The Angle of Coincidence

Introduction

In times of stress in the markets, not only does volatility increase for individual assets, cross-asset correlations can increase dramatically as well. This results in a “double whammy” for a typical portfolio because the portfolio’s volatility increases due to both effects. Therefore, the ability to include correlations as well as volatilities in sensitivity analyses is critical when stress-testing portfolios. However crude approaches to stressing volatilities and correlations simultaneously can easily run into problems, as certain mathematical restrictions can quickly surface.

In particular, this comes down to the requirement that a correlation matrix must be non-negative, which is equivalent to saying that a portfolio of assets cannot have a negative variance. Given weights {w_n} and correlations {ρ_mn}, this requirement states that  This can be rewritten as:

Note that we can easily move correlations closer to zero without running into trouble, so the problem is clearly not symmetric. For example if we multiply all correlations by ½, it is even more true that

The problem arises when the absolute values of pairwise correlations is increased. Noting that the absolute value of correlations cannot exceed 1.0, we can’t just double them. Thus we seek an approach that will safely allow us to increase correlations in a stress test while maintaining the non-negativity condition.

We introduce a structure that enables us to safely adjust correlation assumptions from a base level all the way up to 1.0, while maintaining the non-negativity condition. This gives rise to a number of applications, and we explore some of them in this article.

Basic Identities

Let x_n be the return of the n^th asset (assumed to be a normally distributed random variable).

Let F be a unit normal random variable, independent of {x_n}. Consider

We will call α the “angle of coincidence”, restricted to 0 ≤ α ≤ 90^° or “AoC” for short (this is, of course, unrelated to the “AOC” designation, for Appelation d’Origine Contrôlée, that is awarded to certain French wines of distinction).

A Note Regarding Principal Components
If Ρ is the base correlation matrix, then Ρ_α = sin² αΡ + cos² α1 is the correlation matrix of the angle of coincidence α. It follows that if is an eigenvector of Ρ (a.k.a. “principal component”) such that P=λ  then Ρ_α = λ_α where λ_α = sin² αλ+ cos² α. That is, the principal components remain the same but the eigenvalues converge to 1.

Parametric VaR

Next, consider parametric VaR, determined by the usual variance calculation

Then

where is the weighted sum of the σ_m‘s. Note that as we reduce σ from 90^° to 0, the variance changes from the base case, σ², to the “perfectly coincident” case, σ²_max.

A Refinement

In practice, some return correlations may be pushed towards –1.0; for example,

• Bond prices and equity returns;
• Growth stocks versus value (due to momentum and other factors);
• Stress testing of Long-short portfolios where w_m and w_n have opposite signs;
• Gold versus everything else (when investors avoid all risk and seek a safe haven).

How can we accommodate this?

In this case we write , where

Now we have

Examples

Here, we provide two examples of how the AoC approach can be applied in practice.

#1 – The “Double Whammy”

Consider a portfolio of ten equally weighted stocks. We maintain all of the individual stock volatilities at their current levels but adjust the pairwise correlations between them by decreasing α from 90° to 0°.

The following table shows the effect on various correlations when we change the AoC angle, α.

So, for example, at an angle (α) of 45°, a correlation of 0.40 increases to 0.70, and a correlation of 0.60 increases to 0.80. This has the overall effect of increasing portfolio volatility (and hence VaR) by about 25%, while keeping the individual volatilities constant.

One implication is that if, in a period of high volatility, investors decide to quickly and significantly reduce their equity exposures, or rotate out of equities entirely, both volatilities and correlations will be affected. In running stress tests for risk management, if we only increase volatilities, risk could be under-estimated by more than 25%. Even a modest shift in α of 30° (from 90° down to 60°) increases the portfolio’s VaR (pVaR) by 12%.

If we combine both a shift in volatility levels of +20% with a shift in the AoC to 30°, pVaR increases by 34%
(since 1.20 × 1.12 = 1.34).

Example #2 – A 60-40 portfolio of stocks and bonds

We examine the scenario of a high negative correlation between stocks and bond prices, and contrast it with the base case in which interest rates are unchanged.

In this case, we see that including the bonds produces a small reduction in VaR when the AoC is high, and a much greater comparative reduction as the AoC approaches zero.

Conclusion

The AoC approach provides a useful way to stress a pVaR calculation, and highlights its dependence upon correlation assumptions that can vary significantly, depending upon the measurement window used to estimate them. Even a modest shift in the AoC (say, of 30°, from 90° to 60°) produces a significant change in VaR without increasing volatility. This approach makes it possible to immediately assess the effects on pVaR of combining shifts in both volatilities and correlations without violating the non-negativity requirement for the correlation matrix.

In an upcoming blog we will show how to extend the AoC concept to Historical VaR.

About the Author

As a co-founder of TS Imagine, Dr. Smith sets strategic direction, guides development of TS Imagine’s award-winning analytics, and lends his expertise to sales efforts. Dr. Smith brings a rare combination of business and academic experience to his role as CEO, with thirty-plus years of direct involvement in trading and risk management practices, mathematical modeling, and system development.

Dr. Smith has a PhD in mathematics, and is a former Assistant Professor of Mathematics at Columbia University, as well as a former Courant Instructor at the Courant Institute. He began his finance career at Salomon Brothers in 1986, where he served as the chief quant for the equity derivatives business. He then headed Sakura Global Capital’s Equity Derivatives Trading desk, prior to founding TS Imagine in 1993.