Recover From Drawdown

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

Marcos López de Prado

Lawrence Berkeley National Laboratory Computational Research Division

HOW LONG DOES IT TAKE TO RECOVER FROM A DRAWDOWN?


Key Points

2

• 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.


SECTION I The Need for Performance Evaluation

Why Performance Evaluation?

4

• 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.

Investors’ funds

Allocation 1

Allocation N

…

How much is Performance Evaluation worth?

5

• 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!!

How are PMs Stopped-Out?

6

• 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).

SECTION II Stop-Outs under the IID Normal Assumption

The IID Normal Framework (1/2)

8

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

∆𝜋𝜏 = 𝜇 + 𝜎𝜀𝜏

such that the random shocks are IID distributed 𝜀𝜏~𝑁 0,1 .

• Let us define a function 𝜋𝑡 that accumulates the outcomes ∆𝜋𝜏 over t bets.

𝜋𝑡 = ∆𝜋𝜏

𝑡

𝜏=1

where 𝑡 ∈ 0,1,… ,∞ and 𝜋0 = 0.

The IID Normal Framework (2/2)

9

• Because 𝜋𝑡 is the aggregation of t IID random variables ∆𝜋𝜏~𝑁 𝜇, 𝜎2 , we know that 𝜋𝑡~𝑁 𝜇𝑡, 𝜎2𝑡 .

• For a significance level 𝛼 <1

2, we define the quantile

function for 𝜋𝑡

𝑄𝛼,𝑡: = 𝜇𝑡 + 𝑍𝛼𝜎 𝑡

where 𝑍𝛼 is the critical value of the Standard Normal distribution associated with a probability 𝛼 of performing

worse than 𝑄𝛼,𝑡, i.e. 𝛼:= 𝑃𝑟𝑜𝑏 𝜋𝑡 ≤ 𝑄𝛼,𝑡 . Then, quantile

loss is defined as

𝑄𝐿𝛼,𝑡: = max 0,−𝑄𝛼,𝑡

Maximum Drawdown

10

• PROPOSITION 1: Assuming IID outcomes ∆𝜋𝜏~𝑁 𝜇, 𝜎2 , and 𝜇 > 0, the maximum quantile-loss associated with a

significance level 𝛼 <1

2 is

𝑀𝑎𝑥𝑄𝐿𝛼 =𝑍𝛼𝜎

2

4𝜇

which occurs at the time (or bet)

𝑡𝛼∗ =

𝑍𝛼𝜎

2𝜇

2

Time under Water

11

• PROPOSITION 2: Assuming IID outcomes ∆𝜋𝜏~𝑁 𝜇, 𝜎2 , and 𝜇 > 0, the time under water associated with a

significance level 𝛼 <1

2 is

𝑇𝑢𝑊𝛼 =𝑍𝛼𝜎

𝜇

2

• PROPOSITION 3: Given a realized performance 𝜋 𝑡 < 0 and assuming 𝜇 > 0, the implied time under water is

𝐼𝑇𝑢𝑊𝜋 𝑡 =𝜋 𝑡2

𝜇2𝑡− 2

𝜋 𝑡𝜇+ 𝑡

• It does not only matter how much money a PM has lost, but critically, for how long.

The Triple Penance Rule (1/2)

12

• THEOREM 1: Under IID Normal outcomes, a strategy’s maximum quantile-loss 𝑀𝑎𝑥𝑄𝐿𝛼 for a significance level 𝛼 occurs after 𝑡𝛼

∗ observations. Then, the strategy is expected to remain under water for an additional 3𝑡𝛼

∗ after the maximum quantile-loss has occurred, with a confidence 1 − 𝛼 .

• If we define 𝑃𝑒𝑛𝑎𝑛𝑐𝑒: =𝑇𝑢𝑊𝛼

𝑡𝛼∗ − 1, then the “triple

penance rule” tells us that, assuming independent ∆𝜋𝜏 identically distributed as Normal (which is the standard portfolio theory assumption), 𝑷𝒆𝒏𝒂𝒏𝒄𝒆 = 𝟑, regardless of the Sharpe ratio of the strategy.

The Triple Penance Rule (2/2)

13

-5000000

-4000000

-3000000

-2000000

-1000000

0

1000000

2000000

3000000

4000000

5000000

-0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5

Qu

anti

le (

in U

S$)

Time Under the Water

𝑀𝑎𝑥𝑄𝐿𝛼

𝑇𝑢𝑊𝛼

𝑡𝛼∗ 3𝒕𝜶

∗

It takes three time longer to recover from the maximum quantile-loss (𝑇𝑢𝑊𝛼) than the time it took to produce it (𝑡𝛼

∗ ),

for a given significance level 𝛼 <1

2, regardless of the PM’s

Sharpe ratio.

Example 1

14

-10000000

-5000000

0

5000000

10000000

15000000

20000000

0 0.5 1 1.5 2 2.5 3

Qu

an

tile

(in

US

$)


PM1 PM2

PM1 has an annual mean and standard deviation of US$10m (SR=1), and PM2 has an annual mean of US$15m and an annual standard deviation of US$10m (SR=1.5). For a 95% confidence level, PM1 reaches a maximum drawdown at US$6,763,859 after 0.676 years, and remains up to 2.706 years under water. PM2 reaches a maximum drawdown at US$4,509,239 after 0.3 years, and remains 1.202 years under water.

𝑇𝑢𝑊0.05[PM1]

𝑀𝑎𝑥𝑄𝐿0.05[𝑃𝑀1]

𝑇𝑢𝑊0.05[PM2]


-8000000

-7000000

-6000000

-5000000

-4000000

-3000000

-2000000

-1000000

0

1000000

2000000

0 0.5 1 1.5 2 2.5

Qu

an

tile

(in

US

$)


PM1 PM2

Example 2

15

PM1 has an annual mean and standard deviation of US$10m (SR=1), and PM2 has an annual mean of US$15m and an annual standard deviation of US$10m (SR=1.5). For a ~92% confidence level, PM1 reaches a maximum drawdown at US$5,000,000 after 0.5 years, and remains up to 2 years under water. For a ~98% confidence level, PM2 reaches a maximum drawdown at US$7,500,000 after 0.5 years, and remains up to 2 years under water.

𝑇𝑢𝑊0.08 𝑃𝑀1 = 𝑇𝑢𝑊0.02 𝑃𝑀2



3𝒕𝜶∗ 𝑡𝛼

∗

Implications of the Triple Penance Rule

16

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.

SECTION III The IID Normal Assumption

The IID Normal Assumption (1/2)

18

– Returns are Independent. • However, a test of “runs” shows that

negative returns occur in sequences.

– Returns are Identically Distributed. • However, squared returns exhibit

positive autocorrelation (𝜎-clustering).

– The distribution is Gaussian (or Normal). • However, hedge fund returns exhibit

asymmetry and fat tails.

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1

pdf1 pdf2 pdf Mixture pdf Normal

• In general, traditional performance statistics assume that returns are IID Normal:

• Unfortunately, the “IID Normal” assumption is not supported by the data. Then, why is it used?

The IID Normal Assumption (2/2)

19

• 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.

SECTION IV Stop-Outs under first-order auto-correlated outcomes

First-order auto-correlation

21

• 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.

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

22

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

∆𝜋𝜏 = 1 − 𝜑 𝜇 + 𝜑∆𝜋𝜏−1 + 𝜎𝜀𝜏

such that the random shocks are IID distributed 𝜀𝜏~𝑁 0,1 .

• These random shocks 𝜀𝜏 follow an independent and identically distributed Gaussian process, however ∆𝜋𝜏 is neither an independent nor an identically distributed process. This is due to the parameter 𝜑, which incorporates a first-order serial-correlation effect of auto-regressive form.

• ∆𝜋𝜏 is stationary IIF 𝜑 ∈ −1,1 .


23

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

𝜋𝑡~𝑁 𝜑𝑡+1 − 𝜑

𝜑 − 1∆𝜋0 − 𝜇

+ 𝜇𝑡,𝜎2

𝜑 − 1 2

𝜑2 𝑡+1 − 1

𝜑2 − 1− 2

𝜑𝑡+1 − 1

𝜑 − 1+ 𝑡 + 1


24

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

𝑄𝛼,𝑡 =𝜑𝑡+1 − 𝜑

𝜑 − 1∆𝜋0 − 𝜇 + 𝜇𝑡

+ 𝑍𝛼𝜎

𝜑 − 1

𝜑2 𝑡+1 − 1

𝜑2 − 1− 2

𝜑𝑡+1 − 1

𝜑 − 1+ 𝑡 + 1

12

• PROPOSITION 6: For 𝜇 > 0, 𝑄𝛼,𝑡 is unimodal, a global minimum exists (𝑀𝑖𝑛𝑄𝛼) and 𝑀𝑎𝑥𝑄𝐿𝛼 = 𝑚𝑎𝑥 0,−𝑀𝑖𝑛𝑄𝛼 can be computed.

SECTION V

The cost of assuming that returns are IID Normal

Drawdown Stats assuming IID Normal returns

26

Drawdown stats for hedge fund indices in the HFR database, computed on the a sample between 01/01/1990 and 01/01/2013, for 𝛼 = 0.05. As stated in Theorem 1, Penance = 3: Regardless of the Sharpe ratio of the hedge fund, it takes three time longer to recover from the bottom of the drawdown.

The IID Normal assumption implies that 𝜑 = 0. An 𝛼 = 0.05 means that hedge funds are willing to accept a 5% probability of firing skillful PMs (false positives).

Code Mean Phi Sigma MaxQL t* MaxTuW Penance

HFRIFOF Index 0.0055 0.0000 0.0170 3.53% 6.3996 25.5985 3.0000

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

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

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

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

HFRIMTI Index 0.0085 0.0000 0.0216 3.69% 4.3218 17.2870 3.0000

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

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

HFRIEMA Index 0.0080 0.0000 0.0382 12.38% 15.4963 61.9851 3.0000

HFRISHSE Index -0.0017 0.0000 0.0535 -- -- -- --

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

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

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

HFRIFWIJ Index 0.0084 0.0000 0.0363 10.58% 12.5579 50.2317 3.0000

HFRISTI Index 0.0111 0.0000 0.0464 13.17% 11.8933 47.5731 3.0000

Drawdown Stats accepting serial dependence

27

The t-Stat of 𝜑 is inconsistent with the IID Normal assumption in 21 out of 26 strategies, with a 95% confidence level.

• Properly modeling the first-order serial auto-correlation gives: − 𝑀𝑎𝑥𝑄𝐿𝛼 is on average 65% greater than in the IID Normal case. − 𝑡𝛼

∗ is on average 125% greater than in the IID Normal case. − 𝑇𝑢𝑊𝛼 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 𝝋 is greater. − Penance is lower when the Sharpe ratio is greater.

Code Mean StDev Phi Sigma t-Stat(Phi) MaxQL t* TuW 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

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

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

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

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

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

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

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

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

The high cost of simplified Math (1/4)

28

• 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.


29

• We would like to understand whether hedge funds intending to accept a probability 𝛼1 of firing a truly skillful portfolio manager (a “false positive”) are effectively taking a different probability 𝛼2 as a result of assuming returns independence.

• Combining Propositions 1 and 4 we can compute the 𝛼2

associated with 𝜋𝑡 = 𝑀𝑎𝑥𝑄𝐿𝛼1 = −𝑍𝛼1𝜎

2

4𝜇 as

𝛼2 = 𝑍−

𝑍𝛼1𝜎2

4𝜇−𝜑𝑡𝛼1

∗ +1 − 𝜑𝜑 − 1

∆𝜋0 − 𝜇 + 𝜇𝑡𝛼1∗

𝜎2

𝜑 − 1 2𝜑2 𝑡𝛼1

∗ +1 − 1𝜑2 − 1

− 2𝜑𝑡𝛼1

∗ +1 − 1𝜑 − 1

+ 𝑡𝛼1∗ + 1


30

Suppose that PMs or strategies are stopped-out under the IID Normal assumption, at levels consistent with 𝛼1 = 0.05. Because the IID Normal assumption is wrong, the effective probability of false positives (𝛼2) is much greater. Thus, most firms evaluating their PM’s performance through Sharpe ratio etc. are improperly stopping-out skillful PMs.

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.

Code MaxQL t* Alpha1 Mean2 Phi2 Sigma2 Alpha2

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

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

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

HFRIEMNI Index 0.0116 2.2389 0.0500 0.0052 0.1644 0.0093 0.0728

HFRIFOFC Index 0.0190 3.9492 0.0500 0.0048 0.4557 0.0103 0.1331

HFRIEDI Index 0.0263 2.7554 0.0500 0.0095 0.3916 0.0177 0.1114

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

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

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

HFRIEM Index 0.1098 10.6100 0.0500 0.0104 0.3593 0.0383 0.1243

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 --

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

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

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

HFRIFWIJ Index 0.1058 12.5579 0.0500 0.0084 0.0954 0.0361 0.0667

HFRISTI Index 0.1317 11.8933 0.0500 0.0111 0.1608 0.0458 0.0795


31

HFRIFOF Index

HFRIFWI Index

HFRIEHI IndexHFRIMI Index

HFRIFOFD Index

HFRIDSI Index

HFRIEMNI Index

HFRIFOFC Index

HFRIEDI Index

HFRIFIHY Index

HFRIFI Index

HFRIRVA Index

HFRIMAI Index

HFRICAI Index

HFRIEM Index

HFRIEMA IndexHFRIEMLA Index

HFRIFOFS IndexHFRIENHI Index

HFRIFWIG Index

HFRIFOFM Index

HFRIFWIC IndexHFRIFWIJ Index

HFRISTI Index

1.5

1.7

1.9

2.1

2.3

2.5

2.7

2.9

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Pe

na

nce

Phi

This chart plots Penance for hedge fund indices with various 𝜑 . Although positive serial correlation leads to greater drawdowns, longer 𝑡𝛼

∗ and longer periods under water, Penance may be substantially smaller. In particular, Penance is smaller the higher 𝜑 (Phi) and the higher

the ratio 𝜇

𝜎 (Mean

divided by Sigma).

SECTION VI Conclusions

Conclusions (1/2)

33

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.

Conclusions (2/2)

34

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 𝟐𝟎% 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.

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

http://ssrn.com/abstract=1821643

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 Senior Managing Director at Guggenheim Partners. He is also a Research Affiliate at Lawrence Berkeley National Laboratory's Computational Research Division (U.S. Department of Energy’s Office of Science).

Before that, Marcos was Head of Quantitative Trading & Research at Hess Energy Trading Company (the trading arm of Hess Corporation, a Fortune 100 company) and Head of Global Quantitative Research at Tudor Investment Corporation. In addition to his 15+ years of trading and investment management experience at some of the largest corporations, he has received several academic appointments, including Postdoctoral Research Fellow of RCC at Harvard University and Visiting Scholar at Cornell University. Marcos earned a Ph.D. in Financial Economics (2003), a second Ph.D. in Mathematical Finance (2011) from Complutense University, is a recipient of the National Award for Excellence in Academic Performance by the Government of Spain (National Valedictorian, 1998) among other awards, and was admitted into American Mensa with a perfect test score.

Marcos is the co-inventor of four international patent applications on High Frequency Trading. He has collaborated with ~30 leading academics, resulting in some of the most read papers in Finance (SSRN), three textbooks, publications in the top Mathematical Finance journals, etc. Marcos has an Erdös #3 and an Einstein #4 according to the American Mathematical Society.

41

Disclaimer

• The views expressed in this document are the authors’ and do not necessarily reflect those of the organizations he is affiliated with.

• 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:


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

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