# The positive correlation stress test

When it comes to stress testing market portfolios one idea has been finding increasing traction with risk management teams. The 100% positive correlation stress test.

The concept is simple. We opt for a diversified portfolio because we want to reduce the risk of unfortunate events and market downturns impacting our portfolios at the same time. For instance we would assume that as oil prices and oil stocks go high, airline and automobile stocks wouldn’t do so well. Or as technology and growth stocks do well in times of economic growth, traditional defensive stocks wouldn’t be as appealing and vice versa. Small caps go up, large caps come down. We mix and match portfolios to cater to a mix of economic conditions, our own outlook and performance benchmarks.

At least that is what the theory says. But what if the theory is wrong. What if there are events that cut across the diversification benefit?

If we were to stress test this design feature what is the worst case scenario? Remember we are testing the diversification payoff. The worst case scenario is when the diversification benefits (and correlations) go away. The entire market, irrespective of economic segments and growth outlooks takes a dive. An unanticipated event that the market needs time to digest and understand leads to a rush away from risk and into safety. So oil, automobile and airline stocks all take a dive together. Tech stocks and defensive demographic based stocks go down at the same time.

In recent memory we have seen such instances in 2008-2009 (US Financial crisis and extended post slump), August 2011 (US Credit rating cut by a notch and the resulting free fall in prices) and June 2016 (Brexit). If you have exposure to North American and European securities, these three time slots provide great data sets. Just run your portfolio through these data sets and gauge how well they would have weathered the storm.

But if these markets don’t do a good job of representing your exposure or you don’t think correlations were stressed far enough, you could always try the perfect correlation test.

If you run multi asset portfolios stress testing correlation is now a requirement. For illustrative purposes we want to run two separate correlation stress tests. The first is a 100% positive correlation test (everything goes under) while the second is a zero correlation stress test (correlation drop to zero). The two tests provide interesting contrast for each other.

The methodology we use can be applied to a portfolio of any size however our example is limited to a portfolio with four positions.

## The problem set

Here is the problem definition from a recent portfolio management exam we ran for a group of MBA students.

As a portfolio manager you have created a diverse portfolio comprised of long positions in **Gold** (hedge against inflation and economic downturn), Crude Oil – **WTI** (hedge against a growing global economy), a long position in **Yen** (a hedge against falling dollar) and **Google** (a bet on technology sector and online ecommerce doing better than expected in coming years).

You have optimized your portfolio and your portfolio allocations using the simple criteria of maximizing return per unit of risk using Excel Solver. No single position in your portfolio can be greater than 30% of the size of the total portfolio. No short sales are allowed.

Your board of directors have asked you to perform two simple stress tests. For the first stress test assume zero correlation. For the second stress test assume 100% positive correlation (i.e. correlation = 0 and 1) between all securities. The board wants to compare our standard risk metrics against both scenarios.

## Correlation stress test – the game plan.

- Review data set
- Calculate individual security volatility
- Build correlation matrix
- Build volatility matrix
- Calculate portfolio volatility
- Run first stress test
- Run second stress
- Compare results

## The positive correlation stress test – Step by step

- The data set includes the price series for two separate portfolios. We cover one in this example, you can use the second one for practice. You can download the dataset here.

- See the calculating value at risk with or without the VCV matrix post to review the methodology we use to attack the problem. Using the material in the post you should be able to calculate both the VCV matrix as well as the correlation matrix for a portfolio with n securities. For the first portfolio both matrices are presented (above and below)

- Now that we have the original VCV matrix as well as the correlation matrix, we want to generate an alternate VCV matrix using our correlation matrix. We need to double check and verify that the original VCV matrix and the new alternate VCV matrix have the same values. By deriving volatility from correlation we can control the level of correlation used in our portfolio volatility calculation. We can use the original figures, plug in correlation goes to one or correlation goes to zero or any other set of values in between.

- Calculate portfolio volatility using the alternate VCV matrix. See the post above for directions about estimating portfolio volatility using VCV matrix. Make sure that you use references and don’t hard code any values during the steps used to generate portfolio volatility.
- To calculate revised portfolio volatility under the correlation goes to one scenario all we have to do is to change the values in the correlation matrix to 1. This is our perfect correlation scenario. The portfolio volatility generated by this approach represents the impact of 100% positive correlation.

- To calculate the zero correlation portfolio volatility change all the values to zero as shown below and repeat the process used in step 5. Remember the diagonal remains one. Gold will still remain perfectly correlated with gold under the zero correlation scenario.

## Positive correlation stress test – The results

For our illustrative portfolio the original annual volatility with the original correlation was** 11.5%**.

With the 100% positive stress test portfolio volatility jumps to **20.89%**. In Value at Risk terms this means that under the correlation stress test VaR would approximately double.

With the zero correlation stress test portfolio volatility declines to **11.06%**. Not a significant drop.

The results are likely to vary across different portfolios but the technique we have used above will easily scale to any portfolio with n number of securities.