Hi

@ziminli1228 Yes, indeed, use of log returns is arguably more consistent (and would be fine). But Hull explains (in both textbooks I think) that the simple (aka, arithmetic) return is

**approximating **the log return. For one day (daily) periods, the difference tends to be negligible. It does matters for longer periods (per Kevin Dowds points). So, I do not believe we can say that

*either *should be used. It is an issue of distributional assumption, and as practical matter, log returns can be added over time, but simple returns can be summed across the portfolio. Re:

*What's the assumption for the test per se?* **The exam must state the assumption**. This is the definition that we have most given GARP feedback on, over the past decade (literally dozens of examples have been used). They are well aware that this assumptions need to be given. Actually this and the related issue of compound frequency, which is an assumption that must be given.

In regard to Hull's question 10.10, EWMA volatility is a recursive formula that is the solution to a series in which the weights (assigned to squared returns) are declining. If λ = 0.95, then in reverse chronological order, the weights are

- 1 day ago (the previous day) the weight is (1 - 0.95) = 5.0%
- 2 days ago = 5%*0.95 = 4.75%
- 3 days ago = 4.75%*0.95 = 4.513%; or using the direct formula for the weight that can be used for any prior (t - n) day: (1-λ)*λ^(n-1) = (1-0.95)*0.95^2 = 4.513%
- etc ... notice that consecutive weights have a ratio = λ.

If we dial down λ to 0.85, then the weights are:

- 1 day ago = 15.0%
- 2 days ago = 15%*0.85
- 3 days ago (directly) = .15*.85^2 = 10.84%

Comparing the two, λ = 0.85 is assigning greater weights to the more recent returns. This is the point of a lower "smoother parameter:" it is more reactive to the newest returns and less persistently clinging to older returns. Effectively the lower lambda captures a smaller window; e.g., at λ = 0.85, over half the weight is assigned in only the five recent returns, but λ = 0.95 does get to half the weight until the 14th return.

(minor note: Hull's use of "forecast" is confusing here. unlike GARCH, EWMA does not

*really *forecast. I think better here is "updating volatility" and "impact on updates"). I hope that helps,

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