Daily Multi-Asset Portfolio Variance Formula
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emilioalzamora1
Well-Known Member
Hi,
I am not sure what you are referring to with 'multi assset portfolio variance'? EWMA is used for volatility forecasting.
The EWMA estimator has a long history in business and economic forecasting but has only more recently adopted for vol. forecasting. It is an alternative approach to weight observations in the sample in a such way that their importance declines smoothly into the past.
Note that in order to use the EWMA estimator, we must specify sigma^2! and so the estimated variances are not
unique; However, owing to the exponential weighting, our choice of sigma^2 rapidly becomes irrelevant as we move through the sample; We can set sigma^2 to zero, for example. (As it is often done by Christoffersen for example).
To remove the effect of our specification of sigma^2 , it is common to ignore the first, say, 100 variance estimates.
The only unknown parameter in the EWMA model is the decay factor, lambda ; There are a number of ways lambda can be estimated. For instance, it could be set to the value that minimises the one step ahead forecast error variance; Or it couldbe set so that it provides the optimal conditional coverage when used in a value at risk model; Alternatively, it
could be estimated by by maximum likelihood after specifying a conditional distribution for returns, such as the normal distribution.
For daily financial data, lambda is typically estimated to be between 0.92 and 0.96. Risk Metrics has chosen lambda to be 0.94 (daily data) or 0.97 (monthly data).
The EWMA model places geometrically declining weights on past observations, assigning greater weight to more recent observations. The weights decrease at a geometric rate. The lower lambda, the more quickly older observations are forgotten.
HIGH LAMBDA (close to 1) implies: SLOW decay and LESS weight to recent data (this implies that the last return has a small impact).
Instead, if lambda is reduced (e.g. from 0.97 to for example 0.88), then there is HIGH decay and MORE weight to recent data (this implies that the last return has a greater impact).
{GARCH vs. EWMA: EWMA is a special (RESTRICtED) case of GARCH, both use exponential smoothing. GARCH is a generalized case of EWMA (Risk Metrics). THE main difference between GARCH and EWMA: GACRH adds the parameter that weighs the long-run average and therefore incorporates mean-reversion, but GARCH has the drawback of NON-linearity and greater model risk when forecasting out-of-sample.}
There are two terms on the right-hand side of the EWMA equation:
1. (1-lambda)*r^2(t-1)
determines the INTENSITY of REACTION of volatility to market events. The smaller we set lambda, the more volatility reacts to the market information in yesterday's return.
2. lambda*sigma^2(t-1)
determines the PERSISTENCE in volatility. Irrespective of what happens in the market, if vol. was high yesterday it will be still high today. The closer lambda to 1, the more persistent is vol. following a market shock. Thus a high lambda gives little reaction to actual market events but great persistence in vol.
A low lambda gives high reactive volatilities that quickly die away. A restriction of the EWMA model is that REACTION and PERSISTENCE are NOT INDEPENDENT.
The strength of reaction is determined by (1-lambda) while the persistence of shocks is determined by lambda.
{This last few sentences follow Carol Alexander 'Market Risk', "Interpretation of lambda"}.
Hope this is useful to a certain extent.
I am not sure what you are referring to with 'multi assset portfolio variance'? EWMA is used for volatility forecasting.
The EWMA estimator has a long history in business and economic forecasting but has only more recently adopted for vol. forecasting. It is an alternative approach to weight observations in the sample in a such way that their importance declines smoothly into the past.
Note that in order to use the EWMA estimator, we must specify sigma^2! and so the estimated variances are not
unique; However, owing to the exponential weighting, our choice of sigma^2 rapidly becomes irrelevant as we move through the sample; We can set sigma^2 to zero, for example. (As it is often done by Christoffersen for example).
To remove the effect of our specification of sigma^2 , it is common to ignore the first, say, 100 variance estimates.
The only unknown parameter in the EWMA model is the decay factor, lambda ; There are a number of ways lambda can be estimated. For instance, it could be set to the value that minimises the one step ahead forecast error variance; Or it couldbe set so that it provides the optimal conditional coverage when used in a value at risk model; Alternatively, it
could be estimated by by maximum likelihood after specifying a conditional distribution for returns, such as the normal distribution.
For daily financial data, lambda is typically estimated to be between 0.92 and 0.96. Risk Metrics has chosen lambda to be 0.94 (daily data) or 0.97 (monthly data).
The EWMA model places geometrically declining weights on past observations, assigning greater weight to more recent observations. The weights decrease at a geometric rate. The lower lambda, the more quickly older observations are forgotten.
HIGH LAMBDA (close to 1) implies: SLOW decay and LESS weight to recent data (this implies that the last return has a small impact).
Instead, if lambda is reduced (e.g. from 0.97 to for example 0.88), then there is HIGH decay and MORE weight to recent data (this implies that the last return has a greater impact).
{GARCH vs. EWMA: EWMA is a special (RESTRICtED) case of GARCH, both use exponential smoothing. GARCH is a generalized case of EWMA (Risk Metrics). THE main difference between GARCH and EWMA: GACRH adds the parameter that weighs the long-run average and therefore incorporates mean-reversion, but GARCH has the drawback of NON-linearity and greater model risk when forecasting out-of-sample.}
There are two terms on the right-hand side of the EWMA equation:
1. (1-lambda)*r^2(t-1)
determines the INTENSITY of REACTION of volatility to market events. The smaller we set lambda, the more volatility reacts to the market information in yesterday's return.
2. lambda*sigma^2(t-1)
determines the PERSISTENCE in volatility. Irrespective of what happens in the market, if vol. was high yesterday it will be still high today. The closer lambda to 1, the more persistent is vol. following a market shock. Thus a high lambda gives little reaction to actual market events but great persistence in vol.
A low lambda gives high reactive volatilities that quickly die away. A restriction of the EWMA model is that REACTION and PERSISTENCE are NOT INDEPENDENT.
The strength of reaction is determined by (1-lambda) while the persistence of shocks is determined by lambda.
{This last few sentences follow Carol Alexander 'Market Risk', "Interpretation of lambda"}.
Hope this is useful to a certain extent.
emilioalzamora1
Well-Known Member
I would like to add another important ingredient when setting up the equation and modelling/forecasting the variance one day ahead.
As I was pointing to Christoffersen 'Elements of Fin. Risk Management' there are different approaches used in the industry how start off with the variance at t=0. What I came across so far is the approach of setting the variance for t=0 to zero and for the for one-day-ahead the variance should be set to the 'mu' squared (using the mean return for the whole historical data).
J. Hull writes under chapter 20 'VaR' for problem 20.28 in the solutions:
Set the variance forecast at the end of the first day equal to the square of the average mean return on that day to start the EWMA calculations.
Hence, we do have the following set-up:
1.) sigma^2 (t=0) = 0
2.) sigma^2 (t=1) = mu^2 (the mean return!)
3.) sigma ^2 (t=2) = (1-lambda)*mu^2 (the mean return!) + lambda*sigma^2(t=1)
4.) sigma^2 (t=3) = (1-lambda)*mu^2 (the actual return in t=2) + lambda*sigma^2(t=2)
From 4.) onwards we do always use the actual return of the previous period for the forecast one-step ahead!
As I was pointing to Christoffersen 'Elements of Fin. Risk Management' there are different approaches used in the industry how start off with the variance at t=0. What I came across so far is the approach of setting the variance for t=0 to zero and for the for one-day-ahead the variance should be set to the 'mu' squared (using the mean return for the whole historical data).
J. Hull writes under chapter 20 'VaR' for problem 20.28 in the solutions:
Set the variance forecast at the end of the first day equal to the square of the average mean return on that day to start the EWMA calculations.
Hence, we do have the following set-up:
1.) sigma^2 (t=0) = 0
2.) sigma^2 (t=1) = mu^2 (the mean return!)
3.) sigma ^2 (t=2) = (1-lambda)*mu^2 (the mean return!) + lambda*sigma^2(t=1)
4.) sigma^2 (t=3) = (1-lambda)*mu^2 (the actual return in t=2) + lambda*sigma^2(t=2)
From 4.) onwards we do always use the actual return of the previous period for the forecast one-step ahead!
Hi @emilioalzamora1 brilliant, thank you! FWIW, by "multi-asset" I interpret @burgers to be asking if EWMA can be applied to a portfolio rather than just a single asset. Answer: yes, of course it can. Rather than a single vector of historical returns, we have a matrix of (a) factors by (n) returns. In this case, the EWMA update generates a covariance matrix (rather than the single covariance value which we study in FRM P1). EWMA covariance matrix is really the default, a good introduction (no surprise here) is by Carol Alexander in her Vol II, see https://forum.bionicturtle.com/reso...ractical-financial-econometrics-volume-ii.91/
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