4. What is Risk?
• The chance that an investment's
actual return will be different than expected.
Risk includes the possibility of losing some or
all of the original investment.
5.
6. What is Return?
• The gain or loss of a security in a particular
period. The return consists of the income and
the capital gains relative on an investment. It
is usually quoted as a percentage.
19. Measures of Risk in Academics
1. Absolute Measures
• Variance
• Standard deviation
2. Relative Measures
• Covariance
• Correlation
20. Variance of Return
• Variance is used in statistics for probability
distribution. Since variance measures the variability
(volatility) from an average or mean, and volatility is a
measure of risk, the variance statistic can help determine
the risk an investor might take on when purchasing a
specific security.
• A variance value of zero indicates that all values within a set
of numbers are identical; all variances that are non-zero will
be positive numbers. A large variance indicates that
numbers in the set are far from the mean and each other,
while a small variance indicates the opposite.
21. Standard Deviation
• Standard deviation is a statistical
measurement that sheds light on historical
volatility. For example, a volatile stock will
have a high standard deviation while the
deviation of a stable blue chip stock will be
lower. A large dispersion tells us how much
the return on the fund is deviating from the
expected normal returns.
23. Covariance
• A measure of the degree to which returns on
two risky assets move in tandem. A positive
covariance means that asset returns move
together. A negative covariance means returns
move inversely.
24. Correlation
• Correlation is computed into what is known as the correlation
coefficient, which ranges between -1 and +1. Perfect positive
correlation (a correlation co-efficient of +1) implies that as one security
moves, either up or down, the other security will move in lockstep, in
the same direction. Alternatively, perfect negative correlation means
that if one security moves in either direction the security that is
perfectly negatively correlated will move in the opposite direction. If
the correlation is 0, the movements of the securities are said to have
no correlation; they are completely random.
• In real life, perfectly correlated securities are rare, rather you will find
securities with some degree of correlation.
31. Modern Portfolio Theory
– Modern portfolio theory (MPT)—or portfolio theory—
was introduced by Harry Markowitz with his paper
"Portfolio Selection," which appeared in the 1952
Journal of Finance. 38 years later, he shared a Nobel
Prize with Merton Miller and William Sharpe for what
has become a broad theory for portfolio selection.
– MPT has had most influence in the practice of portfolio
management
– MPT commonly referred to as Mean – Variance Analysis
– MPT is a normative theory
– In its simplest form, MPT provides a framework to
construct and select portfolios based on the expected
performance of the investments and the risk appetite of
the investors.
32. Applications of MPT
• Asset Allocation
• Asset-liability management
• Optimal Manager Selection
• Index funds/Mutual funds
• Fund of Funds
33. Modern Portfolio Theory - In Nut Shell
• Due to scarcity of resources, all economic decisions are made in the
face of trade-offs
• Markowitz identified the trade-off facing the investor : risk versus
return
• The investment decision is not merely which securities to own, but
how to divide investor’s wealth amongst securities – the problem of
portfolio selection
• Markowitz extends the techniques of linear programming to
develop the critical line algorithm
• The critical line algorithm identifies portfolios that minimize risk for
a given level of expected return
• MPT quantified the age old wisdom of diversification by
introducing the statistical notion of covariance or correlation.
MPT and CAPM
34. Assumptions of
Markowitz Portfolio Theory
• Investors consider each investment alternative as
being presented by a probability distribution of
expected returns over some holding period.
• For a given risk level, investors prefer higher
returns to lower returns. Similarly, for a given
level of expected returns, investors prefer less
risk to more risk.
• Your portfolio includes all of your assets and
liabilities
35. Markowitz Portfolio Theory
• Quantifies risk
• Derives the expected rate of return for a portfolio of
assets and an expected risk measure
• Shows that the variance of the rate of return is a
meaningful measure of portfolio risk
• Derives the formula for computing the variance of a
portfolio, showing how to effectively diversify a
portfolio
36. Covariance of Returns
• A measure of the degree to which two variables “move
together” relative to their individual mean values over time
• For two assets, i and j, the covariance of rates of return is
defined as:
• Covij = E{[Ri - E(Ri)][Rj - E(Rj)]}
37. Covariance and Correlation
• The correlation coefficient is obtained by
standardizing (dividing) the covariance by the
product of the individual standard deviations
• Correlation coefficient varies from -1 to +1
jt
iti
ij
Rofdeviationstandardthe
Rofdeviationstandardthe
returnsoftcoefficienncorrelatiother
:where
Cov
r
j
ji
ij
ij
38. Correlation Coefficient
• It can vary only in the range +1 to -1.
• A value of +1 would indicate perfect positive correlation. This
means that returns for the two assets move together in a
completely linear manner.
• A value of –1 would indicate perfect correlation. This means
that the returns for two assets have the same percentage
movement, but in opposite directions
39. Portfolio Standard Deviation
Calculation
• Any asset of a portfolio may be described by
two characteristics:
– The expected rate of return
– The expected standard deviations of returns
• The correlation, measured by covariance,
affects the portfolio standard deviation
• Low correlation reduces portfolio risk while
not affecting the expected return
40. Portfolio Standard Deviation Formula
ji
ijij
ij
2
i
i
port
n
1i
n
1i
ijj
n
1i
i
2
i
2
iport
rCovwhere
j,andiassetsforreturnofratesebetween thcovariancetheCov
iassetforreturnofratesofvariancethe
portfolioin thevalueofproportionby thedeterminedareweights
whereportfolio,in theassetsindividualtheofweightstheW
portfoliotheofdeviationstandardthe
:where
Covwww
41. Portfolio Risk-Return Plots for
Different Weights
-
0.05
0.10
0.15
0.20
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12
Standard Deviation of Return
E(R)
Rij = +1.00
1
2
With two perfectly
correlated assets, it is only
possible to create a two
asset portfolio with risk-
return along a line between
either single asset
42. Portfolio Risk-Return Plots for
Different Weights
-
0.05
0.10
0.15
0.20
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12
Standard Deviation of Return
E(R)
Rij = 0.00
Rij = +1.00
f
g
h
i
j
k
1
2
With uncorrelated assets it is
possible to create a two
asset portfolio with lower
risk than either single asset
43. Portfolio Risk-Return Plots for
Different Weights
-
0.05
0.10
0.15
0.20
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12
Standard Deviation of Return
E(R)
Rij = 0.00
Rij = +1.00
Rij = +0.50
f
g
h
i
j
k
1
2
With correlated assets it is
possible to create a two
asset portfolio between the
first two curves
44. Portfolio Risk-Return Plots for
Different Weights
-
0.05
0.10
0.15
0.20
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12
Standard Deviation of Return
E(R)
Rij = 0.00
Rij = +1.00
Rij = -0.50
Rij = +0.50
f
g
h
i
j
k
1
2
With
negatively
correlated assets it
is possible to
create a two asset
portfolio with
much lower risk
than either single
asset
45. Portfolio Risk-Return Plots for
Different Weights
-
0.05
0.10
0.15
0.20
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12
Standard Deviation of Return
E(R)
Rij = 0.00
Rij = +1.00
Rij = -1.00
Rij = +0.50
f
g
h
i
j
k
1
2
With perfectly negatively correlated assets it is possible to
create a two asset portfolio with almost no risk
Rij = -0.50
46. • Suppose two securities have perfect negative
Correlation. Would it make sense to invest in
these two securities only?
48. The Efficient Frontier
• The efficient frontier represents that set of
portfolios with the maximum rate of return for
every given level of risk, or the minimum risk
for every level of return
• Frontier will be portfolios of investments
rather than individual securities
– Exceptions being the asset with the highest return
49. The Efficient Frontier
and Investor Utility
• An individual investor’s utility curve specifies
the trade-offs he is willing to make between
expected return and risk
• These two interactions will determine the
particular portfolio selected by an individual
investor
50. Selecting an Optimal Risky Portfolio
)E( port
)E(Rport
X
Y
U3
U2
U1
U3’
U2’
U1’
• The optimal portfolio has the highest utility for a given investor
• It lies at the point of tangency between the efficient frontier and
the utility curve with the highest possible utility
51. The MPT Investment Process
Expected Return
Model
Volatility & Correlation
Estimates
Constraints on
Portfolio Choice
Optimal
Portfolio
Investor
Objectives
Risk-Return
Efficient Frontier
PORTFOLIO
OPTIMIZATION
52. Implementation of MPT
• Though the theory is straight forward ….
Implementation is a complication process.
53. Estimation Issues
• Results of portfolio allocation depend on
accurate statistical inputs
• Estimates of
– Expected returns
– Standard deviation
– Correlation coefficient
• Among entire set of assets
• With 100 assets, 4,950 correlation estimates
• Estimation risk refers to potential errors
54. The Proof Is In the Pudding
• The MPT has been around for 40 years and is
still going strong
– But, it has flaws
• There are alternatives, but none has yet to
prove itself better
54
55. An Introduction to Capital Asset
Pricing Model
• The capital asset pricing model is an idealized
portrayal of how financial markets price
securities and thereby determine expected
return on capital investments.
• The model provides a methodology for
quantifying risk and translating that risk into
estimates of expected return on equity.
57. Risk-Free Asset
• An asset with zero standard deviation
• Zero correlation with all other risky assets
• Provides the risk-free rate of return (RFR)
• Will lie on the vertical axis of a portfolio graph
58. Risk-Free Asset
• Covariance between two sets of returns is
n
1i
jjiiij )]/nE(R-)][RE(R-[RCov
Because the returns for the risk free asset are certain,
0RF Thus Ri = E(Ri), and Ri - E(Ri) = 0
Consequently, the covariance of the risk-free asset with any risky asset or portfolio will
always equal zero. Similarly the correlation between any risky asset and the risk-free asset
would be zero.
59. Combining a Risk-Free Asset
with a Risky Portfolio
• Expected return
• the weighted average of the two returns
))E(RW-(1(RFR)W)E(R iRFRFport
This is a linear relationship
60. Combining a Risk-Free Asset
with a Risky Portfolio
• The expected variance for a two-asset
portfolio is
211,221
2
2
2
2
2
1
2
1
2
port rww2ww)E(
Substituting the risk-free asset for Security 1, and the risky asset for Security 2, this
formula would become
iRFiRF iRF,RFRF
22
RF
22
RF
2
port )rw-(1w2)w1(w)E(
Since we know that the variance of the risk-free asset is zero and the correlation
between the risk-free asset and any risky asset i is zero we can adjust the formula
22
RF
2
port )w1()E( i
61. Combining a Risk-Free Asset
with a Risky Portfolio
• Given the variance formula 22
RF
2
port )w1()E( i
22
RFport )w1()E( i The standard deviation is
i)w1( RF
Therefore, the standard deviation of a portfolio that combines the risk-free asset
with risky assets is the linear proportion of the standard deviation of the risky
asset portfolio.
62. Combining a Risk-Free Asset
with a Risky Portfolio
Since both the expected return and the standard deviation of
return for such a portfolio are linear combinations, a graph of
possible portfolio returns and risks looks like a straight line
between the two assets
63. Combining a Risk-Free Asset
with a Risky Portfolio
)E( port
)E(Rport
RFR
M
C
A
B
D
Since both the expected return and the standard deviation of
return for such a portfolio are linear combinations, a graph of
possible portfolio returns and risks looks like a straight line
between the two assets
64. Combining a Risk-Free Asset
with a Risky Portfolio
)E( port
)E(R port
RFR
M
C
A
B
D
Since both the expected return and the standard deviation of
return for such a portfolio are linear combinations, a graph of
possible portfolio returns and risks looks like a straight line
between the two assets To attain a higher
expected return than is
available at point M (in
exchange for accepting
higher risk) either invest
along the efficient
frontier beyond point M,
such as point D or, add
leverage to the portfolio
by borrowing money at
the risk-free rate and
investing in the risky
portfolio at point M
66. • Suppose you pay a higher rate of Interest to
buy stock on margin than you receive when
you invest at the risk free rate. What does the
efficiency frontier look like?
67. The Market Portfolio
• Because portfolio M lies at the point of tangency, it has
the highest portfolio possibility line
• Everybody will want to invest in Portfolio M and
borrow or lend to be somewhere on the CML
• Therefore this portfolio must include ALL RISKY ASSETS
• Because the market is in equilibrium, all assets are
included in this portfolio in proportion to their market
value
• Because it contains all risky assets, it is a completely
diversified portfolio, which means that all the unique
risk of individual assets (unsystematic risk) is diversified
away
68. Statement of CAPM
A model that describes the relationship between
risk and expected return and that is used in the
pricing of risky securities.
69. Systematic Risk
• Only systematic risk remains in the market
portfolio
• Systematic risk is the variability in all risky
assets caused by macroeconomic variables
• Systematic risk can be measured by the
standard deviation of returns of the market
portfolio and can change over time
70. Number of Stocks in a Portfolio and the
Standard Deviation of Portfolio Return
Standard Deviation of Return
Number of Stocks in the Portfolio
Standard Deviation of the Market
Portfolio (systematic risk)
Systematic Risk
Total Risk
Unsystematic
(diversifiable) Risk
71. SML
Security market line (SML) is the representation of the capital asset pricing model. It displays the expected rate of return of an
individual security as a function of systematic, non-diversifiable risk
An investor with a low risk profile would choose an investment at the beginning of the security market line. An investor with a higher risk
profile would thus choose an investment higher along the security market line.
72. Slope of SML
• Given the SML reflects the return on a given
investment in relation to risk, a change in the slope of
the SML could be caused by the risk premium of the
investments. Recall that the risk premium of an
investment is the excess return required by an investor
to help ensure a required rate of return is met. If the
risk premium required by investors was to change, the
slope of the SML would change as well.
• Risk premium=Rm-Rf
73. Properties of any asset on , above or
below the SML
• On the line assets are fairly valued.
• Above the line they are undervalued.
• Below the line they are overvalued.
74. Assumptions of CAPM
• Aim to maximize economic utilities (Asset quantities are given and
fixed).
• Are rational and risk-averse.
• Are broadly diversified across a range of investments.
• Are price takers, i.e., they cannot influence prices.
• Can lend and borrow unlimited amounts under the risk free rate of
interest.
• Trade without transaction or taxation costs.
• Deal with securities that are all highly divisible into small parcels (All
assets are perfectly divisible and liquid).
• Have homogeneous expectations.
• Assume all information is available at the same time to all investors.
75. Arbitrage Pricing Theory
The arbitrage pricing theory (APT) describes the price where a
mispriced asset is expected to be. It is often viewed as an
alternative to the capital asset pricing model (CAPM), since the
APT has more flexible assumption requirements. Whereas the
CAPM formula requires the market's expected return, APT uses
the risky asset's expected return and the risk premium of a
number of macro-economic factors. Arbitrageurs use the APT
model to profit by taking advantage of mispriced securities. A
mispriced security will have a price that differs from the
theoretical price predicted by the model. By going short an over
priced security, while concurrently going long the portfolio the
APT calculations were based on, the arbitrageur is in a position
to make a theoretically risk-free profit.
76. Competition may be
stronger or weaker
than anticipated
Exchange rate
and Political risk
Projects may
do better or
worse than
expected
Entire Sector
may be affected
by action
Interest rate,
Inflation &
News about
economy
Firm-specific Market
A Break Down of Risk
Actions/Risk that
Affect only one
firm
Affects few
firms
Affects many
firms
Actions/Risk
that affect all
investments
77. Multi-factor Model
• Multi factor models explain risk/reward
relationships based on two or more factors:
78. Alternatives
– Single Index Market Model
• With assumption that stock returns can be
described by a single market model, the
number of correlations required reduces to
the number of assets
• Single index market model:
imiii RbaR
bi = the slope coefficient that relates the returns for security i to the returns for the
aggregate stock market
Rm = the returns for the aggregate stock market
80. Random Walk Theory
The idea that stocks take a random and unpredictable
path. A follower of the random walk theory believes it's
impossible to outperform the market without assuming
additional risk. Critics of the theory, however, contend
that stocks do maintain price trends over time - in other
words, that it is possible to outperform the market by
carefully selecting entry and exit points for equity
investments.
81. Implications of Randomness
• The fact that the deviations from true value are random implies, in a rough
sense, that there is an equal chance that stocks are under or over valued
at any point in time, and that these deviations are uncorrelated with any
observable variable.
• For instance, in an efficient market, stocks with lower PE ratios should be
no more or less likely to under valued than stocks with high PE ratios.
• If the deviations of market price from true value are random, it follows
that no group of investors should be able to consistently find under or
over valued stocks using any investment strategy
82. What is an efficient market?
• Efficient market is one where the market price is an unbiased
estimate of the true value of the investment.
• Implicit in this derivation are several key concepts –
– Market efficiency does not require that the market price be
equal to true value at every point in time. All it requires is
that errors in the market price be unbiased, i.e., that prices
can be greater than or less than true value, as long as these
deviations are random.
– Randomness implies that there is an equal chance that
stocks are under or over valued at any point in time.
83. Definitions of Market Efficiency
• Definitions of market efficiency have to be specific not only about the market
that is being considered but also the investor group that is covered.
• It is extremely unlikely that all markets are efficient to all investors, but it is
entirely possible that a particular market (for instance, the New York Stock
Exchange) is efficient with respect to the average investor.
• It is possible that some markets are efficient while others are not, and that
a market is efficient with respect to some investors and not to others.
• This is a direct consequence of differential transactions costs, which confer
advantages on some investors relative to others.
84. Information and Market Efficiency
• Under weak form efficiency, the current price reflects the information
contained in all past prices, suggesting that charts and technical analyses that
use past prices alone would not be useful in finding under valued stocks.
• Under semi-strong form efficiency, the current price reflects the information
contained not only in past prices but all public information (including
financial statements and news reports) and no approach that was predicated
on using and massaging this information would be useful in finding under
valued stocks.
• Under strong form efficiency, the current price reflects all information, public
as well as private, and no investors will be able to consistently find under
valued stocks.
85. Implications of Market Efficiency
• No group of investors should be able to consistently beat the market using a
common investment strategy.
• An efficient market would also carry very negative implications for many
investment strategies and actions that are taken for granted –
– In an efficient market, equity research and valuation would be a costly task that provided no benefits.
– The odds of finding an undervalued stock should be random (50/50). At best, the benefits from
information collection and equity research would cover the costs of doing the research.
– In an efficient market, a strategy of randomly diversifying across stocks or indexing to the market,
carrying little or no information cost and minimal execution costs, would be superior to any other
strategy, that created larger information and execution costs.
– There would be no value added by portfolio managers and investment strategists.
– In an efficient market, a strategy of minimizing trading, i.e., creating a portfolio and not trading unless
cash was needed, would be superior to a strategy that required frequent trading.
86. What market efficiency does not
imply..
• An efficient market does not imply that –
– (a) stock prices cannot deviate from true value; in fact, there can be large
deviations from true value. The deviations do have to be random.
– no investor will 'beat' the market in any time period. To the contrary,
approximately half of all investors, prior to transactions costs, should beat the
market in any period.
– no group of investors will beat the market in the long term. Given the number of
investors in financial markets, the laws of probability would suggest that a fairly
large number are going to beat the market consistently over long periods, not
because of their investment strategies but because they are lucky.
– In an efficient market, the expected returns from any investment will be
consistent with the risk of that investment over the long term, though there may
be deviations from these expected returns in the short term.
87. Necessary Conditions for Market
Efficiency
• Markets do not become efficient automatically. It is the actions of investors, sensing bargains
and putting into effect schemes to beat the market, that make markets efficient.
• The necessary conditions for a market inefficiency to be eliminated are as follows –
• The market inefficiency should provide the basis for a scheme to beat the market and earn
excess returns. For this to hold true –
– (a) The asset (or assets) which is the source of the inefficiency has to be traded.
– (b) The transactions costs of executing the scheme have to be smaller than the expected
profits from the scheme.
• There should be profit maximizing investors who
– (a) recognize the 'potential for excess return'
– (b) can replicate the beat the market scheme that earns the excess return
– (c) have the resources to trade on the stock until the inefficiency disappears
88. Efficient Markets and Profit-seeking Investors:
The Internal Contradiction
• There is an internal contradiction in claiming that there is no possibility of
beating the market in an efficient market and then requiring profit-
maximizing investors to constantly seek out ways of beating the market and
thus making it efficient.
• If markets were, in fact, efficient, investors would stop looking for
inefficiencies, which would lead to markets becoming inefficient again.
• It makes sense to think about an efficient market as a self-correcting
mechanism, where inefficiencies appear at regular intervals
90. Firm Size Effect
The theory holds that smaller companies have a greater
amount of growth opportunities than larger companies.
Small cap companies also tend to have a more volatile
business environment, and the correction of problems -
such as the correction of a funding deficiency - can lead
to a large price appreciation. Finally, small cap stocks tend
to have lower stock prices, and these lower prices mean
that price appreciations tend to be larger than those
found among large cap stocks.
91. Monday Effect
A theory that states that returns on the stock
market on Mondays will follow the prevailing
trend from the previous Friday. Therefore, if the
market was up on Friday, it should continue
through the weekend and, come Monday,
resume its rise.
92. January Effect
The January effect is said to affect small caps more
than mid or large caps. This historical trend,
however, has been less pronounced in recent years
because the markets have adjusted for it. Another
reason the January effect is now considered less
important is that more people are using tax-
sheltered retirement plans and therefore have no
reason to sell at the end of the year for a tax loss.
94. Sharpe Ratio
• Sharpe ratio= (Rp-Rf)/standard deviation of
the portfolio
• Rp is portfolio return
• Rf is risk free rate of return
95. Treynor Portfolio
Performance Measure
• Treynor recognized two components of risk
– Risk from general market fluctuations
– Risk from unique fluctuations in the securities in the
portfolio
• His measure of risk-adjusted performance
focuses on the portfolio’s undiversifiable risk:
market or systematic risk
96. Treynor Portfolio
Performance Measure
• The numerator is the risk premium
• The denominator is a measure of risk
• The expression is the risk premium return per unit of risk
• Risk averse investors prefer to maximize this value
• This assumes a completely diversified portfolio leaving
systematic risk as the relevant risk
i
i RFRR
T
97. Treynor Portfolio
Performance Measure
• Comparing a portfolio’s T value to a similar measure for the market
portfolio indicates whether the portfolio would plot above the SML
• Calculate the T value for the aggregate market as follows:
m
m
m
RFRR
T
98. Treynor Portfolio
Performance Measure
• Comparison to see whether actual return of
portfolio G was above or below expectations
can be made using:
RFRRRFRRE miG
100. Treynor versus Sharpe Measure
• Sharpe uses standard deviation of returns as the
measure of risk
• Treynor measure uses beta (systematic risk)
• Sharpe therefore evaluates the portfolio
manager on the basis of both rate of return
performance and diversification
• The methods agree on rankings of completely
diversified portfolios
• Produce relative not absolute rankings of
performance
102. Jensen Portfolio
Performance Measure
• Also based on CAPM
• Expected return on any security or portfolio is
Where: E(Rj) = the expected return on security
RFR = the one-period risk-free interest rate
j= the systematic risk for security or portfolio j
E(Rm) = the expected return on the market portfolio of risky assets
RFRRERFRRE mjj
