# The other way around: from correlations to returns

### Enrique Millán

#### 08/04/2020

In one way or another, most quantitative models somehow seek to find and exploit relationships between two or more series of returns.

Therefore, the usual pipeline has a time-series go through mathematical procedures which condensate in a couple of figures meaningful information: the expected mean, volatility, drawdowns, runups, correlations, among others. That is, the space of returns, large and noisy, is projected into a smaller space: performance and risk metrics.

Today’s post begins a series whose aim is to climb the ladder the other way around: generate returns with specific risk metrics with the least amount of complexity needed. Why would someone want to do that? To take test-driven development to the next step: acceptance testing.

Acceptance testing proves you actually implemented the concepts you built the code for.

Traditional unit tests validate deterministic code, by establishing a link between the expected and resulting outcome of code instructions with respect to some inputs. This is great, and every serious production-ready code should have them. Additionally, they can be used when migrating code from one language to another.

However, they do not validate if those instructions actually encode the idea that gave birth to them in the first place. If you are clustering assets according to some given risk metrics and you want to test your code, you need to know in advance which assets are close within that cluster. That is, you would need to generate the inputs such that the distance matrix of your cluster is the one you would expect.

## Correlation as a distance

To start off, we selected a famous distance metric: the Pearson correlation between returns.

To generate randomly distributed returns with specific correlations between them, one can leverage the Cholesky decomposition of a matrix [1].

Given a correlation matrix $$\Sigma$$ and its Cholesky factorization $$\Sigma = LL^{T}$$, one can define the mapping $$Y = LX$$ which will give the vector $$Y$$ of correlated vectors according to $$\Sigma$$ (provided $$X$$ is formed by independently distributed variables, can you prove it?).

The algorithm steps are:

1. Build uncorrelated and mean-centered random variables $$X$$ .
2. Build/fetch a correlation matrix $$\Sigma$$ .
3. Obtain the Cholesky factorization.
4. Apply the mapping $$Y = LX$$ to get $$Y$$.
5. Validate by checking the differences between the empirical correlation matrix given by $$Y$$ and the original target correlation $$\Sigma$$ matrix.

### Market returns vs. synthetic returns

A numerical showcase is carried out: the target correlation matrix is obtained from several stock indices; normally distributed returns $$X$$ are built to be used in the projection.

Pretty good match. A cool fact to point out: see how the original indices had two interest rate series. with very slow and constant growth? Our synthetic series have none of that, and yet they replicate those matrix rows and columns (side note: if you increase the number of days in the synthetic series you get better results).

Yet, the correlation matrix here was a given, uncontrollable by the designer of the trading strategy willing to test the code.

The previous algorithm, despite its apparent simplicity, hides a small dose of complexity. As I mentioned at the beginning of the post, we would like to generate specific situations for which we want to test our trading algorithm.

That means that we have to build a correlation matrix with the specific structure we want to exploit. By doing so, we could anticipate which assets our code should pick for us, since we created their returns to fit the distance in the first place.

And this ain’t so easy. When building a correlation matrix one must comply with the following properties:

1. Unit diagonal elements.
2. Symmetric.
3. Off-diagonal values in the interval [-1, 1].
4. Semi-definite positive (… mmm?).

The first three properties are easy to obtain by construction, but the fourth is not necessarily given.

For symmetric matrices, one can check their positiveness through the sign of their eigenvalues (which will always be real by the way). If all of them are larger or equal to zero, you have a semi-positive definite matrix.

Let’s have a look for 1000 randomly generated candidates to be a correlation matrix.

Oh, oh. Positiveness is not so easy to achieve after all.

And these are randomly generated matrices, chances are we find a similar pattern if we imposed some structure to the correlation matrix.

Luckily, there are procedures to find the closest semidefinite matrix to a given matrix. It is the field of Matrix Nearness Problems [2], and we will talk about them and compare two of them in the next session. Many interesting facts about the meaning of correlation arise when the closest matrix is found.

As a final take away: take into consideration that, opposed to methods that replicate what has already happened, this can also lead you to build new scenarios, and see what your code would have done in such situations.

Thanks for reading and see you soon to finish the implementation steps of an acceptance test based on these ideas.

### References

• [1] G. Golub, and C. van Loan. Matrix Computations. JHU Press, Fourth edition (2013)

• [2] Higham, Nicholas. (2000). Matrix Nearness Problems and Applications.