Unique Risk Series – Article 2 out of 3
In the first article of the Unique Risk series, we saw how portfolio diversification can help you achieve higher returns without necessarily exposing you to more risk. We concluded that it is not the riskiness of an individual stock that matters, it’s actually how the stock affects the riskiness of your entire portfolio that determines whether you should invest or not.
In this article, I will describe what a minimum variance portfolio is and will show how investing in a minimum variance portfolio allows you to maximize the benefits of diversification.
Let’s revisit the example used in the last article. You are currently 100% invested in Stock A, which has an expected return of 4% and a standard deviation of 6%. Since Stock B is negatively correlated to Stock A and has a higher expected return, we determined it was beneficial to invest in Stock B so we decided to invest 50% of the portfolio in Stock A and 50% of the portfolio in Stock B. This combination produced a portfolio with an expected return of 6% and a standard deviation of 5.81%.
We were happy that we increased the expected return and lowered the standard deviation by investing in Stock B but is this 50/50 portfolio optimal? To find out, let’s head to the lab to yet again bathe in the glory of mathematics.
The following graph shows the expected return and standard deviation of different portfolios invested in varying amounts of Stock A and Stock B. Each consecutive dot on the line represents a 5% increase in the amount of the portfolio invested in Stock B. For instance, dot 1 represents a portfolio invested 100% in Stock A and 0% in Stock B, dot 2 represents a portfolio invested in 95% in Stock A and 5% in Stock B, and so on until you reach dot 21, which represents a portfolio invested 0% in Stock A and 100% in Stock B.
You can see from the graph that it doesn’t make sense to invest in the portfolios represented by dots 1, 2, 3, 4, or 5 because you’d achieve a higher return while maintaining roughly the same standard deviation if you instead invested in 11, 10, 9, 8, or 7, respectively. The portfolios represented by dots 6 through 21 are considered “efficient” because for those portfolios, it is not possible to obtain a higher expected return without also increasing the standard deviation.
Looking at this graph, it is easy to see which portfolio is the minimum variance portfolio: dot 6. For these two assets, investing 25% in Stock A and 75% in Stock B would allow you to nearly achieve a minimum variance portfolio for these two assets.
If we want to find the exact minimum variance portfolio allocation for these two assets, we can use the following equation:
x = (σb²-ρabσaσb) / (σa² + σb² – 2ρabσaσb)
Plugging in the values from the first article in this series, we can see that x = 74.42%. This means, to achieve a minimum variance portfolio that is invested in Stocks A and B, you should invest 74.42% in Stock A and 25.58% in Stock B.
But I Want a Higher Return!
When we planned on investing 50% in Stock A and 50% in Stock B, we computed that the portfolio should have an expected return of 6%. Now that we are investing in the minimum variance portfolio, however, our portfolio’s expected return is only 5% (see graph). What if you would rather take on more risk in order to earn 6%? Should you just use the dot 11 ratio to achieve the 6% expected return? Not exactly.
Rather than invest your money in a dot 11 portfolio, why not invest in dot 6 but just borrow some money in order to invest more? This would allow you to increase your expected return while still allowing you to invest in the minimum variance portfolio. To invest more, you can borrow money at the risk-free rate by selling the risk-free asset and can use that extra money to invest more in the minimum variance portfolio.
Since the risk-free asset is “risk-free”, it has zero variance. When adding the risk-free asset to the graph (assuming a risk-free rate of 3%), additional investment options become available that are more appealing than simply investing in Stocks A and B alone.
As you can see, to achieve the 6% expected return you are looking for, you’d be better off borrowing at the risk free rate and increasing the amount invested in the minimum variance portfolio (this scenario is represented by letter a on the graph) than you would be investing in the mixture of Stocks A and B represented by dot 11.
If you are more risk-adverse, buying the risk-free asset allows you to lower your standard deviation without forcing you to invest in a non-efficient portfolio. You could achieve a standard deviation of 2% while still earning a 4% expected return by lending some of your money at the risk-free rate and investing the rest in the minimum variance portfolio (see letter b on the graph).
Now, I know this article was a bit complicated and you’re probably thinking, “Mad Fientist, all this theory is great but now I don’t want to invest in the market at all because it seems too difficult. How do I know which stocks to buy? How can I borrow and lend at the risk-free rate?”
I understand your concerns. Next time, in the final article of this series, it will all become clear and I promise, the practical application of all of this theory is shockingly simple. In fact, my conclusions will most likely be similar to investment advice you have heard many times before. The only difference is, now you’ll have a better understanding of the financial theory supporting the advice and may feel more confident to act on it.
If you have any questions about what has already been covered, feel free to ask in the comments. Otherwise, give that calculator of yours a much needed rest and I’ll see you next time for the grand finale of the Unique Risk series!
Next article in the Unique Risk series: Market Portfolio