Download Portfolio Management: Course Introduction

Portfolio Construction
01/26/09
Portfolio Construction
• Where does portfolio construction fit in the portfolio
management process?
• What are the foundations of Markowitz’s MeanVariance Approach (Modern Portfolio Theory)? Twoasset to multiple asset portfolios.
• How do we construct optimal portfolios using Mean
Variance Optimization? Microsoft Excel Solver.
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Portfolio Construction
• How do we incorporate IPS requirements to
determine asset class weights?
• What are the assumptions and limitations of the
mean-variance approach?
• How do we reconcile portfolio construction in
practice with Markowitz’s theory?
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Portfolio Construction within the larger
context of asset allocation
• IPS provides us with the risk tolerance
and return expected by the client
• Capital Market Expectations provide us
with an understanding of what the
returns for each asset class will be
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Portfolio Construction within the larger
context of asset allocation
C1: Capital
Market Conditions
I1: Investor’s Assets,
Risk Attitudes
C2: Prediction
Procedure
I2: Investor’s Risk
Tolerance Function
C3: Expected Ret,
Risks, Correlations
I3: Investor’s Risk
Tolerance
M1: Optimizer
M2: Investor’s
Asset Mix
M3: Returns
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Portfolio Construction within the larger
context of asset allocation
• Optimization, in general, is
constructing the best portfolio for the
client based on the client
characteristics and CMEs.
• When all the steps are performed with
careful analysis, the process may be
called integrated asset allocation.
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Mean Variance Optimization
• The Mean-Variance Approach, developed by
Markowitz in the 1950s, still serves as the
foundation for quantitative approaches to
strategic asset allocation.
• Mean Variance Optimization (MVO) identifies
the portfolios that provide the greatest
return for a given level of risk OR that
provide the least risk for a given return.
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Mean Variance Optimization
• TO develop an understanding of MVO,
we will derive the relationship between
risk and return of a portfolio by looking
at a series of three portfolios:
• One risky asset and one risk-free asset
• Two risky assets
• Two risky assets and one risk-free asset
• We will then generalize our findings to
portfolios of a larger number of assets.8
MVO: One risky and one risk-free asset
• For a portfolio of two assets, one risky (r) and
one risk-free (f), the expected portfolio return
is defined as:
E ( RP )  wr * E ( Rr )  (1  wr ) * R f
• Since, by definition, the risk-free asset has
zero volatility (standard deviation), the
portfolio standard deviation is:
 P  wr * r
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MVO: One risky and one risk-free asset
• With the portfolio return and standard
deviation equations, we can derive the Capital
Allocation Line (CAL):
E ( RP )  R f 
[ E ( Rr )  R f ]
r
* p
• Notice that the slope of this line represent the
Sharpe ratio for asset r. It represents the
reward-to-risk ratio for asset r.
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MVO: One risky and one risk-free asset
• With one risky and one risk-free asset, an
investor can select a portfolio along this CAL
based on his risk / return preference.
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MVO: Two risky assets
• With two risky assets (1 and 2), as long as
the correlation between the two assets is less
than 1, creating a portfolio with the two
assets will allow the investor to obtain a
greater reward-to-risk ratio than either of the
two assets provide.
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MVO: Two risky assets
• Portfolio expected return and standard
deviation can be calculated as follows:
E( RP )  w1 * E( R1 )  w2 * E( R2 )
w2  1  w1
 P  w12 12  w22 22  2w1w2 1 2 12
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MVO: Two risky assets
• Remember that the correlation coefficient can be
calculated as:
12 
Cov1, 2
 1 2
Where
Cov1, 2
1 n

( R1i  R1 )( R2i  R2 )

n  1 i 1
and n = number of historical returns used in the
calculations.
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MVO: Two risky assets
• These values (as well as asset returns and
standard deviations) can be easily calculated
on a financial calculator or Excel.
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MVO: Two risky assets
• By altering weights in the two assets, we can
construct a minimum-variance frontier (MVF).
• The turning point on this MVF represents the global
minimum variance (GMV) portfolio. This portfolio
has the smallest variance (risk) of all possible
combinations of the two assets.
• The upper half of the graph represents the efficient
frontier.
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MVO: Two risky assets
• The weights for the GMV portfolio is
determined by the following equations:
   1 2 12
w1  2
2
 1   2  2 1 2 12
2
2
w2  1  w1
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MVO: Two risky and one risk-free asset
• We know that with one risky asset and the
risk-free asset, the portfolio possibilities lie on
the CAL.
• With two risky assets, the portfolio
possibilities lie on the MVF.
• Since the slope of the CAL represents the
reward-to-risk ratio, an investor will always
want to choose the CAL with the greatest
slope.
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MVO: Two risky and one risk-free asset
• The optimal risky portfolio is where a CAL is
tangent to the efficient frontier.
• This portfolio provides the best reward-to-risk ratio
for the investor.
• The tangency portfolio risky asset weights can be
calculated as:

E ( R )  r *  E ( R )  r * Cov
w 
E(R )  r *  E(R )  r *  E(R )  r  E(R )  r * Cov
1
1
1
f
2
2
2
2
f
2
f
2
2
1
f
1
1, 2
f
2
f
1, 2
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MVO: All risky assets (market) and one
risk-free asset
• We can generalize our previous results by
considering all risky assets and one risk-free
asset. The tangency (optimal risky) portfolio
is the market portfolio. All investors will
hold a combination of the risk-free asset and
this market portfolio.
• In this context, the CAL is referred to as the
Capital Market Line (CML).
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Investor Risk Tolerance and CML
• To attain a higher expected return than
is available at the market portfolio (in
exchange for accepting higher risk), an
investor can borrow at the risk freerate.
• Other minimum variance portfolios (on
the efficient frontier) are not
considered.
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Portfolio Possibilities Combining the Risk-Free
Asset and Risky Portfolios on the Efficient Frontier
E(R p )
M
RFR
p
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Assumptions / Limitations of
Markowitz Portfolio Theory
• Investors take a single-period perspective in
determining their asset allocation.
• Drawback: Investors seldom have a single-period
perspective. In a multiple-period horizon, even
Treasury bills exhibit variability in returns
• Possible Solutions:
• Include the “risk-free asset” as a risky asset class.
• If investors have a liquidity need, construct an efficient
frontier and asset allocation on the funds remaining after
the liquidity need is satisfied.
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Assumptions / Limitations of
Markowitz Portfolio Theory
• Investors base decisions solely on expected
return and risk. These expectations are
derived from historical returns.
• Drawback: Optimal asset allocations are highly
sensitive to small changes in the inputs,
especially expected returns. Portfolios may not be
well diversified.
• Potential solutions:
• Conduct sensitivity tests to understand the effect on
asset allocation to changes in expected returns.
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Assumptions / Limitations of
Markowitz Portfolio Theory
• Investors can borrow and lend at the riskfree rate.
• Drawback: Borrowing rates are always higher
than lending rates. Certain investors are
restricted from purchasing securities on margin.
• Potential solutions:
• Differential borrowing and lending rates can be easily
incorporated into MVO analysis. However, leverage may
be practically irrelevant for many investors (liquidity,
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regulatory restrictions).
Practical Application of MVO
• MVO can be used to determine optimal
portfolio weights with a certain subset
of all investable assets.
• An efficient frontier can be constructed
with inputs (expected return, standard
deviation and correlations) for the
selected assets.
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Practical Application of MVO
• MVO can be either unconstrained, in
which case we do not place any
constraints on the asset weights, or it
can be constrained.
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Practical Application of MVO
• Unconstrained Optimization
• The simplest optimization places no
constraints on asset-class weights except
that they add up to 1.
• With unconstrained optimization, the asset
weights of any minimum variance portfolio
is a linear combination of any other two
minimum variance portfolios.
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Practical Application of MVO
• Constrained Optimization
• The more useful optimization for strategic
asset allocation is constrained
optimization.
• The main constraint is usually a restriction
on short sales.
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Practical Application of MVO
• Constrained Optimization
• We can determine asset weights using the
corner portfolio theorem. This theorem
states that the asset weights of any
minimum variance portfolio is a linear
combination of any two adjacent corner
portfolios.
• Corner portfolios define a segment of the
efficient frontier.
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Practical Application of MVO
• Excel Solver is a powerful tool that can
be used to determine optimal portfolio
weights for a set of assets.
• To use the tool, we need expected
returns and standard deviations for our
assets as well as a set of constraints
that are appropriate for the portfolio.
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Readings
• RB 7
• RB 8 (pgs. 229-239)
• RM 3 (5, 6.1.1 – 6.1.4)
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