Drivers in the Case of Two...
Transcript of Drivers in the Case of Two...
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INTRODUCTION TO PORTFOLIO ANALYSIS
Drivers in the Case of Two Assets
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Introduction to Portfolio Analysis
Future Returns Are Random In Nature
Optimizing Portfolio requires expectations: • about average portfolio return (mean) • about how far off it may be (variance)
Portfolio Return Is A Random Variable
Why?
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Introduction to Portfolio Analysis
Past Performance to Predictions
Portfolio Return Variance
Computed on a sample of T Historical Returns
When the return is a random variable
Mean Portfolio Return
Computed on a sample of T Historical Returns
When the return is a random variable
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Introduction to Portfolio Analysis
Drivers of Mean & Variance● Assume two assets:
● Portfolio Return P = w1 * R1 + w2* R2
● Thus: E[P] = w1* E[R1]+ w2* E[R2]
Asset 1 Asset 2
Weight: w1 Weight: w2
Return: R1 Return: R2
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Introduction to Portfolio Analysis
Portfolio Return VarianceAgain, for a portfolio with 2 assets
Variance of Portfolio Return
Covariance between return 1 and 2
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Introduction to Portfolio Analysis
Correlations
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Introduction to Portfolio Analysis
Take Away Formulas
● var(Portfolio Return) =
● E[Portfolio Return] =
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INTRODUCTION TO PORTFOLIO ANALYSIS
Let’s practice!
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INTRODUCTION TO PORTFOLIO ANALYSIS
Using Matrix Notation
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Introduction to Portfolio Analysis
Variables at Stake for N Assets
● R: the N x 1 column-matrix of asset returns
● μ: the N x 1 column-matrix of expected returns
● w: the N x 1 column-matrix of portfolio weights
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Introduction to Portfolio Analysis
Variables at Stake for N Assets● Σ: The N x N covariance matrix of the N asset returns:
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
=Σ
2N2N1N
N22221
N11221
σσσ
σσσ
σσσ
!"#""
!
Covariance: Outside Diagonal Variance: On Diagonal
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Introduction to Portfolio Analysis
Generalizing from 2 to N Assets
Portfolio Expected Return
Portfolio Return
Portfolio Variance
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Introduction to Portfolio Analysis
Matrices Simplify the Notation● Avoid large number of terms by using matrix notation
● We have 4 matrices:
● weights (w), returns (R), expected returns (μ), and covariance matrix (Σ)
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Introduction to Portfolio Analysis
Simplifying the Notation
Portfolio Variance
Portfolio Expected Return
Portfolio Return
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INTRODUCTION TO PORTFOLIO ANALYSIS
Let’s practice!
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INTRODUCTION TO PORTFOLIO ANALYSIS
Portfolio Risk Budget
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Introduction to Portfolio Analysis
Who Did It?
Asset 1 Asset 2 Asset 3 Asset 4
Capital Allocation Budget Portfolio Volatility Risk
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Introduction to Portfolio Analysis
Portfolio Volatility In Risk Contribution● Portfolio Volatility =
● risk contribution of asset i depends on
● Where:
1. the complete matrix of weights
2. the full covariance matrix
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Introduction to Portfolio Analysis
Percent Risk Contribution
where
Relatively more risky assets:
Relatively less risky assets:
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INTRODUCTION TO PORTFOLIO ANALYSIS
Let’s practice!