Market Risk « Phorgy Phynance

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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

$\text{Risk}\leftrightarrow\text{Geometry of Space}$

$\text{Risk-Return}\leftrightarrow\text{Geometry of Space-Time}.$

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

$X = \sum_i \omega_i X_i$

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

Each of the differentials may be expressed as

$dX_i = \mu_i dt + \sigma_i dW_i$

where $dW_i$ is a standard Brownian motion with $\mu = 0$ and $\sigma = 1$ .

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

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

$\Sigma_{i,j} = \text{cov}(dW_i,dW_j).$

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

$g_{i,j} = \text{cov}(de_i,de_j) = \delta_{i,j}.$

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

Now, we can re-express any Gaussian process via

$X_i = \mu_i dt + \sum_j \sigma_{i,j} de_j.$

Comparing this with the previous expression we see that

$\sigma_i dW_i = \sum_j \sigma_{i,j} de_j.$

Furthermore,

$\sigma^2_i = \sum_j \sigma^2_{i,j}$

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

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

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

$g_{t,i} = g(dt,de_i) = 0$

and

$g_{t,t} = g(dt,dt) = -\frac{1}{c^2}.$

With this extension of the metric, we have

$|X|^2 = g(X,X) = \sigma^2 - \frac{\mu^2.}{c^2}$

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

$c = \frac{\mu}{\sigma}.$
Risk Return Cone
Note that $\sigma$ is the radius of the risk-return cone at the particular security.

If you have two processes $X$ and $X_0$ , we can also look at their difference

$\bar{X} = X - X_0.$

Then the relative process

$d\bar{X} = (\mu-\mu_0) dt + \bar{\sigma} d\bar{W}$

gives rise to a relative risk-return cone

Relative Risk Return Cone II

Note that $\bar{\sigma}$ is the radius of the relative risk-return cone.

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

$\bar{c}= \frac{\mu-\mu_0}{\bar{\sigma}}.$

Now, this applies to finance by letting $X_i$ be the log of the price of some security and letting $X_0$ be the log of the price of some benchmark. With this financial interpretation, the process $dX_i$ is the return of the security and $d\bar{X}_i$ is the excess return.

The “velocity” of the security $X_i$

$c_i = \frac{\mu_i}{\sigma_i}.$

is known as the Sharpe ratio and the “relative velocity” of the relative security $\bar{X}_i$

$\bar{c}_i= \frac{\mu_i-\mu_0}{\bar{\sigma}_i}.$

is known as the information ratio. The radius $\bar{\sigma}_i$ 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.

Written by Eric

October 11, 2009 at 10:27 am

Posted in Market Risk, Mathematical Finance

Daily S&P 500 Value-at-Risk Estimates

with 3 comments

A few people have commented about the methodology used to produce the charts in my last post. Keep in mind, I threw those together quickly for Felix based on charts already put together for a seminar at UCLA. If you want to see what I actually look at on a regular basis, I put the following chart together:

SP500VaR_10yr_linear_annotated

This is the 99%, 1-day VaR using a weighting scheme that places more weight on the most recent data.

Again, note the divergence between the two charts in recent months. Risk systems (like most third party vendors) based on normal distributions are likely indicating that risk continues to decrease. However, the stable distribution indicates the opposite, i.e. risk has begun increasing again.

Written by Eric

August 8, 2009 at 9:49 am

Posted in Market Risk, Stable Distributions, VaR

Visualizing Market Risk: A Physicist’s Perspective

with 3 comments

Physicists learn at an early age (sometimes while still in diapers) about vectors. In fact, I supported myself financially through undergrad largely by explaining vectors to physical therapy students. Physics was the “weeder” course and PT students basically needed an “A” to get into the program. Tutoring them was quite lucrative, but that is another story. Here, I present a very neat way to visualize market risk in terms of arrows… err… I mean vectors.

What is a vector?

A vector can be thought of conceptually as an “arrow”.

What is the information contained in an arrow? There are two basic bits of information contained in an arrow

Length or magnitude
Direction

In terms of direction, probably what is more useful than “absolute” direction, is the “relative” direction, which can be quantified as the angle between two arrows.

Multiplying a Vector by a Number

The next thing you need to know about vectors is that you can multiply a vector $A$ by a number $\alpha$ to get another vector $\alpha A$ .

Multiplying a vector by a number merely “rescales” the arrow, i.e. changes its length, while keeping the direction unchanged.

Adding Vectors

Next, given two vectors $A$ and $B$ , there is a rule for adding the arrows to get a new vector $X = A+B$ . There are a couple different ways to visualize the addition of arrows. One involves drawing the vectors “tail to tail” and forming a parallelogram. The new vector is the arrow going across the diagonal as shown below

Alternatively, but equivalently, you can draw the vectors “head to tail” and the new vector is the arrow pointing from the tail of the first arrow to the head of the second arrow as shown below

It doesn’t matter which you choose. Just remember that the addition of arrows is a little funky and involves parallelograms.

If I haven’t lost anyone up to this point, I will be quite impressed if you stick around for what comes next.

The Dot Product

Both the length and relative direction of a vector, or arrow, can be determined by something that we lovingly refer to as the “dot product”. Later in academic life, I learned that the “dot product” was actually a twice covariant symmetric non-degenerate tensor, but I digress. For our purposes, we just need know that the dot product takes two arrows and spits out a number (in a bilinear fashion). In other words, if

$X = \alpha A + \beta B$ and $Y = \gamma C + \delta D$ ,

then

$X\cdot Y = \alpha \gamma A\cdot C + \alpha\delta A\cdot D + \beta \gamma B\cdot C + \beta \delta B\cdot D$

A fundamental property of the dot product is that when you take the dot product of a vector with itself, the result is the square of the length of the vector, i.e.

$\text{Length of } A = |A| = \sqrt{A\cdot A}$ .

Therefore, the dot product provides us with one of the most important characteristics of a vector: it’s length. To whet your appetite a little, I hope that the expression above makes you think of “variance” or more specifically “standard deviation/volatility”. That is no coincidence!

The dot product of two distinct vectors also gives some very useful information

$A\cdot B = |A||B| \cos\theta$

$\cos\theta = \frac { A\cdot B } { |A||B| }$ .

Note that $\cos\theta$ lies between +1 and -1. Kind of like correlation. Again, not a coincidence!

Projecting One Arrow Onto Another

Once you start doodling with a bunch of arrows, you may start to think about the relationship between two arrows and, like a shadow, how might one arrow project onto another as shown below

If you recall a bit of trigonometry, the length of $B_{\text{Along }A}$ is given by

$|B_{\text{Along }A}| = |B|\cos\theta$ ,

which may be written in terms of the dot product as

$|B_{\text{Along }A}| = \frac{A\cdot B}{|A|}$ .

How about the ratio of that projection? That is determined by simply dividing the above by |A|, i.e.

$\text{Ratio of the Projection} = \frac{ A\cdot B }{ |A|^2 }$ .

This ratio is particularly of interest in finance, as I will show below, and is related to “portfolio beta”.

Portfolio Return as an Arrow

Now the fun starts and I hope at least one person in the universe reads this far down

The return of a portfolio is a vector.

What?!?

Well, hear me out. Consider a portfolio consisting on $N$ securities. The return of the portfolio can be expressed in terms of the returns of the securities as follows

$R_{\text{Portfolio}} = \sum_{i=1}^N \omega_i R_i$ ,

where $\omega_i$ is the weight of the $i$ th security in the portfolio.

Hmm, the security returns are multiplied by numbers and then added together. That sounds a whole lot like a vector to me!

Ok. If returns can be thought of as arrows, what are the meanings of the length and relative directions of two returns?

As we saw above, to determine the length and relative direction, we needed a dot product. What would be a good dot product for the vector space of security returns?

Covariance as a Dot Product

Technically speaking, covariance does not satisfy all the axioms of a dot product, but fortunately physicists like to shoot from the hip and tend to be right under most practical circumstances and in exotic cases where we are not right, we can usually change things around a bit so that we are right. So technical arguments aside, I am simply going to tell you that covariance can be thought of as a dot product on the vector space of security returns.

Woohoo! That is great. Why? Because all the nice pictures above can be re-interpreted in the context of portfolios of securities.

Here is an arrow depicting the return of a portfolio

What information is conveyed by the length of this arrow? Recall that

$\text{Length of }R_{\text{Portfolio}} = \sqrt{R_{\text{Portfolio}}\cdot R_{\text{Portfolio}}}$ ,

but if the dot product is covariance, then we get the very interesting association

$\text{Length of }R_{\text{Portfolio}} = \sigma_{\text{Portfolio}}$ ,

i.e. the length of the arrow is the volatility of the return.

Next, consider the return of a benchmark against which this portfolio is to be measured. The benchmark return can similarly be represented as an arrow and we have

What is the meaning of the angle between the “portfolio arrow” and the “benchmark arrow”? Recall that

$\cos\theta = \frac{ R_{\text{Portfolio}}\cdot R_{\text{Benchmark}} } {|R_{\text{Portfolio}}||R_{\text{Benchmark}}|}$ .

However, we also just learned that

$|R_{\text{Portfolio}}| = \sigma_{\text{Portfolio}}$

so that the above expression can be rewritten as

$\cos\theta = \frac{ \text{cov}(R_{\text{Portfolio}}, R_{\text{Benchmark}}) }{\sigma_{\text{Portfolio}} \sigma_{\text{Benchmark}} }$ ,

which you might recognize as the definition of correlation. The correlation between the portfolio return and the benchmark return is the cosine of the angle between the portfolio arrow and the benchmark arrow.

That is pretty neat, eh? We now have a nice way to visualize both the volatility and correlation of security returns. The length of a “return arrow” is its volatility and the angle between two returns relates to their correlation. I couldn’t make up a better story if I tried

But what can we do with this knowledge?

Tracking Error

The excess return of a portfolio of its benchmark is simply

$\Delta R = R_{\text{Portfolio}} - R_{\text{Benchmark}}$ ,

which we can trivially rearrange as

$R_{\text{Portfolio}} = R_{\text{Benchmark}} + \Delta R$

The right hand side of the equation of above represents the addition of two arrows, which we already described above. Therefore, we can represent the expression visually as

This little trick seems kind of neat, but fairly trivial, so what did it give us? Recall that the length of an arrow is its volatility. Recall also that tracking error is defined as the volatility of the excess return, therefore we have the very cool consequence

$\text{Tracking Error} = \text{Length of }\Delta R$ .

We have a very concise way to visualize tracking error that captures both the volatility of the portfolio and the volatility of the benchmark as well as the correlation between the two in one simple diagram.

Portfolio Beta

“Now how much would you pay? But wait! There’s more!”

The same diagram above that gave you tracking error also gives you your portfolio beta. To see this, simply recall the definition of the projection of an arrow along another arrow and reinterpret that in terms of returns

$|R_{\text{Portfolio along Benchmark}}| = |R_{\text{Portfolion}}| \cos\theta$

$\sigma_{\text{Portfolio along Benchmark}} = \sigma_{\text{Portfolion}} \times\rho$ ,

where $\rho$ is the correlation between the portfolio and benchmark returns.

Next, if we consider the ratio of this projection to the length of the benchmark arrow, we have

$\frac{ \sigma_{\text{Portfolio along Benchmark}} }{ \sigma_{\text{Benchmark}} } = \frac{ \sigma_{\text{Portfolion}} }{ \sigma_{\text{Benchmark}} }\times\rho$ ,

which you might recognize as the definition of the portfolio beta! Therefore,

$\text{Portfolio Beta} = \text{Ratio of the Projection}$ .

Conclusion

We’ve demonstrated, first, that returns can be visualized as arrows where the length of the arrow represents its volatility and the angle between two arrows represents the correlation of the two respective returns. Second, by comparing the portfolio and benchmark returns pictorially, we automatically get a very informative picture of both tracking error and portfolio beta (in one shot) that also contains information about the absolute market risk in terms of the volatilities (lengths) of the portfolio and benchmark arrows as well as the correlation between them.

Written by Eric

February 10, 2008 at 12:34 am

Posted in Market Risk, Visualization

Phorgy Phynance

Archive for the ‘Market Risk’ Category

Market Risk at the Federal Reserve – History books will not be kind

Einstein meets Markowitz: Relativity Theory of Risk-Return

Daily S&P 500 Value-at-Risk Estimates

Visualizing Market Risk: A Physicist’s Perspective

What is a vector?

Multiplying a Vector by a Number

Adding Vectors

The Dot Product

Projecting One Arrow Onto Another

Portfolio Return as an Arrow

Covariance as a Dot Product

Tracking Error

Portfolio Beta

Conclusion

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