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I have been reading about factor models recently. One of the ways in which the developer of these models (Barra/ Axioma) measure the accuracy of their models is by calculating the bias statistic for the risk forecasts provided by these models.

Basically, the process to calculate bias statistic has four legs -

a. At each time period t, forecast the risk of the portfolio. Let this be $sigma_t$ b. Calculate the return of the portfolio over the forecasting horizon (from time t to t+1). Let this be $r_t$ c. Calculate the standardized returns of the portfolio, $Z_t = r_t / sigma_t$ d. Do this T number of times. Bias Statistic will be the standard deviation of the standardized returns $Z_t$

What I fail to understand here is why should the expected value of the bias statistic, given that returns are assumed to be normally distributed and the risk forecasts are accurate, be equal to one?

Everywhere I have read, this is just given as an article of faith but I am just not able to wrap my head around it. Can someone please help me understand why is this true?

Snapshot of bias statistic description

Source: Barra Equity Model Empirical Notes (various editions 2011,2012)

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If $r_t\sim N(\mu, \sigma)$, where $\sigma$ is the "true" standard deviation of $r_t$ (no $t$ subscript), then $\dfrac{r_t}{\sigma}\sim N(\frac \mu \sigma, 1)$.

Assuming perfect risk forecasts (i.e. $\sigma_t = \sigma$ for all $t$), we get $\text{Std}\left(\dfrac{r_t}{\sigma_t}\right) = \text{Std}\left(\dfrac{r_t}{\sigma}\right) = 1$.

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    $\begingroup$ Usually the Forecast Error is defined as Forecast - Actual, and you hope that over the long run the average FE will be zero. Here, though, it is defined in ratio form, so the Unbiased case is when the ratio is 1. If consistently > 1 it would mean the forecast is biased low , and the outcome is consitently above the forecast. $\endgroup$
    – nbbo2
    Commented Jun 2, 2023 at 13:53

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