Unsmoothing Illiquid Asset Returns

A private equity fund reports a Sharpe ratio that makes public equities look reckless by comparison. Part of that is real — but part of it is an accounting artifact. When an asset trades every second, its price reflects the market's freshest opinion. When an asset trades once a quarter, or not at all, its "price" is usually an appraisal — a smoothed estimate that only partially updates each period. That smoothing quietly shrinks reported volatility and correlation, and inflates risk-adjusted returns, without the underlying risk having changed at all.

Why appraisals smooth returns

An appraiser (or a fund manager marking a private position) rarely marks fully to the latest information. They anchor partly on the last reported value and partly on new evidence, producing a reported return that is a blend of this period's true return and last period's reported return:

$$r_t^{observed} = (1-\theta) \, r_t^{true} + \theta \, r_{t-1}^{observed}$$

In words: the return you see this period is mostly the true return, but a weight $\theta$ of it is just carried over from last period's already-smoothed number. A $\theta$ near zero means prices update fully every period, like a liquid stock. A $\theta$ near one means almost nothing new gets reflected, and the series looks artificially calm.

Unsmoothing the series

If you can estimate $\theta$ — usually from the autocorrelation of the reported returns — you can invert the smoothing formula to back out an estimate of the true returns:

$$r_t^{true} \approx \frac{r_t^{observed} - \theta \, r_{t-1}^{observed}}{1-\theta}$$

In words: subtract off the carried-over portion and rescale, which amplifies each period's genuine move back to its full size.

Worked example

A real estate fund reports quarterly returns of 2%, 1%, 3%, and 0.5%, and its return series shows a first-order autocorrelation consistent with $\theta = 0.5$. Unsmoothing the third quarter:

$$r_3^{true} = \frac{3\% - 0.5 \times 1\%}{1 - 0.5} = \frac{2.5\%}{0.5} = 5\%$$

The reported 3% quarter was really a 5% quarter once the drag from the prior quarter's smoothed number is stripped out. Repeating this across the series and computing volatility on the unsmoothed numbers typically doubles or triples the reported standard deviation, and pushes correlations with public markets up — because illiquid assets aren't actually less correlated with equities, they just report their bad news later.

What this means in practice

Allocators comparing a private equity or real estate sleeve against public markets should never take reported Sharpe ratios and correlations at face value. Unsmoothing is standard practice in institutional due diligence precisely because unadjusted numbers make illiquid strategies look like a diversifying free lunch, when much of that benefit is a reporting lag rather than genuine risk reduction.