Risk Capacity as a Function of Time

When it comes to determining how much risk to take in an investment portfolio, it depends on two factors:

1.       Risk Capacity

2.       Risk Tolerance

Risk capacity is a measure of your ABILITY to take risk.

Risk tolerance is a measure of your WILLINGNESS to take risk.

Risk capacity is a quantitative measure of your situation while risk tolerance is more of a qualitative measure of your intestinal fortitude.

While there are a few factors that go into determining risk capacity, the greatest of these is time. In other words, how long do you have until you need to spend the money?

Once you know your time horizon, you can determine how much volatility risk you might be able to withstand.

Volatility risk is measured in standard deviation (SD) and standard deviation helps you determine your range of expected outcome.

For instance, let’s assume you are investing in a certificate of deposit (CD) that pays a rate of 3% per year. Since you have a contractually guaranteed return and there is no volatility, you can determine the exact amount of money you will have when the CD matures.

When we introduce stocks, bonds, or other risky assets to the mix, we now must consider a range of outcomes due to the presence of volatility.

If we have a portfolio with an expected return of 5% and an expected volatility of 5%, we can determine our range of outcomes over a 1-year period by plotting these numbers on a bell curve.

In this case, our average expected return (μ) would be 5%. The 1 standard deviation events on either side of the average (-σ, + σ) would be 0% and 10% respectively. The 2 standard deviation events (-2σ, +2σ) would be -5% and 15% respectively.

If we compare these outcomes with the 3% CD, we can see that we will end up ahead over half the time if we go with the riskier portfolio. However, the breakeven occurs before the 1 standard deviation downside event, so there is still a good chance that the CD would be the better option.

This analysis starts to get a little more complicated when we start considering time horizons beyond 1 year. To analyze multi-year time horizons, we need to determine the range of potential outcomes over multi-year periods.

Time for an equation…

As this equation shows, the volatility over a given time period is determined by multiplying annualized volatility by the square root of the number of years of your investment time horizon.

This multi-year standard deviation can then be used to determine the range of outcome on either side of an expected average outcome.

It’s important to note that expected average outcome is not just a compounding of your expected average annual return (i.e. arithmetic mean). There is a drag that occurs on returns due to annualized volatility. Therefore, we need to determine the annualized return (i.e. geometric mean) by subtracting this expected drag from the average. An approximation for this is:

Now that we have an equation for expected annualized return and multi-year volatility, we can determine our expected ranges of outcomes for a variety of portfolios.

Below is screenshot of a volatility calculator that shows the range of outcomes over a 10-year period for a risk-free portfolio (our 3% CD) and two risky portfolios. The expected annualized returns are compounded by 10 years to determine the average (μ) expected total gain. The 10-year standard deviation is then added/subtracted from this to determine the standard deviation events on either side of the averages.

As expected, the portfolio with the highest expected annualized return has the highest average (μ) expected total return. It also has the highest expected total return when considering a 1 standard deviation movement to the downside. That is why it gets the blue highlight in the cell.

However, when we start to consider less-likely downside events, we can see that the less risky portfolios end out ahead in the 2 and 3 standard deviation events.

Now that we have quantitatively determined the range of outcomes, this is where risk capacity analysis ends and risk tolerance determination begins.

Math can help you create a range of outcomes of a portfolio, but only you can determine whether to base your selection on a 1, 2, 3+ standard deviation event. You have to decide how much risk you are willing to take.