A log revenue forecast needs more than an exponentiation

When an analyst forecasts the logarithm of revenue and then exponentiates the result, the output may represent the median of the revenue distribution rather than its mean. Those values differ when the distribution is skewed. A finance plan that treats the output as expected revenue can therefore use the wrong summary even though the spreadsheet formula correctly reverses the logarithm.

Ask what the forecast is intended to support. A median describes a central outcome with half the modeled distribution on either side under the relevant assumptions. A mean is the probability-weighted average. A downside percentile answers another question. The choice should follow the decision and the loss from forecasting too high or too low, not whichever label the software exports by default.

Forecasting: Principles and Practice explains back-transformation and bias adjustment after forecasting on a transformed scale. The numerical example here uses an explicitly assumed lognormal distribution so the calculation can be checked. It is not evidence that a particular company's revenue follows that distribution.

Suppose the natural logarithm of next month's revenue is modeled as normally distributed with mean equal to the natural logarithm of $100,000 and standard deviation 0.30. Under those assumptions, exponentiating the log-scale mean gives a revenue median of $100,000.

The mean of that lognormal distribution is exp(mu + sigma squared divided by two). Here sigma squared is 0.09, so expected revenue is $100,000 multiplied by exp(0.045), approximately $104,603. The roughly $4,603 difference is a property of the assumed distribution, not an extra sale or a discretionary management uplift.

If the log-scale standard deviation were 0.60 with the same log-scale mean, the revenue mean would be about $119,722. Greater uncertainty creates a larger gap between the mean and median in this particular model. That makes the variance assumption financially consequential and worth inspecting, especially at longer horizons.

Check the logarithm base as well as the units. The expression exp(mu) reverses a natural logarithm. A model fitted using base-ten logarithms needs the matching inverse and a variance adjustment derived for that transformation. A copied formula can return a number without an obvious spreadsheet error while measuring something entirely different from the intended revenue forecast.

The variance in a back-transformation adjustment must match the forecast distribution being summarized. A residual variance from a fitted historical model is not automatically the variance of a six-month-ahead prediction. Parameter uncertainty, future predictors and accumulated process uncertainty can affect the result, depending on the model.

Ask the analyst to show where the variance comes from and whether the forecasting package already performs the adjustment. Applying another correction to a forecast that is already a mean can inflate the result. Conversely, relabeling an exported median as a mean leaves the underlying difference unresolved.

For models outside the lognormal assumptions used above, a simple formula may be unsuitable. An analyst may obtain the original-scale mean from a supported distribution or from simulated outcomes, with the method and assumptions recorded. The finance reviewer should be able to trace the reported number to that calculation.

The natural logarithm of zero is not defined, and ordinary real-valued logarithms cannot be applied to negative revenue observations. Credits, reversals and inactive periods can therefore create problems for a model built on net revenue. Dropping those periods without explanation changes the population being forecast.

Adding a constant before taking logs is sometimes proposed, but the constant changes the transformation and its interpretation. It must also be reversed correctly. A constant chosen in dollars behaves differently when the same data is later expressed in thousands of dollars, unless the implementation accounts for the unit change.

Consider whether gross sales and credits should be modeled separately, or whether another model better fits the measure. The answer depends on the business process and the data. Keep the modeling choice separate from the accounting records; a statistical convenience is not a reason to remove valid credits from the ledger.

A monotonically increasing transformation such as exponentiation preserves quantile ordering. Transforming the endpoints of a log-scale prediction interval gives the corresponding original-scale interval under that model. The resulting dollar interval will generally be asymmetric around the median.

Do not replace that interval with a visually symmetric dollar band merely to simplify a chart. A lower endpoint cannot cross zero in the assumed lognormal model, while the upper tail can extend much farther. The shape communicates an assumption about uncertainty that readers should see.

Also distinguish a range for the model's estimated mean from a prediction interval for an actual future month. The latter includes outcome uncertainty and answers the operational planning question more directly. Confirm which object a chart shows before management uses it to choose a cash or capacity buffer.

Expected values add across components, but medians generally do not. Adding the median forecasts of several product lines does not necessarily produce the median of total revenue. If the plan needs a total distribution, the relationships between product outcomes also matter; summing unrelated percentile endpoints does not solve that problem.

Use a small reproducible calculation to check the model export before relying on it. Retain the forecast target, transformation, unit, horizon, summary statistic and any bias adjustment in the model documentation. Reconcile the original-scale forecast to the values loaded into the financial plan.

Review forecast performance on the original currency scale as well as the transformed scale. A model can fit logarithms well while missing dollar outcomes that matter to payroll or inventory commitments. Decide which errors the business needs to reduce and evaluate the selected summary against that purpose.

When handing the model to another finance owner, include an example that reproduces one forecast from its distribution parameters. A reviewer should not have to infer whether a column labeled revenue contains a median, a mean or a manually adjusted target.