All Investments Have The Same Maximum Risk-adjusted Return

Published on Jun 10th, 2019 by Kyle Piira

All investments have an identical maximum risk-adjusted return. Regardless if it's bonds, stocks, savings accounts, etc. they all have the same maximum return per unit of risk. Now, that doesn't mean that all investments will earn the same absolute return. Obviously if all investments have the same maximum risk-adjusted return then those that take on more risk will yield higher absolute returns on average. It also doesn't mean that taking on high risk investments will always yield higher returns since they could produce a risk-adjusted return below the maximum.

Below is my take on the common risk vs. return graph for investments. On the y-axis is the return (in some abstract unit of return) and similarity on the x-axis is the risk (in some abstract unit of risk) required to receive that return. There are 4 different types of returns that I have graphed: maximum return, average return, average loss, and maximum loss/minimum return.

• Maximum return - is the greatest possible return that can be earned from any investment of a particular risk level. For example, in the graph below, any investments that take on 6 units of risk will earn at most 6 units of return.
• Average return - is the average return on all investments of a particular risk level. So if an investor held a diversified portfolio of investments that assumed 4 units of risk then their portfolio should yield about 2 units of return.
• Average loss - is the average loss on investments of a particular risk level that took a loss (excluding assets that had a positive return). For example, of the investments that assumed a risk level of 5 and yielded negative returns the average return (loss) would be -0.5 units of return.
• Maximum loss / Minimum return - is the maximum amount that an investor can lose at any particular risk level. This metric is different from the previous three because the maximum loss is always 100% of your principal (the money you invested) and is thus constant. For the purposes of the graph I have selected -100% return to equal -1 units of return. Below is the graph of the risk adjusted returns for each of the absolute returns in the graph above. To adjust the returns for the risk required to obtain them I used the formula: Risk Adjusted Return = Absolute Return / Minimum Risk Required.

The end result is that the risk adjusted graph is more-or-less the graph of the derivative of the risk vs. return graph above. I have also excluded the maximum loss since it does not change as a function of risk. As you can see, the risk adjusted returns are constant regardless of what level of risk you assume. This is consistent with the typical advice given to investors of taking on more risk when young (and having a long time-horizon) and less risk when older.

Up to this point, everything has been theoretical, so I wanted to look at some actual market data from publicly traded stocks. I used historical data from 6,200 different equities dating back to their IPOs. For each equity I measured four things: risk (volatility/standard deviation), maximum daily return, average daily return, average daily loss (on days when the asset lost value).

One flaw in this approach is that the maximum daily return may still be under the maximum possible return on a securities risk level, however, it should be a good approximation.

Below is a scatter plot of the results with risk (volatility) on the x-axis and return (% gain) on the y-axis. The graph is somewhat crowded with so many data points, so I decided to plot a linear regression line for each set and hide the rest of the data. Below is the same graph with just the linear regression lines. You'll notice that the trend lines for both maximum return and average return both move upwards as you assume more risk in your stock picks. Similarly, the average loss slopes downwards as a function of risk.

The average return slope is not quite as pronounced as it was in my theoretical graphs, but it is positive 0.0066 nonetheless, so still consistent with the theory.