Why Standardized Portfolio Returns Have Unit Volatility Under Accurate Risk Forecasts
Summary
The note explains the bias statistic used to assess portfolio risk forecasts in factor models. For each period, the method divides the realized portfolio return by its forecast volatility, then calculates the standard deviation of those standardized returns across observations. The central result is that this statistic has a population value of one when returns have a constant true standard deviation and the risk forecast matches it at every period.
The reasoning is that dividing a normally distributed return by its true standard deviation produces a standardized variable with unit standard deviation. A nonzero mean changes the standardized variable's mean, but not its standard deviation. The explanation assumes the same underlying volatility across periods and perfect forecasts; it does not address estimation error, changing volatility, serial dependence, or finite-sample variation. Thus, an observed statistic need not equal one exactly, even when forecasts are broadly sound.
Key ideas
- Dividing returns by their true standard deviation gives a variable with unit standard deviation.
- The bias statistic is the sample standard deviation of returns scaled by forecast risk.
- A nonzero return mean does not change the standardized variable's standard deviation.
- The unit benchmark assumes constant volatility and perfectly accurate forecasts.
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# Why is the expected value of bias statistic one?
# Why is the expected value of bias statistic one?
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?
Source: Barra Equity Model Empirical Notes (various editions 2011,2012)
## Answer by J-F (score 3, accepted)
https://quant.stackexchange.com/a/41197
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 $\text{Std}\left(\dfrac{r_t}{\sigma_t}\right) = \text{Std}\left(\dfrac{r_t}{\sigma}\right) = 1$.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.