Correlation and Risk/Return in Portfolio Management
Today I wanted to discuss the nature of correlative assets within a portfolio, and how the pairing of risky assets can actually reduce the overall risk of the portfolio.
In today’s piece, we are going to be using some quantitative calculations to demonstrate the points I am going to be making, but don’t worry too much about understanding the mathematics, they are more so to illustrate the themes of today’s discussion.
So, first of all it would be important to understand the concept of portfolio return, which is the weighted average returns of the assets within the portfolio.
Think about a portfolio that contains two assets, Asset 1 and Asset 2.
We can illustrate the calculation for the portfolio return as per the below equation.
Rp = Portfolio Return
W1 = Weight of Asset 1
(1-W1) = Weight of Asset 2
R1 = Return of Asset 1
R2 = Return of Asset 2
What this equation says in plain English is; the portfolio return is equal to the weight of asset 1, multiplied by the return of asset 1, plus the weight of asset 2 multiplied by the return of asset 2.
Okay, so lets say we have two assets in our portfolio. Both assets have returns of 10%, and both are equally weighted at 50%.
In this case, our equation will look like:
The portfolio return would equally 10% in this instance, as both assets are equally weighted and have the same return, the weighted average return (the portfolio return) is equal to the return of both assets.
Okay, now imagine that everything stays the same, but asset 2 has a 20% return instead of 10%. Then the average weighted return would be 15%.
Pretty simple right? Now lets add on one more layer, before we get to explaining the impacts of correlation and risk management.
As well as calculating the portfolio return, we can also calculate the variance of the portfolio, sometimes call portfolio risk.
Portfolio variance is simply the measure of the dispersion of returns within a portfolio. It is the aggregate of the actual returns of a given portfolio over a set period of time. We can calculate portfolio variance using the standard deviation of each security in the portfolio and the correlation between securities in the portfolio. We will get to all of that in a little while.
Lets stick with our example of a two-asset portfolio for the entirety of this piece, it makes life easier.
The formula for the portfolio variance calculation can be shown below:
This looks confusing right?
Well lets try make it clearer. Don’t worry, this will become a little more obvious later, and the reason for me using the formulas will make sense in the end.
Essentially this reads as; the portfolio variance is equal to the weight of asset one multiplied by the variance of asset 1, plus the weight of asset 2 multiplied by the variance of asset 2, plus 2 X the weight of asset 1, multiplied by the weight of asset 2, multiplied by the correlation between the two assets, multiplied by the standard deviation of asset 1, multiplied by the standard deviation of asset 2.
Note here that the units for the variance formula are written using the ‘squared’ function, but this is just how they are measured, similar to squared feet. The numbers of not actually squared.
For those familiar with variance and standard deviation, we can display the standard deviation of the portfolio simply by using the same formula but displaying the squared root of the variance.
For those unfamiliar with these nuances of quant, the standard deviation is the squared root of the variance.
Same formula, but we are just finding the square root of the variance.
Note that in the standard deviation formula, we would actually square the units here.
Lets try and make this a little clearer by using another example.
Okay, so imagine we have a two-asset portfolio once more. This is all going to make sense by the end of it, I assure you.
Okay, so lets imagine we are based in the UK, and you also want some exposure to the Indian stock market. So, you decide to create a portfolio with an 80% allocation towards the UK stock index, and 20% weighted towards the Indian stock index.
Now let us assume the expected returns of the UK stock index is 9.93% and the expected returns for the Indian stock index is 18.20%.
The standard deviation of the UK index is 16.21% and 33.11% for the Indian index.
Lastly, lets say that the covariance between these two assets is 0.005.
Covariance is the part of the formula at the end, in this example, but I wanted to avoid introducing it to keep things simple, just know that covariance is equal to the part at the end of the equation.
So, we now have enough information on each of the individual assets to now calculate the expected return and risk of the portfolio as a whole.
So, we already covered the formula for expected return, and if we plug the numbers into the formula, we get the below output, and the expected return of the portfolio is 11.58%.
We also have the information available to calculate the risk (standard deviation) of the portfolio, and once we do that, we are going to come to the conclusion that I am trying to explain to you all in this newsletter.
We can use the same formula that we introduced a little earlier on, but substitute the 1s and 2s for UK and IN.
This then breaks down to the below, and recall that covariance is 0.005 in this example, which is equal to the end part of the formula, as we mentioned earlier.
Okay, so after punching in all those numbers, we work out that the standard deviation (Risk) of the portfolio is 15.1%.
Now HERE is the first point I wanted to portray to you all, I just thought it would be easier to convey with examples.
So what have we discovered here? Think about it for a second.
So think about each of the assets again, the UK and Indian Index.
The UK index had an expected return of 9.93% and a with a risk of 16.21%.
The Indian index had an expected return of 18.20% with a risk of 33.11%.
The combined expected return of the portfolio, weighted at 80% UK and 20% India, had an expected return of 11.58% and risk of 15.1%.
Remember that we said we are a UK based investor.
So, if we had stayed within the UK, we would have earned the returns of the UK Index, with the corresponding risk (standard deviation).
Instead, we invested 20% of our allocation in the seemingly riskier asset, the Indian Index. I say riskier, because the standard deviation was 33.11% compared to the UK’s 16.21%.
The result was that we achieved a greater return (11.58%) versus the UK index (9.93%) and we managed to achieve that with a lower risk of 15.1% versus the UK’s 16.21%.
So, in effect, by combining two risky assets, harnessing the power of diversification, we improved our risk/reward profile.
Do you understand now?
The point of sharing the calculations was more so to kind of illustrate how the underlying levers of this concept works, as well as explaining the idea of risk and portfolio management.
This leads us on to the final point I wanted to make in this newsletter, and that is correlation and how this impacted risk and return within a portfolio. So let us now move onto that idea.
So the last point I wanted to make was one surrounding the idea of correlation and how this impacts the relative risk and return characteristics of a portfolio of assets.
Correlation is simply the tendency for two assets to move in a similar fashion. The range of correlation oscillates between -1 and +1. Again, lets assume we have a portfolio of two assets.
• Correlation of ‘+1’ would imply that both assets move in perfect unison 100% of the time. Perfect correlation.
• Correlation of ‘0’ would imply that the movement of one asset has zero inference on the movement of the other asset. They are un-correlated.
• Correlation of ‘-1’ would imply negative correlation. The assets move in the opposite direction, 100% of the time.
Okay, so now we are going to do a few more calculations, but don’t worry, there is nothing new here. We have already covered the necessary steps for what we are about to do, but this time we are going to focus on the effects of correlation.
So, we are going to run through three identical examples, but one variable will change, and that is the correlation factor from the below formula.
So, we will run through this fairly quickly, as all the calculations must be draining.
Lets go back to that first example again. Two assets, both equally weighted at 50% each. This time the return on both assets will be 20%, and the risk will also be 20% for both assets.
We are now going to calculate the portfolio risk and the portfolio return of both assets, but using three different correlations; +1, 0 and -1.
Correlation = +1
So, for the portfolio return calculation, this does not include the correlation element, thus the portfolio return is 20%, as calculated below, for two equally weighted assets with equal return.
Remember that the portfolio return is just the weighted average return of the two assets.
In terms of the portfolio risk, after punching in the numbers, with a correlation of +1 we end up with a risk of 20%. This makes sense, seeing how the two assets are perfectly correlated.
Correlation = 0
In the case of correlation equaling 0 we would perform the portfolio return calculation and arrive at the same output, 20%.
In terms of the portfolio risk calculation, we replace the 1 with a 0 and we attain a portfolio risk of 14%, which is the same return, but a lower risk than the perfectly correlated assets.
Correlation = -1
For the last case, we would achieve the same return once more, at 20%.
In terms of the portfolio risk calculation, we replace the 0 with a -1 and we attain a portfolio risk of 0%, demonstrating that in some instances the pairing of two negatively correlated assets can set up a ‘riskless’ risk/reward scenario.
So, what can we derive from that?
Essentially, you should consider the correlation of the assets within your portfolio when engaging in portfolio management.
If all of your positions zig and zag in the same direction, with the same triggers, then your correlation will be relatively higher. In this case, it is likely that you will be undertaking a level of risk that you are unaware of.
If the portfolio goes down, then the whole thing is going to go down together, and vis a vis. Great when things are moving upwards, but terrible when the going gets tough.
Conversely, holding a portfolio constructed of assets that exhibit a lower level of correlation, will enable the investor to yield similar returns, with a lower level of implied risk, much like we saw when we calculated from a correlation of +1 to 0.
In some instances, holding assets that are negatively correlated, from 0 to -1, can lead to a significant reduction in your portfolio’s level of implied risk.
Lets say, just hypothetically, that stocks and gold are negatively correlated to such an extent that they are -1. When one goes up, the other always goes down.
Holding a 50/50 portfolio of these assets could lead to a ‘riskless’ portfolio.
Of course, that is a hypothetical, for illustrative purposes. These two assets are not so strongly negatively correlated that they are at -1.
Point being, if you hold a basket of equities, and all those equities are in a similar industry, or prone to the same catalysts, or negative triggers, then you might be undertaking a greater level of risk that you are even aware of. During a raging bull market, this fact is often ignored. Things are going well, so who is really thinking about risk?
Markets are not always in raging bull mode though, and sometimes a great way to preserve capital is by holding some uncorrelated assets.
I think the main takeaway from this discussion should be that diversification is not simply buying a larger number of assets to fill your portfolio. You might think you are diverse, but if these assets are all correlated to a high degree, then you are not diversifying your risk.
Thus, risk diversification is the crucial insight I would hope you take away from this.
Clearly, in the extreme, one can go so far as to mitigate their own returns if they insist in extreme diversification. After all, returns typically favour those willing to take on a little more risk.
This is by no means a recommendation, you know, you have to do what feels best for you. I more so wanted to create awareness around the impacts of correlation on return and risk dynamics, with relation to portfolio management.
Anyway, I hope that was interesting for you.
Until next time,