# De-smoothing and Backfilling Quarterly Returns

The return series of Private Equity and Real Estate instruments typically have significant positive serial correlations and artificially low standard deviations.

For some of our available time series, we treat this by de-smoothing the return series to correct for serial correlations with autoregressive models. For private equity and direct real estate, we estimate an $AR(1)$ model

$R_t=\rho_{0}+\rho_{1}R_{t-1}+ \epsilon_{t}$

where $\epsilon_{t}$ is normally distributied with a zero mean. We then use the model parameter $\rho_{1}$ to de-smooth the time series returns by applying the following

$R_{t}' =\frac{(R_{t}-\rho_{1}R_{t-1})}{(1-\rho_{1})}$

In addition to de-smoothing the returns of private equity and real estate, we backfill quaterly returns into monthly returns by assuming the assets' returns follow a multi-variate normal distribution whose parameters are defined by the available monthly and corresponding quarterly data.

This methodology not only preserves the correlations between the asset classes, but also constrains the monthly returns of Private Equity and Real Estate within the same quarter to sum to the known quarterly return observation. This maintains the statistical features of the quarterly returns but also recovers a monthly total return index that closely corresponds to the actual quarterly total return index.