Topic 5
Modern Portfolio
Concepts I: Markowitz
Portfolio Theory
Source for this PPT:
• Smart’s Fundamentals of Investing,
14th edition Chapter 5
Learning Goals
1. Understand portfolio objectives and the procedures
used to calculate portfolio return and standard
deviation.
2. Discuss the key aspects of correlation and
diversification.
3. Discuss the concepts of Efficient Frontier, Capital
Market line and Optimal Risky Portfolio.
4. Understand the Theorem of Separation.
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What is a Portfolio?
Portfolio is a collection of investments assembled to meet one or
more investment goals.
Growth-Oriented Portfolio: primary objective is long-term price
appreciation
Income-Oriented Portfolio: primary objective is to produce
regular dividend and interest income
Setting portfolio objective involves tradeoffs such as tradeoffs
between risk and return or between potential price appreciation
and income. This depends on your tax bracket, current income
needs, and ability to bear risk.
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Principles of Portfolio Planning
Modern portfolio theory was first developed by Harry
Markowitz in 1952.
Portfolio Objectives
◦ Ultimate goal is an efficient portfolio
Efficient portfolio
◦ A portfolio that provides the highest return for a given level
of risk
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Portfolio Return and Risk Measures
The return on a portfolio is simply the weighted average
of the individual assets’ returns in the portfolio
The standard deviation of a portfolio’s returns is more
complicated (not the weighted average of individual
standard deviation), and is a function of the portfolio’s
individual assets’ weights, standard deviations, and
correlations with all other assets (weighted average of
individual’s standard deviation, take into account
correlation between the assets).
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Return on Portfolio
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Table 5.1 Individual and Portfolio Returns and Standard Deviation of
Returns for International Business Machines (IBM) and Celgene (1 of 2)
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Finding from the above calculations:
So, investor who owns nothing but IBM stocks (average 9% return
with 23.6% std dev) holds an inefficient portfolio – meaning
there exists an alternative portfolio with more return but less
risk! (average 13.4% return with 18.4% std dev.)
By selling a few IBM stocks and using the proceeds to buy a few
Celgene shares, an investor can have more return and less risk at
the same time.
This is known as the ‘power of diversification’.
In diversification, the key factor is the correlation between the
stocks in the portfolio.
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Principles of Portfolio Planning
Rather than using formula for standard deviation, you can
construct an Excel spreadsheet to do the calculations.
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Correlation: Why Diversification Works
Correlation is a statistical measure of the relationship between two
series of numbers representing data
Positively Correlated items tend to move in the same direction
Negatively Correlated items tend to move in opposite directions
Correlation Coefficient is a measure of the degree of correlation
between two series of numbers representing data: -0.43 for IBM and
Celgene during 2005 to 2014. Can you explain a possible reason for the
negative correlation? Refer Text-Smart page 205 → actually unusual
(not normal) because most stocks will have positive correlation as they
are affected in the same way by large macroeconomic forces and
market sentiment (bullish or bearish).
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Correlation Coefficients
Perfectly Positively Correlated describes two positively
correlated series having a correlation coefficient of +1
Perfectly Negatively Correlated describes two negatively
correlated series having a correlation coefficient of -1
Uncorrelated describes two series that lack any relationship
and have a correlation coefficient of nearly zero (both assets
don’t have any common factors affecting their price movement)
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Figure 5.1 The Correlation Between Series M, N, and P
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Correlation: Why Diversification Works!
Assets that are less than perfectly positively correlated tend to
partly offset each others movements, thus reducing the overall
risk in a portfolio.
(As long as correlation between 2 stocks is not +1, some of the fluctuations
in stock A will cancel out fluctuations in stock B)
The lower the correlation, the more the overall risk in a portfolio
is reduced.
◦ Assets with +1 correlation eliminate no risk
◦ Assets with less than +1 correlation eliminate some risk
◦ Assets with less than 0 correlation eliminate more risk
◦ Assets with -1 correlation eliminate all risk
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Figure 5.2 Combining Negatively Correlated Assets
to Diversify Risk
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How to compute covariance of 2 securities? – Example:
Question: If std dev of both securities are 5.56% and 1.22% respectively, what is
the correlation of both securities? Answer: -5.06/(5.56 x 1.22) = -0.746
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Table 5.2 Portfolio Returns and Standard Deviations for International
Business Machines (IBM) and Celgene (CELG)
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Table 5.3 Expected Returns, Average Returns, and Standard Deviations for Assets
X, Y, and Z and Portfolios XY and XZ
-Returns of X and Y are negatively correlated
-Returns of X and Z are positively correlated
-Both portfolios (XY and XZ) have same portfolio expected return (13.3%) but very
different portfolio risk, why? (risk of Portfolio XY is zero but risk of Portfolio XZ is the
weighted average risk of X and Z).
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Efficient Diversification with
Many Risky Assets
We extend the two-risky portfolio methodology to the case
of many risky assets and a risk-free asset in 3 steps:
1. Extend the two-risky assets opportunity set to many
assets.
2. Determine the optimal risky portfolio that supports the
steepest CAL (that maximizes Sharpe ratio).
3. Choose a complete portfolio on optimal CAL based on
the investor’s risk aversion by mixing the risk-free asset
with the optimal risky portfolio.
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Step 1: Finding the Efficient Frontier of Risky Assets
If we examine different two-asset or more combinations and derived
the curves assuming all the possible weights, we would have a graph
like below:
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The envelope curve that contains the best of all these possible combinations is
referred to as the efficient frontier (EF) or Markowitz efficient frontier.
Specifically, the efficient frontier represents that set of portfolios that has the
maximum rate of return for every given level of risk, or the minimum risk for
every level of return.
An example of such
Frontier is shown next:
B
A
C
Individual assets
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Every portfolio that lies on the efficient frontier has either a
higher rate of return for equal risk or lower risk for an equal rate
of return than some portfolio beneath the frontier.
Thus, we would say that Portfolio A in the above diagram
dominates Portfolio C because it has an equal rate of return but
substantially less risk. Similarly, Portfolio B dominates Portfolio C
because it has equal risk but a higher expected rate of return.
Because of the benefits of diversification among imperfectly
correlated assets (as long as correlation not +1), we would expect
the efficient frontier to be made up of portfolios of investments
rather than individual securities. (in other words, individual stock
cannot be on the efficient frontier)
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Step 2: Combining Risk Preference of Investor
with the Efficient Frontier to find “Optimal
Risky Portfolio”
Some investors are more risk-averse, some are less risk averse.
The risk preference of investors can be represented by their
Indifference Curve (or Utility Curve).
An individual investor’s utility curves specify the trade-offs he or
she is willing to make between expected return and risk.
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Exhibit 7.16 shows two sets of utility curves along with an efficient frontier
of investments. The curves labeled U1, U2, and U3 are for a strongly riskaverse investor. These utility curves are quite steep, indicating that the
investor will not tolerate much additional risk to obtain additional
returns. The investor is equally disposed toward any return and risk
combinations along the specific utility curve.
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Exhibit 7.16 continued:
The curves labeled (U3′, U2′, U1′) characterize a less riskaverse investor. Such an investor is willing to tolerate a bit more
risk to get a higher expected return.
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Exhibit 7.16 continued:
The optimal portfolio is the efficient
portfolio that 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.
A conservative investor’s highest utility
is at Point X, where the U2 curve just
touches the efficient frontier. A less
risk-averse investor’s highest utility
occurs at Point Y, which represents a
portfolio on the efficient frontier with
higher expected returns and higher risk
than the portfolio at X.
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Step 3: Capital Allocation Line (CAL)– Combining a
Risky Portfolio with Risk-free Return to form a
“Complete Portfolio”
Now, if we introduce a risk-free asset into our universe of
available assets, we will consider a risk and return
characteristics of a portfolio that combines a portfolio of
risky assets and the risk-free asset.
Standard deviation of such portfolio is →
σp = wxσx
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Capital Allocation Line (CAL)
The line representing all possible combinations of risk
free assets and risky assets is called capital allocation
line (CAL).
It represents the portfolios available to an investor.
Slope of CAL is Sharpe Ratio of Risky Portfolio.
Optimal Risky Portfolio (to be explained later)
◦ Best combination of risky and safe assets to form
portfolio
◦ σp = w x σx
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CAL3 = CML
Optimal risky
portfolio
CAL2
CAL1
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Because Portfolio M lies at the point of tangency, it has the
highest portfolio possibility line, and everybody will want to invest
along the CML line (because you get the highest reward-to-risk
(Sharpe) ratio).
M portfolio must, therefore, include all risky assets or investments
(stocks, bonds, real estate, cash equivalents, derivatives, etc)
when you have considered all the correlations among these assets
for max diversification benefits. This portfolio that includes all
risky assets is referred to as the Market Portfolio.
Market portfolio is a completely diversified portfolio, which
means that all the risk unique to individual assets (unsystematic
risk) in the portfolio is diversified away. This implies that only
systematic risk (market risk), which is defined as the variability in
all risky assets caused by macroeconomic variables, remains in the
market portfolio.
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Components of Risk
Diversifiable (Unsystematic) Risk
◦ Firm-specific or industry-specific risk
◦ Can be eliminated through diversification
◦ Examples: labor strikes, lawsuits, decrease in the world price of
crude palm oil (IOI, KL Kepong), a new competitor, a
management change, a product recall.
◦ Careful selection of 8 to 15 securities can eliminate most
diversifiable risk (but 20-30 stocks are more promising)
Non-diversifiable (Systematic) Risk/Market Risk
◦ Attributable to forces that affect all similar investments
◦ Cannot be eliminated through diversification
◦ Examples: war, economic slowdown/recession, political events,
inflation, interest rate risk.
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Portfolio Risk and Diversification
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The CML and The Separation
Theorem
The CML leads all investors to invest in the same risky asset portfolio,
the M portfolio (investment decision).
Individual investors should only differ regarding their position on the
CML, which depends on their risk preferences (financing decision).
How they get to a point on the CML is based on their financing decisions
(moving along CML line). If you are relatively risk averse, you will lend
some part of your portfolio at the RFR by buying some risk free
securities and investing the remainder in the market portfolio of risky
assets (Point A). In contrast, if you prefer more risk, you might borrow
funds at the RFR and invest everything (all of your capital plus what you
borrowed) in the market portfolio, building the portfolio at Point B. This
financing decision provides more risk but greater returns than the
market portfolio.
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Specifically, to be somewhere on the CML efficient frontier, you
initially decide to invest in the market Portfolio M, which means that
you will be on the CML. This is your investment decision.
Subsequently, based on your risk preferences, you make a separate
financing decision either to borrow or to lend to attain your
preferred risk position on the CML.
Tobin called this division of
the investment decision
from the financing decision
the separation theorem.
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