Simple, Log and Excess Returns Need Different Calculation Contracts

1. What is the difference between simple and log returns?

A simple price return is the ending price divided by the starting price, minus one. A log price return is the natural logarithm of that same positive price ratio. Simple returns express the change relative to starting value; log returns express the logarithm of the gross factor. They are close for small movements but not identical. PerformanceAnalytics exposes discrete and log methods explicitly, reinforcing that the representation is a caller's choice rather than an automatic formatting detail. Price-only formulas also omit distributions unless the input series already incorporates them consistently. Label representation and income treatment before comparing statistics or passing observations into a model.

2. Work the same synthetic path both ways

The constructed price path one hundred, one hundred twenty, one hundred eight gives simple returns twenty percent and minus ten percent. Multiplying their gross factors produces 1.08, hence cumulative simple return eight percent. The corresponding log returns are approximately 0.18232156 and minus 0.10536052; their sum is approximately 0.07696104. Exponentiating that sum recovers the same gross factor. Adding the simple returns instead gives ten percent, which does not reconcile the path. The table keeps both representations beside the original endpoints. These are invented prices in arbitrary units, not historical returns, executions or a record of an investable strategy.

3. Time addition does not imply portfolio addition

Log returns add across consecutive compatible periods for one wealth path. They generally do not average across assets to produce a portfolio's log return. For a single period with equal starting capital in assets returning twenty percent and minus ten percent, portfolio simple return is five percent before costs and flows. Its log return is log of 1.05, approximately 0.04879016. The equal-weight average of the two asset log returns is approximately 0.03848052, a different quantity. Define whether weights describe starting capital, continuously changing exposures or another convention. Otherwise a valid log transformation can be combined with an invalid portfolio aggregation rule.

4. Excess return needs a matching cash interval

A conventional arithmetic excess return subtracts the corresponding cash return from the asset return for the same interval. It is not computed by subtracting an annual yield from every monthly observation. PerformanceAnalytics documents this subtraction with a supplied risk-free series. A relative gross return, dividing the asset gross factor by the cash gross factor and subtracting one, is a different object. A log excess return subtracts compatible log returns. These quantities can serve different analytical purposes; they should not share one unlabeled output field. State the cash instrument proxy, currency, quotation conversion, observation interval and treatment of missing benchmark observations.

5. Arithmetic excess does not automatically link as relative wealth

For the synthetic two-period asset path, assume cash simple returns of two percent and one percent. Arithmetic excess returns are eighteen percent and minus eleven percent. Linking those excess observations as if they were ordinary portfolio returns gives 5.02 percent. Actual relative wealth is instead 1.08 divided by 1.0302, minus one, approximately 4.8340 percent. Both calculations are possible, but they answer different questions. State which is intended before using an excess series in a summary. The cash assumptions are invented for unit reconciliation and are not prevailing interest rates, financing charges or a recommendation of a cash benchmark.

6. Domains and cash flows can break the shortcut

Finite log returns require positive gross factors; a complete loss has a simple return of minus one but no finite logarithm. Zero or negative quoted prices also invalidate the elementary price-ratio log formula without a separately specified economic accounting framework. External contributions are another boundary: taking the log of an account balance ratio treats deposits as performance unless the wealth path has first been adjusted for flows. Preserve these rejection conditions rather than inserting small positive constants or dropping troublesome rows. A representation conversion cannot repair an undefined economic return, an incomplete total-return series or an account history without the necessary valuations.

7. Choose the contract from the question

Ask whether the output concerns one-period capital growth, a linked wealth path, statistical increments or performance relative to cash. Are returns decimal values, and do all series cover matching intervals? Are distributions, fees and external flows treated consistently? Does a claimed portfolio log return come from portfolio wealth rather than weighted asset logarithms? Frequent failures include adding simple returns through time, averaging asset logs across capital weights, and mixing annual yields with period returns. Preserve the original observations and conversion formula in the result record. That makes representations inspectable without suggesting that one convention by itself improves investment performance or supplies a forecast.

Primary references

These sources support methods and conventions. The example values are original constructions, not observations extracted from these documents.

• PerformanceAnalytics: Return.calculate, discrete and log return representations
• PerformanceAnalytics: Return.excess, subtracting an aligned risk-free return

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