A General Framework for Portfolio Theory. Part II: Drawdown Risk Measures

The aim of this paper is to provide several examples of convex risk measures necessary for the application of the general framework for portfolio theory of Maier-Paape and Zhu, presented in Part I of this series (arXiv:1710.04579 [q-fin.PM]). As alternative to classical portfolio risk measures such as the standard deviation we in particular construct risk measures related to the current drawdown of the portfolio equity. Combined with the results of Part I (arXiv:1710.04579 [q-fin.PM]), this allows us to calculate efficient portfolios based on a drawdown risk measure constraint.


Introduction
Modern portfolio theory due to Markowitz [13] has been the state of the art in mathematical asset allocation for over 50 years. Recently, in Part I of this series (see Maier-Paape and Zhu [12]), we generalized portfolio theory such that efficient portfolios can now be considered for a wide range of utility functions and risk measures. The so found portfolios provide an efficient trade-off between utility and risk just as in the Markowitz portfolio theory. Besides the expected return of the portfolio, which was used by Markowitz, now general concave utility functions are allowed, e.g. the log utility used for growth optimal portfolio theory (cf. Kelly [6], Vince [16], [17], Vince and Zhu [19], Zhu [21,22], Hermes and Maier-Paape [5]). Growth optimal portfolios maximize the expected log returns of the portfolio yielding fastest compounded growth.
Besides the generalization in the utility functions, as a second breakthrough, more realistic risk measures are now allowed. Whereas Markowitz and also the related capital market asset pricing model (CAPM) of Sharpe [15] use the standard deviation of the portfolio return as risk measure, the new theory of Part I in [12] is applicable to a large class of convex risk measures.
The aim of this Part II is to provide and analyze several such convex risk measures related to the expected log drawdown of the portfolio returns. Drawdown related risk measures are believed to be superior in yielding risk averse strategies when compared to the standard deviation risk measure. Furthermore, empirical simulations of Maier-Paape [10] have shown that (drawdown) risk averse strategies are also in great need when growth optimal portfolios are considered since using them regularly generates tremendous drawdowns (see also van Tharp [20]). A variety of examples will be provided in Part III [1].
The results in this Part II are a natural generalization of Maier-Paape [11], where drawdown related risk measures for a portfolio with only one risky asset were constructed. In that paper, as well as here, the construction of randomly drawn equity curves, which allows the measurement of drawdowns, is given in the framework of the growth optimal portfolio theory (see Section 3 and furthermore Vince [18].) Therefore, we use Section 2 to provide basics of the growth optimal theory and introduce our setup.
In Section 4 we introduce the concept of admissible convex risk measures, discuss some of their properties and show that the "risk part" of the growth optimal goal function provides such a risk measure. Then, in Section 5 we apply this concept to the expected log drawdown of the portfolio returns. It is worth to note that some of the approximations of these risk measures yield, in fact, even positively homogeneous risk measures, which are strongly related to the concept of deviation measures of Rockafellar, Uryasev and Zabarankin [14]. According to the theory of Part I [12] such positively homogeneous risk measures provide -as in the CAPM model -an affine structure of the efficient portfolios when the identity utility function is used. Moreover, often in this situation even a market portfolio, i.e. a purely risky efficient portfolio, related to drawdown risks can be provided as well.
Finally, note that the main Assumption 2.3 on the trade return matrix T of (2.1) together with a no arbitrage market provides the basic market setup for application of the generalized portfolio theory of Part I [12]. This is shown in the Appendix (Corollary A.11). In fact, the appendix is used as a link between Part I and Part II and shows how the theory of Part I can be used with risk measures constructed here. Nonetheless, Parts I and II can be read independently.

Acknowledgement:
We thank René Brenner for support in generating the contour plots of the risk measures and Andreas Platen for careful reading of an earlier version of the manuscript.

Setup
For 1 ≤ k ≤ M, M ∈ N, we denote the k-th trading system by (system k). A trading system is an investment strategy applied to a financial instrument. Each system generates periodic trade returns, e.g. monthly, daily or the like. The net trade return of the i-th period of the k-th system is denoted by t i,k , 1 ≤ i ≤ N, 1 ≤ k ≤ M . Thus, we have the joint return matrix period (system 1) (system 2) · · · (system M) and we denote For better readability, we define the rows of T , which represent the returns of the i-th period of our systems, as Following Vince [17], for a vector of portions ϕ := (ϕ 1 , . . . , ϕ M ) , where ϕ k stands for the portion of our capital invested in (system k), we define the Holding Period Return (HPR) of the i-th period as where ·, · is the scalar product in R M . The Terminal Wealth Relative (TWR) representing the gain (or loss) after the given N periods, when the vector ϕ is invested over all periods, is then given as Since a Holding Period Return of zero for a single period means a total loss of our capital, we restrict TWR N : G → R to the domain G given by the following definition: Moreover, we define Note that in particular 0 ∈ • G (the interior of G) and R = ∂G, the boundary of G. Furthermore, negative ϕ k are in principle allowed for short positions.
Lemma 2.2. The set G in Definition 2.1 is polyhedral and thus convex, as is defines a half space (which is convex). Since G is the intersection of a finite set of half spaces, it is itself convex, in fact even polyhedral. A similar reasoning yields that In the following we denote by S M −1 1 := ϕ ∈ R M : ϕ = 1 the unit sphere in R M , where · denotes the Euclidean norm.

Assumption 2.3. (no risk free investment)
We assume that the trade return matrix T in (2.1) satisfies In other words, Assumption 2.3 states that no matter what "allocation vector" θ = 0 is used, there will always be a period i 0 resulting in a loss for the portfolio. implies that −θ ∈ S M −1 1 , Assumption 2.3 also yields the existence of a period j 0 resulting in a gain for each θ ∈ S M −1 (c) It is not important whether or not the trading systems are profitable, since we allow short positions (cf. Assumption 1 in [5]).
Lemma 2.5. Let the return matrix T ∈ R N ×M (as in (2.1)) satisfy Assumption 2.3. Then the set G in (2.3) is compact.
Proof. Since G is closed the lemma follows from (2.5) yielding HPR i 0 (sθ) < 0 for s > 0 sufficiently large. Thus G is bounded as well.

Randomly drawing trades
Given a trade return matrix, we can construct equity curves by randomly drawing trades.
Setup 3.1. (trading game) Assume trading systems with trade return matrix T from (2.1). In a trading game the rows of T are drawn randomly. Each row t i · has a prob- Drawing randomly and independently K ∈ N times from this distribution results in a probability space Ω (K) := ω = (ω 1 , . . . , ω K ) : ω i ∈ {1, . . . , N } and a terminal wealth relative (for fractional trading with portion ϕ is used) In the rest of the paper we will use the natural logarithm ln.
is the weighted geometric mean of the holding is independent of j because each ω j is an independent drawing. We thus obtain Next we want to split up the random variable Z (K) (ϕ, ·) into chance and risk parts. Since TWR K 1 (ϕ, ω) > 1 corresponds to a winning trade series t ω 1 · , . . . , t ω K · and TWR K 1 (ϕ, ω) < 1 analogously corresponds to a losing trade series we define the random variables corresponding to up trades and down trades: Up-trade log series: Down-trade log series: . Hence by Theorem 3.2 we get holds.
As in [11] we next search for explicit formulas for E U (K) (ϕ, ·) and E D (K) (ϕ, ·) , respectively. By definition Assume ω = (ω 1 , . . . , ω K ) ∈ Ω (K) := {1, . . . , N } K is for the moment fixed and the random variable X 1 counts how many of the ω j are equal to 1, i.e. X 1 (ω) = x 1 if in total x 1 of the ω j 's in ω are equal to 1. With similar counting random variables X 2 , . . . , X N we obtain integer counts x i ≥ 0 and thus Hence for this fixed ω we obtain Therefore the condition on ω in the sum (3.6) is equivalently expressed as To better understand the last sum, Taylor expansion may be used exactly as in Lemma 4.5 of [11] to obtain G be a vector of admissible portions where θ ∈ S M −1 1 is fixed and s > 0. Then there exists some ε > 0 (depending on x 1 , . . . , x N and θ) such that for all s ∈ (0, ε] the following holds: and ∂ 2 ∂s 2 h(0, θ) < 0. With Lemma 3.5 we hence can restate (3.9). For θ ∈ S M −1 1 and all s ∈ (0, ε] the following holds (3.10) Note that since Ω (K) is finite and S M −1 1 is compact, a (maybe smaller) ε > 0 can be found such that (3.10) holds for all s ∈ (0, ε] , θ ∈ S M −1 1 and ω ∈ Ω (K) . Remark 3.6. In the situation of Lemma 3.5 furthermore After all these preliminaries, we may now state the first main result. For simplifying the notation, we set N 0 := N ∪ {0} and introduce  . Then there exists an ε > 0 such that for all s ∈ (0, ε] the following holds: and with H (K,N ) from (3.12).
Proof. E U (K) (sθ, ·) ≥ 0 is clear from (3.3) even for all s ≥ 0. The rest of the proof is along the lines of the proof of the univariate case Theorem 4.6 in [11], but will be given for convenience. Starting with (3.6) and using (3.7) and (3.10) we get for s ∈ (0, ε] Theorem 3.8. We assume that the conditions of Theorem 3.7 hold. Then: i.e. d (K) (s, θ) is always an upper bound for the expectation of the down-trade log series. Proof. of Theorem 3.8: The arguments given in the proof of Theorem 3.7 apply similarly, where instead of (3.10) we use Lemma 3.5 (b) to get for s ∈ (0, ε] ad(b) According to the extension of Lemma 3.5 in Remark 3.6, we also get for all ω with (3.20). Therefore, no matter how large s > 0 is, the summands of d (K) (s, θ) in (3.15) will always contribute to E D (K) (sθ, ·) in (3.18), but -at least for large s > 0 -there may be even more (negative) summands from other ω. Hence (3.17) follows for all s > 0.
Remark 3.10. Using multinomial distribution theory and (3.12) holds and yields (again) with Theorem 3.7 and 3.8 for s ∈ (0, ε] Remark 3.11. Using Taylor expansion in (3.15) we therefore obtain a first order approximation in s of the expected down-trade log series D (K) (sθ, ·) (3.4), i.e. for s ∈ (0, ε] and θ ∈ S M −1 1 the following holds: In the sequel we call d (K) the first and d (K) the second approximation of the expected down-trade log series. Noting that ln(1 + x) ≤ x for x ∈ R when we extend ln is continuous in s and θ (in s even positive homogeneous) and Proof. (a) is already clear with the statement above. To show (b), the continuity in s of the second approximation is clear. But even continuity in θ follows with a short argument: Using (3.16) x n t n · , θ is continuous in θ , L (K,N ) (θ) is continuous, too, and clearly d (K) is non-positive.

Admissible convex risk measures
For the measurement of risk, various different approaches have been taken (see for instance [3] for an introduction). For simplicity, we collect all for us important properties of risk measures in the following three definitions.   Remark 4.4. It is easy to see that an admissible strictly convex risk measure automatically satisfies (c) in Definition 4.1 and thus it is also an admissible convex risk measure. In fact, if u > s > 0 then s = λu for some λ ∈ (0, 1) and we obtain for θ ∈ S M −1   In order to define a nontrivial ACRM, we use the down-trade log series of (3.4).
stemming from the down-trade log series in (3.4), is an admissible convex risk measure (ACRM).
Proof. We show that r down has the three properties (a), (b), and (c) from Definition 4.1.
is continuous in ϕ, and therefore the same holds true for r down . Moreover, again for ω ∈ Ω (K) fixed, ϕ → ln TWR K 1 (ϕ, ω) = K j=1 ln 1 + t ω j · , ϕ is a concave function of ϕ since all summands are as composition of the concave ln-function with an affine function also concave. Thus D (K) (ϕ, ω) is concave as well since the minimum of two concave functions is still concave and therefore r down is convex.
ad (c) It is sufficient to show that r down from (4.1) is strictly convex along the line sθ 0 : Therefore, let θ 0 ∈ S M −1 1 be fixed. In order to show (4.2) we need to find at least one which is a strictly concave function in s > 0.
It is easy to see that bets in the first system (win 1 with probability 0.5 or lose − 1 2 ) and bets in the second system (win 1 with probability 0.75 or lose −2) are stochastically independent and have the same expectation value 1 4 . The contour levels of r down for K = 5 are shown in Figure 1.
the risk measure r down in (4.1) for K = 1 is not strictly convex. Consider for example Then for ϕ ∈ B ε (ϕ 0 ) , ε > 0 small, in the trading game only the third row results in a loss, i.e.
which is constant along the line ϕ s = ϕ 0 + s · 1 −1 ∈ B ε (ϕ 0 ) for small s and thus not strictly convex.
(b) We refrain from giving a complete characterization for trade return matrices T for which (4.1) results in a strictly convex function, but only note that if besides Assumption 2.3 the condition then this is sufficient to give strict convexity of (4.1) and hence in this case r down in (4.1) is actually an ASCRM. Now that we saw that the negative expected down-trade log series of (4.1) is an admissible convex risk measure, it is natural to ask whether or not the same is true for the two approximations of the expected down-trade log series given in x i t i · , θ = 0, but unlike in (3.25) for d (K) , the sum over the log terms may not vanish. Therefore d (K) (s, θ) is in general also not continuous. A more thorough discussion of this discontinuity can be found after Theorem 4.10. On the other hand, d (K) of (3.22) was proved to be continuous and non-positive in Corollary 3.12. In fact, we can obtain: is concave because the right hand side is concave (see Theorem 4.7). Hence is also concave. Note that right from the definition of d (K) (s, θ) in (3.15) and of L (K,N ) (θ) in (3.25) it can readily be seen that for θ ∈ S M −1 Therefore, some further calculation yields uniform convergence x i t i · , θ = K t i 0 · , θ < 0 giving L (K,N ) (θ) < 0 as claimed.
We illustrate the contour of r downX for Example 4.8 in Figure 2. As expected, the approximation of r down is best near ϕ = 0 (cf. Figure 1). In conclusion, Theorems 4.7 and 4.10 yield two ACRM stemming from expected downtrade log series D (K) of (3.4) and its second approximation d (K) from (3.22). However, the first approximation d (K) from (3.15) was not an ACRM since the coefficients D   x j ln 1 + s t j · , θ < 0 (4.7) and using the characteristic function of a set A, χ A , we obtain for all sθ ∈ x n · ln 1 + s t n · , θ (4.8) giving (4.6).
Observe that d (K) (s, θ) has a similar representation, namely, using we get right from the definition in (3.15) that for all sθ ∈ x n ln 1 + s t n · , θ (4.10) holds. So the only difference of (4.8) and (4.10) is that Ξ x 1 ,...,x N (s) is replaced by Ξ x 1 ,...,x N (with the latter being a half-space restricted to S M −1 1 ). Observing furthermore that due to (3.21) the discontinuity of d (K) clearly comes from the discontinuity of the indicator function x n ln 1 + s t n · , θ = 0 and the "mystery" is solved since Lemma 3.5(b) implies equality in (4.11) for sufficiently small s > 0. Finally note that for large s > 0 not only the continuity gets lost, but moreover d (K) (s, θ) is no longer concave. The discontinuity can even be seen in Figure 3.

The current drawdown
We keep discussing the trading return matrix T from (2.1) and probabilities p 1 , . . . , p N from Setup 3.1 for each row t i · of T . Drawing randomly and independently K ∈ N times such rows from that distribution results in a terminal wealth relative for fractional trading depending on the betted portions ϕ = (ϕ 1 , . . . , ϕ M ), see (3.1). In order to investigate the current drawdown realized after the K-th draw, we more generally use the notation TWR n m (ϕ, ω) := n j=m 1 + t ω j · , ϕ .
The idea here is that TWR n 1 (ϕ, ω) is viewed as a discrete "equity curve" at time n (with ϕ and ω fixed). The current drawdown log series is defined as the logarithm of the drawdown of this equity curve realized from the maximum of the curve till the end (time K). We will see below that this series is the counter part of the run-up (cf. Figure  4). Definition 5.1. The current drawdown log series is set to and the run-up log series is defined as The corresponding trade series are connected because the current drawdown starts after the run-up has stopped. To make that more precise, we fix that where the run-up reached its top.
where * should be minimal with that property.
As discussed in Section 4 for the down-trade log series, we also want to study the current drawdown log series (5.2) with respect to admissible convex risk measures. is an admissible convex risk measure (ACRM).
Proof. It is easy to see that the proof of Theorem 4.7 can almost literally be adapted to the current drawdown case.
Confer Figure 5 for an illustration of r cur . Compared to r down in Figure 1 the contour plot looks quite similar, but near 0 ∈ R M obviously r cur grows faster. Similarly, we obtain an ACRM for the first order approximationd is an admissible convex risk measure (ACRM) according to Definition 4.1 which is moreover positive homogeneous.
Proof. We use (5.14) to derive the above formula for L (K,N ) cur (θ). Now most of the arguments of the proof of Theorem 4.10 work here as well once we know that L (K,N ) cur (θ) is continuous in θ. To see that, we remark once more that for the first topping point * = * (θ, ω) ∈ {0, . . . , K} of the linearized equity curve n j=1 t ω j · , θ , n = 1, . . . , K, the following holds (cf. Definition 5.5 and (5.18)): Thus Although the topping pointˆ * (θ, ω) for ω ∈ Ω (K) may jump when θ is varied in case  A contour plot of r curX can be seen in Figure 6. The first topping point of the linearized equity curve will also be helpful to order the risk measures r cur and r curX . Reasoning as in (5.10) (see also Lemma 3.5) and using that (5.18) we obtain in case * < K for s ∈ (0, ε] and k = * + 1, . . . , K that However, since ln is concave, the above implication holds true even for all s > 0 with ϕ = sθ ∈ • G. Hence for k = * + 1, . . . , K and ϕ = sθ ∈ Looking at (5.9) once more, we observe that the first topping point of the TWR equity curve * necessarily is less than or equal to * . Thus we have shown: we have The second inequality in (5.30) follows as in Section 3 from ln(1 + x) ≤ x (see (5.12) and (5.24)) and the third inequality is already clear from Remark 5.7.

Conclusion
Let us summarize the results of the last Sections. We obtained two down-trade log series related admissible convex risk measures (ACRM) according to Definition 4.1, namely All four risk measures can be used in order to apply the general framework for portfolio theory of [12]. Since the two approximated risk measures r downX and r curX are positive homogeneous, according to [12], the efficient portfolios will have an affine linear structure. Although we were able to prove a lot of results for these for practical applications relevant risk measures, there are still open questions. To state only one of them, we note that convergence of these risk measures for K → ∞ is unclear, but empirical evidence seems to support such a statement (see Figure 7). Assumption A.2. To avoid redundant risky assets, often the matrix is assumed to have full rank M , in particular N ≥ M .
A portfolio is a column vector x ∈ R M +1 whose components x m represent the investments in the m-th asset, m = 0, . . . , M . In order to normalize that situation, we consider portfolios with unit initial cost, i.e.
Since S 0 0 = 1 this implies Therefore the interpretation in Table 1 is obvious.  So if an investor has an initial capital of C ini in his depot, the invested money in the depot is divided as in Table 2.
Clearly (S 1 − R S 0 )·x = S 1 ·x−R is the (random) gain of the unit initial cost portfolio relative to the riskless bond. In such a situation the merit of a portfolio x is often measured by its expected utility E u (S 1 · x) , where u is an increasing concave utility function (see [12], Assumption 3.3). In growth optimal portfolio theory the natural logarithm u = ln is used yielding the optimization problem The following discussion aims to show that the above optimization problem (A.4) is an alternative way of stating the Terminal Wealth Relative optimization problem of Vince (cf. [5], [16]).
Assuming all α ∈ A N have the same probability (Laplace situation), i.e.
we furthermore get This results in a "trade return" matrix ϕ m portion of capital invested in m-th risky asset, m = 1, . . . , M Table 3: Investment vector ϕ for the TWR model whose entries represent discounted relative returns of the m-th asset for the i-th realization α i . Furthermore, the column vector ϕ = (ϕ m ) 1≤m≤M ∈ R M with components ϕ m = S m 0 x m has according to Table 1 the interpretation given in Table 3. Thus we get which involves the usual Terminal Wealth Relative (TWR) of Ralph Vince [16] and therefore under the assumption of a Laplace situation (A.5) the optimization problem (A.4) is equivalent to Furthermore, the trade return matrix T in (A.7) may be used to define admissible convex risk measures as introduced in Definition 4.1 which in turn give nontrivial applications to the general framework for portfolio theory in Part I [12].
To see that, note again that by (A.6) any portfolio vector x = (x 0 , x) T ∈ R M of a unit cost portfolio (A.2) is in one to one correspondence to an investment vector for a diagonal matrix Λ ∈ R M ×M with only positive diagonal entries Λ m,m = S m 0 . Then we obtain: (r2) r is convex in x.
(r3) The two approximations r downX and r curX furthermore yield positive homogeneous r, i.e. r (t x) = t r ( x) for all t > 0.
Remark A.4. It is clear that therefore r = r down , r downX , r cur or r curX can be evaluated on any set of admissible portfolios A ⊂ R M +1 according to Definition 2.2 of [12] if and the properties (r1), (r1n), (r2) (and only for r downX and r curX also (r3)) in Assumption 3.1 of [12] follow from Theorem A.3. In particular r downX and r curX satisfy the conditions of a deviation measure in [14] (which is defined directly on the portfolio space).
Remark A.5. Formally our drawdown or down-trade is a function of a TWR equity curve of a K-period financial market. But since this equity curve is obtained by drawing K times stochastically independent from one and the same market in Setup A.1, we still can work with a one-period market model.
We want to close this section with some remarks on the often used no arbitrage condition of the one-period financial market and Assumption 2.3 which was necessary to construct admissible convex risk measures.
Definition A.6. Let S t be a one-period financial market as in Setup A.1.
(a) A portfolio x ∈ R M +1 is an arbitrage for S t if it satisfies S 1 − RS 0 · x ≥ 0 and S 1 − RS 0 · x = 0 , (A. 12) or, equivalently, if x ∈ R M is satisfying The market S t is said to have no arbitrage, if there exists no arbitrage portfolio.
Once we consider the above random variables as vector S 1 (α i ) − R S 0 · x 1 ≤ i ≤ N ∈ R N , (A.13) may equivalently be stated as where we used the positive cone K := y ∈ R N : y i ≥ 0 for i = 1, . . . , N in R N . Observe that the portfolio x = 0 can never be an arbitrage portfolio. Hence we get: market S t has no arbitrage where we used the negative cone (−K) = y ∈ R N : y i ≤ 0 for i = 1, . . . , N in R N .
Note that the last equivalence leading to (A.16) follows, because with x ∈ R M \ {0} always (− x) ∈ R M \ {0} also holds true. According to Setup A.1, the matrix T S has full rank, and therefore S 1 − R S 0 · x = 0 for all x = 0 anyway. Hence we proceed and some α j 0 ∈ A N with S 1 (α j 0 ) − R S 0 · x > 0 . A very similar theorem is derived in [12]. For completeness we rephrase here that part which is important in the following.  [12], Theorem 3.9, and Assumption A.2 is exactly the point (iii) of that theorem. Therefore, under the no arbitrage assumption again by [1], Theorem 3.9, the claimed equivalence follows.
Together with Theorem A.7 we immediately conclude: Corollary A.9. (two out of three imply the third) Let a one-period financial market S t according to Setup A.1 be given. Then any two of the following conditions imply the third: (c) Assumption A.2 holds, i.e. T S from (A.1) has full rank M . Remark A. 10. The standard assumption on the market S t in Part I [12] is "no nontrivial riskless portfolio", where a portfolio x = (x 0 , x) T ∈ R M +1 is riskless if (S 1 − R S 0 ) · x ≥ 0 and x is nontrivial if x = 0.
Using this notation we get: Corollary A.11. Consider a one-period financial market S t as in Setup A.1. Then there is no nontrivial riskless portfolio in S t if and only if any two of the three statements (a), (b), and (c) from Corollary A.9 are satisfied.
To conclude, any two of the three conditions of Corollary A.9 on the market S t are sufficient to apply the theory presented in Part I [12].