## Archive for **October 2009**

## Einstein meets Markowitz: Relativity Theory of Risk-Return

When working with Gaussian processes, the observation that you can interpret covariance geometrically comes in very handy (see Visualizing Market Risk: A Physicist’s Perspective). Of course, when you’re given a new toy, you’ll want to take it apart. In this post, we’ll extend the analogy

to

The basic idea is to recall that Gaussian processes form a vector space. For instance, given Gaussian processes , the linear combination

is also a Gaussian process. Gaussian distributions are particularly nice because everything you can know about them is encoded in the two parameters (mean of the process) and (standard deviation of the process).

Each of the differentials may be expressed as

where is a standard Brownian motion with and .

We can interpret the as spanning a cotangent space of some **risk manifold**.

If you have such processes, the dimension of the space they span depends on the rank of the covariance matrix

The matrix is symmetric and positive semi-definite. However, if the rank of is , we can find uncorrelated differentials and construct a covariance matrix

This matrix is symmetric and positive definite. Therefore, on the space spanned by the , we can think of the covariance matrix as a **metric tensor**.

Now, we can re-express any Gaussian process via

Comparing this with the previous expression we see that

Furthermore,

which is just the familiar expression for the variance of the sum of uncorrelated processes.

The neat thing comes when you bring back into the picture.

Since we can interpret the covariance matrix as a metric tensor on a **space**, we can extend this to a Lorentzian metric on spacetime by specifying

and

With this extension of the metric, we have

Therefore at each point of our manifold, we have a **risk-return cone** in analogy to the light cone of special relativity with a velocity-like value given by

Note that is the radius of the risk-return cone at the particular security.

If you have two processes and , we can also look at their difference

Then the relative process

gives rise to a **relative risk-return cone**

Note that is the radius of the relative risk-return cone.

The relative risk-return cone has a velocity-like value given by

Now, this applies to finance by letting be the log of the price of some security and letting be the log of the price of some benchmark. With this financial interpretation, the process is the **return** of the security and is the **excess return**.

The “velocity” of the security

is known as the **Sharpe ratio** and the “relative velocity” of the relative security

is known as the **information ratio**. The radius is the **tracking error**.

To further the analogy, you could define an absolute velocity for which all other light cones are compared. This absolute light cone is closely related to an investor’s **risk aversion**.