# RiskMetrics VaR Volatility Sample Size

RiskMetrics calculates volatility using an exponentially weighted moving average. For a decay factor of 0.94, they advise a sample size of 74 past returns. Does this mean the entire calculation should have a total of 74 days of data, including today, or a total of 75 days of data (today and the previous 74 days)?

Realistically this won't affect the calculation very much but I would like to be as precise as possible and faithful to the document's method.

• re volatility forecasting - are you referring to rm1994 or rm2006 methodology? what is your forecast horizon - 1day; 20days? also, i see nothing bad in considering full year history (or above) since decay factor will "filter out" old points by assigning low weights – Nicholas Feb 3 '16 at 22:12

Depending of $\lambda$, pasts observations will be weighted differently, if you compute the volatility at time $t$ , the $t-1$ observation will be weighted by $(1-\lambda)*\lambda^{0}$, the $t-2$ observation by $(1-\lambda)*\lambda^{1}$ and so on so forth.

For $\lambda= 0.94$ :

1. The first observation is weighted by = $(1-0.94) * 0.94^0 =0.06%$

2. The second observation is weighted by = $(1-0.94) * 0.94^1 = 0,0564%$

...

1. The 74 observation is weighted by = $(1-0.94) * 0.94^{73} = 0,00065537%$

2. The 75 observation is weighted by = $(1-0.94) * 0.94^{74} = 0,00061604%$

For each observations taken into account you may compute the cumulative weights , and thus the weights not assigned :

Cumulative weights:

1. cumulative first observation= $0.06$ $\implies$ left $0.94\%$ of weights to assign (ie $1- 0.06 = 0.94$)

2. cumulative second observation = $(0.06+ 0.564 )= 0.1164$ $\implies$ left $0.8836 \%$ of weights to assign (ie $1- (0.1164 ) = 0.8836$ )

....

1. cumulative 74th observation = $(0.06+0.564+….) = 0,98973258$ $\implies$ left $0,01026742 \%$ to assign
2. cumulative 75th observation = $(0.06+0.564+….) = 0,99034863$ $\implies$ left $0,00965137 \%$ to assign

So if you truncate your computation at the 74 th observations (if you compute EWMA over 74 total observations), with lambda of 0.94, you lose more than 1% of weights.

If your tolerance rate is 1% this is not acceptable and you will need to truncate at the 75 th day (because with this number of observation you lose less than 1% of weights).

So for a 1% tolerance rate you need at least 75 past days to compute today volatility.

From Riskmetrics document : the formula wich returns the number of observations needed is given by :

$K = \frac{\ln(to)}{\ln(\lambda)}$

where $to$ is the tolerance level.

Thus for $\lambda = 0.94$ and tolerance $=0.01$ we indeed found a value lightly superior to 74 days (indicating we need 75 days):

$\frac{\ln(0.01)}{\ln(0.94)} = 74,4265073$

Rismetrics in their documents round this value to 74 but in doing so they violate their tolerance rate.

For details : See page 93 and 94 RiskMetrics — Technical Document.(1996)

• if your initial question is not just about decay factor but also about volatility forecasting in riskmetrics then i highly advise to familiarize yourself with RM2006 methodology msci.com/resources/research/technical_documentation/RM2006.pdf as it has improvements over legacy RM1994 – Nicholas Feb 4 '16 at 22:23
• how does RM2006 volatility estimation differ from RM1996? do they not use the same EWMA volatility estimation technique? – beeba Feb 29 '16 at 18:35