Calculating Hedge Fund Risk: The Draw Down and the Maximum Draw Down∗ Alessio Sancetta** and Steve E. Satchell*** **Department of Applied Economics, Cambridge, UK ***Faculty of Economics and Politics, Cambridge, UK March 23, 2004
Abstract Hedge funds, defined in this context as geared financial entities, frequently use some measure of point loss as a risk measure. We consider the statistical properties of an uninterrupted fall in a security price. This is called a draw down. We derive the distribution of the draw downs in an N -trading period together with an approximation to the distribution of the maximum. We also provide complementary results which are useful for risk calculations. We include a brief empirical study of the S&P futures in order to highlight some of the limitations in the presence of extreme events. Keywords: Characteristic function, Downside risk, KST distribution. Aknowledgements: Comments from the referee are gratefully acknowledged. Both authors acknowledge financial support from the ESRC award RES-000-23-0400. ∗
Running title: Draw Downs. Corresponding author: Alessio Sancetta, Department of Applied
Economics, University of Cambridge, CB3 9DE, E-mail:
[email protected].
1
1
Introduction
In this paper we study the statistical properties of draw downs. We define a draw down as the price difference in a continuous fall of a security over a period of N trades. However, other definitions are used by hedge fund managers; we discuss them later in the paper. A draw down is a measure of risk which captures features different from other global and standard measures as Sharpe’s ratio. It is in the spirit of Sortino’s ratio, as it looks at downside risk, but from a conceptually different point of view. In fact, the expected draw down and the probability of the maximum draw down are concerned with a continuous fall in a price. This is clearly of interest for hedge funds and all those funds whose trading is based on electronic trading systems. Indeed, firms that assess hedge fund performance, such as Global Fund Analysis, report, as a matter of course, the maximum draw down. Likewise, quantitative trading systems usually mention it, calculated over different time periods. For example, a draw down can trigger very quickly a stop loss, which in turn may lead to a cascade if many traders use similar systems. Modelling herd behaviour necessary to understand cascades takes us beyond the issues in this paper. Nevertheless, it is of great interest to have some statistics of the draw down (even under simplified conditions). Many hedge funds are highly geared; thus a continuous fall in their basket of contracts may lead to margin calls and possibly to bankruptcy. Despite their great importance, draw downs have not received much attention as other statistics in the context of risk management. Further, there appear to be other definitions of the term draw down, i.e. the local peak to bottom drop (e.g. Grossman and Zhou, 1993, Maslov and Zhang, 1999, Chekhlov et al., 2000, Duady et al., 2000, and Magdon-Ismail et al., 2002). In this case, most of the attention has been restricted to the case of Brownian motion (Maslov and Zhang, 1999, are an exception). For the definition we are mainly concerned with, we found references to draw downs in Johansen and Sornette (2001) and Johansen et al. (2000). We record our indebtedness to these authors for stimulating our interest in this topic. 2
They provide a discussion of draw downs in an appendix. Our plan is as follows. In Section 2, we formally define a set of statistics which loosely speaking are all different variations of a draw down. In particular, we consider the next period draw down, the conditional next period draw down, and the maximum draw down. For these statistics, whose definition is made clear below, we derive their distributions under the condition of iid observations. Further, we extend the analysis to the case of dependence in the sign of the returns. We also consider the distribution of the maximum draw down. Its exact distribution is complex, so we derive an approximation which can be easily used in practice. All these distributional results are given in Section 3. In Section 4 we briefly show how the results can be extended to the case of dependence in the sign of the returns. In Section 5, we consider two specific examples where the distribution of the returns is assumed to follow, respectively, the double Gamma and a double exponential (which is a special case of the former) within the class of KST distributions of Knight et al. (1995). Since the Gamma function is closed under convolutions, this choice is particularly convenient. In Section 6 we use a short simulation to asses the accuracy of the Poisson approximation to the maximum draw down. Section 7 considers issues related to the distribution of the number of draw downs in N trades. Section 8 considers the almost sure and in probability asymptotic behaviour of the maximum draw down. In Section 9 we remark on the distinction between large market falls, whose probabilities are estimated using the draw downs, and rare financial crashes which have to be considered as outliers. We use data on the S&P futures together with the double Gamma of Knight et al. (1995) to illustrate this point. Conclusions follow in Section 10.
2
Draw Downs: Some Definitions
Let (Xt )t∈Z+ be an arbitrary process defined on the real line. We are interested in computing statistics which relate to continuous negative realizations of (Xt )t∈Z+ . We use the term draw down as a synonym for continuous negative realization. 3
In particular, we consider the following cases: the next period draw down, the conditional next period draw down, and the maximum draw down. Each of these statistics is of interest for hedge fund risk calculations. Therefore, we shall treat each of them separately, discussing their practical relevance as risk tools. While for convenience we use the term draw down, the statistics will also be applied in conceptually different ways by virtue of their abstract definition. Before proceeding any further, we state under what conditions we derive our results. Condition 1. (Xt )t∈Z+ is an iid process, such that Xt = εt Xt+ − (1 − εt ) Xt− , where (εt )t∈Z+ is iid Bernoulli and Xt+ and Xt− are, respectively, the positive and negative part of Xt , i.e. Xt = Xt+ − Xt− , Xt+ , Xt− ∈ R+ . Trivially, Pr (Xt ≤ c) = Pr (Xt ≤ c, Xt < 0) + Pr (Xt ≤ c, Xt ≥ 0) = Pr (Xt ≤ c|Xt < 0) Pr (Xt < 0) + Pr (Xt ≤ c|Xt ≥ 0) Pr (Xt ≥ 0) . (1) Let p = Pr (Xt ≥ 0), then 1− p = Pr (Xt < 0), and F − (x) be the distribution
function of Xt− ; then Condition 1 means that
Pr (Xt ≥ − |c| , Xt < 0) = (1 − p) F − (c) , and
¶ µ Pr Xt ≥ − |c| Xt < 0 = F − (c) .
|
Further, we will also consider the following extension. Condition 1a. (Xt )t∈Z+ is a stationary process such that Xt = εt Xt+ − (1 − εt ) Xt− ,
4
(2)
¡ ¢ ¡ ¢ where Xt+ t∈Z+ and Xt− t∈Z+ are iid processes, while (εt )t∈Z+ is a Markovian
Bernoulli process with invariant distribution πI {ε = 1} + (1 − π) I {ε = 0} , e.g.
Pr (εt = x0 , εt−1 = x1 , εt−2 = x2 ) = Pr (εt = x0 |εt−1 = x1 ) Pr (εt−1 = x1 |εt−2 = x2 ) × (πI {εt−2 = x2 } + (1 − π) I {εt−2 = x2 }) , xj = 0, 1, j = 0, 1, 2. Let q11 = Pr (Xt ≥ 0|Xt−1 ≥ 0), and q00 = Pr (Xt < 0|Xt−1 < 0) and F − (x) be
the distribution function of Xt− ; then Condition 1a implies
Pr (Xt ≥ − |c| , Xt < 0, Xt−1 < 0) = (1 − π) q00 F − (c) , and
µ ¶ Pr Xt ≥ − |c| Xt < 0, Xt−1 < 0 = F − (c) ,
where π is as in Condition 1a.
2.1
|
The Next Period Draw Down
We use the term next period draw down to define the following:
YN = −
T X
Xt ,
(3)
t=1
where T ≡ N ∧ (inf {t ≥ 1 : Xt ≥ 0} − 1) and N is the total number of trades. If T = 0, we set YN ≡ 0. For notational convenience, we do not consider X0 in the definition of YN , i.e. we assume that YN is defined unconditionally of previous history. This is not restrictive under Condition 1, while very minor modifications would be required if we wanted to condition on the sign of X0 , under Condition 1a. We call (3) the next period loss because we only consider what can happen in the next period (imagine we are at time t = 0): either there is no loss, or there is one. If there is no loss we stop, if there is one we compute its magnitude. Informally, we may think of squeezing the time dimension: we can either have no loss in the next period or a loss in our squeezed time dimension (squeezed because its random real time is T ). 5
Formally, our sample space, say S, comprises of the following N + 1 elements {H1 , (T1 H2 ) , (T1 T2 H3 ) , ..., (T1 · · · TN −1 HN ) , (T1 · · · TN )} ,
(4)
where (resembling the head, tail coin toss) H stands for no loss (i.e. in the next period Xt ≥ 0) and T stands for a loss. We use the subscripts to exactly identify their order (i.e. order matters). Therefore, in our next period squeezed time dimension, we can have N + 1 different lengths of draw downs, as the case of no draw down is an element of the sample space.
2.2
The Conditional Next Period Draw Down
The conditional next period draw down is (clearly) the conditional version of (3):
YN = −
T X
Xt ,
(5)
t=1
where T ≡ N ∧ (inf {t ≥ 2 : Xt ≥ 0|X1 < 0} − 1) . As before N is the total number of trades. In this case, (5) is called the conditional next period draw down as we condition on the fact that we know that a loss occurs at time 1, though we do not necessarily know what X1 is (apart from the fact that X1 < 0). Formally, the sample space comprises of the following N elements {(T1 H2 ) , (T1 T2 H3 ) , ..., (T1 · · · TN −1 HN ) , (T1 · · · TN )} ,
(6)
i.e. we obviously reduce the size of the sample space by excluding the element H1 . For this reason, (5) can also be interpreted as the size of a draw down when zero is excluded. The above definition is the one considered by Johansen et al. (2001).
2.3
Practical Implications of the Draw Down
An application of (3) and (5) is in evaluating the risk of the following trading strategy: take a position in a security and close it when we are stopped, i.e., using the above notation, when H occurs. In this case, it is more plausible to think of 6
T as a gain, but this is irrelevant for the definitions used in this paper. Therefore, we shall always refer to T as a loss. For example, (3) can be used to evaluate the probability of a stop loss, or a stop profit. A stop loss is an order given to close a position whenever a price is hit. The term loss refers to the fact that closing the position results in a loss (similarly, for stop profit, i.e. we have a profit). Similar comments apply to (5) where we have already considered the fact that we are in a ”losing position”.
2.4
The Maximum of the First Period Draw Down and the Maximum Draw Down in N Trades
For risk management purposes, we may be interested in the maximum draw down. In this case, two cases arise. In the first one, we may ask what is the maximum of the first period draw down in N trades. The second case covers the maximum in a sequence of draw downs in N trades. We discuss both cases separately. 2.4.1
The Maximum of the Next Period Draw Down
The maximum of the next period draw down is directly computed from (4). In fact, the last element in the sample space (4) is T1 · · · TN . Since this last element is just the sum of the previous draw down with TN and so on backwards, the maximum of the next period draw down is given by the event T1 · · · TN . Therefore, our statistic of interest is YN =
N X
Xt− ,
t=1
which is the N-convolution of the negative part of X.
2.5
The Maximum Draw Down in N Trades
There are cases in which we are not just interested in one draw down, but in the total sequence of draw downs over a period of length N. For example, using the
7
previous notations, we may have, for N = 10, H1 T2 T3 T4 H5 T6 H7 T8 T9 T10 , where we have the following three draw downs T2 T3 T4 , T6 , T8 T9 T10 . It is of interest to know what is the probability that all these draw downs are below a certain level, say c. Let DN ∈ Z+ be the number of draw downs in N trades, i.e. a non
negative integer valued random variable. Define YN,j to be the j th draw down out
of N trades. In the above example, for N = 10, we have that Y10,1 is associated to T2 T3 T4 , Y10,2 to T6 and Y10,3 to T8 T9 T10 , i.e. D10 = 3. Then, in N trades, the natural statistic of interest is SN ≡ max YN,j , j≤DN
(7)
where DN ∈ [0, [(N + 1) /2]] : the number of draw downs is bounded above by [(N + 1) /2] , where [z] is the integer part of z (i.e. out of N trades, we can have, at most, [(N + 1) /2] different draw downs). 2.5.1
Practical Implications of the Maximum Draw Down
The maximum draw down is a pure risk indicator. It tells us what is the maximum loss triggered by a draw down. For example, it may be used to estimate the probability of total loss using stop orders in N trades. Being able to assess the maximum draw down in N trades can be helpful to devise optimal stop orders.
2.6
Other Useful Definitions
The usual definition of draw downs is: a percentage decrease in equity prior to recovery. Our definition considers recovery in a broad sense, i.e. a price increment which can also be equal to zero, and not necessarily strictly positive. Considering a recovery as a strict positive price increment would not affect our theoretical analysis. In particular, since we will use continuous random variables, excluding sets of Lebesgue measure equal to zero is irrelevant. 8
An alternative, and frequently used definition of draw downs is to compute from any high point of the series the maximum loss experienced subsequently. Considering sequences of non-overlapping time intervals, we arrive at yet another definition of maximum draw down. Other definitions are also possible. For the sake of completeness, we formally define some of these alternative definitions. In this case it is convenient to consider the continuous time price process (St )t∈T , T ≡ [0, 1] our unit interval. Then, following Duady et al. (2000), we have D0 ≡
sup (St − St0 )
t,t0 ∈[0,1]
D1 ≡ Sτ − inf St t∈[τ ,1]
D2 ≡
sup St − Sτ 0 ,
t∈[0,τ 0 ]
where Sτ ≡ sup St , t∈[0,1]
Sτ 0 ≡ inf St , t∈[0,1]
so that D0 corresponds to the range over a fixed interval; D1 corresponds to the maximum loss from an absolute maximum; D2 is the maximum loss from the highest point proceeding an absolute minimum. For risk management, D0 and D1 are of most practical interest. Results for (St )t∈T being standard Brownian motion, say (St )t∈T = (Bt )t∈T , are given in Duady et al. (2000). In this case, we just recall the following interesting result Law
D0 = sup |Bt | t∈[0,1]
Law
Law
D2 = D1 = sup |Bt | t∈[γ,1]
γ ≡ sup {t ≤ 1 : Bt = 0} . Unfortunately, Brownian motion does not appear to be a suitable process for risk management calculations. Extensions of these results to general Levy processes do not appear to be known.
9
3
Confidence Intervals: Exact, Approximate and Asymptotic Distributions
In this Section we derive distributional results for (3), (5) and (7). Throughout we shall use Condition 1. This simplifies the notation and the discussion. We shall consider the extension to Condition 1a in Section 4. In order to find the distribution of the draw down statistics defined above, we recall (1) Pr (X ≤ c) = Pr (X ≤ c|X < 0) Pr (X < 0) + Pr (X ≤ c|X ≥ 0) Pr (X ≥ 0) . Under Condition 1, this implies that Pr (X < 0) and Pr (X ≥ 0) are Bernoulli random variables. Therefore, to compute the draw downs distributions we need to take expectation of Pr (−X ≤ c|X < 0) = F − (c) with respect to the Bernoulli random variables. We consider each case separately as we have just done above. Notice that Condition 1 implies the framework described by the KST family of distributions of Knight et al. (1995): ¡ ¢ F (x) = (1 − p) 1 − F − (−x) + pF + (x) I {x ≥ 0} ,
where I {...} is the indicator function, F − (x) is the distribution of X − , F + (x) is
the distribution of X + , and X = X + − X − . Therefore, the characteristic function of X, ϕx (t) ≡ E (exp {ixt}), is given by ϕx (t) = (1 − p) ϕx− (−t) + pϕx+ (t) , where ϕx− (t) ≡
Z∞
exp {ixt} dF − (x) ,
Z∞
exp {ixt} dF + (x) .
0
and ϕx+ (t) ≡
0
10
Finding the distribution of the various statistics defined above consists of working with the negative part of the distribution function decomposition or the characteristic function one. In some cases, considerable simplifications can be achieved by working with characteristic functions. For these reason, after directly dealing with distribution functions, we will also consider characteristic function representations.
3.1
The Distribution of the Next Period Draw Down
From the definition of (3), taking expectation over the elements of its sample space as given in (4), we have Pr (YN ≤ c) = p + =
N−1 X n=0
N −1 X n=1
p (1 − p)n F −⊕n (c) + (1 − p)N F −⊕N (c)
p (1 − p)n F −⊕n (c) + (1 − p)N F −⊕N (c) ,
(8)
where F −⊕n (c) is the nth convolution of F − (x), and F −⊕0 (c) ≡ 1. To see that this is true, we consider the sample space. We can have H1 with probability p, and then the process stops, or T1 with probability (1 − p) . If T1 is true, we can have H2 with probability p, and the process stops, or T2 with probability (1 − p). We proceed likewise until time period N − 1, in which case we only have the two alternatives: either T1 · · · TN−1 HN or T1 · · · TN . As we mentioned, the sample space comprises of N + 1 elements, which is reflected in the fact that we have N + 1 elements in the summation above. It is worth noticing, that the first term in (8) is a discrete component in the Lebesgue decomposition of the distribution function. Therefore, care has to be taken when computing integral transforms of the density (i.e. the first p will drop after differentiation). If we let N → ∞, i.e. we allow for an infinite number of trades, we easily see that Pr (Y∞ ≤ c) =
∞ X n=0
p (1 − p)n F −⊕n (c) ,
(9)
so that trading never stops, and we do not have the second term in (8). If we are interested in the distribution of the next period draw down, (8) is clearly the 11
right distribution to use as it is exact. However, (9) provides a good theoretical approximation for large N.
3.2
The Distribution of the Conditional Next Period Draw Down
We can derive the distribution of (5) directly from (8) by dividing by Pr (X1 < 0) = (1 − p) and noticing that (in the above notation) H1 is not an element of the sample space (i.e. the first term in (5) drops down). Therefore, ÃN−1 ! X Pr (YN ≤ c) = (1 − p)−1 p (1 − p)n F −⊕n (c) + (1 − p)N F −⊕N (c) =
ÃN−1 X n=1
n=1
n−1
p (1 − p)
F
−⊕n
N−1
(c) + (1 − p)
F
−⊕N
!
(c) .
(10)
Again, letting N → ∞, Pr (Y∞ ≤ c) =
3.3
∞ X n=1
p (1 − p)n−1 F −⊕n (c) .
(11)
The Distribution of the Maximum Draw Down Conditional on the Number of Draw Downs
We have considered the distribution of one draw down over a period of N trades. However, for risk management purposes, interest usually lies in the distribution of the maximum of a random variable. This distribution in the case of draw downs is easily computed once we realize that despite the fact that the observations are independent, the length of the next draw down depends on the time the previous one occurred (recall we have N trades at most). This problem is different than the usual calculation of the maximum of an iid process due to the fact that the number of draw downs in N -trade periods is stochastic. At first we condition on the number of draw downs. In Section 7 we will consider the distribution of the number of draw downs. Using Pr (A ∩ B) = Pr (A|B) Pr (B), this allows us to find also the 12
unconditional distribution of the maximum draw down by integration of the mass function of the number of draw downs with respect to the counting measure. Now, condition on DN = dN , a large positive integer such that limN→∞ dN = ∞. Then, an expression can be derived as N → ∞. In this case, µ ¶ lim Pr max YN,j ≤ c =
N→∞
j≤dN
=
lim
N→∞
lim
N→∞
"N −1 X
n−1
−⊕n
N −1
n=1
p (1 − p)
n=1
p (1 − p)n−1 F −⊕n (c)
"N X
F
(c) + (1 − p) #dN
F
−⊕N
(c)
,
as the process never ends so that notwithstanding the time spent in the occurrences of the previous draw downs, there will always be infinite time left. Since we are conditioning on a fixed number of draw downs, we use the distribution of the conditional next period draw down inside the braces. It is well known that the distribution of the maximum of a dependent series tends to the distribution of the maximum of iid random variables for large values of the argument under non-long range dependence (see Condition D’ in Leadbetter et al., 1983, p. 58, for a formal statement). Here we recover a result which is formally similar, though conceptually different. The draw downs over a finite number of trades are not iid (despite Condition 1), but as the trading horizon N → ∞, the draw downs become iid. Therefore, for large N and c, Ã " !#dN µ ¶ ∞ X 1 Pr max YN,j ≤ c dN − dN ' 1− p (1 − p)n−1 F −⊕n (c) j≤dN dN n=1 Ã ( !) ∞ X ' exp −dN 1 − p (1 − p)n−1 F −⊕n (c) , (12) n=1
which shows the Poisson character of the maximum draw down when we condition on the number of draw downs. Notice that (12) is not (loosely speaking) the extreme value distribution of the draw down. For practical purposes, the infinite summation in (12) can be truncated after a few terms and the error incurred in including only r terms is less than (1 − p)r (this can be seen by using a geometric progression as, ∀n, F −⊕n (c) is bounded above by one). The accuracy of (12) will be 13
#dN
assessed in Section 6. The condition limN→∞ dN = ∞ could be formally justified by the almost sure behaviour of DN , i.e. the number of draw downs increases almost surely as N → ∞.
3.4
The Characteristic Function of the Draw Downs Over an Infinite Horizon
Since all our expressions are in terms of convolutions, it seems natural to work with the Fourier transform, as some results can be simplified in some cases (e.g. see the second example in Section 5). For convenience, we restrict attention to the case N → ∞. Then, under Condition 1, the characteristic function of the random variable with distribution given by (9) is given by ϕy (t) = p +
∞ X n=1
¢n ¡ p (1 − p)n ϕx− (t) .
The first term follows from the fact that (9) has a discrete component in its Lebesgue decomposition. For this reason, when YN = 0, the distribution is not absolutely continuous at 0 (it has mass at 0 equal to p), but we shall still account for this component when calculating the characteristic function of YN . Further, if ∀δ > 0
¯ ¯ sup ¯ϕy (t)¯ < 1, t≥δ
i.e. the Cram´er condition is satisfied, we can use the fact that x/ (1 − x) = P∞ n n=1 x , when |x| < 1. Therefore, ¶ µ ¶ µ (1 − p) ϕx− (t) (1 − p) =p 1+ . (13) ϕy (t) = p 1 + 1 − (1 − p) ϕx− (t) 1/ϕx− (t) − (1 − p) A similar expression can be derived when YN has its distribution given by (11). In this case, (11) does not have a discrete component in its Lebesgue decomposition. Therefore, similar calculations show that ϕy (t) =
pϕx− (t) p = . 1 − (1 − p) ϕx− (t) 1/ϕx− (t) − (1 − p) 14
(14)
We notice that the above expression together with (11) is also considered in Johansen and Sornette (2001). Further, there is some formal similarity between (13) and the characteristic function derived by Acar and Satchell (1997) for the two period moving average trading rule. Intuitively this is simply explained by the fact that a trading rule where we take a position and close it at some random time (i.e. when stopped) corresponds to an application of the next period draw down. 3.4.1
The Finite Horizon Case: N Trading Periods
¡ ¢ P n Using the fact that x − xN+1 / (1 − x) = N n=1 x , when |x| < 1 and (8) we see
that the finite horizon equivalent of (13) is given by à £ ¤N ! ¤N £ (1 − p) ϕx− (t) − (1 − p) ϕx− (t) + (1 − p) ϕx− (t) . ϕy (t) = p 1 + 1 − (1 − p) ϕx− (t)
Similar calculations using (10) show that the finite horizon version of (14) is given by
¤N−1 £ ¤N £ 1 − (1 − p) ϕx− (t) ϕy (t) = pϕx− (t) + (1 − p)N−1 ϕx− (t) , 1 − (1 − p) ϕx− (t)
due to conditioning.
3.5
The First Few Cumulants of the Next Period Draw Down Over an Infinite Horizon
Once we have an expression for the characteristic function, we can easily find the first few cumulants. To this end, it is more convenient to take logs and work with the cumulant generating function, say Ky (t) ≡ ln ϕy (t) . We consider (13) and (14) separately.
15
3.5.1
The Cumulants of the Next Period Draw Down Over an infinite Horizon
We calculate the first two cumulants for the next period draw down over an infinite horizon, which we denote by Y∞ , as before. Taking logs of (13), we have ¶ µ p (1 − p) ϕx− (t) Ky (t) = ln p + . 1 − (1 − p) ϕx− (t) The derivatives are easily found and can be simplified by symbolic computer software. Therefore, (1 − p) ϕ0x− (t) , 1 − (1 − p) ϕx− (t) (1 − p) ¡ − ¢ E (Y∞ ) = Ky0 (0) = E X , p Ky0 (t) =
and
(15)
n £ ¤ £ ¤2 o (1 − p) ϕ00x− (t) 1 − (1 − p) ϕx− (t) + (1 − p) ϕ0x− (t) Ky00 (t) = , ¤2 £ 1 − (1 − p) ϕx− (t) var (Y∞ ) = Ky00 (0) =
where σ 2x−
i o n h (1 − p) E (X − )2 p + (1 − p) [E (X − )]2
p2 ¸2 ∙ (1 − p) 2 E (X − ) σ x− + (1 − p) = , p p i h = E (X − )2 − [E (X − )]2 .
Intuitively, (15) show that the expected draw down is the expected loss times the probability of a loss divided by the probability of a gain. This is clearly a decreasing function of p, decreasing at a decreasing rate. 3.5.2
The Cumulants of the Conditional Next Period Draw Down Over an Infinite Horizon
We compute the same cumulants for the conditional next period draw down. Taking logs of (14), we have £ ¤ Ky (t) = ln p + ln ϕx− (t) − ln 1 − (1 − p) ϕx− (t) . 16
Therefore, Ky0 (t) =
ϕ0x− (t) £ ¤, ϕx− (t) 1 − (1 − p) ϕx− (t)
E (Y∞ ) = Ky0 (0) =
E (X − ) , p
(16)
and £ ¤ £ 0 ¤2 £ ¤ 00 (t) ϕ (t) 1 − (1 − p) ϕ (t) − ϕ (t) (t) 1 − 2 (1 − p) ϕ ϕ x− x− x− x− x− Ky00 (t) = , £ ¤ª2 © ϕx− (t) 1 − (1 − p) ϕx− (t)
∙ ¸2 σ 2x− E (X − ) (0) = , + (1 − p) var (Y∞ ) = p p i h where, as before, σ 2x− = E (X − )2 − [E (X − )]2 . The above result (16) is reported Ky00
in Johansen and Sornette (2001).
The Mean of the Next Period Draw Down in the Finite Horizon Case It is interesting to understand the behaviour of the mean of the draw down in the finite horizon case. We report the mean of the next period draw down in order to unveil its dependence on N. The mean of the next period draw down is given by
E (YN ) =
i (1 − p) ¡ − ¢ h E X 1 − (1 − p)N . p
(17)
The correction for considering a finite horizon is seen to go to zero exponentially fast; (17) is increasing in N. The Mean of the Conditional Next Period Draw Down in the Finite Horizon Case The mean of the conditional next period draw down in the finite N case is given by E (YN ) =
i E (X − ) h 1 − (1 − p)N , p
where, again, the correction has exponentially decreasing behaviour in N, and E (YN ) is again increasing in N.
17
4
Extension to Condition 1a
Under Condition 1a we have the following Pr (−Xt ≤ c|Xt < 0, ..., X0 < 0) = F − (c) , Pr (Xt < 0|Xt−1 < 0) = q00 , and lim Pr (Xt ≥ 0|Xt−j ≥ 0) = Pr (Xt ≥ 0) = π.
j→∞
With this notation, we can directly derive all the results in the previous section taking expectation with respect to the sign distribution.
4.1
The Next Period Draw Down under Sign Dependence
By direct use of π and q00 in (8) instead of p and 1 − p, we have Pr (YN ≤ c) = π +
N−1 X n=1
n−1 N−1 −⊕N (1 − π) q00 (1 − q00 ) F −⊕n (c) + (1 − π) q00 F (c) ,
and Pr (YN ≤ c) = π +
4.2
∞ X n=1
n−1 (1 − π) q00 (1 − q00 ) F −⊕n (c) .
The Conditional Next Period Draw Down under Sign Dependence
Similarly, but using (11) and noticing that conditioning is now achieved dividing by (1 − π) and no by (1 − p) (as done in (11)), we have Pr (YN ≤ c) =
N −1 X n=1
n−1 N−1 −⊕N q00 (1 − q00 ) F −⊕n (c) + q00 F (c) ,
and Pr (YN ≤ c) =
∞ X n=1
n−1 q00 (1 − q00 ) F −⊕n (c) .
18
4.3
The Distribution of the Maximum under Sign Dependence
We consider the approximation to the distribution of the maximum conditional on DN = dN . Formally, we have Pr
µ
max YN,j
j≤≤dN
à ( !) ¶ ∞ X n−1 ≤ c ' exp −dN 1 − q00 (1 − q00 ) F −⊕n (c) . n=1
By the Markovian condition on the sign random variables (εt )t∈Z+ , the above expression could be formally justified. In general, we need to rule out long range dependence of (εt )t∈Z+ . Formally, mixing conditions like strong mixing are sufficient (e.g. Loynes, 1965), but weaker conditions can be used (Leadbetter and Rootz´en, 1988, p. 437, and Lemma 2.1.1, p. 438). All these conditions imply Condition 1a.
5
Examples
In order to highlight the practical aspects involved in the computation of the distribution of the draw downs statistics, we look at two examples of distributions which are of interest. All our definitions have been tailor made for the KST family of distributions of Knight et al. (1995). Within this family, we consider two common examples, the double Gamma distribution and the exponential one. Both examples are quite convenient as they are convolution closed. For economy of space we only consider the case of the next period draw down when N → ∞. For this case, we also give the expression for the approximate distribution of the maximum.
5.1
The Double Gamma
Recall that the KST Gamma distribution is given by
19
ab11 b1 −1 pdf (x) = p exp {−a1 x} , if x ≥ 0 x Γ (b1 ) ab2 = (1 − p) 2 |x|b2 −1 exp {−a2 |x|} , if x < 0 Γ (b2 )
(18)
where aj , bj > 0, j = 1, 2. Each side of the above distribution is a Gamma distribution. Its characteristic function ϕx (t) is given by ϕx (t) = pϕx+ (t) + (1 − p) ϕx− (−t) , where ϕx+ (t) =
µ
a1 a1 − it
¶b1
(19)
and similarly for ϕx− (t), with parameters a2 , b2 . Therefore, the distribution is convolution closed with parameter nbj (j = 1, 2), where n is the number of variables that we convolve. Using the KST distribution, we have that (9) is given by ⎛ c ⎞ Z ∞ nb 2 X a2 Pr (YN ≤ c) = p + p (1 − p)n ⎝ xnb2 −1 exp {−a2 x} dx⎠ Γ (nb ) 2 n=1 ⎛ 0a c ⎞ Z2 nb2 −1 ∞ X s exp {−s} ⎠ p (1 − p)n ⎝ ds = p+ Γ (nb ) 2 n=1 ⎛0 ⎞ Z∞ nb2 −1 ∞ X s exp {−s} ⎠ p (1 − p)n ⎝1 − ds = p+ Γ (nb2 ) n=1 a2 c
µ ¶ Γ (nb2 , a2 c) n p (1 − p) 1 − , = p+ Γ (nb2 ) n=1
where Γ (θ, x) ≡
R∞ x
∞ X
(20)
exp {−t} tθ−1 dt is the incomplete Gamma function. Inserting
the conditional version of (20) into (12), we also have à ( µ ¶ µ ¶!) ∞ X , a c) Γ (nb 2 2 ' exp −dN 1 − p (1 − p)n−1 1 − Pr max YN,j ≤ c j Γ (nb 2) n=1 à !) ( ∞ X Γ (nb , a c) 2 2 p (1 − p)n . = exp −dN 1 − 2p + Γ (nb ) 2 n=1 20
5.2
The Double Exponential Distribution
The double exponential distribution is a special case of the KST distribution where b1 = b2 = 1. In this case, it is more convenient to work with the characteristic function. From (13) and (mutatis mutandis) from (19), it follows that the characteristic function of the draw downs, when X − is exponential with parameter a, is given by Ã
¡
¢ !
a a−it ¡ a ¢ p) a−it
(1 − p)
ϕy (t) = p 1 +
1 − (1 − ¶ µ pa , = p + (1 − p) pa − it
where the second step follows by simple algebra. Since the first term in the last display (i.e. p) is just the Fourier transform of the discrete component, we only need to invert the second term. But the second term is just (1 − p) times the characteristic function of an exponential random variable with parameter pa. It follows that the distribution of the draw downs is given by the following expression Pr (YN ≤ c) = p + (1 − p) (1 − exp {−pac}) = 1 − (1 − p) exp {−pac} ,
(21)
and, inserting the conditional version of (21) into (12), µ ¶ Pr max YN,j ≤ c ' exp {−dN exp {−pac}} , j≤dN
for large c.
6
Assessing the Quality of the Approximation for the Extreme Distribution of the Draw Down
The approximation for the distribution of the maximum of a sequence of random variables is commonly approximated using a Poisson distribution (e.g., Aldous, 1989, Vanmarcke, 1983). Further, the Poisson approximation is asymptotically 21
exact (e.g. Pickands, 1969). Notwithstanding the above, it would be reassuring to compare the behaviour of the approximate distribution with simulated data. For this purpose, we consider conditional next period draw downs generated by 1 − exp {−apx} , which would correspond to negative returns approximately generated by an exponential distribution with parameter a. Notice that we are not concerned with the data generating process of the returns. We are interested in the approximation of the distribution of the maximum draw down when the distribution of the draw downs is available. This does not introduce any extra source of uncertainty, but simplifies the simulations. In particular, we choose a = 1, p = .5 and dN =50, 200, 800, 1600. In each case we compute the maximum of the distance between the simulated distribution of the maximum draw down and the approximate one. The maximum is computed over a grid of equispaced points of 1/10. The support for the calculation is given by the largest subset in the range of the simulated maxima, such that the end points have their second decimals rounded to the nearest first decimal point. The simulated distribution is obtained by 1000 simulated samples of size dN . Results are in Table I. In order to visualize the results, Figure I provides the plot of the simulated distribution and the approximate one when dN = 50 (they are almost indistinguishable). [T ableI] [F igureI]
7
The Distribution of the Number of Draw Downs: Theory, Practice and Empirical Evidence
The distribution of the number of draw downs in N trades is equivalent to the distribution of the number of runs of tails (or heads) in N coin tossings. This and 22
related statistics have been studied extensively, and two main approaches are possible: the combinatorial approach (e.g. Mood, 1940), and more recently, the Markov chain approach (e.g. Fu and Koutras, 1994, Stefanov, 2000). Unfortunately, the analytic formulae are not simple and are not necessarily convenient in practice. Below we will consider a few known results obtained using combinatorics. Then we will shift attention to the actual behaviour of the draw downs in practice and the approximation of the distribution via Monte Carlo simulation.
7.1
Some Known Results Using Combinatorics
We state a few results obtained by combinatorics. Let N = N+ + N− , where P PN N+ = N I {X ≥ 0} and N = t − t=1 t=1 I {Xt < 0} . Then, ¡n− −1¢¡N −n− +1¢ Pr (DN = dN |N− = n− ) =
dN −1
¡ N ¢dN n−
(e.g. Mood, 1940). Let e+ ≡ n+ /N, and e− ≡ n− /N. If limN→∞ e+ = O (1) and similarly for e− , from Theorem 1 in Mood (1940), we have ¡ ¢ DN − n− e+ e− d → N 0, σ 2− , √ n−
¡ ¢ d where → stands for convergence in distribution, and N 0, σ 2− is the Gaussian distribution with variance
σ 2− ≡ e2+ e2− − e+ e2− − e2+ e− + e+ e− . In particular, for Bernoulli trials, e+ = p and e− = 1 − p. Therefore,
where
¡ ¢ DN − n− p (1 − p) d → N 0, σ 2− , √ n−
σ 2− = p2 (1 − p)2 − p (1 − p)2 − p2 (1 − p) + p (1 − p) . Since N− is binomial (N, 1 − p) , the mass function is given by Pr (N− = n− ) = ¡N ¢ (1 − p)n− pN−n− , which implies n− ¶ ¶µ µ n− − 1 N − n− + 1 Pr (DN = dN , N− = n− ) = (1 − p)n− pN−n− , dN − 1 dN 23
and summing over n− = dN , ..., N + 1 − dN Pr (DN = dN ) =
N+1−d XN j=dN
µ
¶ ¶µ j −1 N −j +1 (1 − p)j pN−j , dN − 1 dN
for dN ∈ [0, [(N + 1) /2]]. Unfortunately this expression is a bit complex for practical use. The Markov chain approach considerably mitigates this problem (e.g. Fu and Koutras, 1994). However, the dimension of the chain, despite its simplicity, grows exponentially. Fortunately, it is straightforward to obtain the mass function of the number of draw downs out of N trades using simple simulations. Therefore, from a practitioner’s point of view, this seems to be the best solution.
7.2
The Empirical and Simulated Behaviour of the Distribution of the Number of Draw Downs
It is constructive to consider the behaviour of the number of draw downs for real securities. We consider the number of draw downs for the S&P futures (January 1965-August 1999). We look at daily, weekly, and monthly returns. Interesting, the average number of draw downs seems to be almost invariant of low levels of aggregation; see Table II.
[T ableII] Further, Figure II plots the empirical frequency derived from 173 samples of 50 trades using the same data. [F igureII] It is easy to see that we have a mode at 11. Consequently, the apparent symmetry gives that 11/50 = .22 which is very close to the mean number of draw downs for daily data, over a unit interval. Since it is quite straightforward to simulate the distribution of the number of draw downs, it is simple to use simulated distributions. To give an idea of the 24
behaviour of the number of draw downs in the case of iid returns, Figure III plots their mass function when p = .05, .25, .5, .75, .95 for a sample of 50 trades. [F igureIII] It is apparent that we get a bell shaped distribution for p ' .5, a number we would expect to get for daily data.
8
Almost Sure Behaviour of the Maximum Draw Down under Condition 1
The previous Sections have considered the draw downs from a distributional prospective. In some cases, it is of great interest to infer the asymptotic behaviour of the maximum draw down; in particular, convergence in probability and almost sure convergence. Under the assumption of iid random variables, possibly with lattice distribution, this problem has been considered by Frolov et al. (1998). In particular we are interested in their Theorem 2, which provides rates of convergence for the maximal gain in N trials. While the conditions of this theorem are automatically satisfied for the cases of interest of this paper, we state the relevant condition first, but using our notation. Condition 2. Let ϕ (t) = E (exp {X − t}). The following holds t0 = sup {t : ϕ (t) < ∞} > 0, ¯ ¯ E ¯X − ¯ < ∞,
¢ ¡ Pr X − = x < 1, ∀x.
Remark. The last condition is automatically satisfied in our context as we are considering non-degenerate continuous distributions, i.e. Pr (X − = x) = 0, ∀x. Theorem A. (Frolov et al., 1998, Theorem 2) Let ¾ ½ 1 ∗ > 0. t = sup t ≥ 0 : ϕ (t) ≤ 1−p 25
Then, under Conditions 1-2, if t∗ < t0 , p
max YN,j → j
ln N − (1/2) ln ln N t∗ a.s
lim sup max YN,j → j
N →∞
a.s
lim inf max YN,j → j
N →∞ p
ln N + (1/2) ln ln N t∗ ln N − (1/2) ln ln N , t∗
a.s
where → and → stand for convergence in probability and almost surely.
8.1
Examples (cont’d)
We consider the application of Theorem A to the examples in Section 5, i.e. when X − has a Gamma or exponential distribution. In both cases, Condition 2 is satisfied. Further, the moment generating function is given by µ
a a−t
¶b
,
where b = 1 in the exponential case. Therefore, we only consider the Gamma distribution. By monotonicity, we have t∗ < t0 , so that the condition of the Theorem is satisfied. In particular, t∗ is given by the solution of the following equation µ which implies
a a−t
¶b
=
1 , 1−p
i h 1 b t = a 1 − (1 − p) . ∗
Therefore,
p
max YN,j → j
and
ln N − (1/2) ln ln N i , h 1 a 1 − (1 − p) b
ln N + (1/2) ln ln N ln N − (1/2) ln ln N h h i ≤ lim max YN,j ≤ i , 1 1 N→∞ j b b a 1 − (1 − p) a 1 − (1 − p)
almost surely.
26
(22a)
9
Draw Downs Versus Large Crashes
The distribution of the draw downs and the maximum draw downs allow us to assess the probability of some large losses of a portfolio. However, here we want to remark that there are unexpected events which cannot be captured by a frequentist approach (i.e. an approach based on the frequency of events). We refer to large financial crashes. In the second half of the 20th century, we recall the October 1987 crash. Despite the probability distribution of losses being correctly specified, there is no way to consistently include such a crash in the tail behavior of the losses. We use the distribution of the maximum draw down to corroborate this claim. We consider a period of roughly 35 years (January 1965-August 1999) for the S&P futures. While large falls may be explicable, the largest draw down (33% related to the 1987 crash) is, in any sense of the term, an outlier. Table III reports the 40 largest draw downs in descending order (out of a total of 1,943). It is clear that the largest value does not compare to the others in any meaningful sense using a straight frequentist approach. [T ableIII] To further explore this claim, we use the KST distribution to estimate the left tail of the distribution of the S&P log returns. We use this because it usually gives a very good fit for the tails of the distribution of financial returns, under all but extreme conditions. We obtain the following values: a = 2.034, b = 1.338 and p = .5450. With these values, we have that Pr (max YN ≤ 33) = .9999995700, with dN = 1947 (the actual number of draw downs out of N = 8685 trades). This implies that a draw down equal or greater than 33% may happen once every 2325, 581 draw downs which is equivalent to once every 20.59 centuries (assuming 252 trading days in a year and .2232 draw downs per day; see Table II). Similar conclusions have been given in Johansen et al. (2001). However, these authors did not use the distribution for the maximum draw down as a means for their inference. On the other hand they directly used the distribution of the draw dawns (which 27
is not the correct distribution to use in this case). Even using the distribution of the maximum, the prediction of the worse day in 22 centuries of daily trading contradicts empirical evidence. To further investigate this issue, we use the results in Section 8. Plugging our estimated values of a, b and p into (22a) we have t∗ = .905; then, Theorem A, for N = 8685, implies the following bounds almost surely 8.80 ≤ max YN,j ≤ 11.24. j∈DN
Again, the largest draw down in Table III is inconsistent with the assumed Gamma distribution. Given the good fit that this distribution can provide, especially on the tails, this shows that the largest draw down has to be considered as an outlier. There have been other two large market falls in the 20th century which are roughly comparable with the October 1987: the crash following the outbreak of the first world war and the great crash of 1929. However, using the distribution of the draw downs to correctly assess the probability of such events is totally misleading. Consequently, such events should be considered as outliers. Two explanations are plausible. First, the data are not iid, for which there is much evidence. Second, within the class of non-iid phenomena, regimes occur so that the usual history of returns cannot explain the non-typical behaviour of crises.
10
Conclusion
This paper considered a measure of point loss which is particularly relevant to hedge fund risk. This is called the draw down. Using a number of definitions of draw downs, we provided a fairly exhaustive analysis of its properties and its maximum under iid conditions and dependence in the sign of the series, but not in its magnitude. Our results are a combination of exact analytical approaches together with approximate and simulated ones. Our purpose was to provide the simplest easy to use results which could have direct applications in risk evaluation. For this reason, simple approximations may be of much value as opposed to complex 28
exact symbolic results. Furthermore, results that do not assume Brownian motion seem to be the relevant ones. As we developed a taxonomy for these measures of risk and loss, our analysis tried to cover each case separately to provide a clear picture of their statistical properties and applications. As mentioned in the paper, there is a clear link between the assessment of trading rules based on stop limit orders and the statistics defined in terms of the draw downs. We also combined our theoretical results with an empirical analysis for the log returns on the S&P 500 futures. This unveiled interesting features like the aggregation invariance property in the mean number of draw downs over a time interval. We also remarked on the limitations of the straight frequentist approach when trying to capture extreme rare events like the October 1987 crash.
References [1] Acar, E. and S.E. Satchell (1997) A Theoretical Analysis of Trading Rules: An Application to the Moving Average Case with Markovian Returns. Applied Mathematical Finance 4, 165-180. [2] Aldous, D. (1989) Probability Approximations via the Poisson Clumping Heuristic. New York: Springer-Verlag. [3] Chekhlov, A., S. Uryasev and M. Zabarankin (2000) Portfolio Optimization with Drawdown Constraints. Preprint, Department of Industrial and System Engineering, University of Florida. [4] Duadi, R., A.N. Shiryaev and M. Yor (2000) On the Probability Characteristics of ”Downfalls” in a Standard Brownian Motion. Theory of Probability and Applications 44, 29-38. [5] Frolov, A., A. Martikainen and J. Steinebach (1998) On the Maximal Gain over Head Runs. Studia Scientiarum Mathematicarum Hungarica 34, 165-181. 29
[6] Fu, J.C. and M.V. Koutras (1994) Distribution Theory of Runs: A Markov Chain Approach. Journal of the American Statistical Association 89, 10501058. [7] Grossman, S and Z. Zhou (1993) Optimal Investment Strategies for Controlling Drawdowns. Mathematical Finance 3, 241-276. [8] Johansen, A and D. Sornette (2001) Large Stock Market Price Drawdowns are Outliers. Journal of Risk 4, 69-110. [9] Johansen, A., O. Ledoit and D. Sornette (2001) Crashes as Critical Points. International Journal of Theoretical and Applied Finanace 3, 219-255. [10] Knight, J., S. Satchell and K. Tran (1995) Statistical Modelling of Asymmetric Risk in Asset Returns. Applied Mathematical Finance, 2, 155-172. [11] Leadbetter, M.R., G. Lindgren and H. Rootz´en (1983) Extremes and Related Properties of Random Sequences and Processes. New York: Springer. [12] Leadbetter, M.R. and H. Rootz´en (1988) Extremal Theory for Stochastic Processes. The Annals of Probability 16, 431-478. [13] Loynes, R.M. (1965) Extreme Values in Uniformly Mixing Stationary Stochastic Processes. Annals of Mathematical Statistics 36, 993-999. [14] Magdon-Ismail, M., A.F. Atiya, A. Pratap and Y.S. Abu-Mostafa (2002) The Sharpe Ratio, Range and Maximal Drawdown of a Brownian Motion. Rensselaer Polytechnic Institute, Computer Science Technical Report. [15] Maslov, S. and Y.C. Zhang (1999) Probability Distribution of Drawdowns in Risky Investments. Physica A262, N1-2, 232-241. [16] Mood, A.M. (1940) The Distribution Theory of Runs. Annals of Mathematical Statistics 11, 367-392.
30
[17] Pickands, J. (1969) Asymptotic Properties of the Maximum in a Stationary Gaussian Process. Transactions of the American Mathematical Society 145, 75-86. [18] Stefanov, V.T. (2000) On Run Statistics for Binary Trials. Journal of Statistical Planning and Inference 87, 177-185. [19] Vanmarcke, E.H. (1983) Random Fields. Cambridge, MA: MIT Press.
31
Table I. Error in the Poisson Approximation dN
50
200
800
1500
max 0.021 0.020 0.017 0.036
32
Table II. Mean Number of D.D. Daily
Weekly Monthly
0.2232
0.2418
33
0.2292
.
Table III. 40 Largest D.D. for the S&P 500 33.49 13.22 10.32 10.32 10.2 10.08 9.87 9.59 9.08 8.66
8.17 7.84 7.81 7.56 7.56 7.29 6.98 6.83 6.81 6.77
6.72 6.45 6.39 6.29 6.28 6.24 6.21 6.15 6.15 6.03
34
5.99 5.92 5.91 5.89 5.87 5.86 5.86 5.85 5.8 5.76
Figure I. Empirical Distribution and its Approximation (dN = 50)
1.0
Probability
0.8
0.6
0.4
0.2
0.0
5
10
15 Draw Down
35
20
Figure II. Empirical Mass Function for the Number of D.D. 173 Samples of size N = 50 of S&P Daily Data
0.25
Frequency
0.20
0.15
0.10
0.05
0.00
7
9
11 13 Number of Draw Downs
36
15
Figure III. Simulated Mass Function For the Number of D.D. out of 50 Trades p = .05
p = .25
0.25
0.20
0.20
Frequency
Frequency
0.15 0.15
0.10
0.10 0.05 0.05
0.00
0.00
1
3 5 Number of Draw Downs
7
0
2
0.15
0.15 Frequency
0.20
0.10
0.05
0.00
0.00
8
8 10 Number of Draw Downs
12
14
0.10
0.05
6
6
p = .75
0.20
10 12 14 Number of Draw Downs
16
18
2.5
5.0
p = .95
0.25
0.20
Frequency
Frequency
p = .5
4
0.15
0.10
0.05
0.00
1
3 5 Number of Draw Downs
37
7
7.5 10.0 12.5 Number of Draw Downs
15.0
17.5
16
