# Modern Portfolio Theory: Beating the Index

How do you beat the market without superior (inside) information? One would immediately start arguing that beating the market needs superior skill, informational or hardware advantage. For example in trading public news one need superior information processing skills. If information is relatively easy to process, then everybody can do that, therefore one might need faster hardware etc. Well, what if I told you that you do not need any of those to beat the market?

According to Sun Tzu (The Art of War) knowing your enemy is important. So if we intend to beat the market, the market is the enemy we should investigate closer. Asset management industry is accustomed to beating specific indices. In other words, what is the market benchmark we intend to beat? To analyze this, let’s take Dow Jones as an example of the market. Dow Jones index includes 30 companies and is price-weighted. To put in layman’s words the index value is portfolio holding one stock of each company. Companies with higher share prices will account for larger share of portfolio assets.

Such price-weighted portfolio might be less than efficient according to modern portfolio theory. What is modern about modern portfolio theory? The quantitative approach is modern. To summarize, the theory assumes that risk is variance of the returns – more variance more risk. The efficient portfolio is one that achieves the required return with minimal risk. Portfolio risk can be computed as:

$$w^{\prime}Qw=\sigma^2_{portfolio}$$

with restrictions

$$\Sigma_i w_i=1$$

$$ER^{\prime}w\geq r$$

where $$Q$$ is variance-covariance matrix of the returns, $$ER$$ is vector of expected (historical average) return, $$w$$ column vector of portfolio allocations and $$r$$ is scalar minimal required return. By varying $$r$$ we get efficient frontier (blue line: no shorting, no leverage allowed; black line: unrestricted):

Anything above the frontier is unattainable and anything below the frontier is inefficient. In the figure I mark the location of the DJIA index risk-return. As expected, the index portfolio is inefficient, thus can be improved. With 11.1% risk we get only 15% return. Following vertical red line we can select long-only portfolio on the efficient frontier (blue line) with the same amount of risk. With the same risk we could harvest 22.6% return – an outperformance of the index of 7.6% per annum. It seems like beating the market (index) is not so difficult. Or is it? Note that covariance matrix $$Q$$ used in the optimization is in-sample, i.e. not known beforehand. The matrix can be forecasted, albeit the task is not easy, but this is a topic for another time. (Some discussion on variance-covariance matrix forecasting and accompanying R code could be find here)

Interested reader could find R code below (click to expand)

## 1 Comment

1. Hi Justinas, thanks for the source code. We did a related study on efficient frontiers with intraday data and real-world portfolio sizes in this tutorial for our PortfolioEffectHFT package.