We're talking about the lack of constancy in parameters extracted from historical data and ... >Like Volatility and Correlation of stock returns? Yes. We often get some numbers, like Mean Return and Volatility, then base our expectations for the subsequent evolution of our portfolio on those numbers ... for several years into the future. In Monte Carlo simulations it's common to assume "fixed" parameters, in spite of historical variations. (See Figure 1.) In the Capital Asset Pricing Model (CAPM) you may be rewarded for taking systematic risks ... meaning your portfolio differs from the market as a whole. This difference depends upon the correlation between the assets in your portfolio and the market. In Value At Risk (VAR) one assumes certain "fixed" parameters to estimate the probability that you'll lose $X in the next N months.

Figure 1

In ...

>I get it! Could we continue?
Suppose that x₁, x₂ ... x_n is a collection of variables and we assume that another variable, y_k, depends upon the x's.
We might take as our best estimate of y_k:

[1A]             Expected value of y_k, given the values of the x_j = E[y_k | x]
where x stands for the set of x's.

Of course, this Expected value will be just an estimate of y_k (given x, the set of x's). There'll be some error, so we write:

[1B]             y_k = E[y_k | x] + e_k

Now it's typical to assume that the error terms, e_k, are random variables with zero Mean so that sometimes the estimate is high and sometimes low, but their average, over time, is zero. In particular, it's normally assumed that the errors e₁, e₂, e₃ ... are independent and identically distributed. (So-called iid variables.)

However, to account for observed variations over time and the fact that (often) large (small) errors are followed by large (small) errors, Engle assumed that the error is represented by:

[2A]             e_k = z_k h_k^1/2
              where z_k are iid variables with Mean = 0 and Volatility (or SD = Standard Deviation) = 1, BUT
              h_k = a₀ + a₁e_k-1²+ a₂e_k-2² + ... + a_qe_k-q² ... depending upon the squares of "q" past errors

Note the lag (identified by "q") may be large or small though it's often taken to be q = 1 ... a "first-order" model (I think!)
Then, if we have a recent series of large (or small) errors, then we'd expect another large (or small) error.

>Yeah! When my favourite stock tanks, if keeps going down and ...
Pay attention.

Figure 2 shows the daily returns for the S&P 500 over the past five years (1999-2003). That's the upper chart. The lower chart shows the variance (that's the volatility2) of these daily returns over a moving 2-week period. Notice that when the variance is large it tends to stay that way for a while and ... >And when it's small it stays that way for a while, right? Right. Alternating periods of volatility and relative calm. The distribution of these daily returns looks like this: The moral of this story is:

Figure 2

Do not assume that selecting returns from this "fixed" distribution is an adequate ritual for estimating future portfolio evolution.

Indeed, if you just pick random values from this distribution you're unlikely to get those periods of calm and high volatility.

>And that funny prescription does it?
Yes, in fact a variation on the above scheme is:

[2B]             e_k = z_k h_k^1/2
              where h_k = a₀ + [a₁e_k-1²+ a₂e_k-2² + ... + a_qe_k-q²] + [ b₁h_k-1+ b₂h_k-2 + ... + b_ph_k-p] ... depending upon past h-values as well

The scheme [2A] is associated with ARCH and [2B] is with GARCH (a Generalized ARCH). For example, suppose we start a series with e1 = h1 = 0 (or any other arbitrary pair of numbers) and plot: [3A]          en = zn hn1/2 ... for n = 1, 2, 3 etc.           where hn = a0 + a1en-12 + b1hn-1           and zn = Normal[0,1] ... selected from a Normal distribution: Mean = 0, Standard Deviation = 1 Look at what happens to the hn-sequence after a while, in Figure 3. You can imagine what that does to the error terms en, eh? >Periods of high volatility, then low then ... Yes, just as one often observes in stock returns. Of course, Figure 3 shows just one possibility (for the chosen parameters).

Figure 3 : a0=0.001, a1=0.11, a2=0.89

The next sequence I generate will undoubtedly be different, because the z's are random variables.

>Yeah, but how does equation [3A] gives such an up-and-down behaviour? I mean ...
I know exactly what you mean.
In [3A], let's put e_n-1² = z_n-1² h_n-1 (from the first equation) into the second equation, like so:

[3B]          h_n = a₀ + (a₁z_n-1² + b₁)h_n-1

Now, that z_n-1² guy is always positive (or at least not negative), and since the z's are Normally distributed with Mean = 0 and SD = 1,
then z² will have a Mean of 1. (See Stat Stuff #3 & #4.)

Suppose (for example) z_n-1² were stuck at the value 1/2. Then equation [3B] gives a recursive relation between successive values of h, namely
h_n = a₀ + (0.5*a₁ + b₁)h_n-1, as illustrated in Figure 4a.

See what happens?

Start at some x = h1 on the x-axis and move UP to the line y = a0 + (0.5*a1 + b1) x The y-coordinate of that point will be h2. Now move horizontally to the line y = x and the x-coordinate of that point will also be h2 (since y = x) Now move UP again to the line y = a0 + (0.5*a1 + b1) x The y-coordinate of that point will be h3. Now move horizontally to the line y = x and the x-coordinate of that point will also be h3.

Figure 4a

Repeating these steps, moving UP then horizontally, then UP etc. etc. and you generate the sequence of h's.
They'll be the sequence of x-coordinates.

>So the h's move toward that point you've coloured yellow ... in Figure 4a?
Aah, but only if the z's were constant. In fact, they vary from one iteration to the next, so the slope of that line keeps changing ... as in Figure 4b.

Then, every once in a while (since z is a random variable), that z-value will be huge ... and we'd get Figure 4c. Figure 4c

Figure 4b

>Big change in h, eh?
Yes indeed ... hence a big change in the error e_n = z_n h_n^1/2.

for Part III