# Time-varying betas in Risk Management

Time-varying betas are a better alternative to regression-estimated betas because in risk management monitoring is done on a frequent basis – daily and even intra-daily. Indeed, parameters estimated by OLS at these frequencies will not reflect the actual market conditions because they just represent an average value over time on the sample.

Thus, for the purpose of mapping a portfolio and assessing its risks, higher frequency data (e.g. daily) could be used to estimate a time-varying portfolio beta for the index model in which the systematic and specific risks are not considered constant over time. Specifically, such estimates better reflect the current risk factor sensitivity for daily risk management purposes. The simplest possible time-varying parameter estimates are based on an exponentially weighted moving average (EWMA) model based on the only assumption that returns are independent and identically distributed. The EWMA beta estimates vary over time, even though the model specifies only a constant, unconditional covariance and variance. More advanced techniques include the class of generalized autoregressive conditional heteroscedasticity (GARCH) model, where we model the conditional covariance and variance and so the true parameters, as well as the parameter estimates, change over time. A time-varying beta is estimated as the covariance of the asset and factor returns divided by the variance of the factor returns and denoting the EWMA smoothing constant by λ. The EWMA beta estimate is the ratio of the EWMA covariance estimate to the EWMA variance estimate with the same smoothing constant. The analyst must choose a value for between 0 and 1, and values are normally in the region of 0.9–0.975. We assume = 0 95, which corresponds to a half-life of approximately 25 days (or 1 month, in trading days).

Various alternative implementations have been attempted during the years; for instance, by linking time-varying betas with coefficients related to the business cycle.

#### References

Elton, E., Gruber, M., Brown, S., Goetzmann, W. (2007). Modern Portfolio Theory and Investment Analysis, 11th edition.