Mean Portfolio Return

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)

Why?

Portfolio Return Is A Random Variable

Introduction to Portfolio Analysis

Past Performance to Predictions Mean Portfolio Return Computed on a sample of T Historical Returns When the return is a random variable

Portfolio Return Variance 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: Asset 1

Asset 2

Weight: w1

Weight: w2

Return: R1

Return: R2

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Portfolio Return P = w1 * R1 + w2* R2

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Thus: E[P] = w1* E[R1]+ w2* E[R2]

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Portfolio Return Variance Again, for a portfolio with 2 assets Variance of Portfolio Return

Covariance between return 1 and 2

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Correlations

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Take Away Formulas ●

E[Portfolio Return] =

●

var(Portfolio Return) =

INTRODUCTION TO PORTFOLIO ANALYSIS

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INTRODUCTION TO PORTFOLIO ANALYSIS

Using Matrix Notation

Introduction to Portfolio Analysis

Variables at Stake for N Assets ●

w: the N x 1 column-matrix of portfolio weights

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R: the N x 1 column-matrix of asset returns

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μ: the N x 1 column-matrix of expected returns

Introduction to Portfolio Analysis

Variables at Stake for N Assets ●

Σ: The N x N covariance matrix of the N asset returns: 2 1

⎡σ σ 12 ! σ 1N ⎤ ⎢ ⎥ 2 σ σ σ 21 2 2N ⎥ ⎢ Σ= ⎢ " ⎥ " # " ⎢ ⎥ 2 ⎢⎣σ N1 σ N 2 ! σ N ⎥⎦ Covariance: Outside Diagonal Variance: On Diagonal

Introduction to Portfolio Analysis

Generalizing from 2 to N Assets Portfolio Return

Portfolio Expected Return

Portfolio Variance

Introduction to Portfolio Analysis

Matrices Simplify the Notation ●

Avoid large number of terms by using matrix notation

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We have 4 matrices: ●

weights (w), returns (R), expected returns (μ), and covariance matrix (Σ)

Introduction to Portfolio Analysis

Simplifying the Notation Portfolio Return

Portfolio Expected Return

Portfolio Variance

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INTRODUCTION TO PORTFOLIO ANALYSIS

Portfolio Risk Budget

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Who Did It? Capital Allocation Budget

Asset 1

Portfolio Volatility Risk

Asset 2

Asset 3

Asset 4

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Portfolio Volatility In Risk Contribution ●

Portfolio Volatility = ●

●

Where:

risk contribution of asset i depends on 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:

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