Heteroscedasticity: Only downside being "Hard to spell"

As mentioned in my last post, volatility of stock returns vary over time (Heteroscedasticity) and they happen in a systematic way so as to ensure mean reversion. The concept is appealing well beyond the regular homoscedastic models which assume constant volatility. And so the best book I had that provided hands on model calibration experience was the elementary “Time Series Analysis by Cryer and Chan”.

Wiki says on the subject:

“The possible existence of heteroscedasticity is a major concern in the application of regression analysis, including the analysis of variance, because the presence of heteroscedasticity can invalidate statistical tests of significance that assume that the modeling errors are uncorrelated and normally distributed and that their variances do not vary with the effects being modeled.”

And also that the basic Ordinary Least Square (OLS) methods in determining parameters cannot work:

“When using some statistical techniques, such as ordinary least squares (OLS), a number of assumptions are typically made. One of these is that the error term has a constant variance. This might not be true even if the error term is assumed to be drawn from identical distributions.”

And so I shall jump straight to the GARCH model. Cryer and Chan illustrate the very convenient method of Maximum Likelihood Estimation (MLE) for this model and based on the praise given to it by the Investment Guarantees by Hardy book, I’ll only explore the GARCH (1,1) model. The equations take the following form:

Cryer and Chan model daily returns on the CREF (College Retirement Equities Fund), which is probably something. The equation they use is more simplified:

And hence the probability function used is:

The excel solver is appropriate for the maximization of the likelihood function under certain limitations:

1. w/(1-a-b) = Sample Variance
2. a + b < 1
3. a + b > 0
4. w > 0

Of course for the first instance of modeled data, i.e. t = 1, the required variance at t = 0 is assumed as the sample variance which implicitly is also the assumed long term variance along with the return at t = 0 equal to the average sample return.

The parameters easily matched with Cryer and Chan, I moved to fitting the model on KSE-100 daily closing index level from 1^st January 2009 to 30^th June 2014 (http://www.quotenet.com/index/historical-prices/KSE_100/30.6.2014_1.1.2009). This period hasn’t seen many systemic downturns or crashes and mutual funds have boasted performance during this period with annual returns over 30%. Fitted parameters came in as:

ω	0.0228912040
β	0.89440
α	0.08742
σ^2 (Sample Variance)	1.258909991
µ (Sample Mean)	0.1205

A couple 1-year length (251 business days) simulations illustrate Moving Volatility vs Log Returns as under:

Assorted percentile growths out of 1000 simulations are illustrated as under:

Notice how the 50^th percentile converges to the straight line which illustrates uniform portfolio growth at a uniform mean of 0.1205 (sample mean), and this happens relentlessly no matter how many times the simulation is run. Also, a total of 36 simulations yielded negative growth implying a 3.6% probability of losing money in a year.

If this was a Variable Annuity Product with a principal preservation guarantee, at a 99.5% VaR level the portfolio size reduces to 81.35% requiring a 18.65% capital assuming no service charges in order to achieve a target rating of A or AA.

WHAT I LEARNED HERE

I was more pleased with the variety of portfolio performance and tails this model produced than the log-normal and RSLN models. Also I can think of many uses in pricing products. However, I still seek a modeling approach that models all economic variables consistently, an Economic Scenario Generator (ESG) if you will. I need to read more on that material as Solvency II draws closer and focus of investors on Risk Management and Economic Capital grows.