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8/10/2019 5. Asset Allocation
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Dr. Denis Schweizer Associate Professor of Finance John Molson School of Business, Concordia UniversityMailing address: 1455 de Maisonneuve Boulevard West, Montreal, Quebec H3G 1M8Office: MB 11.305Phone: +1(514)-848-2424, ext. 2926Fax: +1(514)-848-4500E-mail: [email protected]
5. Asset Allocation
Investment Analysis
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Page 2Investment AnalysisDenis Schweizer
Agenda
I. Ambiguity Risk
II. Strategic Asset Allocation with Alternative Investments
III. Application to Hedge Funds
IV. Appendix: Estimation of Ambiguity Aversion
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Page 3Investment AnalysisDenis Schweizer
Zeitraum: 12.94-12.04
A n n u a l i s i e r t e R e n d i t e
4%
6%
8%
10%
12%
100% Hedgefonds
100% trad. Portfolio
15% Hedgefonds
4,2 % 4,4 % 4,6 % 4,8 % 5,0 %
14%
Annualisierte Volatilität
10% 5% Hedgefonds
z Ein Investment in diversifizierten Hedgefonds-Strategien bietet die Möglichkeit,
das Risiko-/ Ertragsverhältnis zu verbessern
z Zahlreiche wissenschaftliche Studien belegen vorgenanntes Ergebnis
Trad. Portfolio*
76% Renten
10% Aktien
10% Immobilien
4% Liquidität
7.0%
9.0%
11.0%
13.0%
15.0%
17.0%
0% 2% 4% 6% 8% 10% 12% 14% 16%
volatility
r e t u r n
equity, bonds, without hedgefundsequities, bonds plus hedge funds
8,5%
9,0%
9,5%
10,0%
10,5%
11,0%
11,5%
12,0%
0,0% 2,0% 4,0% 6,0% 8,0% 10,0% 12,0% 14,0% 16,0% 18,0%
Volatilität p.a.
R e n
d i t e p . a .
Rohstoffe: 40%
US-Aktien: 40%
US-Anleihen: 20%
Rohstoffe: 10%
US-Aktien: 10%
US-Anleihen: 80%
US-Anleihe: 100%
US-Aktien: 100%
Is an investmen t in tradit ional investmen ts st i l l reasonable?
Or is the re levant qu estion: “H ow m uch should be invested in w hich Alternative
Investme nt c lass?”
Do Alternative Investments add Value to
traditional Portfolios?
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Page 4Investment AnalysisDenis Schweizer
How do portfolio of German institutional investors
look like and why is this the case?*
0,005
0,0055
0,006
0,0065
0,007
0,0075
0,008
0,0085
0,009
0,006 0,011 0,016 0,021 0,026 0,031
E x p
e c t e d R e t u r n p . m .
Standard Deviation p. m.
Empirischer Rand Impliziter Rand investiertes Portfolio
0,005
0,0055
0,006
0,0065
0,007
0,0075
0,008
0,0085
0,006 0,011 0,016 0,021 0,026 0,031
e r w a r t e t e R e n d i t e p . m .
Standardabweichung p. m.
Empirischer Rand
investiertes Portfolio
Results are based on a study by Funke, Johanning und Rudolph (2006). They asked147 German institutional investors for their asset allocation.
*
Summary
• The average invested portfolio is not onthe efficient frontier
• Diversification benefits are lost
• For the given expected return the in
investment weight for……bonds is close to be identical…shares is too high (about 6%)…Alternative Investments is too low
(about 5%)
The reasons for this investment behavior can be found in the Behavioral-Finance-Research and aresummarized under the term „ambiguity risk“
The more familiar an investor is with a certain asset class the less is his „ambiguity risk“
The „ambiguity aversion“ for German institutional investors corresponds with an risk increase of
87%, which means that the implicit monthly standard deviation is 6.38% compared with an empirical
3.41%!
Bonds Stocks Alternative Inv.
Average Allocation 80.02% 13.71% 6.28% Optimal Allocation 81.36% 7.74% 10.90%
Empirical Frontier Implicit Frontier Invested Portfolio
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Page 5Investment AnalysisDenis Schweizer
Agenda
I. Ambiguity Risk
II. Strategic Asset Allocation with Alternative Investments
III. Application to Hedge Funds
IV. Appendix: Estimation of Ambiguity Aversion
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Page 6Investment AnalysisDenis Schweizer
Agenda
II. Strategic Asset Allocation with Alternative Investments
a. Distribution Properties
b. Risk Measures
c. Risk Adjusted Performance Measures
d. Optimization Procedure
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Page 7Investment AnalysisDenis Schweizer
The normal distribution is the classical reference
for the measurement of risk and return
x
f(x)
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Page 8Investment AnalysisDenis Schweizer
Are the properties of the normal distribution
sufficient to evaluate return distributions?
The normal distribution can be described completely by the parameters µ (expected value)and σ (standard deviation)
The focal point of the distribution is the expected value – this is where the mostobservations are realized
The distribution is symmetric around the expected value
− The skewness is zero
The “width” of the distribution is determined by the standard deviation – the higher thestandard deviation, the flatter the curve of the distribution
− The kurtosis of the normal distribution is three, i.e. the excess kurtosis is zero
− The higher the kurtosis, the more of the realized returns are at the edges of the
distribution Normally distributed returns are frequently assumed in financial economics / portfolio
theory
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Page 9Investment AnalysisDenis Schweizer
Are returns normally distributed?
The normal distribution and the “68-95-rule”
H ow large i s the skewness kurtos is and excess-kurtos is of the normal
distr ibut ion?
H ow can w e evalua te if returns a re in fac t norm ally distributed?
x
f(x)
68% 95%
μ +2 σ μ - 2 σ
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Page 10Investment AnalysisDenis Schweizer
Typical Shapes of “Normal” Distributions
normal distribution right (positive) skewed distribution
left (negative) skewed distribution leptocurtic (fat teiled) distribution
platykurtic distribution
Return
P r o b a b i l i t y m a s s
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Page 11Investment AnalysisDenis Schweizer
Typical hedge fund strategy distributions
-40% -33% -25% -17% -9% -1% 7%
Historical Return
Equity Market Neutral
Frequency
Normal Distribution
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Page 12Investment AnalysisDenis Schweizer
The skewness and its influence as the third
moment of the return distribution
FSD=FPSD
FSD=FPSD
FSD>FPSD
FSD<FPSD
FSD>FPSD
Symmetrical Distribution (SD)
Positively Skewed
Distribution (PSD)
Mean
FSD=FPSD
FSD=FPSD
FSD>FPSD
FSD<FPSD
FSD>FPSD
Symmetrical Distribution (SD)
Positively Skewed
Distribution (PSD)
Mean
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Page 13Investment AnalysisDenis Schweizer
Risk-averse investors have a preference for right-
skewed distributions
The symmetrical distribution features more returns at the left edge of the distributionthan a positively (right) skewed distribution
At the vertical lines, both distributions have the same number of realizations above andbelow these lines
This implies that between the two vertical lines more returns are realized in the positively(right) skewed distribution
In the positively (right) skewed distribution, more realizations are below the
expected value
− But: extreme losses can be avoided
− Furthermore, the probability of high profits is higher than for a symmetricdistribution that is identical in all other respects
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Page 14Investment AnalysisDenis Schweizer
The kurtosis and its influence as fourth moment
of the return distribution
FEK=FNEKFEK=FNEK
FEK>FNEK
FEK<FNEEK
FEK>FNEK
Distribution without
Excess-Kurtosis (NEK)
Distribution with
Excess-Kurtosis (EK) FEK<FNEEK
Mean=Median=Modus
FEK=FNEK
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Page 15Investment AnalysisDenis Schweizer
Investors with High Loss Aversion Prefer Return
Distributions without Excess-Kurtosis
Return distributions with positive Excess-Kurtosis (leptokurtic) feature extreme
returns more often than distributions with negative Excess-Kurtosis (platykurtic)
Expected absolute losses as well as probability of profits is higher for leptokurticdistributions
Investors aiming at avoiding high losses do not invest in Asset classes with leptocurticdistributions
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Page 16Investment AnalysisDenis Schweizer
Testing for Normal Distribution
The Jarque-Bera Test uses the estimators for skewness and kurtosis of a distribution. Thetest statistic is calculated as follows:
=6 2 +
3 2
4, = ~ 2 2
The test parameter is asymptotically 2 2 -distributed with two degrees of freedom
H0 = data is normally distributed
H1 = data is not normally distributed
The null hypothesis can be rejected at the 5% (1%) confidence level, if the test parameter( JB ) is higher than 5.99 (9.21)
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Page 17Investment AnalysisDenis Schweizer
Agenda
II. Strategic Asset Allocation with Alternative Investments
a. Distribution Properties
b. Risk Measures
c. Risk Adjusted Performance Measures
d. Optimization Procedure
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Page 18Investment AnalysisDenis Schweizer
Variance
The variance σ 2 is the most frequently used variability measure
In order to calculate the variance, the differences of all returns with their average aresquared, summed up, and divided by the number of observations
2 =
=1
1 1 ̅ 2 + 2 ̅ 2 + ⋯ + ̅ 2
=1
1 ̅ 2
=1
The variance is the average squared deviation from the average!
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Page 19Investment AnalysisDenis Schweizer
Standard Deviation
The standard deviation σ is the square root of the variance Advantage: the standard deviation is denominated in the same dimension as the average
(the return) and is therefore easier to interpret.
The variance is denominated in the dimension of return-square.
= =
=1
1 ̅ 2
=1
Annualization: .. = .. ∙
Empirical examples for the standard deviation
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Page 20Investment AnalysisDenis Schweizer
Semi-Volatility
When calculating the semi-volatility, only the negative deviations from the average aretaken into account:
=
1
� 2
=1
For being below the average
Annualization is conducted analogously to the standard deviation
.. = .. ∙ 252
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Page 21Investment AnalysisDenis Schweizer
Maximum Draw Down
Maximum Draw Down (MaxDD) indicates the highest loss ever realized
0
50
100
150
200
250
300
J a n 9 1
J a n 9 2
J a n 9 3
J a n 9 4
J a n 9 5
J a n 9 6
J a n 9 7
J a n 9 8
J a n 9 9
J a n 0 0
J a n 0 1
J a n 0 2
J a n 0 3
J a n 0 4
J a n 0 5
J a n 0 6
W e r t
Crude oil
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Page 22Investment AnalysisDenis Schweizer
Short Fall Risk (Lower Partial Moments)
What is the probability of missing the target (loss)?
How large are the expected losses in the case of losses?
What is the variability of the losses?
=1
∙
=1 = 1,
<
= 0,
= ()
0 =
1 = 2 = ( )
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Page 23Investment AnalysisDenis Schweizer
Exercise: LPM and Semi-Volatility
You observe following prices for an asset:
Calculate following risk measures for τ = 0% and τ = 5%:
− (0)
− (1)
− (2)
− Semi-Volatility
Date Price
Jan 1, 2013 $100.00
Feb 1, 2013 $105.00
Mar 1, 2013 $98.00
Apr 1, 2013 $102.30
May 1, 2013 $96.25
Jun 1, 2013 $108.00
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Page 25Investment AnalysisDenis Schweizer
Value-at-Risk (VaR)
The Value-at-Risk denotes the loss that will not be exceeded with a probability ( α ) and
within a certain time horizon (t).
Example: A portfolio with a total value of $ 5 million has a VaR of $ 50,000 for an a of99% and a holding period of t=1 day
That means, that with a certainty of 99%, for an invested sum of $ 5 million, there will notoccur a loss larger than $ 50,000 within the next day.
What are the problems associated with using the Value-at-Risk ?
Probability,e.g. 1%
Value distribution for a giventime horizon,e.g. 1 day
Expected Value
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Page 26Investment AnalysisDenis Schweizer
Problems with using the VaR
What is the risk and the VaR of a short put?
S t +1 Bt =S t −VaR(95%)t stock
Value-at-Risk for a normally distributed underlying
Probability
p=5 %
∆S
Profit-and-loss profile of ashort put at maturityduration = holding period
Strike Price
VaR(95%) underlying(10 days holding period)
∆S =0S t =current price of the underlying
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Page 27Investment AnalysisDenis Schweizer
-15,0% -10,0% -5,0% 0,0% 5,0% 10,0% 15,0%
Historical Data
Fitted Distribution
Max Loss CVaR VaR
F r e q u e
n c y
Probabilityα=0.01
Return-15.0% -10.0% -5.0% 0.0% 5.0% 10.0% 15.0%
Conditional Value at Risk
The volatility measures the
symmetric scattering of the expectedreturns around the average
For asymmetrical returndistributions the volatility does notmeasure the risk adequately
VaR as a risk measure cuts thereturn distribution at the alpha-quantile and disregards all lossesbeyond that threshold –theoretically, total default risks canbe neglected by this risk measure
CVaR indicates the average expectedloss beyond the VaR and thereforedoes not only account for themagnitude, but also the probabilityof extreme risks
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Page 28Investment AnalysisDenis Schweizer
Agenda
II. Strategic Asset Allocation with Alternative Investments
a. Distribution Properties
b. Risk Measures
c. Risk Adjusted Performance Measures
d. Optimization Procedure
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Page 29Investment AnalysisDenis Schweizer
Sharpe Ratio
TheSharpe Ratio
measures the excess return of a portfolio over the risk free rate per riskunit (volatility)
= =
Volatility
Performance
Risk Free Rate
(MoneyMarket)
Portfolio
α
SR m easures the s lope
Th e high er the s lopte the
highe the sha rpe rat io
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Page 30Investment AnalysisDenis Schweizer
Sortino Ratio
The Sortino Ratio measures the excess return of a portfolio over the risk free rate per riskunit (LPMtau (2))
= = 2
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Page 31Investment AnalysisDenis Schweizer
Sterling Ratio and Reverse Calmar Ratio
The Sterling Ratio measures the excess return of a portfolio over the risk free rate perrisk unit measured by Maximum Drawdown (MaxDD)
=
=
What is the average time period an investor has to wait (measured in years) until the loss iscaught up – Reverse Calmar Ratio?
= �(. )
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Page 32Investment AnalysisDenis Schweizer 32
Return on Value-at-Risk
The Return on Value-at-Risk measures the excess return of a portfolio over the risk free
rate per risk unit measured by Value-at-Risk (in absolute terms) for a givenα
-Quantile andtime period
() = ()
© Denis Schweizer
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Page 33Investment AnalysisDenis Schweizer
Autocorrelation
Abnormality(?):
High Sharpe Ratios for Hedge Fund indices (market inefficiencies?)
Returns of Hedge Funds are in general not normally distributed
Negative skewness and high kurtosis, especially when the Sharpe Ratio is high
High autocorrelation (AC) of first order for the returns
High autocorrelation yields to an underestimation of the volatility ( Smoothing ) and
therefore the Sharpe Ratio is overestimated
⇒ Reasons are e.g. illiquid trading strategies for which a daily appraisal is not possible
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Page 34Investment AnalysisDenis Schweizer
Basic De-Smoothing of Autocorrelation
De-Smoothing of the returns – first order AC (Brooks / Kat 2002, Geltner 1993):
= ∗
−1
∗
1
− = unobservable , true return in t
− ∗ = observable Return in t
− = Autocorrelation
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Page 35Investment AnalysisDenis Schweizer
Advanced De-Smoothing Techniques
New method introduces by Getmansky, Lo und Makarov (2004, JFE) which isappropriate for Alternative Investments
Returns of Alternative Investments often show high levels of autocorrelation because ofappraisal smoothing, stale pricing, illiquidity, etc.
The observable return is the weighted sum of the past returns
− 0 = Θ0 ∙ + ⋯ + Θ ∙ −
− Θ 0,1 and = 0, … ,
− 1 = Θ0 + ⋯ + Θ
Therefore:− 0 =
− 0 = 2 ∙ 2 ≤ 2 with 2=Θ02+,…, Θ2
Advanced De-Smoothing Techniques
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Page 36Investment AnalysisDenis Schweizer
Advanced De-Smoothing Techniques
(continued)
For solving the problem Getmansky, Lo und Makarov (2004) suggest a de-smoothing ofthe returns:
− 1. step: Estimation of the parameters Θ0 + ⋯ + Θ with MaximumLikelihood (MLE)
− 2. step: Calculation of the true returns and volatilities
Example:
− 0 = 0.25 ∙ + 0.25 ∙ −1 + 0.25 ∙ −2 + 0.25 ∙ −3
−
2=
4
∙0.252 = 0
.25
Therefore, the true (de-smoothed) variance is four times higher then the observed variance
Descriptive Statistics for the Different HF
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Page 37Investment AnalysisDenis Schweizer
Descriptive Statistics for the Different HF
Strategies with and without Autocorrelation
Without Autocorrelation
Convertible
Arbitrage Distressed
Emerging
Markets
Equity Market
Neutral Event Driven
Fixed Income
Arbitrage Global Macro L/S Equity
Mean p.m. 0,31% 0,67% 0,50% 0,51% 0,50% 0,09% 0,91% 0,72%
Standard Diviation
p.m. 3,98% 3,84% 8,55% 3,73% 2,66% 3,55% 5,48% 6,24%
Skewness -3,45 -2,63 -1,14 -8,84 -0,92 -4,15 -0,50 -0,32
Excess Kurtosis 18,14 15,94 5,08 97,83 5,42 25,48 4,42 3,19
arque-Bera 2370,74 1772,01 194,97 62186,87 206,12 4518,47 129,18 66,64
P-Value 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00%
With Autocorrelation Convertible
Arbitrage Distressed
Emerging
Markets
Equity Market
Neutral Event Driven
Fixed Income
Arbitrage Global Macro L/S Equity
Mean p.m. 0,45% 0,80% 0,68% 0,51% 0,54% 0,24% 1,02% 0,82% Standard Diviation
p.m. 2,08% 1,97% 4,31% 3,45% 1,31% 1,84% 2,87% 3,09%
Skewness -3,59 -2,73 -1,29 -11,29 -1,05 -4,50 -0,28 0,05
Excess Kurtosis 18,74 14,01 6,61 134,79 4,68 27,26 3,71 3,41
arque-Bera 2534,27 1423,88 316,87 117522,78 165,71 5185,03 88,65 73,17
P-Value 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00% 0,00%
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Page 38Investment AnalysisDenis Schweizer
Agenda
II. Strategic Asset Allocation with Alternative Investments
a. Distribution Properties
b. Risk Measures
c. Risk Adjusted Performance Measures
d. Optimization Procedure
Validity and Applicability of Modern Portfolio
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Page 39Investment AnalysisDenis Schweizer
Validity and Applicability of Modern Portfolio
Theory (MPT) Depends on Strict Assumptions
Assumptions of Markowitz’ portfolio selection (1952)
Perfect and complete capital market
Risk averse investors
Investments by the µ-σ-princpile
Time horizon is 1 period n linear independent and risk-bearing securities
Number of risk-bearing securities is 2 (or more)
Securities have finite variances and expected values
Normative recommendations
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Page 40Investment AnalysisDenis Schweizer
Optimization Problem
The multiple-assets-portfolio has to be optimized by using different risk measures(volatility, semi-variance, conditional value at risk (5%) and maximum drawdown and fordifferent commodity benchmarks
The goal of the analysis is to answer the question if commodities are useful componentsfor portfolios under various levels of risk appetite and different expectations towards thefuture return of multiple-assets-portfolios?
Subject to constraint
for , , , , .
= ; 0 < ≤
1 + ⋯ + = 1 ∀ = 1, . . ,
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Page 41Investment AnalysisDenis Schweizer
Agenda
I. Ambiguity Risk
II. Strategic Asset Allocation with Alternative Investments
III. Application to Hedge Funds
IV. Appendix: Estimation of Ambiguity Aversion
Descriptive Statistic for the Different Hedge
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Page 42Investment AnalysisDenis Schweizer
Descriptive Statistic for the Different Hedge
Fund Strategies with AC
With
AutocorrelationConvertible
Arbitrage Distressed
Emerging
Markets
Equity
Market
Neutral
Event
Driven
Fixed
Income
Arbitrage
Global
Macro
L/S
Equity
Mean p.m. 0.45% 0.80% 0.68% 0.51% 0.54% 0.24% 1.02% 0.82%
Standard
Diviation p.m. 2.08% 1.97% 4.31% 3.45% 1.31% 1.84% 2.87% 3.09%
Skewness -3.59 -2.73 -1.29 -11.29 -1.05 -4.50 -0.28 0.05
Excess Kurtosis 18.74 14.01 6.61 134.79 4.68 27.26 3.71 3.41
Jarque-Bera 2,534.27 1,423.88 316.87 117,522.78 165.71 5,185.03 88.65 73.17
P-Value 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%
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Autocorrelation Structure
Lag 1 Lag 2 Lag 3 Lag 4 Lag 5 Lag 6 Lag 7 Lag 8 Lag 9 Lag 10
Convertible
Arbitrage AC 0.557 0.270 0.180 0.141 -0.004 0.175 0.302 0.180 0.126 0.175
Distressed AC 0.388 0.281 0.155 0.119 0.072 0.005 -0.088 0.029 -0.014 -0.025
Emerging
Markets AC 0.281 0.081 0.110 0.033 0.016 -0.011 0.119 0.047 0.008 -0.018
Equity MarketNeutral
AC 0.074 0.257 0.157 0.105 0.051 -0.008 0.091 0.153 -0.001 0.024
Event Driven AC 0.336 -0.011 -0.102 -0.068 0.199 0.091 -0.024 -0.080 -0.036 0.097
Fixed Income
Arbitrage AC 0.503 0.134 0.070 0.059 -0.095 -0.015 0.283 0.221 0.093 0.094
Global Macro AC 0.159 0.072 0.108 -0.106 0.103 0.078 -0.126 0.150 -0.112 0.030
L/S Equity AC 0.202 0.104 -0.017 -0.088 -0.191 0.178 0.121 0.123 0.116 0.039
Numbers in „bold“ mean that the is a s ignificant autocorrelation for the respective lag (95%-level)
Descriptive Statistic for the Different Hedge
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Page 44Investment AnalysisDenis Schweizer
p g
Fund Strategies without AC
Without Autocorrelation
Convertible
Arbitrage Distressed
Emerging
Markets
Equity
MarketNeutral
Event
Driven
Fixed
Income
Arbitrage
GlobalMacro
L/S Equity
Mean p.m. 0.45% 0.80% 0.68% 0.51% 0.54% 0.24% 1.02% 0.82%
StandardDiviation p.m. 3.98% 3.84% 8.55% 3.73% 2.66% 3.55% 5.48% 6.24%
Skewness -3.45 -2.63 -1.14 -8.84 -0.92 -4.15 -0.50 -0.32
Excess Kurtosis 18.14 15.94 5.08 97.83 5.42 25.48 4.42 3.19
Jarque-Bera 2,370.74 1,772.01 194.97 62,186.87 206.12 4,518.47 129.18 66.64
P-Value 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%
With
AutocorrelationConvertible
Arbitrage Distressed
Emerging
Markets
Equity
Market
Neutral
EventDriven
Fixed
Income
Arbitrage
GlobalMacro
L/S Equity
Mean p.m. 0.45% 0.80% 0.68% 0.51% 0.54% 0.24% 1.02% 0.82%
Standard
Diviation p.m. 2.08% 1.97% 4.31% 3.45% 1.31% 1.84% 2.87% 3.09%
Skewness -3.59 -2.73 -1.29 -11.29 -1.05 -4.50 -0.28 0.05
Excess Kurtosis 18.74 14.01 6.61 134.79 4.68 27.26 3.71 3.41
Jarque-Bera 2,534.27 1,423.88 316.87 117,522.78 165.71 5,185.03 88.65 73.17
P-Value 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%
-Optimization and -Efficiency Line
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Page 45Investment AnalysisDenis Schweizer
Use the Excel file to play around with weight restrictions. What is the impact on the efficient frontier
for different sets of restrictions? What are the „important“ HF strategies? Why does the portfolio for
the minimum variance portfolio look different from the maximum return portfolio? What is the
impact of the financial crisis? (Hint: The „green“ cells can be changed)
Convertible Arbitrage
Distressed
Emerging Markets
Equity Market Neutral
Risk Arbitrage(Event Driven)
Fixed Income Arbitrage
Global Macro
L/S Equity
0,0%
2,0%
4,0%
6,0%
8,0%
10,0%
12,0%
14,0%
0,0% 2,0% 4,0% 6,0% 8,0% 10,0% 12,0% 14,0% 16,0% 18,0% 20,0%
R e t u r n p . a .
Volatility
Mean-Variance-Optimal without AC Mean-Variance-Optimal with AC
Convertible Arbitrage Distressed
Emerging Markets Equity Market Neutral
Risk Arbitrage (Event Driven) Fixed Income Arbitrage
Global Macro L/S Equity
p y
for Funds of Hedge Funds
C a u t i o n : C a p e d o e s n o t e n a b l e u s e r s
t o f l y .
—
m
n
c
o
um
w
n
n
a
b
FoHF Portfolio Optimizer.xlsm
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Page 46Investment AnalysisDenis Schweizer
Agenda
I. Ambiguity Risk
II. Strategic Asset Allocation with Alternative Investments
III. Application to Hedge Funds
IV. Appendix: Estimation of Ambiguity Aversion
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Page 47Investment AnalysisDenis Schweizer
Research Paper
Do Institutional Investors Care About the
Ambiguity of Their Assets? – Evidence From
Portfolio Holdings in Alternative Investments
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Page 48Investment AnalysisDenis Schweizer
Motivation
One of the major contributions of the capital asset pricing theory (CAPM) is an easily-
applicable optimal portfolio strategy – market Portfolio and risk-free asset.
Admittedly, in practice researchers have documented several deviations from the CAPMportfolio strategy like
− disposition effect (see e.g. Odean(1998), Grinblatt/Keloharju (2001) andFeng/Seasholes (2005)),
− home bias effect (see e.g. Lewis (1995, 1999) and French/Poterba (1991)),
− loss aversion bias (see e.g. Kahneman/Tversky (1979)),
− status quo bias (see e.g. Samuelson/Zeckhauser (1998)),
− etc.
Besides behavioral biases, there is a further important reason for why investors might notfollow the optimal portfolio strategy according to the CAPM:
→ When investors do not perfectly know the risk- and return-characteristics of all riskyassets
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Page 49Investment AnalysisDenis Schweizer
Motivation – Risk and Ambiguity
There are two urns:
− Urn A contains 50% red balls and 50% black balls
− Urn B contains red and black balls
(unknown distribution)
Now a gamble: You can win EUR 100 if a red ball is drawn.
Which urn do you choose for the gamble, A or B?
Now the same gamble, with the same urns, but now you can win EUR 100 if a black ball isdrawn.
Again: Which urn do you choose for this gamble, A or B?
A B
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Page 50Investment AnalysisDenis Schweizer
ambiguity: state of
uncertainty withunknown probabilities
for possible outcomes
risk: state of
uncertainty withknown probabilities for
possible outcomes
Motivation – Ellsberg-Paradox
Common observation: people choose urn A for both gambles
But: If you choose urn A in the first place, you implicitly assume that urn B contains moreblack balls than red balls
… which makes urn B the optimal choice in the second gamble so called Ellsberg
paradox
Possible explanation: difference between risk and ambiguity
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Page 51Investment Analysis
Denis Schweizer
Research Questions
In the real world (institutional and private) investors do not know for sure the true retrun-and risk-characteristics of all risky assets.
− Especially for some asset classes such as hedge funds and private equity because
no availibilty of a long history of asset prices,
the investment focus of the managers might considerably change over time,
intransparancy of the portfolio holdings,
etc.
Therefore, two research questions arise:
− Are empirical portfolio holdings of institutional investors effected by ambiguityaversion?
− Which asset classes do exhibit a high ambiguity and which do not?
Relation Between Ambiguity and Optimal
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Page 52Investment Analysis
Denis Schweizer
g
Portfolio Allocation
Optimal Portfolios With and Without Ambiguity – illustrative example
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Page 53Investment Analysis
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Model Framework Used for Calibration Exercise
We use the closed-form solutions provided by Uppal/Wang (2003):
αi represents the asset specific ambiguity parameter
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Page 54Investment Analysis
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Description of Data Sample
Detailed portfolio holdings of 119 institutional investors
Proxy Indices for the asset classes are:
− Stocks: MSCI Europe -- Total Return index
− Bonds: JPM Europe Government Bond -- Total Return index
−
Real Estate: FTSE NAREIT All REITs -- Total Return index
− Private Equity : CepreX - equally weighted index of the US Growth/Small Buyout andUS Venture Capital single fund indices
− Hedge Funds: HFRX Global Hedge Fund Index
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Descriptive Statistics of Return Distributions
• We adjust the returns of the CepreX indices for management fees• We use the Getmansky/Lo/Makarov (2004) approach to unsmooth the private equity time series for significant
autocorrelation up to lag 12.
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Page 56Investment Analysis
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Calibration Exercise
Implied Ambiguity Parameter Values
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Page 57Investment Analysis
Denis Schweizer
Robustness Check
Implied Ambiguity for Different Expected Returns of Alternative Investments
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Page 58Investment Analysis
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Conclusion
Using the formal model introduced by Uppal/Wang (2003) we can implicitly determinethe ambiguity of the assets in the portfolios of institutional investors.
We find that institutional investors do care for ambiguity because they accept aportfolio Sharpe ratio which is strongly below the maximum Sharpe ratio withoutambiguity.
In terms of the asset specific ambiguity, we can show that equity and bond portfolioshave a relatively low ambiguity in contrast to alternative investments
Among the alternative investments hedge funds exhibit a large ambiguity, even muchhigher than that of private equity investments; however, real estate investments ratherhave a relatively moderate ambiguity.
These qualitative results are robust with regard to the used parameters for the expectedreturns of the asset classes.
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References
Feng, L., and Seasholes, M., 2005, “Do Investor Sophistication and Trading Experience Eliminate
Behavioral Biases in Finance Markets?," Review of Finance, 9, 305-51.
French, K. R., and Poterba, J. M., 1991, “Investor Diversification and International Equity Markets," American Economic Review, 81, 222-226.
Grinblatt, M., and Keloharju, M., 2001, “What Makes Investors Trade?," Journal of Finance, 56, 589-616.
Kahneman, D., and Tversky, A., 1979, “Prospect Theory: An Analysis of Decision under Risk,"
Econometrica, 47, 263-291.
Lewis, K. K., 1995, „Puzzles in International Financial Markets," In: Grossman, G. M., and Rogoff, K.
(Eds.), Handbook Of International Economics, Vol. III, North-Holland, Amsterdam.
Lewis, K. K., 1999, “Trying to Explain Home Bias in Equities and Consumption," Journal of Economic
Literature, 37, 571-608.
Odean, T., 1998, “Are Investors Reluctant to Realize Their Losses?," Journal of Finance, 53, 1775-1798.
Samuelson, W., and Zeckhauser, R., 1988, „Status Quo Bias in Decision Making," Journal of Risk and
Uncertainty, 1, 7-59.