Apply the exponentially weighted moving average (EWMA) method to volatility estimation, and outline the other ways in which older return observations can be given weights.

FRM® Part I · Valuation and Risk Models · Measuring and Monitoring Volatility · FRM1.VR.3.e

The idea

The EWMA model updates variance in one step. New variance = λ × old variance + (1 − λ) × latest squared return, usually assuming a zero mean. Here λ is the decay factor. The weight on an observation k days old is proportional to λ raised to the power k. A λ near 1 gives slow decay, a smooth estimate and a long memory, while a smaller λ reacts faster. RiskMetrics used 0.94 for daily data. Forecasts of future variance equal the current estimate at every horizon. The common slip is swapping λ and 1 − λ, or updating volatility rather than variance. Other schemes weight older data differently: equal weights, a window cut-off, and hybrid or weighted historical simulation. In that last scheme, there are K scenarios. Latest weight = (1 − λ) ÷ (1 − λ^K). Each earlier scenario gets λ times the weight of the next, so that the weights sum to one.

Formulas

EWMA variance update

new variance = λ × old variance + (1 − λ) × u²

λ: the decay factor, 0.94 in RiskMetrics for daily data
old variance: the previous variance estimate
u: the latest return, assuming a zero mean

Work in variance, not volatility, and take the square root at the end. Swapping λ and 1 − λ is the common slip.

Weight on the latest scenario in weighted historical simulation

latest weight = (1 − λ) ÷ (1 − λ^(K))

λ: the decay factor
K: the number of scenarios

Each earlier scenario gets λ times the weight of the next, so the weights sum to one. Without the denominator the weights fall short of one.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which expression is the EWMA update for the daily variance, with decay factor lambda and latest return u?

New variance = (1 - lambda) x old variance + lambda x u²Why not: Swaps lambda and 1 - lambda, giving the new return the large weight.
New volatility = lambda x old volatility + (1 - lambda) x |u|Why not: Updates volatility rather than variance; the recursion is linear in variance.
New variance = lambda x old variance + (1 - lambda) x u² - correct
New variance = omega + lambda x old variance + (1 - lambda) x u²Why not: Adds a GARCH-style constant; EWMA has none, and the weights would not then sum to the stated form.

EWMA weights the previous variance by lambda and the latest squared return by 1 - lambda, so the weights sum to one and a lambda near 1 gives a slowly changing estimate.

A harder one

An EWMA model with lambda = 0.94 has a daily volatility estimate of 1.20% at the close. The next day's return is -2.5%. Assuming a zero mean, what is the updated daily volatility estimate?

1.28%Why not: Averages volatility and the absolute return instead of updating the variance.
2.44%Why not: Swaps the weights, giving the new squared return 0.94 and the old variance 0.06.
1.31% - correct
1.73%Why not: Reports the updated variance (in %²) as if it were the volatility; the square root is missing.

New variance = 0.94 x 1.44 + 0.06 x 6.25 = 1.7286 (%²); volatility = square root = 1.315%.

The hardest tier

A risk manager wants an EWMA in which an observation 23 days older than another carries half its weight. Which decay factor lambda is closest?

0.9703 - correct
0.9783Why not: Treats the decay as linear, 1 - 0.5/23, rather than geometric.
0.9565Why not: Sets 1 / (1 - lambda) = 23, which gives the average age of the weights, not the half-life.
0.9400Why not: Uses the RiskMetrics daily default without checking it against the required half-life (about 11 days at 0.94).

Weights fall by lambda per day, so lambda^23 = 0.5 and lambda = 0.5^(1/23) = 0.9703. A lambda of 0.94 has a half-life of ln 0.5 / ln 0.94 = 11.2 days.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

Start free →