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