I was reading an old Business Week and noticed an article about the 2003 Nobel prize being awarded to a couple of guys who, among other things, had done work on time series (like stock prices or returns) where the volatility changed over time and that's always interested me and ...

>A couple of guys?
Yes, these guys:

Robert F. Engle, New York University, USA
“... for methods of analyzing economic time series with time-varying volatility (ARCH)”

Clive W. J. Granger, University of California at San Diego, USA
“... for methods of analyzing economic time series with common trends (Cointegration)”

>ARCH?
Autoregressive Conditional Heteroskedasticity. Scary, eh?

>And you never heard of ARCH and cointegration, right? Well, I'd heard of cointegration but I never heard of that ARCH stuff. Note that a homoskedastic time series has a constant volatility (or standard deviation), over time. A heteroskedastic doesn't. There are periods of high volatility and periods of low. That's volatility clustering, and ...

Figure 1

>What's plotted in Figure 1? It's a time series ... something is plotted against time ... it doesn't matter what. The point is, parameters (like Volatility) need not oscillate about some constant value. That's a so-called non-stationary time series. In particular, if you do a regression between two variables and find that the correlation is high, you'd be tempted to conclude that there was a statistically significant relationship between the variables. >Regression? You plot the values of one against those of the other and get a regression line, as in Figure 1a. If the points lie near the regression line (shown in red) you'd expect an intimate relation between the two sets. (Figure 1a plots "invented" time series values, where gains from one point to the next is determned by selecting a return from a Normal distribution where the Mean Return and Standard Deviation are chosen randomly.) The two time series are shown in Figure 1b, where the correlation is 92%. There's an intimate relation between variables, right? It was Granger who found that apparently significant relationships between "non-stationary" variables could be spurious. >You decided to learn something about this stuff, right? Well ... yes, if I can understand it. >It can't be that tough. Well, to test a certain null hypothesis, I find this in the literature:

Figure 1a Figure 1b

>What does it mean?
I have no idea ...

Changing Volatility ... and correlation (and their effect on Portfolio management)

Figure 2 shows a moving average: the average volatility of GE (and Microsoft) stock returns over the previous year.

To be more precise, it's the average of monthly stock returns over 12 months, multiplied by SQRT(12) to give an "annual" number
... a number which is more familiar.

>The ol' square-root-of-time thing? Yes ... and note that, even for a staid stock like GE, it varies by more than 2-to-1. >Yeah, but look at MSFT! Yes, and ... >So you wouldn't want to assume constant volatility, using Monte Carlo, eh? I wouldn't think so. Besides, even the correlation varies wildly: Figure 2a

Figure 2

Okay, it's 1995 and you look at the past three years and find a negative correlation between these two stocks.

>That's good, eh?
Yes, if you want to allocate resources so as to reduce your portfolio volatility.
We note the following:

• The volatilities over this three year period are: MSFT = 26% and GE = 22%
• You invest 40% in MSFT and 60% in GE, based upon these historical volatilities.
• Had you done that over this time period, your Portfolio volatility would have been 15%

>Whooeee! We've reduced the Risk, right? If you regard volatility as "risk", yes ... we've reduced the risk, yet we'd have got a very respectable return as noted in Figure 3. >Uh ... how'd you get that 40% + 60%? I used the quick-and-dirty stuff noted here, namely           where P and Q are the two volatilities and I'd devote a fraction x to the asset with volatility P. >That's exact?

Figure 3

>It was -11%!
Yes, but remember that we're pretending it's 1995 and we're looking back at the past three years and ...

>And we've picked our portfolio based upon that historical performance.
Exactly. Now we do the 40%+60% allocation, confident that our "risk" is reduced.

It's now 1998 and we look back at the previous three years. We find:

• The volatilities over this three year period are: MSFT = 30% and GE = 19%
• We've invested 40% in MSFT and 60% in GE (rebalancing so as to maintain this allocation).
• Our Portfolio volatility (Jan, 1995 to Jan,1998) has been 19%

>There's goes your reduced "risk". Well, actually, there goes our reduced Volatility. >I thought the "risk" would go down by mixing the two assets. Yes, if you do the right allocation which (for 1995-1998) would have been 20% MSFT. (SeeFigure 3a) >And that 2003 Nobel Prize? What's that got to do with ...? I was just trying to point out that extracting parameters from the past, like Mean Return and Volatility and Correlation - extracting these guys in order to estimate future performance over the next umpteen years ... >Yeah, it's bad ... but what else should you do?

Figure 3a

>And when that history changes, so do you, right?
Well, so must our Portfolio strategy.

>That's the question, but what's the answer?
I haven't finished reading Engle and Granger ...

for Part II