Copula Example: Joint Probability Model for Stock Indices

Last Modified: 2026-09-04 13:14:23.147

HEC-Neptune version alpha was used to created this workshop.

Overview

This workshop was developed to illustrate the use of copulas as joint probability models in applications outside of water resources. While HEC-Neptune is primarily designed for statistical analyses of hydrometeorologic, hydrologic, and hydraulic data, it is, at its core, a general-purpose statistical tool. This workshop demonstrates the Copula Analysis capabilities within HEC-Neptune and applies them to stock market index data, illustrating how the same statistical methods can be used to analyze dependence between financial variables.

This workshop should not be interpreted or used as financial advice.

Background

Copulas have been used in quantitative finance to model tail dependencies between assets. In this tutorial, we will investigate the dependence between two stock market indices: the Standard and Poor's (S&P) 500 (GSPC) and the Financial Times Stock Exchange (FTSE, pronounced "footsie") 100. The S&P 500 tracks the stock performance of the top 500 companies listed on the United States stock exchanges. The FTSE 100 tracks the performance of the 100 largest companies by market capitalization listed on the London Stock Exchange.

Data

Monthly stock index price data was downloaded. The logarithmic return of a stock index is often used as a normalized version of stock returns. The log return is computed as:

$\begin{array}{l}r_t = ln(\frac{P_t}{P_t_-_1})\end{array}$ where P_t is the index value at time t and P_t-1 is the value at the prior period.

Time series plot of monthly log returns of FTSE and GSPC stock indices Scatter plot of monthly log returns of FTSE and GSPC stock indices

Fit a Joint Probability Model

We are interested in describing the joint probability of the each year's worst log returns (i.e. the most extreme losses in each year) of the S&P 500 and FTSE indices. The log return values corresponding to losses were negated so that we could treat them as annual maxima and fit marginal distributions to non-negative values. The annual maxima losses of FTSE and GSPC for 1985-2025 are shown below.

Annual maxima losses based on log returns

We will only parameterize and fit one copula model since the maximum losses of both stock indices are coincident over the period of record.

Copula scenario for annual maximum stock indices losses

We selected the GEV distribution as the marginal distribution for both series of annual maximum losses. The Joe copula had the lowest AIC, indicating the best model fit. This makes sense since we would expect that the losses (when negated) exhibited upper tail dependence.

Copula model fits

Analyze Results

Now that we've fit a copula, we can examine the results. Kendall's $\begin{array}{l}\tau\end{array}$ between the two stock indices losses is 0.512, indicating a moderately strong ordinal association.

The purple Xs on the figure below represent the historic annual maxima losses. We can attribute a probability of exceedance to a particular pair of losses. For example, the purple X that plots furthest to teh upper right corner is from October 1987 and corresponds to the Black Monday losses. This combination of losses has an exceedance probability that is a bit less than 1/200, or 0.005.

Note that the majority of the historic loss pairs have exceedance probabilities more frequent than 1/10, or 0.1.

Isoline plot