Efficiency scoring method picks winning ETFs and sizes each bet

Every investor faces the same two-headed puzzle: which assets deserve a place in a portfolio, and how much money should each one receive. A new study tackles both questions at once using a mathematical technique borrowed from industrial efficiency measurement, applying it to exchange-traded funds traded on the Tehran Stock Exchange. The approach, built on data envelopment analysis, or DEA, treats each fund as a decision-making unit that consumes inputs such as risk and cost to produce outputs such as return, and then scores how well each fund converts the one into the other.

The result is a fully specified pipeline that not only identifies the strongest funds but also assigns each of them an exact share of the portfolio, something the researchers argue has been missing from earlier DEA-based finance studies, which typically stopped at performance evaluation without a procedure for selection and allocation. The classical starting point for portfolio construction remains Harry Markowitz's mean-variance optimization, which maximizes expected return for a given level of variance. Despite decades of dominance, the model carries baggage that troubles practitioners: it assumes returns are normally distributed, it ignores liquidity and transaction costs, and it presumes a linear relationship between risk and reward that real markets frequently violate.

Risk measures such as Value at Risk and Conditional Value at Risk were developed to patch some of these holes, with CVaR capturing the average loss beyond the VaR threshold and offering a more honest view of tail risk. Yet CVaR brings its own burdens, including higher computational complexity and acute sensitivity to data quality in the far reaches of the return distribution, where observations are scarce and estimation errors multiply. Data envelopment analysis entered finance as a fundamentally different kind of yardstick.

Rather than assuming a functional form for the relationship between risk and return, DEA is non-parametric: it constructs an efficiency frontier from the observed data themselves and measures each unit's distance from that frontier. The original CCR model, named for Charnes, Cooper and Rhodes, assumes constant returns to scale and proportional changes in all inputs or outputs, which classifies it as a radial model. The later BCC model relaxes the scale assumption by adding a convexity constraint, allowing variable returns to scale.

The two models differ sharply in how they handle negative data: the CCR model cannot accommodate negative inputs or outputs at all, whereas the input-oriented envelopment form of the BCC model can handle negative outputs, and its output-oriented counterpart can process negative inputs. Even so, financial datasets remain hostile territory for radial models, and ETF data are full of negative values, from negative Sharpe ratios to negative annual returns during bad market stretches. The study's central tool is the slacks-based measure, or SBM, a non-radial DEA model that confronts this problem head-on.

Instead of measuring efficiency by proportional movement toward the frontier, SBM explicitly quantifies slack variables, meaning the excess inputs each fund consumes and the shortfall in outputs each fund delivers, and folds them directly into the efficiency score, which takes values between zero and one. Because the objective function is unit-invariant and translation-invariant with respect to outputs, adding a constant to every value of a negative output leaves the efficiency ranking untouched. The researchers exploited this property by shifting two output columns into positive territory: a uniform constant of 11.320 was added to every Sharpe ratio observation, and 2.504 was added to every annual return observation, preserving the relative differences between funds while satisfying the model's positivity requirements.

One remaining limitation of SBM is that all efficient funds receive the same score of one, leaving them indistinguishable. Earlier attempts to break this tie, such as the AP model, removed efficient units from the reference set but suffered from instability and infeasibility, and the model fails outright when a fund has a zero input, while near-zero inputs can produce extremely large scores. To solve this, the study deploys the super-SBM model, which incorporates super-efficiency features and allows the slack variables to take negative values.

Efficient funds that sit outside the conventional production possibility set are projected back onto it, and the resulting scores exceed one, providing a precise ranking among funds that ordinary DEA would treat as equals. The researchers then convert this ranking into money: each fund's score is reduced by one to isolate its performance advantage beyond the efficiency threshold, and these excess scores are normalized by their sum to produce portfolio weights. A fund with a higher super-SBM score therefore receives a proportionally larger allocation, translating relative efficiency directly into capital distribution, although the authors caution that the score represents relative efficiency and should not be read as a financial return.

The empirical application drew on a universe of 442 investment funds listed on the Tehran Stock Exchange over the period from 23 September 2022 to 23 September 2023, of which 177 were exchange-traded funds and 65 were equity ETFs. Strict eligibility criteria narrowed the field to 34 funds: each needed at least one year of operating history and diversified exposure across multiple sectors, ruling out 31 funds that were too young or concentrated in a single industry such as banking, pharmaceuticals or basic metals. Four inputs characterized each fund: the beta coefficient measuring market sensitivity, the management fee, the expense ratio, and the annualized standard deviation of monthly returns.

Two outputs measured what each fund delivered: the Sharpe ratio, computed with a risk-free rate of 20.5 percent corresponding to the rate published by the Central Bank of Iran during the study period, and the annual return. Quarterly fee figures were averaged into annual values, and all data were cross-checked against official sources including FIPIRAN and CODAL, with no imputation or outlier adjustment required, since modifying valid extreme observations could alter the efficiency frontier and distort the relative ranking. Running the models in GAMS, a general algebraic modeling system built for large-scale mathematical programming, the SBM evaluation identified exactly seven efficient funds out of the 34 candidates: ETF6, ETF8, ETF15, ETF17, ETF20, ETF31 and ETF33.

These seven funds, each converting risk and cost into risk-adjusted return without measurable waste, formed the investable set. The super-SBM model then separated them, with scores ranging from 1.111 for ETF6 to 1.386 for ETF20. After subtracting one from each score and normalizing by the sum of 1.529, the resulting allocation assigned 25.243 percent of the portfolio to ETF20, 23.213 percent to ETF31, 14.519 percent to ETF33, 11.071 percent to ETF17, 9.410 percent to ETF15, 9.268 percent to ETF8 and 7.276 percent to ETF6.

Every efficient fund retained a positive weight, avoiding concentration in a single name, and the allocation mechanism preserved the discriminatory information generated during the ranking stage rather than treating all frontier funds as equally attractive. The framework also anticipates a practical snag: what happens when the number of fully efficient funds falls short of a manager's desired portfolio size? Restricting investment exclusively to frontier funds could produce an investable set too narrow for meaningful diversification.

The researchers propose incorporating quasi-efficient funds, whose efficiency scores sit close to but slightly below the frontier, expanding the candidate pool in controlled steps until the required number of holdings is reached, after which the super-SBM ranking and allocation proceed as before. This extension turns DEA from a one-shot grading exercise into a sequential decision framework encompassing selection, expansion of the opportunity set, discrimination among finalists, and capital allocation, while investor preferences remain embedded in the choice of inputs and outputs rather than in an explicit utility function, and no transaction-cost adjustment, liquidity constraint or maximum-weight restriction is imposed. The authors are candid about the method's boundaries.

SBM and super-SBM are sensitive to data quality and accuracy, and both models are static: they evaluate funds on a single snapshot of inputs and outputs and do not track how efficiency evolves over time, which limits their use for dynamic portfolio management. Because no maximum-weight constraint is imposed, the allocation can concentrate capital in the strongest funds, a feature future versions could temper with investor-specific restrictions. The researchers also flag robustness testing under uncertain or volatile inputs as a priority for follow-up work, along with systematic ranking of near-efficient funds to broaden diversification and the exploration of dynamic optimization methods.

Subject of Research: Applied mathematics and finance

Article Title: Portfolio optimization using data envelopment analysis: Exchange-traded funds portfolio selection and allocation

Article References: Malekahmadi, A., Kordlouie, H., & Abri, A. G. (2026). Portfolio optimization using data envelopment analysis: Exchange-traded funds portfolio selection and allocation. Heliyon, 12(16), Article e45595. https://doi.org/10.1016/j.heliyon.2026.e45595

Image Credits: AI Generated

DOI: 10.1016/j.heliyon.2026.e45595

Keywords: portfolio optimization, data envelopment analysis, exchange-traded funds, slacks-based measure, super-SBM, quantitative finance, risk management, financial modeling, Portfolio, optimization, data, envelopment