝑐𝑎𝑡𝑒𝑠 𝑡ℎ𝑒 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝐼 𝑎𝑛𝑑
𝑄𝑛+1 𝑡ℎ𝑒 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑎𝑟𝑜𝑢𝑛𝑑 𝑡ℎ𝑎𝑡 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒
It is assumed that the covariance between
𝐶𝑖 𝑎𝑛𝑑 𝐶𝑗 𝑖𝑠 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑖 𝑎𝑛𝑑 𝑗 𝑖 ≠ 𝑗

6. The diagonal model requires the following predictions from a
security analyst:
a. Values of 𝐴𝑖 , 𝐵𝑖 𝑎𝑛𝑑 𝑄𝑖 𝑓𝑜𝑟 𝑒𝑎𝑐ℎ 𝑜𝑓 𝑁 𝑠𝑒𝑐𝑢𝑟𝑖𝑡𝑖𝑒𝑠
b. 𝑉𝑎𝑙𝑢𝑒𝑠 𝑜𝑓 𝐴𝑛+1 𝑎𝑛𝑑 𝑄𝑛+1 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑖𝑛𝑑𝑒𝑥 𝐼
7. Once the parameters of the diagonal model have been specified all
the inputs required for the standard portfolio analysis problem can be
derived. The relationships are:

Sharpe’s paper (Diagonal Model)
𝐸𝑖 = 𝐴𝑖 + 𝐵𝑖 𝐴𝑛+1
𝑉𝑖 = 𝐵𝑖
2
𝑄𝑛+1 + 𝑄𝑖
𝐶 = (𝐵𝑖)(𝐵𝑗)(𝑄𝑛+1)

Sharpe’s Paper (PA restated)
𝑅𝑝 =
𝑖=1
𝑛
𝑋𝑖𝑅𝑖
𝑅𝑝 =
𝑖=1
𝑛
𝑋𝑖 𝐴𝑖 + 𝐵𝑖𝐼 + 𝐶𝑖
𝑅𝑝 =
𝑖=1
𝑛
𝑋𝑖 𝐴𝑖 + 𝐶𝑖 +
𝑖=1
𝑛
𝑋𝑖𝐵𝑖 𝐼

𝑋𝑛+1 ≡
𝑖=1
𝑛
𝑋𝑖𝐵𝑖
𝑅𝑝 =
𝑖=1
𝑛
𝑋𝑖(𝐴𝑖 + 𝐶𝑖) + 𝑋𝑛+1 𝐴𝑛+1 + 𝐶𝑛+1
𝑅𝑝 =
𝑖=1
𝑛+1
𝑋𝑖(𝐴𝑖 + 𝐶𝑖)

The expected return of a portfolio is thus:
𝐸 =
𝑖=1
𝑛+1
𝑋𝑖𝐴𝑖
Variance of portfolio:
𝑉 =
𝑖=1
𝑛+1
𝑋𝑖
2
𝑄𝑖

Sharpe’s Paper (PA restated final)
Maximize: λ𝐸 − 𝑉
Where:
𝑉 =
𝑖=1
𝑛+1
𝑋𝑖
2
𝑄𝑖

Sharpe’s Paper (PA restated final)
Subject to:
𝑋𝑖 ≥ 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖 𝑓𝑟𝑜𝑚 1 𝑡𝑜 𝑛
𝑖=1
𝑛
𝑋𝑖 = 1
𝑖=1
𝑛
𝑋𝑖𝐵𝑖 = 𝑋𝑛+1

Introduction
1. Capital Market Theory is concerned with how asset pricing should occur
if investors behaved as Markowitz Suggested.
2. The capital asset pricing model uses the results of capital market theory
to derive the relationship between the expected returns and systematic
risk of individual securities and portfolios.
3. Capital Market Theory is a major extension of the Portfolio Theory of
Markowitz.
4. Portfolio Theory is really a description of how rational investors should
build efficient portfolios.
5. Capital Market Theory tells us how assets should be priced in the capital
markets if, indeed, everyone behaved in the way portfolio theory
suggests. The Capital Asset Pricing Model (CAPM) is a relationship
explaining how assets should be priced in the capital markets.

Assumptions underlying Capital Market
Theory
1. Investors can short sell any amount of shares without limit.
2. Purchases and sales by a single investor cannot affect prices
3. There are no transaction costs. Where there are transaction costs,
returns would be sensitive to whether the investor owned a security
before the decision period.
4. The purchase or sale of securities is done in the absence of personal
income taxes.
5. The investor can borrow or lend any amount of funds desired at an
identical riskless rate.
6. Investors share identical expectations with regard to the relevant
decision period, the necessary decision inputs, their form and size.

The CAPM
1. Fig 6.1 shows the standard efficient frontier ABCD.
2. Borrowing and Lending options transform the efficient frontier into a
straight line.
3. Lending is best thought of as an investment in a riskless security.
4. Borrowing can be thought of as the use of margin.
5. Assume that an investor can lend at the rate of Rf = 0.05. hence, the
point Rf represents a risk free investment (Rf = 0.05, SD = 0).
6. The investor can place all or part of his funds in this riskless asset. If he
placed part of his funds in the risk free asset and part in one of the
portfolios of risky securities along the efficient frontier, he could
generate portfolios along the straight line segment RfB.

The CAPM
7. The shape of efficient frontier in fig 6.1 will change to the right of point B
with the introduction of possibility of borrowing funds.
8. The introduction of borrowing and lending gives us an efficient frontier
that is a straight line throughout. Fig. 6.2 shows the new efficient frontier.
9. Point M now represents the optimal combination of risky securities.
10. The existence of this combination simplifies our problem of portfolio
selection.
11. The investor need only decide how much to borrow or lend as no other
investment or combination of investments available is as efficient as point M

The CAPM
12. The decision to purchase M is the investment decision.
13. The decision to buy some riskless asset (lend) or to borrow (leverage the
portfolio) is the financing decision.
14. These conditions give rise to what has been referred to as the separation
theorem.
15. The theorem implies that all investors, conservative or aggressive, should
hold the same mix of stocks from the efficient set. They should use
borrowing or lending to attain their preferred risk class.
16. This conclusion flies in face of more traditional notions of selection of
portfolios.
17. This analysis suggests that both types of investors should hold identically
risky portfolios. Desired risk levels are then achieved through combining
portfolio M with lending and borrowing.

The CAPM
18. If all investors face similar expectations and the same lending and
borrowing rate , they will face a diagram such as that in Fig 6.3 and further
more all of the diagrams will be identical.
19. The portfolio of assets held by any investor will be identical to the
portfolio of risky assets held by any other investor.
20. If all investors hold the same risky portfolio, then, in equilibrium, it must
be the market portfolio M. [Very important].
21. The market portfolio is a portfolio comprised of all risky assets. Each
asset will be held in the proportion that the market value of the asset
represents in the total market value of all risky assets.
22. This is the key: all investors will hold combinations of only two portfolios,
the market portfolio and a riskless security.

The CAPM
23. The efficient frontier has a corresponding part below the apex A
which can be called the inefficient frontier.
24. On this part of the curve, the returns are lower for the same risk
compared to the returns on the efficient frontier. There are however
some points on the inefficient part of the frontier which may be
relevant in another context. These will be talked about when we
discuss Black’s zero Beta model.
25. The straight line depicted to in fig 6.3 is referred to as the capital
market line (CML).
26. All investors will end up with portfolios somewhere along the CML
and all efficient portfolios would lie along the CML.

The CAPM
27. However, not al securities or portfolios lie along the CML. From the
derivation of the efficient frontier we know that all portfolios, except
those that are efficient, lie below the CML.
28. Observing the CML tells us something about the market price of
risk.
29. The equation of CML (connecting the risk less asset with a risky
portfolio) is as follows:
𝑅𝑒 = 𝑅𝐹 + (𝑅𝑀 − 𝑅𝐹)
𝜎𝑒
𝜎𝑀
Where e denotes an efficient portfolio.

The CAPM
30. The term
𝑅𝑀−𝑅𝐹
𝜎𝑀
can be thought of as the extra return that can be gained
by increasing the level of risk (standard deviation) on an efficient portfolio by
one unit.
31. The entire second term on the right side of the equation is thus the
market price of risk times the amount of risk in the portfolio.
32. The expression 𝑅𝐹 is the price of time.
33. The expected return on an efficient portfolio is:
𝑃𝑟𝑖𝑐𝑒 𝑜𝑓 𝑡𝑖𝑚𝑒 + 𝑃𝑟𝑖𝑐𝑒 𝑜𝑓 𝑅𝑖𝑠𝑘 ∗ (𝐴𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑅𝑖𝑠𝑘)
34. Although this equation sets the return on an efficient portfolio, we need
to go beyond to deal with returns on non-efficient portfolios or on individual
securities.

Security Market Line
1. The CAPM postulates that investors can expect a return only for the
systematic risk of their investment, since the non-systematic risk
can be diversified away.
2. Therefore, the unique risk of securities in a diversified portfolio is
irrelevant in asset pricing.
3. The systematic risk is measured by the beta of the portfolio.
4. When we replace the total risk of the CML with Beta of the
portfolio, the systematic risk, we get the security market line.

5. We have seen that all investments and all portfolios of investments
lie along a straight line in the return to Beta space.
6. To determine this line we need to only connect the intercept (beta of
0 return of riskless security) and the market portfolio (beta of one and
return of RM). These two points identify the straight line shown in Fig
6.4
7. The equation of the straight line is:
𝑅𝑖 = 𝛼 + 𝑏𝛽𝑖

8. The first point on the line is the risk less asset with a beta of 0. So:
𝑅𝐹 = 𝛼 + 𝑏 0
𝑅𝐹 = 𝛼
9. The second point on the line is market portfolio with a beta of 1.
Thus:
𝑅𝑀 = 𝛼 + 𝑏 1
𝑅𝑀 − 𝛼 = 𝑏
𝑏 = (𝑅𝑀 − 𝑅𝐹)
10. Combining the two results gives us:
𝑅𝑖 = 𝑅𝐹 + 𝛽𝑖(𝑅𝑀 − 𝑅𝐹)

11. This is a key relationship. It is called the security market line or the
CAPM.
12. The above equation gives the expected return for a security for a
given level of risk.
13. In practice the actual return may be higher or lower.
14. If the return is higher the security is underpriced.
15. If the return is lower the security is overpriced.
16. Either of the situations would be temporary, and the security price
would correct very soon so that the return would correctly reflect its
beta.

Testing the CAPM
1. The CAPM was developed on the basis of a set of unrealistic
assumptions as a result the SML may not reflect an accurate
description of how investors behave and how rates of returns are
established in the market.
2. Therefore, the CAPM must be tested empirically and validated
before it can be used with any real confidence.
3. Next we look at some of the results of significant work on the
question of validity.

Stability of Beta Coefficients
1. Investigators have calculated betas for individual securities and
portfolios of securities for a range of timespans.
2. Their first conclusion is that the betas of individual stocks are
unstable, this suggests that past betas for individual securities are not
good estimators of their future risks.
3. Their second conclusion is that betas of portfolios are relatively
stable. In other words, past portfolio betas are good estimators of
future portfolio volatility. In effect, the errors in the estimates of beta
for individual securities tend to offset one another when combined in a
portfolio.

Testing the CAPM based on the slope of the
SML
1. CAPM tells us that a linear relationship exists between a security’s
required rate of return and its beta.
2. Further, when the SML is graphed, the vertical axis intercept should
be Rf and the required rate of return for a stock or a portfolio with
beta = 1.0 should be Rm, the required rate of return on the market.
3. Researchers have tried to test the validity of the model by
calculating betas and the realized rates of return, plotting these
values in graphs, and then observing whether the intercept is equal
to Rf, the regression line is linear, and the line passes through the
point B = 1.0. Also, most studies analyze portfolios rather than
individual securities because security betas are so unstable.

Caveat
1. It is critical to recognize that although the CAPM is an ex-ante or
forward-looking model, the data used to test it are entirely
historical.
2. The realized rates of return over past holding periods are not
necessarily equal to the expected rates of return with which the
model should deal. Also, historical betas may or may not reflect
either current or expected future risk. This lack of ex-ante data
makes testing the true CAPM extremely difficult. Some key results
of these studies are notable.

Results
1. The evidence generally shows a significant positive relationship between
realized returns and systematic risk. However, the slope of the
relationship is usually less than that predicted by the CAPM.
2. The relationship between risk and return appears to be linear. There is no
evidence of significant curvature in the risk-return relationship.
3. Tests that attempt to assess the relative importance of market and
company-specific risk do not yield definitive results. The CAPM theory
implies that company-specific risk is not relevant, yet both kinds of risks
appear to be positively related to security returns.
4. If CAPM were completely valid, it should apply to all financial assets.
However, when bonds, for example, are introduced into the analysis,
they do not plot on the SML.

Is CAPM salvageable?
1. The CAPM is appealing in its elegance and logic.
2. However, doubts begin to arise when one examines the assumptions
more closely and these doubts are as much reinforced by the empirical
tests.
3. The model’s focus on the market rather than total risk is clearly a useful
way of thinking about the riskiness of assets in general.
4. Thus, as a conceptual model the CAPM is of truly fundamental
importance.
5. Although CAPM is enticing with its precision of numbers we do not know
precisely how to measure any of the inputs required to implement the
CAPM.

Is CAPM salvageable?
6. These inputs should all be ex-ante, yet we only have ex-post data
available.
7. Historical data for Rm and betas vary greatly depending on the time
period studied and the methods used to estimate them.
8. The estimates used in the CAPM are subject to potentially large errors.
9. Because the CAPM represents the way people who want to maximize
returns while minimizing risk ought to behave, assuming they can get all the
necessary data, the model is definitely here to stay.
10. Attempts will continue to improve the model and to make it more useful.

Arbitrage Pricing Theory
1. The CAPM asserts that only a single number – a security’s beta against
the market – is required to measure risk. At the core of the arbitrage
pricing theory (APT) is the recognition that several systematic factors
affect security returns.
2. The actual return, R, on any security or portfolio may be broken down
into 3 constituent parts, as follows:
R = E + bf + e
E = expected return on the security
b = security’s sensitivity to change in the systematic factor
f = the actual return on the systematic factor
e = returns on the unsystematic, idiosyncratic factors

3. The above equation merely states that the actual return equals the
expected return, plus factor sensitivity times factor movement, plus residual
risk.
4. Empirical work suggests that a three or four-factor model adequately
captures the influence of systematic factors on stock market returns. The
above equation may thus be expanded to:
R = E + (b1)(f1) + (b2)(f2) + (b3)(f3) + (b4)(f4) + e
5. What are these factors? They are the underlying economic forces that are
the primary influences on the stock market.
6. Research suggests that the most important factors are unanticipated
inflation, changes in the expected level of industrial production,
unanticipated shifts in the risk premiums and unanticipated movements in
the shape of the term structure of interest rates.

7. The biggest problems in APT are factor identification and separating
unanticipated from anticipated factor movements in the measurement of
sensitivities.
8. Far more critical is the measurement of the b’s. The b’s measure the sensitivity of
returns to unanticipated movements in the factors. By just looking at how a given
stock relates to, say, movements in the money supply, we would be including the
influence of both anticipated and unanticipated changes, when only the latter are
relevant.
9. Empirical testing of the APT is still in its infancy, and concrete results proving the
APT and disproving the CAPM do not exist. For these reasons, it is useful to regard
CAPM and APT as different variants of the true equilibrium pricing model.
10. Both are therefore useful in supplying intuition into the way security prices and
equilibrium returns are established.

Portfolio Performance Evaluation

Portfolio monitoring & evaluation
1. Performance is evaluated on a risk return basis and popular
measures are-
1. Sharpe’s ratio
2. Treynor’s ratio
3. Jensen’s alpha
4. Fama’s net selectivity

2. Sharpe’s ratio = (RP – Rf) / σP
3. Treynor’s ratio = (RP – Rf) / βP
4. Jensen’s Alpha = RP – [(Rf + βP (Rm – Rf))]
5. Fama’s net selectivity : Fama broke down the return on the portfolio
into 4 components:
1. Risk free rate [Rf]
2. Return due to imperfect diversification[(σP / σM)- βP]*(RM – Rf)
3. Return due to systematic risk βP*(RM – Rf)
4. Return due to superior stock selection [residual]

6. RP = Rf + βP*(RM – Rf) + [(σP / σM)- βP]*(RM – Rf) + NSP
7. The return on a mutual fund portfolio during the last few years was
18%, when the return on the market portfolio was 15%. The standard
deviation of the portfolio return was 28% whereas the standard
deviation of the market portfolio return was 20%. The portfolio beta
was 1.2. The risk free rate was 9%. Decompose using Fama’s net
selectivity.

• Ans.
• Risk free rate [Rf] = 9%
• Return due to imperfect diversification[(σP / σM)- βP]*(RM – Rf) = [(28/20)-
1.20]*(15%-9%)=1.20%
• Return due to systematic risk βP*(RM – Rf)=1.20*(15%-9%) = 7.20%
• Return due to superior stock selection [residual] = 18 – (9% + 1.20% + 7.20%)
= 0.60%

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