The Return-to-Present curve (RTP) puts the 1-month, 3-month, 6-month, 1-year, 3-year and 5-year cells of a trailing-return table on one continuous line. Each point is the total return from a hypothetical purchase date to today. On adjusted prices, Liu writes it as R(h;t) = P(t)/P(t-h) - 1. He varies the look-back horizon h over the full available history, then plots the result against the calendar purchase date u = t - h. At h = 0 every curve returns to zero by construction. Those six trailing-return cells are simply six selected points.
The paper earns its keep through that change in presentation. A normalized cumulative chart requires an origin, so its ranking depends on the date chosen. RTP holds the endpoint fixed and displays the ranking from every origin at once. Liu states the boundary clearly: "Its contribution is representational rather than algebraic".
Liu studies retirement-plan funds that are outside our tradable universe. The specific results therefore do not carry over to anything we would run. Yet the construction itself requires only an adjusted price path and can be applied unchanged to listed equities and liquid ETFs.
Four illustrations make the case. The first compares NVDA with AMD. For some entry dates three to five years earlier, the NVDA-minus-AMD difference curve rises above 1,000 percentage points. Around three years before the endpoint, it crosses zero. AMD outperformed NVDA through a substantial part of the period that followed. The widest gap occurs near entry dates 500 days before the evaluation date, then contracts again as the endpoint approaches.
Unequal inception dates drive the next example, which uses three AI-themed ETFs: AIQ from May 11, 2018, CHAT from May 18, 2023, and AIS from December 3, 2024. Choosing the newest inception, December 3, 2024, as a shared origin would discard more than six years of AIQ history and about eighteen months of CHAT. Third comes a comparison of three large-cap retirement funds. VPMAX stays ahead of JLGMX and VIIIX over most of a five-year entry-date window. Fourth is a hypothetical rotation from DIA to QQQ dated January 2, 2026, with performance evaluated to August 20, 2026. At the marked date, QQQ's RTP is about five percentage points above DIA's.
The difference curve hides a rank
The pairwise sign has a simple interpretation. D_AB is positive at entry date u exactly when P_A(t)/P_A(u) exceeds P_B(t)/P_B(u). Rearrangement gives P_A(t)/P_B(t) > P_A(u)/P_B(u). In plain terms, today's relative price of A against B must exceed its level at u.
The share of entry dates that favor A is therefore the percentile rank of the current A/B ratio within its own history over the displayed window. Liu proposes crossing counts as a secondary summary. Each crossing is where that ratio path passes through the horizontal line set by its terminal value.
Nothing else determines the sign.
Liu presents the share statistic as an input to decisions. He asks whether one asset "produces the larger realized return to the common endpoint for a substantial majority of historical purchase dates". The qualification follows immediately: "This does not establish future superiority, but it provides a direct historical criterion for choosing between investments that may otherwise appear comparable."
The algebra sharpens what this criterion selects. It favors the asset whose relative price stands near a multi-year high against its peer. That belongs to long-horizon relative strength, the family the paper already cites through Jegadeesh and Titman and through Moskowitz, Ooi and Pedersen. A difference curve takes its sign from the change in the A/B price ratio. We did not find that equivalence stated in the discussion. We have encountered the same pattern before, where an apparent signal becomes a monotone transform of trailing relative returns, in an earlier case where a learned sector signal reduced to trailing relative returns.
Magnitude remains separate. It records how far below today's level the ratio stood in the past, so a 1,000-point gap conveys something different from a 5-point gap. The chart's ranking, however, is the rank.
A new endpoint redraws everything
Every point depends on the endpoint through one multiplier. Moving the evaluation date from t to t' multiplies every gross return for an asset by P(t')/P(t). In a pairwise difference, the assets receive two different multipliers. Any week when A gains relative to B lifts the full A-minus-B curve and may shift every zero crossing together.
Liu acknowledges this dependence: "RTP remains conditional on the selected evaluation endpoint." He then defends the choice. "The present provides a natural default for current investment comparison, whereas an alternative historical endpoint may be appropriate when motivated by a specific market, company, or policy event." The default still leaves the entire display tied to one endpoint. Away from today, the only rule supplied is that the choice should be "substantively motivated rather than selected to favor a particular result". Liu also concedes that a large move near the endpoint "affects returns over many look-back horizons". Such sensitivity to exceptional endpoint-adjacent periods, he writes, "warrants further study".
A five-year chart may look persistent, though it contains far less than five years of independent evidence. The paper makes the dependence explicit: "neighboring RTP values are strongly dependent because they share the same endpoint and largely overlapping price histories". As a result, the share-of-entry-dates statistic has no effective sample size behind it. Liu also warns against treating the summaries as probabilities of future outperformance.
The illustrations leave further gaps. Figure 5 alone supplies dated endpoints, namely the January 2, 2026 rotation and the August 20, 2026 evaluation. For every other figure, the paper gives no data source, frequency or evaluation date. Each figure contains Two or three named assets, without a stated selection rule. Liu does offer non-performance reasons for the comparisons. NVDA and AMD are "two major semiconductor companies", while the three funds are alternatives within one employer-sponsored plan. The paper reports no aggregate statistics for a wider universe. It mentions those summaries only as something that "may be reported secondarily".
Where the chart earns a place
The paper's cleanest idea is its treatment of unequal inception dates. Truncating the left edge of each curve avoids choosing between discarded older history and returns measured over mismatched horizons.
Retrospective transaction review also suits the display. A sale on January 2 and a replacement purchase on January 3 can be read from the same two curves because both legs end on August 20. Liu says a conventional fixed-start chart "does not directly represent a transaction pair with different starting dates", and he is right.
The contribution-date example promises more than it delivers. A stream of retirement contributions has a dollar-weighted return, which these single-purchase curves do not calculate. The examples include no transaction costs and no taxes. The NVDA against AMD panel mainly demonstrates the display itself. Liu accepts that reproducing the same pattern with fixed-start charts would require repeated changes to the origin and repeated redrawing.
A pre-specified rule based on the entry-date share would move me from "good chart" to "tradable". Such a rule would need an out-of-sample run across a real cross-section. It would also need to beat the percentile rank of the relative price, whose sign it matches by construction. Liu does not claim that such a rule exists, and he treats prediction as a separate question. I agree it is separate.