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# P1.T2.324. Estimating volatility (Topic Review)

#### Nicole Seaman

##### Director of FRM Operations
Staff member
Subscriber
Questions:

324.1. Consider the following GARCH(1,1) model: Each of the following is true about this GARCH(1,1) model EXCEPT which is false?

a. If the volatility(n-1) is 5.0% and the return(n-1) is 3.0%, then the updated volatility estimate is about 4.6390%
b. The 10-day forward forecast (n+10) is a daily volatility of about 2.980%
c. The model's long-run (unconditional) volatility is 3.3%
d. The model's persistence is 0.880

324.2. Analyst Barbara employs an exponentially weighted moving average (EWMA) volatility model to generate a current daily volatility estimate of 2.15%. This EWMA model updates yesterday's volatility of 2.0% with yesterday's daily return of +3.0%. Which is nearest to her lambda parameter?

a. 0.79340
b. 0.87550
c. 0.91810
d. 0.95430

324.3 Consider the historical series of stock prices below, including the daily returns and the squared daily returns (Return^2). Although only the previous nine days are displayed, the actual horizon includes 55 trading days. The volatility estimates computed for the previous day, t-1, are given for each of the GARCH(1,1), EWMA and moving average (MA) methods. Also given are the lambda for the EWMA model and the GARCH parameters; e.g., the unconditional long-run variance, V(L), is 3.0%^2; the gamma weight applied to this long-run variance is 10%, such that omega = 10%*3.0%^2 = 0.000090: Today's price drop contributes a dramatic -10.00% return to the series! Which are nearest to the updated volatility estimates given by, respectively, GARCH(1,1) and EWMA? Bonus question: what are the WEIGHTS assigned by each of these models to the (t-5) return^2; i.e., the fifth previous return was also dramatic at -11.12%, what weight is assigned to its square, under each method?

a. GARCH = 4.73% and EWMA = 4.90%
b. GARCH = 4.95% and EWMA = 7.00%
c. GARCH = 5.25 % and EWMA = 5.25%
d. GARCH = 5.25% and EWMA = 5.74%

Answers:

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
FYI, we now go on a brief hiatus with respect to the daily practice questions. Why? because in a few weeks, GARP will share with training providers an early draft of the 2014 syllabus readings, and I don't want to start a new sequence of questions for a reading that might be obsolete soon. By waiting now, I get to resume the daily sequence against readings that I'm confident, with some probability if not certainty, will survive into 2014. Also, this liberates some time to focus on our notes/video/XLS content production. Thanks!

#### cash king

##### New Member
1st question confused me.

A) and D) seem true. Long-run variance =w/(1-a-b)=0.000048/(1-0.06-0.82)=0.0004, which corresponds to a long-run volatitity of 2%, so answer C) is false.
Meanwhile, 10-day forward forecast of variance =(1-persistence^10)*0.0004+(persistence^10)*(0.04639^2)=0.00175,
or a forecast of volatility of 4.2%, which makes answer B) false too.

Where did I go wrong? Particularly, did I use wrong formula for the 10 day forecast?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
HI @cash king where is that long run variance formula from? We use Hull's: V(n) = V(long-run) + persistence^n*[V(current) - V(long-run)], which is intuitive: the gap [V(current) - V(long-run)] is "compounding forward" by the persistence which, if less than 1.0, tends toward zero such that V(n) = V(n) --> V(long-run). In this case, V(10) = 0.00040 + 0.880^10*(4.6390%^2 - 2%^2) = 0.00088793 and volatility = SQRT(0.00088793) ~= 2.980%.
(i.e., you are correct than long run volatility is 2.0% such that C is false). Thanks,

#### cash king

##### New Member
I realized our formula are essentially the same:
mine: variance (n)=(1-persistence^10)*0.0004+(persistence^10)*(0.04639^2)=0.00175
yours: variance(n)=0.0004+persistence^10*(0.04639^2-0.0004)

But admitted yours is easier to memorize. Thanks a lot for your time!