Portfolio Construction

01/26/09

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Portfolio Construction

• Where does portfolio construction fit in the portfolio management process?

• What are the foundations of Markowitz’s Mean-Variance Approach (Modern Portfolio Theory)? Two-asset 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: CapitalMarket Conditions

I1: Investor’s Assets,Risk Attitudes

C2: PredictionProcedure

C3: Expected Ret,Risks, Correlations

I2: Investor’s RiskTolerance Function

I3: Investor’s RiskTolerance

M1: Optimizer

M2: Investor’sAsset 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.

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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:

• Since, by definition, the risk-free asset has zero volatility (standard deviation), the portfolio standard deviation is:

frrrP RwREwRE *)1()(*)(

rrP w *

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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):

• Notice that the slope of this line represent the Sharpe ratio for asset r. It represents the reward-to-risk ratio for asset r.

pr

frfP

RRERRE

*])([

)(

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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:

)(*)(*)( 2211 REwREwRE P

12212122

22

21

21 2 wwwwP

12 1 ww

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MVO: Two risky assets

• Remember that the correlation coefficient can be calculated as:

Where

and n = number of historical returns used in the calculations.

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2,112

Cov

n

iii RRRR

nCov

122112,1 ))((

1

1

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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:

122122

21

122122

1 2

w

12 1 ww

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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:

2,121

212

221

2,12221

1 *)()(*)(*)(

*)(*)(

CovrRErRErRErRE

CovrRErREw

ffff

ff

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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 free-rate.

• 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

p

)E(R p

RFR

M

CML

Borrowing

Lending

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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 risk-free 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, regulatory restrictions).

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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)