AASSET SSET AALLOCATIONLLOCATION

Portfolio Management

Ali Nejadmalayeri

Asset Allocation

• Strategic Asset Allocation– Set weights for general asset classes to meet

return and risk objectives

• Tactical Asset Allocation– Short-term changes to SAA to take advantage

of expected relative performance of different asset classes

Strategic Asset Allocation

• Set target and permissible target ranges for asset class weights

• Meet return and risk objectives– Specifies the desired “systematic” risk

• Evidence suggest that large fraction of total return variation is due to asset allocation

SAA Approaches

• Asset Only – Black-Litterman model

• Start with global value-weighted index

• Deviate from those weights to reflect investor’s view of expected returns and variations

• Asset/Liability Approaches– Cash flow matching

• Inflows and outflows are matched

– Immunization• Weighted averages durations are matched

SAA & IPS

• SAA & Return Objectives– Find weights that achieves desired return

• SAA & Risk Objective– Since investors are risk-averse, or

UP = E(RP) – 0.005 λA σP

then, shortfall risk has to be managed• Sharpe ratio, SFRatio (Roy’s Ratio), etc.

What is an Asset Class?

Asset Class should be:1. Assets in the class should relatively homogenous2. Asset classes should be mutually exclusive3. Asset classes should be diversifying4. Asset classes, as a group, should account for the

preponderance of world wealth5. Asset classes should have the capacity to absorb

majority of investor’s portfolio without damaging liquidity

When to Add a New Asset Class?

• Beyond the obvious, the following should hold:

PnewP

FP

new

Fnew RRCorrRRERRE

,

Finding Optimal Portfolio

• Unconstrained MVF– Asset weights of any MVF is a linear

combination of asset weights in two other MVFs

• Sign-Constrained MVF– Find adjacent “Corner Portfolios”– Asset weights of any MVF are positive linear

combination of the corresponding weights in the adjacent corner portfolios

FrontiersReturn

Risk

= 1.0

0 < < 1.0

< 0

Efficient Frontiers

Minimum Variance Frontiers

Corner Portfolios

Global Minimum Variance Portfolio

How to Optimize?

An Algorithmic Approach to

Finding Corner Portfolios

Portfolio Construction

Given a set of selected Securities

Finding Appropriate Asset Weights

Optimizing the Portfolio:

Highest Return for a Given Level of Risk

Optimal Portfolio

• Define the Risk Level

• Given the set of assets, Find the Bundle that Maximizes the Portfolio Return (Markowitz Optimization)– Define Measures of Return and Risk– Account for the Covariation of Asset Returns– Maximize Portfolio Return, or Minimize

Portfolio Risk

Risk and Return

• In finance, we ALWAYS perceive everything in a forward looking way so:– Return and Risk are Expected Measures

• Q: How Does One Make Up Expectation about Future Return and Risk?

• A: Either History tells, or a Model Defines

How Construct EF?

• With Historical Information:– 1st, find asset returns from prices– 2nd, find return on an equally weighted portfolio– 3rd, find the average and std. dev. of returns for

the portfolio– 4th, use SOLVER to determine that given a

level of return, what are the variance minimizing weights

Historical Measures: Return

• Ordinary we know of transaction prices, so:– If Pbeg and Pend are price of an asset at the beginning and end of an

unit period of time, say one month, and CF is the additional cash flow payment to holders of the asset at the end of the period, then:

beg

begend

P

CFPPR

Expected Return by History

Let’s assume for T period we know that returns are given: R1, …, RT, then Expected Return, E(R), is:

T

iiR

TRE

1

1)(

Risk by History

Ordinary we measure risk with variance, Var(R). Let’s assume for T period we know that returns are given: R1, …, RT, then Risk (variance), Var(R), is:

)()(

)(1

1)(

1

22

RVarRStd

RERT

RVar

R

T

iiR

How Construct EF?

• With Non-Historical Expectations:– 1st, use the correlation (variance-covariance)

structure, find average and std. dev. of returns for the portfolio

– 2nd, use SOLVER to determine that given a level of return, what are the variance minimizing weights

Covariation by History

Ordinary we measure covariation with covariance, Cov(R) and correlation, Corr(R). Let’s assume for T period we know that returns for two assets are given: asset X; RX

1, …, RX

T, and asset Y; RY1, …, RY

T then Covariance, Cov(R), is:

the Correlation, Corr(R), is:

T

i

YYi

XXiYX

YX RERRERT

RRCov1

, )()(1

1),(

)()(),(),( ,YXYX

YXYX RVarRVarRRCovRRCorr

Portfolio Variance

Say we have N assets with N expected returns of E(R1), …, E(RN), N variances of Var(R1), …, Var(RN), and N N pairs of correlations, 1,1, …, i,j,…, N,N. Then the variance of portfolio with weights of w1, …, wN is given:

N

i

N

ijjijiji

N

ii

RVarRVarww

RVarwRVari

Portfolio

)()(2

)()(

,

1

2

Implementation:1st, Set-up the Problem

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N

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21

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Implementation:2nd, Simplify Correlations

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1

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2,122

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21

21

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Implementation:3rd, Weights Stdevs

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Implementation:4th, Weights Stdev’s

Corr.’s

221,11

22

22212,121

1,11212,12121

21

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Implementation:Last, Sum All Elements

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