SHARPEning Your Portfolio

In the last article, we reviewed how the equation of a line that we learned in algebra can also be an expression of the expected return of an asset. We established that:

If you want to attempt to increase your expected return, you either need to take on more volatility risk or you need to increase your efficiency of return (i.e. Sharpe ratio).

In this article, we will dive a bit deeper into the Sharpe ratio.

The Sharpe ratio is the slope of the lines in the title image. It is your return above the rate of cash divided by your risk above that of cash. We want to have a higher Sharpe ratio because that means we are generating more return for the amount of risk we are willing to take.

The Sharpe ratio of a single asset portfolio is pretty easy to understand. It is just:

Where things start to get a little more complicated is when we attempt to understand the Sharpe ratio of a multi-asset portfolio. For the sake of this article, we are just going to consider a 2-asset portfolio where both assets have similar expected return and volatility. Here is the full equation for the Sharpe ratio of a 2-asset portfolio:

As a reminder, the goal in all of this is to increase efficiency of return. We want to either increase the numerator (excess return) or decrease the denominator (volatility).

In the numerator, we have the excess return of the portfolio. The excess return of a 2-asset portfolio is the weighted average return of the two assets minus the rate of return of cash (the 3-month US treasury bill). If both assets have the same expected return, it doesn’t matter how we allocate the portfolio. The expected return of the portfolio will be the same regardless of asset weighting.

The denominator is different though. This is where things start to get fun.

The volatility (standard deviation) of a 2-asset portfolio is a function of the weight, volatility, and correlation of the two assets.

For the sake of simplicity, let’s assume a zero correlation between the two assets. By doing this, we can immediately cancel out the third term of the equation. That leaves us with:

If our goal is to decrease the portfolio volatility to increase efficiency and the volatility of each asset is identical, how should we allocate the weightings so that this part of the equation is minimized?

If this equation was just weight times volatility, it wouldn’t matter. Any allocation would result in the same portfolio volatility. ?

However, because the weight terms are squared, we can determine that the minimum volatility will occur when the weight is split 50-50 between the two assets.

You have always heard that diversification is a good thing in investing. This is the mathematical reason why that is the case.

If you take multiple assets with similar return and volatility and limited correlation, you can have the potential to increase the efficiency of return of your portfolio through a reduction in volatility.

An important item to note in all of this is that this math takes a long time to play out in the real world. A single asset portfolio can be more efficient than a multi-asset portfolio for long periods of time.

That is why it is important to understand the math of diversification and be confident in the strategy you are implementing. If, not when, a multi-asset portfolio is underperforming a single asset portfolio, you need to be able to have the mental fortitude to not jump back into owning a single asset that has been doing well recently.

There is no way to know which asset will perform best going forward. Therefore, it's better to own both and benefit from the volatility reduction that tends to come with a more diversified portfolio.