FRM Exam Part II · Illiquid Assets

Smoothed Returns and Return Unsmoothing (Geltner Method)

Updated 11 October 2026 · Fact-checked

Appraisal-based pricing makes reported returns of illiquid assets lag true values, so they look smooth. This creates positive autocorrelation, understated volatility, understated correlation and beta, and inflated Sharpe ratios. To unsmooth, estimate the smoothing parameter α from first-order autocorrelation, then compute r* = (r − α × previous r) ÷ (1 − α).

Understand Smoothed Returns and Return Unsmoothing

Liquid assets trade often, so their prices update constantly. Illiquid assets such as real estate, private equity and some hedge fund positions do not. Their values come from appraisals or manager marks, which use old transactions and are revised slowly. So reported returns reflect today's true return only in part and carry a share of past returns.

This is smoothing. A simple model says the reported return is a weighted average of the true return this period and the reported return last period. The weight on the past is α, the smoothing parameter. The higher α is, the more stale the valuations. Smoothed returns show positive autocorrelation: a good period tends to be followed by another good-looking period.

The damage is to risk measures, not to the long-run average return. Smoothing shrinks period-to-period swings, so volatility is understated. It also delays the response to market moves, so correlation and beta against liquid markets are understated. The asset looks like a strong diversifier with a high Sharpe ratio. VaR and other risk numbers based on reported data come out too low. Maximum drawdowns also look milder.

Unsmoothing tries to reverse this. Under the Geltner approach, you estimate α from the autocorrelation of reported returns, then strip out the lagged part to recover an estimate of the true return. The unsmoothed series has higher volatility and higher correlation with liquid markets. Treat the result as an estimate, not a fact. It relies on assumptions about how the smoothing works.

Key formulas to remember

Smoothing model: r(t) = (1 − α) × r*(t) + α × r(t−1); r(t) is the reported return, r*(t) the true return, α the smoothing parameter between 0 and 1. A larger α means more smoothing.
Geltner unsmoothing: r*(t) = [r(t) − α × r(t−1)] ÷ (1 − α); You lose the first observation because it has no prior return. Use α estimated from the data.
Estimating α: α ≈ ρ1, the first-order autocorrelation of reported returns; Works when true returns are roughly uncorrelated over time and smoothing is first order.
Volatility adjustment: σ* ≈ σ(reported) × √[(1 + α) ÷ (1 − α)]; Holds if true returns are independent over time. Since the factor exceeds 1, true volatility is higher than reported.
Reported variance under smoothing: σ²(reported) = σ*² × (1 − α) ÷ (1 + α); The same relationship rearranged. It shows how much variance smoothing removes.

How to solve Smoothed Returns and Return Unsmoothing questions

Use this order for any question on smoothed returns or unsmoothing.

1Identify the pricing source. Appraisal, manager marks or stale quotes signal smoothing; daily exchange prices do not.
2Look for the evidence: positive first-order autocorrelation in reported returns, unusually low volatility and low correlation with liquid markets.
3Find α. Use the value given, or set α equal to the first-order autocorrelation.
4If asked to unsmooth a return, apply r* = (r − α × previous r) ÷ (1 − α). Use the reported previous return, not the unsmoothed one.
5If asked to adjust volatility, multiply reported volatility by √[(1 + α) ÷ (1 − α)].
6Recompute dependent measures: Sharpe ratio (same excess return, higher volatility), VaR, beta and correlation.
7State the interpretation: reported risk was understated, diversification benefit was overstated, and the result is only an estimate.

Quickest way: Autocorrelation shortcut

When to use it: Use when the question gives autocorrelation or α and asks for true volatility, a Sharpe ratio or one unsmoothed return.

Set α equal to the given autocorrelation.
For volatility, compute the factor √[(1 + α) ÷ (1 − α)] and multiply.
For one return, compute (r − α × previous r) ÷ (1 − α).
Check direction: unsmoothed volatility must be higher and the Sharpe ratio lower than reported.
Eliminate any option that shows true volatility below reported volatility.

Common mistakes in Smoothed Returns and Return Unsmoothing

Saying smoothing lowers the average return.
Students link lower risk with lower return.
Fix: Smoothing mainly changes the timing and variability of returns. The long-run average is not the main effect. The distortion is in volatility, correlation, beta and Sharpe ratio.
Dividing reported volatility by √[(1 + α) ÷ (1 − α)] instead of multiplying.
The factor is mixed up with the variance relationship, which has (1 − α) on top.
Fix: True volatility is larger. Multiply by √[(1 + α) ÷ (1 − α)], which is greater than 1 for positive α.
Using the unsmoothed previous return in the formula.
Students think they are working recursively.
Fix: The formula uses the reported previous return r(t−1). Both returns in the numerator come from the reported series.
Forgetting to divide by (1 − α).
Students stop after subtracting α × previous return.
Fix: The division rescales the new information in the return. Without it the result is too small.
Claiming illiquid assets are great diversifiers based on reported correlations.
Low reported correlation looks like real diversification.
Fix: Stale prices lower measured correlation. After unsmoothing, correlation with liquid markets and beta rise, and the diversification benefit shrinks.
Treating negative autocorrelation as smoothing.
Students ignore the sign.
Fix: Smoothing from appraisals produces positive autocorrelation. Only positive α fits the model.

Worked examples

Example 1

A real estate fund reports returns of 5% last quarter and 3% this quarter. The estimated smoothing parameter α is 0.40. Using the Geltner method, what is the unsmoothed return for this quarter?

Show the solution

Write the formula: r* = (r − α × previous r) ÷ (1 − α).
Compute α × previous r = 0.40 × 5% = 2%.
Numerator: 3% − 2% = 1%.
Denominator: 1 − 0.40 = 0.60.
r* = 1% ÷ 0.60 = 1.67%.

Answer: The unsmoothed return is about 1.67%.

Example 2

A private real estate index has reported annual volatility of 8% and a first-order autocorrelation of 0.50. The excess return is 6% a year. Estimate the true volatility and the Sharpe ratio before and after unsmoothing, assuming true returns are independent over time.

Show the solution

Set α = 0.50.
Volatility factor = √[(1 + 0.50) ÷ (1 − 0.50)] = √3 = 1.732.
True volatility = 8% × 1.732 = 13.86%.
Reported Sharpe ratio = 6% ÷ 8% = 0.75.
Unsmoothed Sharpe ratio = 6% ÷ 13.86% = 0.43.
Interpret: reported risk was understated, so the Sharpe ratio was overstated by a large margin.

Answer: True volatility is about 13.86% and the Sharpe ratio falls from 0.75 to about 0.43.

Exam tips

Expect questions that give autocorrelation and ask for corrected volatility, Sharpe ratio or VaR. Memorise the square-root factor.
Know the direction of every effect: volatility, correlation and beta go up after unsmoothing; Sharpe ratio goes down.
In conceptual questions, link the cause (appraisal-based, stale pricing) to the evidence (positive autocorrelation) and then to the risk consequence.
Be careful with wording. Reported figures are understated, and unsmoothed figures are estimates that depend on model assumptions.
When a case describes a portfolio with a large private asset share, expect risk to be understated and diversification overstated.

Practice questions from Illiquid Assets

Smoothed Returns and Return Unsmoothing: frequently asked questions

Why do appraisal-based returns show positive autocorrelation?

Appraisers rely on past transactions and revise values gradually. So each period's reported value carries part of the previous period's value. A rise tends to be spread over several periods, which creates positive autocorrelation.

How do I choose α in the Geltner method?

Use the value given in the question. If you must estimate it, use the first-order autocorrelation of the reported return series. This works best when true returns have little autocorrelation themselves.

Does smoothing change the average return?

Its main effect is on the pattern of returns, not the long-run average. The impact shows up in volatility, correlation, beta and Sharpe ratio, which are all biased in a flattering direction.

Why does smoothing understate correlation with the market?

Reported returns react to market moves late and only in part. In the same period the asset seems not to move with the market, so measured correlation and beta are too low. Unsmoothing restores a more realistic link.