What is the difference between squared returns and variance?

I am trying to calculate 1-day ahead volatility forecasts using the exponentially weighted moving average, however I am unsure on how to read the formula provided within Risk-Metrics Technical Documentation for one day ahead forecasts. That formula is

$σ_{1,t+1|t}^2=λ σ_{1,t|t-1}^2+(1-λ) r_{1,t}^2$

(This is equation 5.3 on page 81 of this document)

Can someone please explain the difference between the variance and the squared returns? Both of these components are required for the calculation, however I was using squared returns as my variance for the series. Thanks

• Can you remind me what "the formula provided within risk-metrics for one day ahead forecasts" looks like. – Alex C Mar 18 '18 at 12:52
• σ_(1,t+1|t)^2=λσ_(1,t|t-1)^2+(1-λ) r_(1,t)^2. Apologies for the format, a more eligible version can be found on page 81 within the attached document: msci.com/documents/10199/5915b101-4206-4ba0-aee2-3449d5c7e95a – Harry Statman Mar 21 '18 at 12:40

This equation shows how you update your forecast. (In a word: recursively).

At the close of business on day t, when the day's return $r_{1,t}$ becomes available, you take a weighted average of:

• The forecast you had made yesterday for today, $σ_{1,t|t-1}^2$. (Hopefully you wrote that number down yesterday and you still have the piece of paper on which you wrote it down, otherwise you are in trouble).

• and, the square return for today

the number you thus compute is your forecast $σ_{1,t+|t}^2$ for tomorrow.

(So you are computing this time series of $\sigma^2$ values, it is not a variance taken from somewhere else and used in the calculation).

The justification for this method is what @phdstudent said, namely that the expected $r$ is negligible and so is being left out of the calculation.

Usually the formula for the sample variance of a stock is given by:

\begin{equation} Var(R_{i}) = E (R_t - E(R_t))^2 \end{equation}

If you are using daily data to compute the variance then the second term: $E(R_t) \approx 0$, therefore you can drop it from the computation. Which yields:

\begin{equation} Var(R_{i}) \approx E (R_t)^2 \end{equation}

With weekly, monthly, annual data this is no longer a good approximation.

• Just to give a couple of numbers. I've just checked, and since July 1962 the average daily return (i.e. $E[r]$) for the SP500 is 0.03%, the average squared return (i.e. $E[r^2]$) is 0.01%. So indeed $E[r]^2$ is small (about 0.00001%) compared to $E[r^2]$. The daily variance of returns is 0.01% and the daily volatility is about 1%. – fni Mar 18 '18 at 22:12
• Thanks for the responses so far. The expected return for the daily prices dataset t I am using is 0.000012%, so I think its fair to use the assumption that this value is equal to zero for the rest of the study. σ_(1,t+1|t)^2=λσ_(1,t|t-1)^2+(1-λ) r_(1,t)^2. This is the formula that I am using for the out of sample forecasts. Apologies for the format, a more eligible version can be found on page 81, within msci.com/documents/10199/5915b101-4206-4ba0-aee2-3449d5c7e95a. – Harry Statman Mar 21 '18 at 12:35