Trailing error gives climate tilts room after they win

A trailing-error limit gives a climate tilt more room after it has beaten its benchmark. The case for changing mandates turns on whether that room reflects lower risk or simply rewards a run of wins. Dijkstra, Hoepner, Schneider and Słomka-Gołębiowska argue that variance, standard deviation and tracking error treat large gains like large losses. For a portfolio meant to move away from its benchmark, they see that symmetry as costly. Yet tracking error measures uneven active returns: a manager who beats the benchmark by the same margin every week uses almost none of the budget.

Why change the limit?

Dijkstra, Hoepner, Schneider and Słomka-Gołębiowska describe climate risk as two-dimensionally asymmetric. At a given date, more assets lose from climate change than gain; over time, warming and accumulated heat show no mean reversion. They cite NGFS scenarios in which global GDP losses reach up to 2.8% and unemployment rises by up to 1.7 percentage points in the next few years, as warming approaches 1.5°C. Transition portfolios, they argue, must tilt away from benchmarks that decarbonise too slowly. Bolton, Kacperczyk and Samama, whom they cite, find that the common 1% tracking-error limit would be breached by the early 2040s at the latest. A symmetric limit then binds on the portfolio's departure from its benchmark, even when the tilt has made money.

Their proposed replacements are lower partial moments, measures that count shortfalls below a chosen target. Semi-variance sets that target at the portfolio's own mean. Their "standard risk" takes the root of the second lower partial moment with a zero target, counting losing periods only. "Trailing error" sets the benchmark as the target and counts periods of underperformance. For a 100 basis point tracking budget, the authors say favourable outcomes consume roughly half the capacity, leaving risk measurement "diluted by approximately 50%". Trailing error, they write, "roughly doubles the effective risk capacity available for climate transition opportunities."

The evidence offered for the change is one chart. There is no optimised or backtested portfolio, and the 50% and doubling claims are stated rather than calculated. The paper also proposes four qualitative Principles for Risk Aware Investing without calibrating a trailing-error budget.

The arithmetic of active returns

Tracking error measures the standard deviation of active returns around their own mean. A steady weekly premium, however large, scores zero. A 100 basis point budget pays for dispersion around that mean; the share of dispersion above it tells us something different from the share of returns above the benchmark.

The doubling works under a specific assumption. With active returns symmetric around zero, trailing error equals tracking error divided by √2. A 1% trailing limit therefore permits a 1.41% tracking limit, exactly double in variance units. A board could admit the same portfolio by moving its tracking-error allowance from 1% to 1.41%. The choice of measure changes which portfolios qualify when active returns have a nonzero mean or are skewed. Meanwhile, the 50% claim assumes the symmetry that the paper's premise disputes. We did not find that tension addressed.

The authors qualify their argument: a binding limit is "a perverse outcome if a climate transition strategy produces superior returns." Superior returns are the condition the mandate must live with. The Bolton result concerns divergence from a benchmark. If a tilt falls behind steadily, it can use trailing error faster than tracking error; a constant shortfall has zero tracking error and counts in full under trailing error. Calling tracking error the wrong measure presumes the manager knows in advance which way active returns will go.

The authors also state the benefit plainly. An outperforming period "consumes zero trailing error budget for that period. Its full budget is reserved for periods of genuine underperformance." The freed capacity rests on credit for past wins.

What does the Russell 1000 chart show?

Figure 1 covers weekly Russell 1000 returns from Q2 2023 to Q1 2025. Four of the five weeks standard deviation ranks riskiest are gains. Standard risk puts the five largest losses in its top five, as a zero-target downside measure must: winning weeks cannot register as losses.

These are absolute index returns across about two years, a sample the authors describe as mostly positive weeks. With no active returns in the chart, it cannot settle the tracking-error versus trailing-error question. It also compares standard deviation with standard risk, whose zero target differs from the benchmark target proposed for mandates. The rankings show how the measures sort weeks. They leave open what portfolios built under either limit would earn.

Choices a mandate still has to make

• Centring. The appendix defines trailing error as the root of the second lower partial moment at the benchmark return, without removing the mean. Tracking error removes it. Over a trailing window, wins shrink trailing error and losses enlarge it, making the limit partly a trend rule for the tilt's active return.
• Calibration. Equal 1% limits give trailing error √2 (about 1.41 times) as much room under symmetry, before any economics enters. The paper supplies no number.
• Frequency. Figure 1 uses weekly data. For the mandate argument, we did not find a stated frequency or window.
• Forecasting. A mandate's tracking error is usually an ex-ante estimate from a covariance matrix. A lower partial moment draws on shortfall observations alone; we did not find a discussion of forecasting it within a risk model.

Our clean-energy ETF run

We have no emissions or Paris-alignment data and could not build or verify a climate-transition mandate. Instead, we tested the choice of risk limit with a thematic ETF proxy. Clean-energy ETFs cannot establish Paris alignment. This is neither a replication nor a test of the paper's climate claim, and it says nothing about whether climate risk is skewed. The candidate book holds 75% SPY, 12.5% TAN and 12.5% ICLN against a SPY benchmark. On each month's first trading day, we use 252 daily active returns through the prior close. Each branch calculates its own measure, tracking error or trailing error. When that measure exceeds 1% annualised, the branch scales its tilt by 0.01 divided by the measure and puts the rest in SPY. We chose the daily window and the shared 1% limit.

Our figures run from January 2020 to July 2024, after commissions of four tenths of a cent a share. The combined run of both branches returned 64.38%, with a Sharpe of 0.71, volatility of 20.32% and a maximum drawdown of -36.43%. The paper reports no performance figures of its own for comparison. We did not record the branches' individual scaling factors, realised errors or shortfall frequencies. This one automated pass therefore cannot tell us which limit allowed more clean-energy exposure, the question it was built to answer. The gap is ours, and the run is no verdict on the authors' work.

SPY stayed at or above 75% of the book. The window also begins just before the March 2020 crash. With beta of 0.93 to SPY, the 20.32% volatility and -36.43% drawdown read as broad equity exposure.

Should mandates make the swap?

Trailing error can grant extra room through √2 relabelling, which a board can obtain directly by raising a 1% tracking limit to 1.41%. It can also grant credit for recent outperformance, placing a bet on continued wins. Neither establishes higher returns or smaller benchmark shortfalls, and the paper offers no evidence of either. The authors say the change costs no fiduciary discipline. Yet tracking error catches concentrated bets that may reverse: within a 100 basis point budget, a large upside deviation is a warning that a downside-only limit discards.

A comparison of subsequent benchmark shortfalls would change our view. Admit tilts under a trailing limit and under a tracking limit scaled to match it under symmetry. Smaller later shortfalls for the trailing-error group would show the measure doing risk work. Otherwise, the proposal amounts to a looser budget with a better name.

Our backtest stops at 2024-07-01, and everything after that date is deliberately left untouched so the same strategy can be checked out of sample later.

On the first trading day of each month:
  Form 252 aligned, completed daily returns ending at the prior close.
  If the required observations are incomplete, skip both branch rebalances.
  Candidate = 75% SPY + 12.5% TAN + 12.5% ICLN; XLE = 0%.
  Active return = constant-mix candidate return − SPY return.
  For each branch independently:
    R = sqrt(252) × sample_std(active returns, ddof=1)
        OR sqrt(252 × mean(max(0, −active return)^2)).
    f = 1 if R = 0, else min(1, 0.01 / R).
    Target = f × candidate + (1 − f) × 100% SPY.
    Rebalance at that day's adjusted close; let holdings drift until the next trade.