a sset a llocation portfolio management ali nejadmalayeri
TRANSCRIPT
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AASSET SSET AALLOCATIONLLOCATION
Portfolio Management
Ali Nejadmalayeri
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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
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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
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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
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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.
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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
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When to Add a New Asset Class?
• Beyond the obvious, the following should hold:
PnewP
FP
new
Fnew RRCorrRRERRE
,
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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
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FrontiersReturn
Risk
= 1.0
0 < < 1.0
< 0
Efficient Frontiers
Minimum Variance Frontiers
Corner Portfolios
Global Minimum Variance Portfolio
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How to Optimize?
An Algorithmic Approach to
Finding Corner Portfolios
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Portfolio Construction
Given a set of selected Securities
Finding Appropriate Asset Weights
Optimizing the Portfolio:
Highest Return for a Given Level of Risk
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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
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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
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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
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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
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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)(
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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
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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
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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
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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
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Implementation:1st, Set-up the Problem
NNNNNNN
NNNN
N
N
N
w
w
w
www
,,11,
,11,1
2,21,222
,12,11,111
21
21
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Implementation:2nd, Simplify Correlations
1
1
1
1
,1,1
,1
2,122
,12,111
21
21
NNNNN
NN
N
N
N
w
w
w
www
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Implementation:3rd, Weights Stdevs
1
1
1
1
,1,1
,1
2,122
,12,111
2211
NNNNN
NN
N
NN
w
w
w
www
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Implementation:4th, Weights Stdev’s
Corr.’s
221,11
22
22212,121
1,11212,12121
21
NNNNN
NNN
www
www
wwwww
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Implementation:Last, Sum All Elements
N
i
N
ijjijiji
N
iii
RVarRVarww
RVarwRVar Portfolio
)()(2
)()(
,
1