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A nonlinear optimisation model for constructing minimal drawdown portfolios

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A portfolio built to minimise drawdown can beat its market index on return, Sharpe ratio, maximum drawdown, and average drawdown out of sample.

desk verdict A clean, honest formulation for rolling drawdown minimization whose headline out-of-sample edge is gross of costs and probably fragile to realistic frictions. read the letter →

arxiv 1908.08684 v1 pith:J7ERNJA4 submitted 2019-08-23 q-fin.RM math.OCq-fin.PM

C.A. Valle , J.E. Beasley This is my paper · ORCID
classification q-fin.RMmath.OCq-fin.PM MSC 91G1090C30
keywords portfoliodrawdownnonlinearprogrammingoptimisationindexout-performancerebalancingSharperatiopartiallinearisationlong-shortportfolios
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that a portfolio constructed to minimise drawdown—the percentage of value lost from the best recent peak—can beat the market index on four measures at once: return, Sharpe ratio, maximum drawdown, and average drawdown, out of sample. The strategy uses only the previous 30 days of prices, rebalances every 10 trading days, and was tested on three major equity indices over roughly seven years of daily data. The claim matters because drawdown is a path-dependent risk measure that standard mean-variance optimisation does not capture; two portfolios with identical mean and variance can have very different drawdowns. If the reported dominance holds, drawdown minimisation is a tractable, data-driven route to index out-performance.

What carries the argument

The load-bearing object is the path-dependent drawdown sequence $\{d_t\}$, defined against a rolling maximum $M_t$ over the last $D$ periods. The nonlinear parts of the model are the maximum in $M_t$ and the fraction in $d_t$; the paper partially linearises them by writing $M_t \ge P_\tau$ for every $\tau$ in the lookback window and $d_t \ge 100(M_t-P_t)/M_t$. Since the objective is monotone decreasing in drawdown, an optimal solution forces these inequalities to hold with equality, so no maximum operator or division needs to be solved directly. That replacement is what makes the formulation tractable enough for a nonlinear programming solver to find proven global optima in a large fraction of the rebalances, and it is the technical step the computational results rely on.

What would settle it

Recompute the same rebalancing strategy while charging a conservative transaction cost, for example 10 to 50 basis points of traded value, and compare net out-of-sample average daily return and Sharpe ratio with the index; if net performance no longer beats the index, the headline dominance result fails.

Watch

Extended reading notes

Core claim

The central discovery is that minimal-drawdown portfolios, obtained by solving a nonlinear program that minimises either average drawdown, maximum drawdown, or a weighted combination of the two, dominate their market index out of sample. Drawdown at time $t$ is defined as $d_t = 100(M_t-P_t)/M_t$, where $P_t$ is portfolio value and $M_t$ is the maximum value over the current and preceding $D$ periods, so it is the opportunity cost of the single best missed sale-and-repurchase trade. The paper shows that the nonlinear maximum defining $M_t$ can be replaced by linear inequalities $M_t \ge P_\tau$, and the equality defining $d_t$ can be relaxed to $d_t \ge 100(M_t-P_t)/M_t$; because the objective pushes drawdown down, these inequalities are tight at an optimum. With $T=30$, $D=20$, rebalancing every 10 days, and zero transaction costs, the long-only portfolios beat the index on average daily return, annualised Sharpe ratio, maximum drawdown, and average drawdown over roughly 1,800 out-of-sample days. Allowing short positions improves in-sample performance but does not consistently improve out-of-sample returns.

Load-bearing premise

The load-bearing assumption is that transaction costs are zero, while the strategy rebalances every 10 days across up to 500 assets; realistic trading costs would reduce the reported out-of-sample excess return and could overturn the dominance claim.

Editorial extensions

If this is right

  • If the out-of-sample dominance holds, drawdown minimisation is a viable stand-alone index out-performance rule that needs only recent prices and an optimiser, not forecasts or scenario paths.
  • The objective can be tuned between average drawdown, maximum drawdown, or a weighted mix, so an investor can choose whether to penalise frequent shallow dips or rare deep troughs.
  • Long-only minimal-drawdown portfolios appear preferable to allowing short positions under the tested parameter settings, since shorting helped in-sample metrics but did not consistently help out-of-sample returns.
  • The formulation already includes cash inflows and outflows as well as transaction-cost constraints, so it extends to realistic rebalancing settings with modest changes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper does not test realistic transaction costs; because the strategy rebalances every 10 days across up to 500 assets, charging even a few basis points per trade could erode or reverse the reported return edge.
  • The in-sample windows overlap (30 days of data reused every 10 days), so the in-sample drawdown improvements may partly reflect fitting overlapping windows; a non-overlapping robustness check would isolate the effect.
  • The benchmark is the market index only; comparing the same rebalancing rule with an equal-weight or momentum portfolio would show whether the benefit comes from drawdown minimisation itself or from active rebalancing in rising markets.
  • A natural extension is to vary $T$ and $D$ jointly to map how much price history and drawdown memory are needed for the out-of-sample dominance to persist.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper introduces a nonlinear programming formulation for constructing portfolios that minimise average drawdown, maximum drawdown, or a weighted combination of the two, using a rolling in-sample window and periodic rebalancing. The optimisation model includes cash inflows/outflows, transaction-cost variables, short-selling constraints, and a partial linearisation that replaces the maximum-value equality with equivalent linear inequalities. Computational results are reported for EURO STOXX 50, FTSE 100, and S&P 500 daily data over 2010–2016, with a 30-day in-sample window, 20-day drawdown lookback, and rebalancing every 10 days. The paper claims that, on average, the resulting minimal drawdown portfolios dominate the respective market indices out-of-sample on return, Sharpe ratio, maximum drawdown, and average drawdown over roughly 1800 trading days.

Significance. The mathematical contribution is sound and the paper is clearly written: the partial linearisation argument in Section 3.3 is correct, and the use of a global nonlinear solver (SCIP) with reported optimality gaps is a strength. The out-of-sample evaluation is a genuine predictive test rather than a fitted in-sample exercise, because the parameters T=30, D=20, and the 10-day rebalancing interval are fixed by hand and not tuned to the out-of-sample period. If the empirical dominance claim were robust, the paper would be a useful OR contribution to drawdown-based portfolio construction. However, the evidence for the central claim is thin: it rests on three single-path histories over one historical period, is computed under an explicit zero-transaction-cost assumption, and is reported without any statistical error measure. These gaps currently limit the strength of the conclusions that can be drawn.

major comments (3)
  1. [Section 5; Table 2] The headline out-of-sample dominance is computed under the explicit assumption that transaction costs are zero (stated in Section 4.1). With rebalancing every 10 trading days and up to 500 assets, turnover can be substantial, and the model does not penalise turnover because costs are set to zero. The average row of Table 2 shows gross excess daily returns over the index of only about 0.00004–0.00010 (roughly 1–2.5% annualised). A one-way cost of 10 basis points on a portfolio that turns over only 50% per rebalance would cost about 2.5% per year, which is of the same order as the reported excess return. The conclusion in Section 5 states the dominance result without repeating this gross-of-costs qualification. The authors should report turnover per rebalance, perform a sensitivity analysis with realistic cost levels (e.g., 5–20 basis points one-way), or include a turnover penalty in the optimisation; without this, the central empirical claim is not robust.
  2. [Section 5; Table 2] The empirical claim of out-of-sample dominance is based on three indices and one historical period (2010–2016). Each instance provides a single out-of-sample path, and the average row in Table 2 is the mean over only these three paths. No standard errors, confidence intervals, bootstrap resamples, or tests across subperiods are reported. Given the small gross excess returns, sampling variation alone could overturn the dominance conclusion. The authors should at least report the distribution of out-of-sample returns across rebalances, or validate the strategy over additional periods or asset universes, before claiming that the proposed portfolios 'dominated the market indices' on average.
  3. [Section 4.1; Section 4.2] The central computational results depend on several hand-chosen parameters: T=30, D=20, the 10-day rebalancing interval, the proportion limits δi, and the shorting limits. No sensitivity analysis is provided, so the reader cannot judge whether the reported dominance is robust to reasonable changes in these settings. For example, the 10-day rebalancing interval interacts directly with the transaction-cost issue raised above, and the choice of T=30 means the in-sample drawdown is computed from only 30 observations. A sensitivity study over T, D, and rebalancing frequency would substantially strengthen the empirical claims.
minor comments (4)
  1. [Section 3.2, equations (1)–(2)] The formulation assumes Mt > 0 whenever Pt > 0. If Pt = 0 and all Pτ in the lookback window are also zero, drawdown in equation (2) is undefined (0/0). With real price data this does not arise, but a short comment on the assumption would avoid ambiguity.
  2. [Section 3.5, equation (23)] The validity of replacing the equality definition of dt by the inequality dt ≥ 100(Mt−Pt)/Mt is stated to follow by an argument 'very similar' to that for equation (12). For the MINMAX objective, the reasoning is slightly different because dt does not appear directly in the objective; the authors could spell out why alternative optimal solutions make the inequality tight.
  3. [Table 2] Table 2 is dense and somewhat hard to read; the column grouping into in-sample and out-of-sample blocks would benefit from a separator or subheadings, and the 'Proportion limit' column could be labelled more explicitly (e.g., 'δ limit' for long-only and 'δL/δS limit' for shorting cases).
  4. [Section 4.1] The data are described as 'manually curated' to avoid survivor bias, but no details are given on how the index compositions were obtained or verified. A brief description of the curation procedure would improve reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the out-of-sample dominance claim is a genuine predictive test, with parameters fixed before the out-of-sample period.

full rationale

The paper's central claim is that portfolios formed by repeatedly solving the MINAVG/MINMAX nonlinear programs on a trailing 30-day window and holding them for the next 10 days dominate the corresponding market index out-of-sample. This is a genuine walk-forward procedure: the objective (3) is minimized using only data in [1,T], the out-of-sample values in Table 2 are computed from the non-overlapping 10-day holding periods (approximately 1800 days), and no parameter of the model is fitted to the out-of-sample period. The T=30 and D=20 values are chosen by prior reasoning about short-term funds, not by optimizing against the test period. The partial linearisation in Section 3.3 is an equivalence argument (minimization forces Mt down to the maximum of Ptau), not an assumption of the result. The zero transaction-cost assumption in Section 4.1 is an external validity limitation, not a circular step, because the comparison is explicitly gross of costs; if anything it makes the reported dominance conditional, but it does not make the derivation depend on its own conclusion. Self-citations ([8], [14], [43], [44]) appear only in literature-review and cost-discussion contexts and are not used to justify the model's predictive content. No fitted parameter is relabelled as a prediction and no uniqueness theorem or ansatz is imported from the authors' prior work. Hence no circular step is present.

Assumptions & free parameters 5 free parameters · 3 assumptions · 0 invented entities

The models rely on standard mathematical programming constructs and publicly available solver software. The main burdens are the zero-transaction-cost assumption and the unshipped curated dataset. No new physical or financial entities are introduced.

free parameters (5)
  • In-sample period T = 30 trading days
    Chosen by hand to represent a short-term rebalancing horizon; not derived from theory or fitted to out-of-sample data.
  • Drawdown lookback D = 20 trading days
    Chosen by hand to look a limited distance into the past; results may be sensitive to this value.
  • Rebalancing interval = 10 trading days
    Chosen to implement frequent rebalancing; transaction costs are assumed zero, which interacts with this choice.
  • Maximum asset proportion delta_i = 0.1 or 1
    Two variants are tested to limit concentration; higher values allow larger single-asset positions.
  • Shorting limits delta_L, delta_S, Delta_L, Delta_S = 0.1, 0.1, 1.1, 0.1
    Set by hand to allow limited short exposure; only one set of values is tested.
assumptions (3)
  • domain assumption Transaction costs are zero in all computational experiments.
    Stated on page 15 of the manuscript. This is critical because the strategy rebalances every 10 days; without it the out-of-sample returns would be lower.
  • domain assumption The manually curated index compositions are correct and free of survivor bias.
    The authors claim they know the exact composition on each day, but they do not provide the data or verification procedure, so external reproducibility depends on this assumption.
  • domain assumption SCIP solves the nonlinear programs to proven global optimality within the time limits.
    The paper relies on SCIP's global optimization capabilities, and for many instances only a feasible solution with a bounded gap is reported. The reported portfolios may not be truly optimal for those rebalances.

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Cite this review

Pith. "Pith review of A nonlinear optimisation model for constructing minimal drawdown portfolios." pith.science (2026). https://pith.science/paper/J7ERNJA4

@misc{pith2026190808684,
  author       = {Pith},
  title        = {Pith review of: A nonlinear optimisation model for constructing minimal drawdown portfolios},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J7ERNJA4}},
  note         = {Machine review of arXiv:1908.08684}
}
read the original abstract

In this paper we consider the problem of minimising drawdown in a portfolio of financial assets. Here drawdown represents the relative opportunity cost of the single best missed trading opportunity over a specified time period. We formulate the problem (minimising average drawdown, maximum drawdown, or a weighted combination of the two) as a nonlinear program and show how it can be partially linearised by replacing one of the nonlinear constraints by equivalent linear constraints. Computational results are presented (generated using the nonlinear solver SCIP) for three test instances drawn from the EURO STOXX 50, the FTSE 100 and the S&P 500 with daily price data over the period 2010-2016. We present results for long-only drawdown portfolios as well as results for portfolios with both long and short positions. These indicate that (on average) our minimal drawdown portfolios dominate the market indices in terms of return, Sharpe ratio, maximum drawdown and average drawdown over the (approximately 1800 trading day) out-of-sample period.

Figures

Figures reproduced from arXiv: 1908.08684 by the authors.

Figure 1
Figure 1. Drawdown illustrated ● ● ● ● ● ● 0 10 20 30 40 50 60 70 80 90 100 1 2 3 4 5 6 Time Portfolio value tunity cost of the single best missed trading opportunity associated with selling the portfolio at some point before t and then repurchasing it at t. It represents the (percentage) value forgone by not having sold the portfolio at its previous maximum value point (the single best point at which to sell the portfolio) a… view at source ↗
Figure 2
Figure 2. Out-of-sample performance: EURO STOXX 50 [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Out-of-sample performance with shorting: EURO STOXX 50 [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.