Understanding Absolute, Discrete, And Continuous Proportional Returns
Updated: Oct 18
Introduction
A stock moves from $100 to $120. What did it return? There are three correct answers: a $20 absolute gain, a 20% discrete return, and an 18.23% continuous (log) return. None of them is a rounding variant of another. Each answers a different question, and each behaves differently when you add, average, compound, or model it.
Absolute returns answer: how many dollars did I make? Discrete returns answer: by what fraction did my capital grow? Continuous returns answer: at what continuously compounded rate did it grow? The first is about money, the second about proportion, the third about rate.
This distinction sounds academic until it reaches a desk. A performance report, a VaR report, a stress scenario, and an option pricer can all use the same price history and quietly disagree because each uses a different definition of return. Most of the subtle errors in return-based analysis come not from wrong arithmetic but from using the right formula in the wrong place.
This article treats returns as a family of measures, not a single number. The goal is to know what each one is, why it exists, how it aggregates, where it breaks, and which one to reach for in a given task, including how the same ideas reappear as "shocks" in risk management.
Practical Challenges
5 recurring problems force practitioners to keep more than one definition of return.
Scale: A $10,000 position that makes $1,000 and a $1,000,000 position that makes $1,000 show the same absolute P&L. One earned 10%, the other 0.1%. Any comparison of managers, strategies, or assets must remove position size, which absolute returns cannot do.
Time Aggregation: An asset returns +10% and then -5%. Adding gives 5%, but the true two-period return is 1.10 x 0.95 - 1 = 4.5%. Discrete returns compound multiplicatively, so summing them over time overstates growth. Log returns do add over time, which is why multi-period risk models are built on them.
Aggregation Across Assets: The reverse problem appears in a portfolio. Discrete returns combine exactly as a value-weighted average across positions in a single period; log returns do not. A model that adds log returns across positions, or sums discrete returns across days, carries a silent approximation error that grows with volatility and horizon.
Statistical Modelling: Discrete returns are floored at -100% and unbounded above, so they cannot be exactly normal. Log returns live on the whole real line and are the natural variable in geometric Brownian motion, Black-Scholes and most econometric models. But real log returns are still fat-tailed and skewed; "approximately normal" is a modelling convenience, not an empirical law.
Communication: Clients, boards, and regulators think in percentages and dollars. Telling a stakeholder that a portfolio earned "18.23% continuously compounded" when it grew by 20% creates confusion, or worse, the impression that numbers are being massaged.
The consequences are concrete. A portfolio manager who averages discrete returns arithmetically overstates long-run growth. A risk manager who applies relative shocks to near-zero interest rates produces meaningless scenarios. A pricing model that mixes compounding conventions misprices forwards and options by exactly the gap between the two definitions.
What's Covered
The article moves from definitions, to mechanics, to judgement.
What a return measures – returns as outcomes, and why risk managers call them shocks.
Absolute returns – dollar PnL, its uses, and its scale problem.
Discrete returns – percentage change, portfolio aggregation, and multiplicative compounding.
Continuous returns – log returns, continuous compounding, and time additivity.
Example – one 5-day price path through all three approaches: Absolute, Discrete, and Continuous.
Conversions, approximations, and ordering – moving between measures and knowing which is larger.
Aggregation – across time versus across assets, plus volatility drag.
Limitations – where each measure breaks or misleads.
Decision framework – which return to use for which task.
Returns as shocks – absolute, discrete, and log shocks in VaR, stress testing, and FRTB sensitivities.
Best practices and common errors – data, conventions, and the mistakes that recur on desks.
Appendix – derivations of the key relationships.
Readers focused on application can go straight to sections 9 to 11; the appendix is for those who want the proofs.
What a Return Measures
A return is the change in value of a position over a defined interval, expressed in a chosen unit. Three choices define it, and all three must be stated for a return to be unambiguous.
Unit: currency (absolute), fraction of starting value (discrete), or log of the value ratio (continuous).
Interval: daily, weekly, monthly, annual, or the holding period.
Scope: price only, or total return including dividends, coupons, and other cash flows.
With income included, the discrete total return over one period is:
where Dt is the cash distribution paid during the period. Price returns understate the economic return of any income-paying asset, sometimes by several percent a year for high-dividend equities.
In risk management, the same quantity is usually called a "shock": the change applied to a risk factor to revalue a portfolio. A historical VaR model takes past daily returns of each factor and replays them as shocks on today's positions. A stress scenario prescribes shocks directly, such as equities -30% or rates +200bp. Whether a shock is absolute, discrete, or logarithmic is not a presentational choice; it determines how the scenario scales with today's market level, which Section 10 covers in detail.
Absolute Returns
An absolute return is the change in price or value, in currency units:
where St is the price at time t and S{t-1} is the price one period earlier. A share bought at $100 and now at $120 has an absolute return of $20; on 100 shares, $2,000.
Where It Is The Right Measure: Absolute returns are what PnL is. They add exactly across positions at a point in time: the desk's PnL is the sum of its books' PnL, with no weights required. Limits, capital, and losses are ultimately absolute numbers, so every risk report ends in currency even when the model runs in percentages. For risk factors quoted as rates or spreads, the absolute change in basis points is the natural unit: a yield moving from 4.00% to 4.25% is a +25bp absolute move.
Where It Fails: Absolute returns are not scale-free. Investment A makes $1,000 on $5,000 (20%); Investment B makes $1,000 on $100,000 (1%). In dollars, they look identical. A manager judged on absolute PnL can look better simply by running more capital. Absolute price changes also depend on the price level: a $2 move means something different on a $10 stock than on a $500 stock, so a history of absolute equity moves cannot be compared across time once the price has drifted far from where it was earlier.
Across Time: Absolute PnL does add across periods as accounting (total gain is the sum of period gains), but it says nothing about growth rate, because each period's gain was earned on a different capital base.
Discrete Proportional Returns
A discrete (simple, arithmetic) return expresses the price change as a fraction of the starting price:
For $100 to $120, this is 20/100 = 0.20, or 20%.
Properties That Make It The Default:
Scale-Free: 20% means the same on $100 or $100 million, so performance can be compared across positions, managers, and assets.
Exact Portfolio Aggregation: For a portfolio with value weights wi at the start of the period, the portfolio return is exactly the weighted sum of asset returns. This makes discrete returns the basis of performance attribution and of mean-variance portfolio theory.
Intuitive: It is the number clients, boards, and fact sheets expect.
Asymmetry: A long unleveraged position can lose at most 100% but can gain without limit. A -50% move followed by +50% leaves you at 75% of where you started, not 100%. The distribution of discrete returns is therefore bounded on one side and right-skewed over longer horizons.
Compounding Across Time: Discrete returns multiply; they do not add. Take +10%, -5%, +8%:
After period 1: $100 × 1.10 = $110.00
After period 2: $110 × 0.95 = $104.50
After period 3: $104.50 × 1.08 = $112.86
The total return is 12.86%. Adding the three gives 13%, overstating it. In general:
The gap between the sum and the true compounded return grows with volatility. That gap is the root of volatility drag (section 7) and the reason the arithmetic mean of discrete returns overstates the growth an investor actually experiences.
Continuous Proportional (Log) Returns
A continuous return is the natural logarithm of the price ratio:
For $100 to $120, ln(1.20) = 0.1823, or 18.23%.
What It Means? The log return is the rate that, compounded continuously over the period, turns S{t-1} into St:
When compounding 20% once a year, 10% twice, 5% four times, and so on towards infinitely often, each step needs a slightly lower nominal rate to reach the same end value. The limit is 18.23%. Continuous compounding does not create more wealth than discrete compounding; it describes the same growth with a lower rate.
Time Additivity: Because the log of a product is the sum of the logs, log returns add over time. Using the earlier +10%, -5%, +8% path:
Period | Discrete return | Log return |
1 | +10.00% | +9.53% |
2 | -5.00% | -5.13% |
3 | +8.00% | +7.70% |
Total | +12.86% (compounded) | +12.10% (summed) |
When converting back, e^0.1210 - 1 = 12.86%, matching the compounded discrete figure exactly.
Symmetry In Log Space: A +10% log return followed by -10% returns exactly to the start; the discrete equivalents do not. Log returns range over the whole real line, and the price itself is always positive because St = S{t-1} x e^r.
Why Models Use It: If log returns over successive intervals are independent and identically distributed, their sum over n periods has mean nμ and variance nσ². This gives the square-root-of-time rule for scaling volatility, and with normality it gives lognormal prices: the assumption behind geometric Brownian motion and Black-Scholes. Most GARCH and time-series volatility models are also specified on log returns.
Example: 5 Days, 3 Approaches
Over 5 days, a stock moves from $100.00 to $107.25. All three measures agree on the end point; they differ in how the daily pieces add up to it.
Day | Price ($) | Absolute ($) | Discrete | Log |
0 | 100.00 | – | – | – |
1 | 102.50 | +2.50 | +2.50% | +2.47% |
2 | 101.80 | -0.70 | -0.68% | -0.69% |
3 | 105.00 | +3.20 | +3.14% | +3.10% |
4 | 103.50 | -1.50 | -1.43% | -1.44% |
5 | 107.25 | +3.75 | +3.62% | +3.56% |
Total | +7.25 (sum) | +7.25% (compounded) | +7.00% (sum) |
Three things to read from the table:
Absolute returns sum exactly to the total dollar gain of $7.25, because PnL is additive accounting.
Discrete returns must be compounded. 1.0250 x 0.9932 x 1.0314 x 0.9857 x 1.0362 - 1 = 7.25%. Summing them gives about 7.16%, an error of 9bp in just 5 quiet days.
Log returns sum exactly to 7.00%, and e^0.0700 - 1 = 7.25% recovers the discrete total.
On every up day the log return is below the discrete return; on every down day it is more negative. That ordering is not specific to this path; it holds always (section 6).
Conversions, Approximations, and Ordering
All three measures are deterministic transformations of the same price pair, so any one can be converted into the others given the starting price.
Taylor Approximation: Expanding the logarithm around zero:
For small moves, the quadratic term is negligible, and the two measures are interchangeable. For daily returns on liquid assets, the gap is usually a fraction of a basis point; for monthly or annual returns, or volatile assets, it is material.
Discrete Move | Log Return (up) | Discrete Move | Log Return (down) |
+1% | +0.995% | -1% | -1.005% |
+5% | +4.88% | -5% | -5.13% |
+10% | +9.53% | -10% | -10.54% |
+20% | +18.23% | -20% | -22.31% |
+50% | +40.55% | -50% | -69.31% |
Ordering: Because ln(1 + x) ≤ x for all x > -1, with equality only at zero:
For a gain, the log return is smaller than the discrete return: 18.23% versus 20%.
For a loss, the log return is more negative than the discrete return: -22.31% versus -20%.
The absolute return divided by the starting price is the discrete return, so it is not a third quantity to rank.
In short, r^log ≤ r^disc in every case. Log returns understate gains and overstate losses relative to discrete returns. This is not a conservative bias that a risk model can bank on: if a shock is measured in log terms and re-applied in log terms, the resulting price is identical. A bias appears only when conventions are mixed, for example by estimating volatility on log returns and then applying it as a discrete shock.
Aggregation Across Time and Across Assets
The single most useful fact in this article: discrete returns aggregate exactly across assets, log returns aggregate exactly across time, and neither does both.
Measure | Across Assets (one period) | Across Time (one asset) |
Absolute | Exact sum of P&L | Exact sum of P&L, but no growth rate |
Discrete | Exact value-weighted average | Multiply (1 + r), never add |
Log | Not a weighted average | Exact sum |
Why Log Returns Fail Across Assets? The portfolio log return is the log of the weighted sum of gross returns, not the weighted sum of log returns:
A 50/50 portfolio where asset A rises 20%, and asset B falls 20% has a discrete return of exactly 0% and a log return of exactly 0%. The weighted average of the two log returns, however, is (18.23% - 22.31%) / 2 = -2.04%. For daily returns of diversified portfolios, the error is small, which is why the approximation is widely used; for concentrated, long-horizon or high-volatility portfolios, it is not.
Practical Workflow: Compute position and portfolio returns in discrete terms for each period (or as PnL), then convert the portfolio series to log returns to aggregate over time, annualise, or model.
Volatility Drag: Because discrete returns compound, volatility lowers realised growth. A path of +50% then -50% has an arithmetic mean return of 0% but leaves you at $75 from $100, a geometric mean of -13.4% per period. To a good approximation:
The same relationship appears in continuous time: if discrete expected return is μ, the expected log return is μ - σ²/2. Projecting long-run wealth with the arithmetic mean of discrete returns therefore overstates the growth an investor should expect.
Scaling Over Horizons: If daily log returns are i.i.d. with mean μ and volatility σ, the n-day log return has mean nμ and volatility σ√n. This square-root-of-time rule is exact for i.i.d. log returns and only approximate for discrete returns. It also breaks down when returns are autocorrelated or volatility clusters, which they usually do.
Limitations of Each Measure
Each measure breaks in a predictable place. Knowing where is more useful than knowing the formula.
Absolute Returns
Not comparable across position sizes or price levels.
Historical absolute moves become stale as price levels drift: a $5 daily move on a stock that was then $50 implies something very different if replayed on the same stock at $250.
Silent on growth rate and on risk-adjusted performance.
Discrete Returns
Not additive over time; summing overstates growth and the arithmetic mean overstates long-run compounding.
Bounded below at -100% and right-skewed, so a normal distribution is wrong in the tails, especially over long horizons.
Undefined or explosive when the starting value is zero or near zero, and sign-ambiguous when it is negative. Interest rates, spreads, some commodity spreads, and power prices can all be zero or negative.
Leveraged and short positions can lose more than 100% of the margin posted, which breaks the -100% floor intuition.
Log Returns
Not additive across assets; a weighted average of log returns is only an approximation of the portfolio return.
Undefined for zero or negative prices, so unusable for rates in a negative-rate regime, for spreads that cross zero, or for P&L series.
Less intuitive for stakeholders; 18.23% will be heard as "less than 20%".
The common assumption that log returns are normal is convenient, not true. Empirical log returns show fat tails, negative skew in equities, and volatility clustering. Models that rely on normality underestimate tail losses regardless of which return definition they use.
All Measures
Price-only returns ignore dividends, coupons, carry, and corporate actions unless the series is adjusted.
Returns computed from stale or non-synchronous prices (illiquid assets, different market closes) understate volatility and distort correlations.
Decision Framework: Which Return for Which Task
The right measure follows from the operation you are about to perform: summing across positions, chaining across time, fitting a distribution, or reporting a result.
Task | Use | Why |
PnL, limits, capital, loss reporting | Absolute | Money is what is lost or gained; sums exactly across books |
Performance reporting to clients | Discrete (compounded) | Intuitive; matches GIPS-style time-weighted returns |
Portfolio return and attribution in one period | Discrete | Exact value-weighted aggregation |
Multi-period growth, annualisation | Log (or compound discrete) | Exact time additivity |
Volatility estimation, GARCH, EWMA | Log (daily discrete is near-identical) | Time-additive, unbounded, model-consistent |
Option pricing, GBM, Monte Carlo of prices | Log | Lognormal prices stay positive |
Rates, spreads, near-zero or negative factors | Absolute (bp) | Relative changes undefined or unstable near zero |
Equity, FX, commodity price shocks | Relative (discrete or log) | Moves scale with price level |
Comparing assets of different size or price | Discrete or log | Scale-free |
Three rules of thumb cover most cases:
Aggregating across positions? Work in discrete returns or PnL.
Aggregating across time or modelling dynamics? Work in log returns.
Talking to people? Report discrete returns and dollars, and convert from logs before you present.
Returns as Shocks in Risk Management
In VaR, expected shortfall, and stress testing, a return stops being a description of the past and becomes an instruction: move this risk factor by this much and revalue. The return definition becomes the shock type, and it decides how a historical or hypothetical move is translated to today's market level.
where X0 is today's level of the risk factor and Δ or r is the shock taken from history or from the scenario.
Why It Matters: Level Dependence: Suppose a historical day saw a policy rate move from 5.00% to 5.50%: +50bp absolute, +10% relative. Replay that day on today's rate.
Today's Rate | Absolute Shock (+50bp) | Relative Shock (+10%) |
4.00% | 4.50% | 4.40% |
1.00% | 1.50% | 1.10% |
0.10% | 0.60% | 0.11% |
-0.50% | 0.00% | -0.55% |
At low rates, the relative shock collapses towards nothing, and at negative rates it moves in the wrong direction. This is why rates and credit spreads are normally shocked in absolute basis points. Some models use a shifted log approach, adding a constant before taking logs, to keep relative behaviour while tolerating negative levels.
Equities, FX and commodities are the opposite case. A -9% equity day from a crisis period, replayed as -9% on today's index, scales sensibly with the index level. Replayed as an absolute index-point move, it would be far too small or far too large depending on how much the index has drifted since.
Discrete vs. Log Shocks In Scenarios: A stress scenario that says "equities -30%" must state its convention. Applied as a discrete shock, the price falls 30%. Applied as a log shock of -0.30, it falls by e^-0.30 - 1 = 25.9%. Over a large book, that difference is material, and it comes entirely from a definitional choice, not from the scenario's narrative.
Regulatory Conventions: The FRTB standardised approach builds this logic into its sensitivity definitions: delta for interest rates and credit spreads is measured against a 1bp absolute shift, while delta for equity, FX and commodity is measured against a 1% relative shift. The choice tracks the same level-dependence reasoning above.
Consistency Rule: Measure the shock and apply the shock in the same convention. A log return estimated from history should be reapplied with an exponential, a discrete return with (1 + r), and an absolute change by addition. Most shock-related breaks in production come from a mismatch between the engine that generates shocks and the engine that applies them.
Best Practices and Common Errors
Data and Conventions
Label every return series with its unit, interval, and scope: "daily log total return" is unambiguous; "return" is not.
Use adjusted prices for splits and corporate actions, and total-return series where income matters.
Align timestamps across assets before computing returns; mixing India, London, and New York closes distorts correlations.
Treat stale prices and holidays explicitly rather than recording a zero return that never happened.
Choose the frequency for the task: daily for risk, monthly for long-run performance; higher frequency adds microstructure noise.
Calculation
Compound discrete returns over time; never sum them.
Aggregate across positions with discrete returns or PnL; never average log returns across assets unless you accept the approximation.
Annualise log returns by multiplying by the number of periods, and volatility by the square root; state that this assumes i.i.d. returns.
Convert back to discrete returns before presenting results.
Use absolute changes for any factor that can approach zero or go negative.
Errors That Recur on Desks
Quoting the arithmetic mean of monthly returns as the expected annual growth rate.
Building a portfolio log-return series as a weighted average of asset log returns, then compounding it over years.
Estimating volatility on log returns and applying it as a discrete shock, or the reverse.
Applying historical relative shocks to rates in a low-rate regime.
Comparing a manager's price return with a total-return benchmark.
Computing returns from prices that were not refreshed, then concluding the asset is low-volatility.
Appendix: Derivations
A1. Continuous Compounding as a Limit: A nominal rate R compounded m times per period grows $1 to (1 + R/m)^m. As m grows without bound:
Setting e^R = S_t / S_{t-1} and solving gives R = ln(S_t / S_{t-1}), the log return.
A2. Time Additivity: For prices S0, S1, …, Sn the ratios telescope:
A3. Log Return Never Exceeds Discrete Return: Let f(x) = x - ln(1 + x) for x > -1. Then f(0) = 0 and f'(x) = x / (1 + x), which is negative for -1 < x < 0 and positive for x > 0. So f has its minimum of zero at x = 0, giving ln(1 + x) ≤ x, with equality only at zero.
A4. Volatility Drag: Take expectations of the second-order Taylor expansion of ln(1 + r), with E[r] = μ and Var(r) = σ²:
The last step drops μ², which is small relative to σ² at short horizons. In continuous time, Itô's lemma applied to ln S under geometric Brownian motion gives the same drift adjustment exactly.
A5. Error From Averaging Log Returns Across Assets: Expanding e^{ri} and ln(1 + y) to second order:
The bracket is the cross-sectional variance of the asset log returns, which is never negative. Averaging log returns across assets therefore understates the portfolio log return by about half that dispersion. In the 50/50 example of Section 7, the correction is +2.05%, almost exactly offsetting the -2.04% average.
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