 that an investor has and the need to allocate that budget between competing investments. This budget choice could be on a cash basis or on a volatility basis, where a certain percentage of their target volatility is allocated to an investment. Correlation and the risk budget are linked as the risk budget is a choice between investing a proportion of your wealth in different investments, each with a varying degree of risk. This is why the correlation amongst those investment choices is an important input.  

Correlation is a simple measure that explains the relationship between two variables. It normalises the covariance of the two variables, i.e how they move together, by scaling using the standard deviation of the variables. 

If your portfolio consists of two stocks which have a negative correlation, then your overall portfolio volatility will be lower than the weighted average of their volatilities. When one stock rises in price, most likely the other will fall and in aggregate your portfolio volatility will be reduced. Correlation has the following properties

bounded between -1 and 1 and can never be outside that range

represents how strong the relationship is between two sets of data   

can and does change through time and depends on the window used for the correlation calculation

Correlation is a critical component when you consider two or more stocks in your portfolio. Diversification of a stock portfolio only benefits the investor when there is less than perfect correlation between the stocks. We saw this in our post on the 

 where we illustrated the risk versus return benefit of combining stocks with less than perfect correlation.   

When it comes to portfolio management and multiple strategies competing for a risk budget, i.e inclusion in the overall portfolio, then the importance of correlation shifts from the stock level to the strategy level. We will illustrate these concepts in the following sections. 

Correlation moves from stocks to strategies

We have seen the power of modifying correlation between stock A versus stock B. Changing correlation can either reduce the portfolio risk whilst generating the same amount of return or the correlation change improves the returns for the same amount of risk - both methods increase the risk-adjusted returns of the portfolio. We now shift that logic to the strategy level instead of the stock level. We are considering the impact of strategy correlation when combining two strategies together into one final portfolio. 

An investor has thousands of stocks to choose from to form their portfolio and can choose from various investment strategies, each of which generates a set of stocks to invest in. The set of stocks investors can choose from is normally called the 

of stocks. In quantitative terminology, when we consider all stocks and want to rank all stocks against each other we refer to this as the 

In this cross-sectional world where we are attempting to come up with quantitative equity trading signals, we must move away from any toy examples on a small set of 3-5 stocks and instead think about ranking between thousands of stocks. Therefore, our effort is focused on developing strategies that are based on sound mathematical and financial concepts that are applicable across all stocks so we can generate a final score (ranking) for each stock, which represents a stocks future forecasted return or

Imagine we develop two investment strategies, call them strategy A and strategy B. Strategy A is a value strategy where it focuses on a mathematical rules-based method to identify the top 50 value stocks. Strategy B is a momentum strategy that focuses on identifying the top 50 stocks based on some new momentum metrics. The correlation concept we applied to stock A and stock B can likewise be applied at the strategy level so our goal here is to hopefully identify low or even negative correlation between momentum and value and therefore combine the two strategies into a final portfolio. This final portfolio would then provide higher risk-adjusted returns to an end investor, rather than investment in purely strategy A or strategy B independently.

To illustrate the strategy correlation concept we mention above, we can look at equity style factors and their correlation. Equity style factors in general have some overall correlation principles but the correlation is time-varying and does move around over short time horizons. Two factors that have historically tended to have negative correlation are 

Therefore an end investor if they wanted to increase their risk-adjusted returns could consider investing in two strategies, say for example 

, to really benefit from this negative correlation to increase their overall risk-adjusted returns of their final portfolio.

Decrease portfolio volatility by equal-weighting value and momentum 

An investor if they wished to reduce overall portfolio volatility could allocate 50% of their investment to value and 50% to momentum and 

Why does the correlation among assets matter so much. We describe how lower correlation helps lower portfolio risk and increase risk-adjusted returns.

Correlation within your Stock Portfolio

How to Analyse your Quantitative Strategy ?

Give an overview on what a strategy is and why it is important to quantify metrics associated to the strategy such as statistics and distribution 

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1. Strategy Analytics

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Post outlining regression models and illustrate a few different options. After then describe how they are used in finance

8. Regression Models

Portfolio construction is an area of research that occasionally feels neglected in comparison to alpha generation or signal construction. 

What methods are used for portfolio construction ?

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This is a post concerning portfolio construction and why it is important.

5. Portfolio Construction

The efficient frontier is a risk versus return plot that illustrates which portfolio is optimal to hold for your given level of risk. This was inspired by academic work from Harry Markowitz in the 1950s.

A portfolio is just a weighted combination of a set of stocks. By definition, each stock in your portfolio has an expected return and expected risk level associated with it. These expected return and expected risks are key to deciding which stocks you want to hold in your portfolio. The risk and return can be estimated via complex mathematical models and this is what the alpha-generating hedge fund industry focuses on, where alpha refers to future expected returns. 

For this article, what can we do if we do not have access to these sophisticated estimation methods for risk and return ? 

We can use a very simple heuristic which is that we expect future risk and return to equal the past risk and return - so let's go with that model assumption for this article. 

What do we mean when we say Portfolio A dominates Portfolio B ?

The concept of domination relates to the comparison of two portfolios where one portfolio is clearly a better choice than the other. This better choice, assuming A dominates B, will either be based on 

Portfolio A has a greater return than Portfolio B, with both Portfolio A having the same or less risk than portfolio B

Portfolio A has the same return as Portfolio B but it has lower risk 

Mathematically we can write this as a mean-variance condition

Essentially, any portfolio that does not sit on the efficient frontier is dominated by a portfolio that does sit on the efficient frontier.

Choosing a portfolio from a set of N stocks 

There are essentially an infinite number of ways to combine a set of N stocks into a portfolio P if we are allowed to use any weight combinations for our N stocks, no matter how small each weight is. If we take a simple two stock portfolio of stock A and stock B, then the two limits in this case are 100% weight in stock A and 0% in stock B or 100% in stock B and 0% in stock A - but in between these limits we can also invest x% in stock A and y% in stock B. 

It is easy to show a risk-return graph of all the possible portfolios of stock A and B, if we have an assumption for the correlation between stock A and B. To make things simple, we can again suggest that the future correlation between stock A and B is equal to the past correlation. For a specific example, let us take Amazon and Walmart as our two stocks and calculate the average annual return, risk and correlation.   

The average annual return, risk and correlation between Amazon and Walmart, all figures in percentages, are given by:

We can quickly look at the potential portfolios assuming we shift our portfolio in steps from 100% invested in Walmart to 100% invested in Amazon.

Investigating further to test the impact of the correlation parameter, we can vary it from a very unrealistic -70% all the way to 70%. Each point on each line is a potential portfolio for an investor to invest in, under the fixed correlation assumption for that line plot. This hopefully illustrates why a low to negative correlation is so desired from a diversification perspective and allows the investor to obtain superior risk-adjusted returns. 

Clearly the left-most blue line is our preferred portfolio (even though it is based on a non-existent -70% correlation). 

This extreme example was shown to really illustrate the power of negative correlation, or even small positive correlation, and why it is so important at both the stock or portfolio level but also at the strategy level. If an investor can combine one strategy (or portfolio) with another that has low correlation or even negative correlation then the final combined portfolio will offer much better risk-adjusted returns than either of the investments independently.  

Example of the efficient frontier for US stocks 

Let us see a toy example and assume we have a small set of 6 US stocks, covering the technology sector, food and clothing. We can calculate the average annual return and the volatility of each of these stocks. Let us now assume that out of these 6 stocks we want a portfolio of 5 stocks only, then the question is 

What weight should each stock in our portfolio have to ensure we are maximising our return for a given risk level i.e we are on the efficient frontier ?

We show in the table below the returns and risk for each stock where our goal is to obtain the optimal weights i.e the set of weights that lie on the efficient frontier. These sets of weights will be the vector of weights in the last column 

One simple way to do this is to create a large set of random portfolio weights via a Monte Carlo simulation and then calculate the portfolio risk and return. If we plot all of those simulations, which in our case is 10,000 we get a plot like below. For each risk point, we have highlighted the best portfolio return and it is clear to see, a la Markowitz, that there is a frontier where portfolios on that frontier 

Additionally, we have also highlighted with a red start the optimal sharpe ratio portfolio, which is the best risk-adjusted return portfolio a user could obtain theoretically. 

The efficient frontier shows the power of diversification when choosing amongst a portfolio of non-perfectly correlated stocks.

Efficient Frontier

What is VAR and how to compute it ? How to measure drawdown statistics accurately.




2. Drawdowns and VAR

We describe the mathematical concepts crucial to understanding quantitative trading. We also go into detail about mathematical techniques and models deployed to build investment strategies. 

The Math and Models behind the Strategies

Correlation within your Stock Portfolio

Strategy Analytics

Regression Models

Portfolio Construction

Efficient Frontier

Drawdowns and VAR