Intro
Volatility is the price uncertainty of an asset. When I was perusing Twitter a few weeks ago, I saw the following tweet asking a question about what happens to portfolio expectations when adding uncorrelated assets to the portfolio. The answer gets at the benefits of diversification and the effects of volatility. Although this can all be done with matrix math, I thought it would be more fun and intuitive to think about the question using simulations and visualizations. The setup of the problem is such:
Suppose we have a portfolio of N number of assets, and we believe both assets should deliver individually:
Compound annual growth rate (CAGR) of 10%
Annualized volatility of 25%
Correlation between all assets is 70%
Annual Rebalance
Question: What is the best estimate of portfolio CAGR?
Answering this question will help us see the benefits of diversification when building a portfolio of assets (most likely stocks with a 10% CAGR!). There are a couple of important considerations when thinking about this question.
First - Volatility Drag
In investing, average rates of return more closely mimic a geometric than the arithmetic mean because of volatility and the compounding nature of returns. The reason actual return more closely mimics a geometric mean is that you need better return performance to compensate for an asset’s volatility. An easy way to think about this is: what if the investment goes down 25% in the first year? To catch up to that return, you must compound more than 10% - the arithmetic mean. To demonstrate what we are talking about, let’s simulate what an asset with a mean return of 10% with 25% volatility looks like compared to an investment with a CAGR of 10% and no volatility.
End Frame
As we can tell from the graphic above, most of the volatile portfolios underperformed the no-vol portfolio. Only 32% of the portfolios with volatility outperform the 10% CAGR portfolio, despite the average return for both being 10%. This should help us visualize the concept of volatility drag and the impact volatility has on a portfolio.
Here is a histogram for added clarity:
Both the line chart and histogram should also help highlight the range of outcomes when volatility is added to an asset. We should note as volatility increases, so does the range of outcomes.
Intuitively, investors know diversification is good, so this should help. We will simulate all this to prove that adding uncorrelated assets helps portfolio performance.
Effects of Diversification
Using what we learned above, we can now look to answer the question from the tweet: how does diversification help a portfolio when you add assets that aren’t 100% correlated?
If we expect our CAGR to be 10% a year, we need to figure out our return expectations as a geometric mean because we need to account for volatility. Luckily, there is a simple estimation we can use:
A 10% CAGR translates to a ~13.125% geometric mean. In other words, to generate a 10% annual return, we will need the average of the yearly return stream to be 13.125%. We will generate data to simulate portfolio returns of three different-sized portfolios, all with the same characteristics outside of the number of assets. The three portfolios will be:
A single asset portfolio with 25% volatility
A 2-asset portfolio with 25% volatility and 70% correlation between assets
A 10-asset portfolio with 25% volatility and 70% correlation between assets
We will simulate each of these portfolios 500 times and collect the summary information. Based on what we know, we should expect the one asset portfolio to have a return of ~10% and that adding assets should increase the CAGR of the other two portfolios.
After simulating, we have the following distribution of Annual Returns by the number of assets in the portfolio.
Though slightly difficult to see with the image, if you look closely, you can tell the 10-asset portfolio outperforms the one and 2-asset portfolios.
Here is a table of the results after 500 simulations to more easily show the increased performance of adding assets.
In dollar terms, assuming $100 to start:
The single asset portfolio earns: $257
The 10-asset portfolio earns: $282
Conclusions
As expected, the single asset portfolio is ~10%, and adding uncorrelated assets to a portfolio improves returns. This is an important consideration when adding assets to a portfolio. If choosing between two assets with similar return expectations, you would be better off adding the less correlated asset. Again, this could be done with matrix algebra. However, this approach also helps demonstrate the path dependency of returns.
*I am not an expert here, so take this as a fun example.