TS IV Stat701 Fall 1999

RiskMetrics

Todays class.

Jacques Longerstaey. Capital Market Strategies, July 1995. Provides motivation and context for the Risk Metrics methodology - compares with standard methods.

Issues to consider when reading the Longerstaey paper.

• A wider context
• Stochastic component: recognition of uncertainty and the requirement to quantify this uncertainty.
• Improved data processing skills/power.
• Techniques: basic measures mean, s.d./variance, correlation.
• Linearity assumptions.
• Normal distributions, joint movements/distributions.
• Probability statements for describing uncertainty:
  • Parametrically based -- percentiles from the normal distribution..
  • Non-parametrically based -- empirical percentiles.
• The creation of virtual worlds;
  • Assumption based vs. history based.
• Taylor's theorem/series. Locally the world is linear.
• The delta method - not delta hedging but still about linearity!
• Quantify the lack of linearity - curvature corrections.

Forecasts.

Exponential smoothing.

The method described here is for non-seasonal time series showing no systematic trend.

Data: .

Define the 1 step ahead forecast from N points as , then

where the are weights.

Downweight points further back in time.

Use Geometric weights: .

needs an infinite number of observations, but realistically only have a finite number.

Write in recurrence form:

Update using the last observation and last forecast.

It's a weighted average between observed and expected at time N.

Or can express in error correction form;

It turns out that exponential smoothing is ``optimal'' if the underlying process is

which gives the differenced series as MA(1), so that is ARIMA(0,1,1). In particular

• As an example, we could set , then .
• Now simulate an ARIMA(0,1,1) process with , which can be written in the standard form:

  

  (see S-Plus stats manual p.586).

• The code to do this.
• In terms of our original definition of AR processes, this process has and , therefore we expect the auto-correlogram of the differences to have a spike at k = 1, then drop off immediately.
• The value of the spike is in theory .
• Now see which value of is best for making the exponential smooth that has minimum one step ahead forecast error.