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Quantitative methods in Hedge Fund of Fund construction
By Peter Urbani, CIO Infiniti-Capital
Weaknesses of models used to analyse Hedge FundsWeaknesses of models used to analyse Hedge Funds
“Models currently used to analyze hedge funds generally display a number of major weaknesses:
The models do not pay sufficient attention to the asymmetry of hedge fund returns (hedge funds returns are not normally distributed). VaR type models therefore do not measure risk accurately.
The models do not correct for the presence of widespread auto-correlation causing significant understatement of volatility of hedge fund returns.
Benchmarks used are not often significant resulting in spurious comparisons.
The models do not consider the impact of asymmetry on dependence measures such as correlation.
The models do not consider the persistence of any alpha.
The models generally seek to condense all of the relevant detail into one single standardized comparative number that is frequently meaningless.
The weaknesses in existing models mean that the unique characteristics of hedge funds and risks are not captured.”
Satyajit DasAuthor of Traders Guns and Money – p28, Wilmott Magazine August 2007
Some Unique Attributes of Hedge FundsSome Unique Attributes of Hedge Funds
Asymmetry
Autocorrelation
(i)Liquidity
Non-Linear dependence
Hedge Funds v.s. Hedged FundsHedge Funds v.s. Hedged Funds
A Perfectly ‘Hedged’ fund
Fund
Returns
-ve Equity Returns +ve
Hedge Funds v.s. Hedged FundsHedge Funds v.s. Hedged Funds
A Perfect ‘Hedge’ fund
Fund
Returns
-ve Equity Returns +ve
Has 0 or negative downside correlation
and Beta
Has positive alpha in all market regimesHas positive upside
beta
‘PerfectPerfect’ vs. MSCI Daily TR Gross World Free USD, for 31-Jan-93 to 31-Mar-07
-6.00%
-4.00%
-2.00%
0.00%
2.00%
4.00%
6.00%
8.00%
10.00%
12.00%
-16.0% -11.0% -6.0% -1.0% 4.0% 9.0%
BMKs 95% VaR
= -6.75%
BMKs 95% cVaR
= -9.87%
Funds 95% VaR
= -0.45%Funds 95% cVaR
= -0.89%
-16.0% -13.4% -10.8% -8.2% -5.6% -3.0% -0.4% 2.2% 4.8% 7.4% 10.0%
0% - 5%
5% - 15%
15% - 25%
25% - 35%
35% - 45%
45% - 55%
55% - 65%
65% - 75%
75% - 85%
85% - 95%
95% - 100%
Theoretical Empirical
Prob[Fund>0.0%] = 91.70% 91.81%
Prob[Fund>BMK] = 81.50% 80.70%
Prob[Fund>MAX{0.0% & BMK} | BMK=x] = 75.27% 75.44%
Down Up Overall
Beta -0.104 0.972 0.450
Alpha 0.24% 0.43% 2.03%
Correl -0.43 0.97 0.78
RSQ 18.6% 93.4% 60.7%
Piecewise RSQ= 93.5%
Fund BMK
Holding Period Return (HPR) 5741.74% 301.39%
Compound Annual Growth Rate (CAGR) 33.04% 10.24%
Mean (Ann.) 29.17% 10.67%
Standard Deviation (Ann.) 7.61% 13.17%
Skewness 0.948 -0.692
Excess Kurtosis 0.381 0.961
Maximum Drawdown -0.69% -46.31%
95.0% Normal VaR -1.18% -5.36%
95.0% Modified VaR -0.54% -6.01%
Lowest Return -0.69% -13.32%
95.0% Infiniti VaR -0.45% -6.75%
95.0% Infiniti cVaR -0.89% -9.87%
Note Asymmetric payoff
Avg HF vs. MSCI Daily TR Gross World Free USD, for 31-Jan-93 to 31-Mar-07
-4.00%
-2.00%
0.00%
2.00%
4.00%
6.00%
8.00%
10.00%
-16.0% -11.0% -6.0% -1.0% 4.0% 9.0%
BMKs 95% VaR
= -6.75%
BMKs 95% cVaR
= -9.87%
Funds 95% VaR
= -1.18%Funds 95% cVaR
= -1.93%
-16.0% -13.4% -10.8% -8.2% -5.6% -3.0% -0.4% 2.2% 4.8% 7.4% 10.0%
0% - 5%
5% - 15%
15% - 25%
25% - 35%
35% - 45%
45% - 55%
55% - 65%
65% - 75%
75% - 85%
85% - 95%
95% - 100%
Theoretical Empirical
Prob[Fund>0.0%] = 80.44% 83.04%
Prob[Fund>BMK] = 55.61% 54.39%
Prob[Fund>MAX{0.0% & BMK} | BMK=x] = 45.30% 44.44%
Fund BMK
Holding Period Return (HPR) 1104.93% 301.39%
Compound Annual Growth Rate (CAGR) 19.08% 10.24%
Mean (Ann.) 17.75% 10.67%
Standard Deviation (Ann.) 5.72% 13.17%
Skewness 0.683 -0.692
Excess Kurtosis 1.022 0.961
Maximum Drawdown -2.43% -46.31%
95.0% Normal VaR -1.24% -5.36%
95.0% Modified VaR -0.87% -6.01%
Lowest Return -1.95% -13.32%
95.0% Infiniti VaR -1.18% -6.75%
95.0% Infiniti cVaR -1.93% -9.87%
Down Up Overall
Beta 0.047 0.137 0.189
Alpha 0.61% 1.59% 1.31%
Correl 0.12 0.17 0.44
RSQ 1.5% 3.0% 18.9%
Piecewise RSQ= 22.2%
Note Asymmetric payoff
Less than 12% of Hedge Funds ‘Normally’ distributed
Gumbel (Min)5%
Three Parameter Lognormal
13%
Pearson IV1%
Skew-T35%
Normal11%
Gumbel (Max)12%
Modified Normal5%
Uniform4%
Johnson (Lognormal)10%
Mixture of Normals4%
Based on analysis of 5400 Hedge Fund distributions
Impact of Autocorrelation on Volatility
What is it ? ‘Stale pricing’ where prior estimates are revised or where valuation is infrequent and Monthly values are interpolated
Eg. Property Fund
Affects 30% of Hedge Funds
Fix using Blundell – Wald or Kalman filter
Average 28% increase in Volatility after filtering
0%
10%
20%
30%
40%
50%
60%
70%
0 50 100 150 200 250 300 350
Liquidity in Days
CA
GR%
0%
10%
20%
30%
40%
50%
60%
70%
0 50 100 150 200 250 300 350
Liquidity in Days
CA
GR%
0%
10%
20%
30%
40%
50%
60%
70%
0 50 100 150 200 250 300 350
Liquidity in Days
CA
GR%
0%
10%
20%
30%
40%
50%
60%
70%
0 50 100 150 200 250 300 350
Liquidity in Days
CA
GR%
(i)Liquidity a Source of Alpha (i)Liquidity a Source of Alpha ??
Relationship between liquidty and Returns
Our research indicates that longer lock-ups are compensated for by additional alpha of 300 – 400bp per annum
Infiniti’s Single Fund Analysis (SFA) ranking methodologyInfiniti’s Single Fund Analysis (SFA) ranking methodology
“No interest”recorded in MGX
Quantitative Assessment (DD)
SFA
Risk Return Persistence
VaR
Volatility
CAGR
% Positive
Omega
Correlation
30% 40% 30%
Fail < 30% < Marginal > QFL > 70%
Funds cannot be passed onto the Qualified Funds / Buy List (QFL) without the sign-off of the 3 Research Department Heads
Qualitative
QuantitativeForensic
Infiniti SFA Risk score AmaranthInfiniti SFA Risk score Amaranth
Amaranth
-80.00%
-70.00%
-60.00%
-50.00%
-40.00%
-30.00%
-20.00%
-10.00%
0.00%
10.00%
20.00%
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
AmaranthSFA Risk ScoreBuy ThresholdSell Below
First Warning signal 31 May 2005
Second Warning signal 30 April 2006
Outright Sell signal 31 May 2006
Amaranth VaR to 31 Mar 2006
-8.00%
-7.00%
-6.00%
-5.00%
-4.00%
-3.00%
-2.00%
-1.00%
0.00%
1.00%
2.00%Fe
b-0
1
May-
01
Aug-0
1
Nov-
01
Feb-0
2
May-
02
Aug-0
2
Nov-
02
Feb-0
3
May-
03
Aug-0
3
Nov-
03
Feb-0
4
May-
04
Aug-0
4
Nov-
04
Feb-0
5
May-
05
Aug-0
5
Nov-
05
Feb-0
6
Normal VaR
Infiniti 'Best Fit' - VaR
Significant deviation as distribution type changes
in April / May 2005
Infiniti ‘Best Fit’ Value at Risk (VaR) AmaranthInfiniti ‘Best Fit’ Value at Risk (VaR) Amaranth
Analysis of Classic Correlation (top Right Quadrant) and Modified Correlation (bottom Left Quadrant) of sample Portfolio
Fun
d 1
Fun
d 2
Fun
d 3
Fun
d 4
Fun
d 5
Fund 1 1 0.629 0.651 0.357 0.633
Fund 2 1 0.537 0.486 0.428
Fund 3 1 0.548 0.313
Fund 4 1 0.238
Fund 5 1
0.589
0.601 0.470
0.387 0.476 0.553
0.695 0.522 0.306 0.249 0.486
0.428
0.548
0.313
0.238
0.589
0.601
0.470
0.387
0.476
0.553
0.695
0.306
0.249
Fund 1 vs Fund 2 0.629
Fund 1 vs Fund 3 0.651
Fund 1 vs Fund 4 0.629
Fund 1 vs Fund 5 0.633
Fund 2 vs Fund 3 0.537
Fund 2 vs Fund 4
Fund 2 vs Fund 5
Fund 3 vs Fund 4
Fund 3 vs Fund 5
Fund 4 vs Fund 5
0.357
0.522
Portfolios 95% Normal VaR = -0.77%
Portfolios 95% Modified VaR = -0.82%
Linear Analysis of sample PortfolioLinear Analysis of sample Portfolio
0.486
0.428
0.548
0.313
0.238
Fund 1 vs Fund 2 0.629
Fund 1 vs Fund 3 0.651
Fund 1 vs Fund 4 0.629
Fund 1 vs Fund 5 0.633
Fund 2 vs Fund 3 0.537
Fund 2 vs Fund 4
Fund 2 vs Fund 5
Fund 3 vs Fund 4
Fund 3 vs Fund 5
Fund 4 vs Fund 5
0.357
Portfolios 95% Normal VaR = -0.77%
Pearson Correlation
Fund Name Mean StDev
Fund 1 0.84% 0.89%Fund 2 0.80% 0.86%Fund 3 1.04% 1.78%Fund 4 1.33% 2.26%Fund 5 0.64% 1.01%
Sample Portfolio 0.93% 1.03%
VaR cVaR
-0.62% -0.99%-0.62% -0.98%-1.89% -2.63%-2.39% -3.34%-1.03% -1.45%
-0.77% -1.21%
Normal/Gaussian
Descriptives and VaRs
Mean Contributor
StDev Contributor
nVaR Contributor
18.18% 13.15% -0.06%17.17% 11.32% -0.03%22.32% 28.56% -0.28%28.60% 35.20% -0.33%13.72% 11.78% -0.07%
100.00% 100.00% -0.77%
Fund Name
Fund 1Fund 2Fund 3Fund 4Fund 5
Sample Portfolio
Attribution of Portfolio Descriptives
Normal “Type”
DiversifierDiversifierHigh ReturnHigh ReturnDiversifier
Non-Linear Analysis of sample PortfolioNon-Linear Analysis of sample Portfolio
Fund 1 vs Fund 2
Fund 1 vs Fund 3
Fund 1 vs Fund 4
Fund 1 vs Fund 5
Fund 2 vs Fund 3
Fund 2 vs Fund 4
Fund 2 vs Fund 5
Fund 3 vs Fund 4
Fund 3 vs Fund 5
Fund 4 vs Fund 5
Portfolios 95% Modified VaR = -0.82%
Modified Correlation
0.589
0.601
0.470
0.387
0.476
0.553
0.695
0.306
0.249
0.522
Fund Name “Mod SD” Skew Kurtosis
Fund 1 0.84% 0.75% 0.458 6.619Fund 2 0.80% 0.95% -0.685 0.634Fund 3 1.04% 1.68% 0.150 2.425Fund 4 1.33% 2.00% 0.549 1.408Fund 5 0.64% 1.26% -4.041 21.616
Sample Portfolio 0.93% 1.06% -0.254 1.160
VaR cVaR
Modified/Cornish Fisher
-0.38% -1.52%-0.77% -1.27%-1.73% -3.05%-1.96% -2.90%-1.44% -2.75%
-0.82% -1.49%
Descriptives and VaRs
Mean
Attribution of Portfolio Descriptives
Mean Contributor
“Mod SD”Contributor
mVaR Contributor
18.18% -0.06%17.17% -0.06%22.32% -0.27%28.60% -0.32%13.72% -0.11%
100.00% 100.00% -0.82%
Fund Name
Fund 1Fund 2Fund 3Fund 4Fund 5
Sample Portfolio
13.32%12.54%26.95%33.48%13.72%
Skew Contributor
Kurt Contributor
17.90% 15.59%39.94% 9.56%
-10.79% 25.72%-6.34% 33.45%59.28% 15.67%
100.00% 100.00%
DiversifierDiversifierHigh ReturnHigh ReturnDiversifier
Normal “Type”
Attempts to address the non-linear dependence of hedge funds by coming up with an analogue or ‘modified’ correlation matrix using the additional co-skewness and co-kurtosis matrices. This allows for a better understanding of the underlying risk factors within the portfolio
Normal and Cornish Fisher Probability Distribution Functions
-8.00% -6.00% -4.00% -2.00% 0.00% 2.00% 4.00% 6.00% 8.00%
Modified
Normal
Comparison of Normal and Modified DistributionsComparison of Normal and Modified Distributions
Fatter Tails
Negatively Skewed
Normal Modified95% VaR -0.77% -0.82%99% VaR -1.48% -1.93%
Putting it all together – The Infiniti Capital Analytics Suite (IAS)Putting it all together – The Infiniti Capital Analytics Suite (IAS)
Import database of FundsImport database of Funds
Fund DatabaseFund Database
Filter by Infiniti Qualified (QFL) and Invested ListFilter by Infiniti Qualified (QFL) and Invested List
Filter furtherFilter further
Filter further by Fund AUM exclude funds with less than $20mFilter further by Fund AUM exclude funds with less than $20m
Filter further by Fund AUM exclude funds with less than $20mFilter further by Fund AUM exclude funds with less than $20m
Ensure all funds have up to date historyEnsure all funds have up to date history
Load filtered list into Simulated Annealing OptimiserLoad filtered list into Simulated Annealing Optimiser
Set weight constraintsSet weight constraints
Cooling schedule for Annealing and no of iterations - DefaultsCooling schedule for Annealing and no of iterations - Defaults
Fee Information - DefaultsFee Information - Defaults
Drag and Drop standard check-limits or build custom limitsDrag and Drop standard check-limits or build custom limits
Default objective function is Infiniti SFA Total ScoreDefault objective function is Infiniti SFA Total Score
What is SFA Score What is SFA Score ? ? – Ranking system for Risk, Return and – Ranking system for Risk, Return and PersistencePersistence
Risk, Return and Persistence scores made up of multiple factorsRisk, Return and Persistence scores made up of multiple factors
Can also use any other objective functionCan also use any other objective function
Here objective function is maximise CAGR and minimise DrawdownsHere objective function is maximise CAGR and minimise Drawdowns
Run Portfolio improvement routine for 10,000 iterationsRun Portfolio improvement routine for 10,000 iterations
Generates in-sample Returns of 12.65% with volatility of 2.22%Generates in-sample Returns of 12.65% with volatility of 2.22%
Change Benchmark to CSFB TremontChange Benchmark to CSFB Tremont
Show Benchmark Returns and remove fees if investableShow Benchmark Returns and remove fees if investable
Verify all Check-limit constraints satisfiedVerify all Check-limit constraints satisfied
Out of Sample performanceOut of Sample performance
Change Chart to SFA Total Score or any other statisticChange Chart to SFA Total Score or any other statistic
Verify SFA Score matches optimised valueVerify SFA Score matches optimised value
Can be used to build portfolios with any shape distributionCan be used to build portfolios with any shape distribution
DISCLAIMER: This presentation is by Infiniti Capital AG, the Investment Manager of The Infiniti Capital Trust and its portfolio’s. Application for shares can only be made on the basis of the current Prospectuses. The Funds are unregulated collective investment schemes in the UK and Switzerland and their promotion by authorised persons in the UK is restricted by the Financial Services and Markets Act 2000. The price of shares and the income from them can go down as well as up and the value of an investment can fluctuate in response to changes in exchange rates. The following information is intended for institutional investors who are accredited investors and qualified purchasers under the securities laws.Investment in the Fund involves special considerations and risks. There can be no assurance that the Fund’s investment objectives will be achieved. An investment in the Fund is only suitable for sophisticated investors who fully understand and are capable of assuming the risk of an investment in the Fund.
Multi Manager Multi Strategy Fund of Funds