4 Apr 2014

As part of producing a demo for FP Complete's new IAP product, I wound up implementing the Minimum Variance Portfolio calculation for a stock portfolio in R, then in Haskell for the IAP, and finally in Python using the NumPy and SciPy extension libraries. I want to look at the process of writing each of these, and the resulting code in this article. I am not an expert in these things, so if you know a better way to do them, please let me know. In particular, the R example was borrowed from someone else.

## The function

First, we find a version of the formula for MVP that can be converted into those systems. I like the one used by Yue Kuen KWOK:

ω = Ω⁻¹ **1**/ **1** ⋅ Ω⁻¹ **1**

where Ω is the covariant matrix of the returns for the stocks in question.

## R version

The R version is fairly straightforward - we just need to put the pieces together:

```
minvar <- function (px){
Rt <- px[-1,]/px[-nrow(px),]-1
cov.Rt <- cov(Rt)
one.vec <- rep(1,nrow(cov.Rt))
num <- solve(cov.Rt) %*% one.vec
den <- as.numeric(t(one.vec) %*% num)
return(num/den)
}
```

Calculating returns can be done with standard matrix operations and slicing. The covariant function is built in, as is inverting it (`solve`

). Since the numerator Ω⁻¹ **1** appears in the denominator, I reuse it's value there.

All the array operations were documented in the same place. That I only needed one unit vector was a bit of a surprise, but R sized it dynamically to work. That I had to transpose the unit vector and use the cross product operator (`%*%`

) to get a dot product was a a less pleasant surprise, but is apparently a standard R idiom.

## Python version

The Python version is almost a direct copy of the R version.

```
def minvar(prices):
cov = np.cov((prices[1:] / prices[:-1] - 1).transpose())
vu = np.array(cov.shape[1] * [1], float)
num = np.dot(np.linalg.inv(cov), vu)
den = np.dot(vu, num)
return num / den
```

In this case, I passed the returns matrix to the covariant function directly. The NumPy `dot`

function performs both cross products and dot products, and again the unit vector adopts it's length dynamically.

Documentation was a bit more scattered. Being a mix of traditional imperative and object-oriented, some functions are functions in the module, and others are object methods. The biggest surprise was that the returns matrix needed to be transposed before the covariant was calculated.

## Haskell version

Haskell is not quite as obvious a translation of the R and Python versions, but is a more straightforward translation of the original formula - once you notice that Ω⁻¹ **1** has been factored into `tv`

. It reads from top to bottom like the original, with the main formula at the top and the various terms defined below.

Returns again use the standard Haskell idiom for slicing the array. This is a bit more verbose than either R or Python, as they are functions rather than special syntax.

minVariance prices = tv / scalar (tvu <.> tv) where rets = dropRows 1 prices / takeRows (rows prices - 1) prices - (_, cov) = meanCov $ rets vu = constant 1.0 (cols cov) tvu = constant 1.0 (rows cov) tv = inv cov <> vu

The documentation was again straightforward, with everything being a function in the hmatrix package. In this case, both unit vectors were needed, as Haskell does not scale them automatically. It was the least surprising in at least one way - it used a distinct dot product operator for the two vectors rather than transposing - whether implicitly like Python or explicitly like R - the unit vector in a cross product.

### Performance

While performance comparisons with IAP aren't very useful, as it runs in the cloud so doing comparisons on identical systems may be impossible, Haskell does have one interesting advantage.

All three of these systems have the same basic architecture - a high-level language running in an interactive environment, with bindings to low-level, fast implementations of the matrix manipulations. Haskell adds the ability to compile your program into native x86_64 code. Doing so reduced the wall clock time of this short demo by roughly two orders of magnitude.

## Summary

I found the IAP version a little easier to deal with. Having custom operators and functions for everything - and this will only get better with time - along with being able to use the mathematical layout of the *where* statement just made things a little easier to deal with. While not having unit vectors that automatically size themselves - or transpose into matrices - is a little inconvenient, this also exposed a problem in the original R version, in that the unit vector's length was initially set wrong. I'm not sure that will make any real difference, but the thought that the language can catch such errors for me is comforting.