“It has been a cruel summer for one of the trendiest, most innovative investment strategies of the asset management industry”.
This quote was taken from an article last week in the FT.
• Title: Risk parity funds suffer a cruel summer
• Author: Robin Wigglesworth
• Source: Financial Times (FT)
Here I will demonstrate how a risk parity portfolio can be calculated quite easily using R.
By definition, a risk parity portfolio is one for which all percentage contributions to risk (PCTR) are equal. By the same definition, it means that the total contributions to risk (TCTR) are all equal also.
Using some basic calculus, it can be shown that the volatility of a portfolio can be decomposed as the weighted sum of each asset’s marginal contribution to risk (MCTR). MCTR tells us the impact of an infinitesimal increase in an asset’s weight on the total portfolio risk. With each MCTR known, each assets PCTR can be derived.
In general, a risk-parity portfolio needs to be solved by some numerical method, for example a Newton algorithm.
The setup for the Newton method requires writing the risk parity problem as a system of nonlinear equations (imposing the restriction that the weights add up to one), finding the Jacobian (using multivariable calculus) and taking a one-term Taylor expansion approximation. Then the Newton iteration can be formulated by defining a stopping rule and starting point.
For reference, see “Efficient Algorithms for Computing Risk Parity Portfolio Weights”
So using simple arithmetic and matrix operations, I demonstrate how this can be implemented. My example portfolio consists of 9 liquid stocks using end-of-day (EOD) prices sourced from Quandl.
As we can see in the attachment, each asset’s PCTR is 11.11%
Now, is that trendy and innovative or what?
Thanks to Doug Martin for his lectures on portfolio optimization.