INTRODUCTION TO PORTFOLIO ANALYSIS Drivers in the Case of Two Assets
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INTRODUCTION TO PORTFOLIO ANALYSIS
Drivers in the Case of Two Assets
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?
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
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
Introduction to Portfolio Analysis
Portfolio Return VarianceAgain, for a portfolio with 2 assets
Variance of Portfolio Return
Covariance between return 1 and 2
Introduction to Portfolio Analysis
Correlations
Introduction to Portfolio Analysis
Take Away Formulas
● var(Portfolio Return) =
● E[Portfolio Return] =
INTRODUCTION TO PORTFOLIO ANALYSIS
Let’s practice!
INTRODUCTION TO PORTFOLIO ANALYSIS
Using Matrix Notation
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
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
Introduction to Portfolio Analysis
Generalizing from 2 to N Assets
Portfolio Expected Return
Portfolio Return
Portfolio Variance
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)
Introduction to Portfolio Analysis
Simplifying the Notation
Portfolio Variance
Portfolio Expected Return
Portfolio Return
INTRODUCTION TO PORTFOLIO ANALYSIS
Let’s practice!
INTRODUCTION TO PORTFOLIO ANALYSIS
Portfolio Risk Budget
Introduction to Portfolio Analysis
Who Did It?
Asset 1 Asset 2 Asset 3 Asset 4
Capital Allocation Budget Portfolio Volatility Risk
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
Introduction to Portfolio Analysis
Percent Risk Contribution
where
Relatively more risky assets:
Relatively less risky assets:
INTRODUCTION TO PORTFOLIO ANALYSIS
Let’s practice!
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