Maximum Drawdown Is Generally Greater Than in IID Normal Case

THE HIGH COST OF SIMPLIFIED MATH: Overcoming the “IID Normal” Assumption in Performance Evaluation

Marcos López de Prado Hess Energy Trading Company Lawrence Berkeley National Laboratory

Electronic copy available at: http://ssrn.com/abstract=2254668 Key Points

• Investment management firms routinely hire and fire employees based on the performance of their portfolios. • Such performance is evaluated through popular metrics that assume IID Normal returns, like Sharpe ratio, Sortino ratio, Treynor ratio, Information ratio, etc. • Investment returns are far from IID Normal. • If we accept first-order serial correlation: – Maximum Drawdown is generally greater than in IID Normal case. – Time Under Water is generally longer than in IID Normal case. – However, Penance is typically shorter than 3x (IID Normal case). • Conclusion: Firms evaluating performance through Sharpe ratio are firing larger numbers of skillful managers than originally targeted, at a substantial cost to investors. 2 Electronic copy available at: http://ssrn.com/abstract=2254668 SECTION I The Need for Performance Evaluation

Electronic copy available at: http://ssrn.com/abstract=2254668 Why Performance Evaluation?

• Hedge funds operate as banks lending money to Portfolio Managers (PMs): – This “bank” charges ~80% − 90% on the PM’s return (not the capital allocated). – Thus, it requires each PM to outperform the risk free rate with a sufficient confidence level: Sharpe ratio. – This “bank” pulls out the line of credit to underperforming PMs.

Allocation

1 …

Investors’ funds Allocation N

4 How much is Performance Evaluation worth?

• A successful hedge fund serves its investors by: – building and retaining a diversified portfolio of truly skillful PMs, taking co-dependencies into account, allocating capital efficiently. – weeding out unskilled PMs to protect the invested principal. • Investors pay high fees for those services, typically: – 2% management fee. – 20% performance fee. • An accurate performance evaluation methodology is worth a lot of money!!

5 How are PMs Stopped-Out?

• Drawdowns can be the result of – Poor investment skills: The PM should be weeded out. – Bad luck: The PM should be kept on platform. • Stopping-Out a PM is a decision under uncertainty.

An accurate performance evaluation methodology is able to discriminate between both: • maximizing the probability of true negatives (retaining good PMs). • subject to a user-defined probability of false positives (“bad luck” stop-outs).

6 SECTION II Stop-Outs under the IID Normal Assumption The IID Normal Framework (1/2)

• Suppose an investment strategy which yields a sequence of cash inflows ∆휋휏 as a result of a sequence of bets 휏 ∈ 1, … , ∞ , where

∆휋휏 = 휇 + 휎휀휏

such that the random shocks are IID distributed 휀휏~푁 0,1 . • Let us define a function 휋푡 that accumulates the outcomes ∆휋휏 over t bets. 푡

휋푡 = ∆휋휏 휏=1 where 푡 ∈ 0,1, … , ∞ and 휋0 = 0.

8 The IID Normal Framework (2/2)

• Because 휋푡 is the aggregation of t IID random variables 2 2 ∆휋휏~푁 휇, 휎 , we know that 휋푡~푁 휇푡, 휎 푡 . 1 • For a significance level 훼 < , we define the quantile 2 function for 휋푡

푄훼,푡: = 휇푡 + 푍훼휎 푡

where 푍훼 is the critical value of the Standard Normal distribution associated with a probability 훼 of performing worse than 푄훼,푡, i.e. 훼: = 푃푟표푏 휋푡 ≤ 푄훼,푡 . Then, drawdown is defined as

퐷퐷훼,푡: = max 0, −푄훼,푡

9 Maximum Drawdown

2 • PROPOSITION 1: Assuming IID outcomes ∆휋휏~푁 휇, 휎 , and 휇 > 0, the maximum drawdown associated with a 1 significance level 훼 < is 2

2 푍훼휎 푀푎푥퐷퐷 = 훼 4휇 which occurs at the time (or bet)

2 푍훼휎 푡∗ = 훼 2휇

10 Maximum Time under Water

2 • PROPOSITION 2: Assuming IID outcomes ∆휋휏~푁 휇, 휎 , and 휇 > 0, the maximum time under water associated 1 with a significance level 훼 < is 2 2 푍훼휎 푀푎푥푇푢푊 = 훼 휇

• PROPOSITION 3: Given a realized performance 휋 푡 < 0 and assuming 휇 > 0, the implied maximum time under water is 2 휋 푡 휋 푡 푀푎푥푇푢푊 = − 2 + 푡 휋 푡 휇2푡 휇 • It does not only matter how much money a PM has lost, but critically, for how long. 11 The Triple Penance Rule (1/2)

• THEOREM 1: Under IID Normal outcomes, a strategy’s maximum drawdown 푀푎푥퐷퐷훼 for a significance level 훼 ∗ occurs after 푡훼 observations. Then, the strategy is ∗ expected to remain under water for an additional 3푡훼 after the maximum drawdown has occurred, with a confidence 1 − 훼 .

푀푎푥푇푢푊훼 • If we define 푃푒푛푎푛푐푒: = ∗ − 1, then the “triple 푡훼 penance rule” tells us that, assuming independent ∆휋휏 identically distributed as Normal (which is the standard portfolio theory assumption), 푷풆풏풂풏풄풆 = ퟑ, regardless of the Sharpe ratio of the strategy.

12 The Triple Penance Rule (2/2)

5000000 It takes three time longer to recover from the maximum

4000000 drawdown (푀푎푥푇푢푊훼) than the time it took to produce it ∗ 1 (푡훼), for a given significance level 훼 < , regardless of the 3000000 2 PM’s Sharpe ratio. 2000000

푀푎푥푇푢푊훼 1000000

0 -0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5

Quantile (in US$) (in Quantile -1000000

푀푎푥퐷퐷훼 -2000000

-3000000

-4000000 ∗ ∗ 푡 3풕휶 -5000000 훼 Time Under the Water 13 Example 1

20000000 PM1 has an annual mean and standard deviation of

푀푎푥푇푢푊0.05[PM1] US$10m (SR=1), and PM2 15000000 has an annual mean of 푀푎푥푇푢푊0.05[PM2] US$15m and an annual standard deviation of 10000000 US$10m (SR=1.5).

5000000 For a 95% confidence level, PM1 reaches a maximum

Quantile (in US$) (in Quantile drawdown at US$6,763,859

0 after 0.676 years, and

0 0.5 1 1.5 2 2.5 3 ]

] remains up to 2.706 years

푃푀2 푃푀1 [ [ under water.

-5000000

05 05

. . 0 0

PM2 reaches a maximum 푀푎푥퐷퐷 -10000000 푀푎푥퐷퐷 drawdown at US$4,509,239 Time Under the Water after 0.3 years, and remains PM1 PM2 1.202 years under water. 14 Example 2

2000000 PM1 has an annual mean and standard deviation of 1000000 US$10m (SR=1), and PM2 푀푎푥푇푢푊0.08 푃푀1 = 푀푎푥푇푢푊0.02 푃푀2 0 has an annual mean of 0 0.5 1 1.5 2 2.5 US$15m and an annual -1000000 standard deviation of 푀푎푥퐷퐷0.08[푃푀1] US$10m (SR=1.5). -2000000

-3000000 For a ~92% confidence level, PM1 reaches a maximum -4000000 Quantile (in US$) (in Quantile drawdown at US$5,000,000 after 0.5 years, and remains -5000000 up to 2 years under water. -6000000 푀푎푥퐷퐷0.02[푃푀2] For a ~98% confidence level, -7000000 ∗ ∗ PM2 reaches a maximum 푡훼 3풕휶 -8000000 drawdown at US$7,500,000 Time Under the Water after 0.5 years, and remains PM1 PM2 up to 2 years under water. 15 Implications of the Triple Penance Rule

1. It makes possible the translation of drawdowns in terms of time under water [Cf. Proposition 3]. 2. It sets expectations regarding how long it may take to earn performance fee (for a certain confidence level). – The remaining time under water may be so long that withdrawals are expected. This has implications for the firm’s cash management. 3. It shows that the penance period is independent of the Sharpe ratio (in the IID Normal case). – E.g., if a PM makes a fresh new bottom after being one year under water, it may take him 3 years to recover, under the confidence level associated with that loss. This holds true whether that PM has a Sharpe of 1 or a Sharpe of 10.

16 SECTION III The IID Normal Assumption The IID Normal Assumption (1/2) • In general, traditional performance statistics assume that returns are IID Normal:

0.014 – Returns are Independent. 0.012 • However, a test of “runs” shows that negative returns occur in sequences. 0.01

0.008 – Returns are Identically Distributed. 0.006 • However, squared returns exhibit positive autocorrelation (휎-clustering). 0.004 – The distribution is Gaussian (or 0.002 0 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 Normal). pdf1 pdf2 pdf Mixture pdf Normal • However, hedge fund returns exhibit asymmetry and fat tails. • Unfortunately, the “IID Normal” assumption is not supported by the data. Then, why is it used? 18 The IID Normal Assumption (2/2)

• It is a “Hail Mary pass”, a convenient leap of faith that simplifies the math involved (… at a substantial cost to firms and investors!)

“Experience with real-world data, however, soon convinces one that both stationarity and Gaussianity are fairy tales invented for the amusement of undergraduates.” David J. Thomson, Bell Labs (1994) • A popular myth is that Central Limit Theorems (CLTs) justify the IID Normal assumption on a sufficiently large sample. This is false: − CLTs require either independence or weak dependence. − Normality is not recovered over time in the presence of dependence.

19 SECTION IV Stop-Outs under first-order auto-correlated outcomes First-order auto-correlation

• It is well established that hedge fund strategies exhibit significant first-order auto-correlation. E.g., see Brooks and Kat [2002]. • There are various reasons why strategies’ returns exhibit first-order serial-correlation: – Unmonitored risk concentration (quite different from VaR). – Inconsistent profit taking and stop loss rules. – Serially correlated and cointegrated investments. • First-order auto-correlation introduces a serial dependence that explains by itself why returns are: – Non-Identically distributed. – Non-Normal.

21 Non IID Normal Perform. Eval. Framework (1/3)

• Suppose an investment strategy which yields a sequence of cash inflows ∆휋휏 as a result of a sequence of bets 휏 ∈ 1, … , ∞ , where

∆휋휏 = 1 − 휑 휇 + 휑∆휋휏−1 + 휎휀휏

such that the random shocks are IID distributed 휀휏~푁 0,1 . • These random shocks 휀휏 follow an independent and identically distributed Gaussian process, however ∆휋휏 is neither an independent nor an identically distributed process. This is due to the parameter 휑, which incorporates a first-order serial-correlation effect of auto- regressive form.

• ∆휋휏 is stationary IIF 휑 ∈ −1,1 . 22 Non IID Normal Perform. Eval. Framework (2/3)

• PROPOSITION 4: Under the stationarity condition 휑 ∈ −1,1 , the conditional distribution of a cumulative function 휋푡 of a first-order auto-correlated random variable ∆휋휏 follows a Normal distribution with parameters:

휑푡+1 − 휑 휋 ~푁 ∆휋 − 휇 푡 휑 − 1 0

휎2 휑2 푡+1 − 1 휑푡+1 − 1 + 휇푡, − 2 + 푡 + 1 휑 − 1 2 휑2 − 1 휑 − 1

23 Non IID Normal Perform. Eval. Framework (3/3)

• PROPOSITION 5: The distribution of 휋푡 is non-stationary and unconditionally non-Normal.

휑푡+1 − 휑 푄 = ∆휋 − 휇 + 휇푡 훼,푡 휑 − 1 0 1 휎 휑2 푡+1 − 1 휑푡+1 − 1 2 + 푍 − 2 + 푡 + 1 훼 휑 − 1 휑2 − 1 휑 − 1

• PROPOSITION 6: For 휇 > 0, 푄훼,푡 is unimodal, a global minimum exists (푀푖푛푄훼) and 푀푎푥퐷퐷훼 = 푚푎푥 0, −푀푖푛푄훼 can be computed.

24 SECTION V The cost of assuming that returns are IID Normal Drawdown Stats assuming IID Normal returns

Code Mean Phi Sigma MaxDD t* MaxTuW Penance HFRIFOF Index 0.0055 0.0000 0.0170 3.53% 6.3996 25.5985 3.0000 Drawdown stats for hedge HFRIFWI Index 0.0089 0.0000 0.0202 3.10% 3.4905 13.9621 3.0000 HFRIEHI Index 0.0099 0.0000 0.0264 4.80% 4.8667 19.4669 3.0000 fund indices in the HFR HFRIMI Index 0.0095 0.0000 0.0215 3.28% 3.4435 13.7740 3.0000 HFRIFOFD Index 0.0052 0.0000 0.0174 3.96% 7.6477 30.5909 3.0000 database, computed on HFRIDSI Index 0.0096 0.0000 0.0188 2.48% 2.5827 10.3309 3.0000 HFRIEMNI Index 0.0052 0.0000 0.0094 1.16% 2.2389 8.9554 3.0000 the a sample between HFRIFOFC Index 0.0048 0.0000 0.0116 1.90% 3.9492 15.7968 3.0000 HFRIEDI Index 0.0095 0.0000 0.0192 2.63% 2.7554 11.0216 3.0000 01/01/1990 and HFRIMTI Index 0.0085 0.0000 0.0216 3.69% 4.3218 17.2870 3.0000 01/01/2013, for 훼 = 0.05. HFRIFIHY Index 0.0072 0.0000 0.0177 2.95% 4.1164 16.4656 3.0000 HFRIFI Index 0.0069 0.0000 0.0129 1.64% 2.3883 9.5530 3.0000 HFRIRVA Index 0.0080 0.0000 0.0130 1.42% 1.7701 7.0803 3.0000 HFRIMAI Index 0.0071 0.0000 0.0104 1.03% 1.4444 5.7777 3.0000 As stated in Theorem 1, HFRICAI Index 0.0071 0.0000 0.0200 3.79% 5.3200 21.2800 3.0000 HFRIEM Index 0.0104 0.0000 0.0410 10.98% 10.6100 42.4399 3.0000 Penance = 3: Regardless of HFRIEMA Index 0.0080 0.0000 0.0382 12.38% 15.4963 61.9851 3.0000 HFRISHSE Index -0.0017 0.0000 0.0535 ------the Sharpe ratio of the HFRIEMLA Index 0.0111 0.0000 0.0508 15.79% 14.2615 57.0458 3.0000 HFRIFOFS Index 0.0068 0.0000 0.0248 6.09% 8.9046 35.6185 3.0000 hedge fund, it takes three HFRIENHI Index 0.0101 0.0000 0.0367 9.02% 8.9357 35.7430 3.0000 HFRIFWIG Index 0.0094 0.0000 0.0360 9.33% 9.9416 39.7662 3.0000 time longer to recover HFRIFOFM Index 0.0056 0.0000 0.0159 3.05% 5.4422 21.7686 3.0000 HFRIFWIC Index 0.0089 0.0000 0.0390 11.50% 12.8580 51.4319 3.0000 from the bottom of the HFRIFWIJ Index 0.0084 0.0000 0.0363 10.58% 12.5579 50.2317 3.0000 drawdown. HFRISTI Index 0.0111 0.0000 0.0464 13.17% 11.8933 47.5731 3.0000 The IID Normal assumption implies that 휑 = 0. An 훼 = 0.05 means that hedge funds are willing to accept a 5% probability of firing skillful PMs (false positives). 26 Drawdown Stats accepting serial dependence

Code Mean StDev Phi Sigma t-Stat(Phi) MaxDD t* MaxTuW Penance HFRIFOF Index 0.0055 0.0170 0.3594 0.0158 6.2461 6.65% 14.5551 52.1831 2.5852 HFRIFWI Index 0.0089 0.0202 0.3048 0.0192 5.1907 4.74% 7.3222 24.4918 2.3449 The t-Stat of 휑 is HFRIEHI Index 0.0099 0.0264 0.2651 0.0255 4.4601 7.27% 9.0236 32.1120 2.5587 HFRIMI Index 0.0095 0.0215 0.1844 0.0211 3.0419 4.15% 5.4157 19.1093 2.5285 HFRIFOFD Index 0.0052 0.0174 0.3535 0.0163 6.1295 7.52% 16.9638 61.9700 2.6531 inconsistent with HFRIDSI Index 0.0096 0.0188 0.5458 0.0158 10.5612 5.40% 10.7065 30.4208 1.8413 HFRIEMNI Index 0.0052 0.0094 0.1644 0.0093 2.7035 1.33% 3.4722 11.6921 2.3674 HFRIFOFC Index 0.0048 0.0116 0.4557 0.0103 8.3023 4.00% 11.9696 39.0229 2.2602 the IID Normal HFRIEDI Index 0.0095 0.0192 0.3916 0.0177 6.9021 4.34% 7.3855 22.6758 2.0703 HFRIMTI Index 0.0085 0.0216 -0.0188 0.0216 -0.3051 ------HFRIFIHY Index 0.0072 0.0177 0.4838 0.0155 8.9720 6.69% 13.3986 43.7383 2.2644 assumption in 21 HFRIFI Index 0.0069 0.0129 0.5059 0.0111 9.5874 3.12% 8.9080 25.0456 1.8116 HFRIRVA Index 0.0080 0.0130 0.4528 0.0116 8.2430 2.00% 5.9134 15.3920 1.6029 HFRIMAI Index 0.0071 0.0104 0.2982 0.0100 5.0670 1.08% 3.2508 8.9163 1.7428 out of 26 HFRICAI Index 0.0071 0.0200 0.5780 0.0163 11.4865 11.60% 22.1308 74.4170 2.3626 HFRIEM Index 0.0104 0.0410 0.3593 0.0383 6.2431 21.71% 23.4821 87.9134 2.7439 HFRIEMA Index 0.0080 0.0382 0.3112 0.0363 5.3109 22.57% 30.2969 116.2881 2.8383 strategies, with a HFRISHSE Index -0.0017 0.0535 0.0907 0.0533 1.4776 ------HFRIEMLA Index 0.0111 0.0508 0.1969 0.0499 3.2575 22.77% 21.7061 84.0775 2.8735 HFRIFOFS Index 0.0068 0.0248 0.3231 0.0235 5.5360 11.00% 18.2415 67.7961 2.7166 95% confidence HFRIENHI Index 0.0101 0.0367 0.2011 0.0359 3.3299 12.84% 13.8963 52.7651 2.7971 HFRIFWIG Index 0.0094 0.0360 0.2314 0.0350 3.8573 14.15% 16.4723 62.5481 2.7972 HFRIFOFM Index 0.0056 0.0159 0.0422 0.0159 0.6842 3.25% 6.0074 23.5097 2.9135 level. HFRIFWIC Index 0.0089 0.0390 0.0505 0.0390 0.8200 12.59% 14.3295 56.6921 2.9563 HFRIFWIJ Index 0.0084 0.0363 0.0954 0.0361 1.5542 12.55% 15.4084 60.4123 2.9207 HFRISTI Index 0.0111 0.0464 0.1608 0.0458 2.6428 17.61% 16.8089 65.0637 2.8708 • Properly modeling the first-order serial auto-correlation gives: − 푀푎푥퐷퐷훼 is on average 65% greater than in the IID Normal case. ∗ − 푡훼 is on average 125% greater than in the IID Normal case. − 푀푎푥푇푢푊훼 is on average 89% greater than in the IID Normal case. • Penance is on average 17% lower than in the IID Normal case: − Penance is lower when 흋 is greater. − Penance is lower when the Sharpe ratio is greater. 27 The high cost of simplified Math (1/4)

• These results lead to two interesting implications: – Hedge fund strategies are much riskier than implied by the IID Normal assumption. This leads to an over-allocation of capital by Markowitz-style approaches to hedge fund strategies. – PMs and strategies evaluated by those IID-based metrics are being stopped-out much earlier than it would be appropriate. A good PM running a strategy that delivers auto-correlated outcomes may be unnecessarily stopped-out because the firm assumed IID Normal returns. • Wrongly stopping-out a PM is a particularly bad decision, because one positive aspect about strategies with auto- correlated returns is that their Penance is shorter than in the IID Normal case. The firm is taking a 20% loss on the drawdown every time it fires a skillful stopped-out PM.

28 The high cost of simplified Math (2/4)

• We would like to understand whether hedge funds intending to accept a probability 훼1 of firing a truly skillful portfolio manager (a “false positive”) are effectively taking a different probability 훼2 as a result of assuming returns independence.

• Combining Propositions 1 and 4 we can compute the 훼2 2 푍 휎 associated with 휋 = 푀푎푥퐷퐷 = − 훼1 as 푡 훼1 4휇 2 푡∗ +1 푍 휎 휑 훼1 − 휑 − 훼1 − ∆휋 − 휇 + 휇푡∗ 4휇 휑 − 1 0 훼1 훼2 = 푍 2 푡∗ +1 푡∗ +1 휎2 휑 훼1 − 1 휑 훼1 − 1 − 2 + 푡∗ + 1 휑 − 1 2 휑2 − 1 휑 − 1 훼1

29 The high cost of simplified Math (3/4) Code MaxDD t* Alpha1 Mean2 Phi2 Sigma2 Alpha2 Suppose that PMs or HFRIFOF Index 0.0353 6.3996 0.0500 0.0055 0.3594 0.0158 0.1205 HFRIFWI Index 0.0310 3.4905 0.0500 0.0089 0.3048 0.0192 0.1014 strategies are stopped-out HFRIEHI Index 0.0480 4.8667 0.0500 0.0099 0.2651 0.0255 0.0975 HFRIMI Index 0.0328 3.4435 0.0500 0.0095 0.1844 0.0211 0.0796 under the IID Normal HFRIFOFD Index 0.0396 7.6477 0.0500 0.0052 0.3535 0.0163 0.1207 HFRIDSI Index 0.0248 2.5827 0.0500 0.0096 0.5458 0.0158 0.1312 assumption, at levels HFRIEMNI Index 0.0116 2.2389 0.0500 0.0052 0.1644 0.0093 0.0728 consistent with 훼 = 0.05. HFRIFOFC Index 0.0190 3.9492 0.0500 0.0048 0.4557 0.0103 0.1331 1 HFRIEDI Index 0.0263 2.7554 0.0500 0.0095 0.3916 0.0177 0.1114 Because the IID Normal HFRIMTI Index 0.0369 4.3218 0.0500 0.0085 -0.0188 0.0216 -- HFRIFIHY Index 0.0295 4.1164 0.0500 0.0072 0.4838 0.0155 0.1400 assumption is wrong, the HFRIFI Index 0.0164 2.3883 0.0500 0.0069 0.5059 0.0111 0.1224 HFRIRVA Index 0.0142 1.7701 0.0500 0.0080 0.4528 0.0116 0.1029 effective probability of false HFRIMAI Index 0.0103 1.4444 0.0500 0.0071 0.2982 0.0100 0.0814 HFRICAI Index 0.0379 5.3200 0.0500 0.0071 0.5780 0.0163 0.1688 positives (훼2) is much HFRIEM Index 0.1098 10.6100 0.0500 0.0104 0.3593 0.0383 0.1243 greater. Thus, most firms HFRIEMA Index 0.1238 15.4963 0.0500 0.0080 0.3112 0.0363 0.1139 HFRISHSE Index -- -- 0.0500 -0.0017 0.0907 0.0533 -- evaluating their PM’s HFRIEMLA Index 0.1579 14.2615 0.0500 0.0111 0.1969 0.0499 0.0873 HFRIFOFS Index 0.0609 8.9046 0.0500 0.0068 0.3231 0.0235 0.1145 performance through HFRIENHI Index 0.0902 8.9357 0.0500 0.0101 0.2011 0.0359 0.0872 HFRIFWIG Index 0.0933 9.9416 0.0500 0.0094 0.2314 0.0350 0.0940 Sharpe ratio etc. are HFRIFOFM Index 0.0305 5.4422 0.0500 0.0056 0.0422 0.0159 0.0567 HFRIFWIC Index 0.1150 12.8580 0.0500 0.0089 0.0505 0.0390 0.0586 improperly stopping-out HFRIFWIJ Index 0.1058 12.5579 0.0500 0.0084 0.0954 0.0361 0.0667 skillful PMs. HFRISTI Index 0.1317 11.8933 0.0500 0.0111 0.1608 0.0458 0.0795 In some cases, firms may be firing more than three times the number of skillful PMs, compared to the number they were willing to accept under the (wrong) assumption of returns independence. 30 The high cost of simplified Math (4/4)

HFRIFWIC Index This chart plots 2.9 HFRIFWIJ Index HFRIEMLA Index HFRIFOFM Index Penance for hedge HFRISTI Index HFRIEMA Index HFRIFWIG Index fund indices with HFRIENHI Index HFRIEM Index 2.7 HFRIFOFS Index various 휑 . HFRIFOFD Index HFRIFOF Index HFRIEHI Index HFRIMI Index 2.5 Although positive

HFRICAI Index serial correlation leads HFRIEMNI Index HFRIFWI Index 2.3 to greater drawdowns, HFRIFIHY Index ∗

HFRIFOFC Index longer 푡훼 and longer Penance

2.1 periods under water, HFRIEDI Index Penance may be

1.9 substantially smaller. In HFRIDSI Index particular, Penance is HFRIFI Index HFRIMAI Index smaller the higher 휑 1.7 (Phi) and the higher HFRIRVA Index 휇 the ratio (Mean 1.5 휎 0.0 0.1 0.2 0.3 0.4 0.5 0.6 divided by Sigma). Phi

31 SECTION VI Conclusions Conclusions (1/2)

1. Far from being a theoretical argument, wrongly assuming that returns are IID Normal has measurable costs to firms and investors. 2. Assuming IID Normal returns leads to the “Triple Penance” rule: Regardless of the Sharpe ratio of a strategy, it takes 3 times longer to recover from a maximum drawdown than to produce it, with the same confidence level. 3. However, taking serial dependence into account leads to Penance lower than 3x. 4. In particular, under first-order auto-correlation, Penance is lower the greater the Sharpe ratio and also the greater the serial dependence.

33 Conclusions (2/2)

5. In some hedge fund strategies, if the accepted probability of false positives was 5%, the actual rate at which skillful PMs are fired is up to three times greater. 6. This is extremely costly: If two out of three PMs are wrongly fired − the firm will have to replace them. − nothing guarantees that the new PMs have superior skills. − the new PMs will not own the loss, and will be paid for every new dollar they make. 7. There is a ퟐퟎ% loss on the drawdown for each false positive. For a large firm, this amounts to tens of millions of dollars lost annually, as a result of wrongly assuming that returns are IID Normal. 34

THANKS FOR YOUR ATTENTION!

35 SECTION VII The stuff nobody reads Bibliography (1/4)

• Alexander, G. and A. Baptista (2006): “Portfolio selection with a drawdown constraint.” Journal of Banking and Finance, pp. 3171-3189. • Bailey, D. and M. López de Prado (2012): “The Sharpe Ratio Efficient Frontier”. Journal of Risk, 15(2), Winter, 3-44. Available at http://ssrn.com/abstract=1821643. • Brooks, C. and H. Kat (2002): “The statistical properties of Hedge Fund index returns and their implications for investors”, Journal of Alternative Investments, 5(2), pp. 26-44. • Chekhlov, A., S. Uryasev and M. Zabarankin (2003): “Portfolio optimization with drawdown constraints”, in B. Scherer (Ed.): “Asset and liability management tools.” Risk Books. • Chekhlov, A., S. Uryasev and M. Zabarankin (2005): “Drawdown measure in portfolio optimization.” International Journal of Theoretical and Applied Finance, Vol. 8(1), pp. 13-58. • Cherny, V. and J. Obloj (2011): “Portfolio optimization under non-linear drawdown constraints in a semi-martingale financial model.” Technical Report, Mathematical Institute, University of Oxford.

37 Bibliography (2/4)

• Cvitanic, J. and I. Karatzas (1995): “On portfolio optimization under drawdown constraints.” in IMA Lecture Notes in Mathematics and Applications, Vol. 65, pp. 77-88. • Getmansky, M., A. Lo and I. Makarov (2004): “An econometric model of serial correlation and illiquidity in hedge fund returns.” Journal of Financial Economics, Vol. 74, pp. 529-609. • Grinstead, C. and Snell (1997): “Introduction to Probability.” American Mathematical Society, Chapter 7, 2nd Edition. • Grossman, S. and Z. Zhou (1993): “Optimal Investment Strategies for controlling drawdowns.” Mathematical Finance, Vol. 3, pp. 241-276. • Hamilton, J. (1994): “Time Series Analysis.” Princeton, Chapter 4. • Hayes, B. (2006): “Maximum drawdowns of hedge funds with serial correlation.” Journal of Alternative Investments, Vol. 8(4), pp. 26-38. • Jorion, P. (2006): “Value at Risk: The new benchmark for managing financial risk.” McGraw-Hill, 3rd Edition. • Lo, A. (2002): “The Statistics of Sharpe Ratios.” Journal of Financial Analysts, Vol. 58, No. 4, July/August.

38 Bibliography (3/4)

• López de Prado, M. and A. Peijan (2004): “Measuring the Loss Potential of Hedge Fund Strategies.” Journal of Alternative Investments, Vol. 7(1), pp. 7-31. Available at http://ssrn.com/abstract=641702. • López de Prado, M. and M. Foreman (2012): “Markowitz meets Darwin: Portfolio Oversight and Evolutionary Divergence.” Working paper, RCC at Harvard University. Available at http://ssrn.com/abstract=1931734. • Magdon-Ismail, M. and A. Atiya (2004): “Maximum drawdown.” Risk Magazine, October. • Magdon-Ismail, M., A. Atiya, A. Pratap and Y. Abu-Mostafa (2004): “On the maximum drawdown of a Brownian motion.” Journal of Applied Probability, Vol. 41(1). • Markowitz, H.M. (1952): “Portfolio Selection.” Journal of Finance, Vol. 7(1), pp. 77– 91. • Markowitz, H.M. (1956): “The Optimization of a Quadratic Function Subject to Linear Constraints.” Naval Research Logistics Quarterly, Vol. 3, 111–133. • Markowitz, H.M. (1959): “Portfolio Selection: Efficient Diversification of Investments.” John Wiley and Sons.

39 Bibliography (4/4)

• Mendes, M. and R. Leal (2005): “Maximum drawdown: Models and applications.” Journal of Alternative Investments, Vol. 7, pp. 83-91. • Meucci, A. (2005): “Risk and Asset Allocation”. Springer. • Meucci, A. (2010): “Review of dynamic allocation strategies: Utility maximization, option replication, insurance, drawdown control, convex/concave management.” SSRN Working Paper Series. • Pavlikov, K., S. Uryasev and M. Zabarankin (2012): “Capital Asset Pricing Model (CAPM) with drawdown measure.” Research Report 2012-9, ISE Dept., University of Florida, September. • Sharpe, W. (1975) “Adjusting for Risk in Portfolio Performance Measurement.” Journal of Portfolio Management, Vol. 1(2), Winter, pp. 29-34. • Sharpe, W. (1994) “The Sharpe ratio.” Journal of Portfolio Management, Vol. 21(1), Fall, pp. 49-58. • Thomson, D.J. (1994): “Jackknifing multiple-window spectra”, Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing, VI, pp. 73-76.

40 Bio

Marcos López de Prado is Head of Quantitative Trading & Research at Hess Energy Trading Company, the trading arm of Hess Corporation, a Fortune 100 company. Before that, Marcos was Head of Global Quantitative Research at Tudor Investment Corporation, where he also led High Frequency Futures Trading and several strategic initiatives. Marcos joined Tudor from PEAK6 Investments, where he was a Partner and ran the Statistical Arbitrage group at the Futures division. Prior to that, he was Head of Quantitative Equity Research at UBS Wealth Management, and a Portfolio Manager at Citadel Investment Group. In addition to his 15+ years of investment management experience, Marcos has received several academic appointments, including Postdoctoral Research Fellow of RCC at Harvard University, Visiting Scholar at Cornell University, and Research Affiliate at Lawrence Berkeley National Laboratory (U.S. Department of Energy’s Office of Science). He holds a Ph.D. in Financial Economics (Summa cum Laude, 2003), a Sc.D. in Mathematical Finance (Summa cum Laude, 2011) from Complutense University, is a recipient of the National Award for Excellence in Academic Performance by the Government of Spain (National Valedictorian, Economics, 1998), and was admitted into American Mensa with a perfect score.

Marcos is a scientific advisor to Enthought's Python projects (NumPy, SciPy), and a member of the editorial board of the Journal of Investment Strategies (Risk Journals). His research has resulted in three international patent applications, several papers listed among the most read in Finance (SSRN), publications in the Review of Financial Studies, Journal of Risk, Journal of Portfolio Management, etc. His current Erdös number is 3, with a valence of 2.

41 Disclaimer

• The views expressed in this document are the authors’ and do not necessarily reflect those of Hess Energy Trading Company or Hess Corporation. • No investment decision or particular course of action is recommended by this presentation. • All Rights Reserved.

42 Notice:

The research contained in this presentation is the result of a continuing collaboration with

Prof. David H. Bailey, LBNL

The full paper is available at: http://ssrn.com/abstract=2201302

For additional details, please visit: http://ssrn.com/author=434076 www.QuantResearch.info