All industries have their fair share of jargon. It can be frustrating for a potential investor when he/she could not comprehend the investment manager, even when they speak a common language. If you have sat through any presentations by a hedge fund manager or read their monthly reports, you will most certainly have come across terms like Sharpe ratio, Sortino ratio or Calmar ratio. What are these ratios? Yes, you probably have guessed it by now. They are risk-adjusted returns.
First, let us talk about absolute returns. Absolute returns of an investment is simply the dollar return of the investment. It is often expressed as a percentage. If I invest $1000 in a fund, and in a year, its value grows to $1200, then my absolute return for that year is $200 or 20%. This is what most people mainly focused on. After all, we invest with the ultimate aim to make money. However, absolute returns is only one part of the equation when assessing any investments. We cannot only look at returns in isolation without talking about risks and vice versa. Not all “returns” are equal. A risky investment generating a higher return is not necessarily better than one with a lower return but does so at a lower risk.
Taking Risk Into Account
So how can we take risks into account when we compare the performance of different investments? We can use returns per unit risk, instead of absolute returns. In this case, risk becomes the common denominator across all investments considered. Each of the ratios mentioned uses a specific risk measure. Sharpe ratio uses volatility, Sortino ratio uses downside volatility, and Calmar ratio uses maximum drawdown.
Risk is a very complex subject. There is a broad range of risks and some cannot be adequately quantified. Many people also questioned the validity of volatility-based measures as a proxy for risks. But we can discuss this another time. For now, let’s just look at how these risk measures and ratios work.
To facilitate practical understanding, I provided an Excel file to go along here. However, I will not be going through step by step in detail. You can explore the file. All the formulas and scripts are there.
Download Excel File: Risk Adjusted Returns
Note: I am using Microsoft Excel 2013.
Volatility — Mathematical Interpretation
Volatility appears in modern portfolio theory. It has gained wide acceptance and many funds use it as their proxy for risk. You can view it as a measure on the tendency of the fund’s value to swing up and down. The larger those swings are, the more volatile the fund is. Because of that, we are more uncertain whether the fund is going to meet its return target, and hence the riskier it is.
For the mathematically inclined, the volatility of a fund’s return is basically its standard deviation. Standard deviation quantifies the dispersion of the returns around its mean.
Most funds calculate annualized volatility as follows (assuming monthly return data)
n is the total number of months
Rt is the return on investment in month t
RM is the average monthly return on investment over the entire period
Firstly, we get the variance using the squared difference between each month’s return and the average monthly return over the entire period. Then we scale it by a time factor of 12 to get the annualized variance. Finally, we apply a square root to the number to arrive at the annualized volatility. While this formula might look a bit daunting, you can easily calculate this in excel. Please reference the Volatility sheet in the excel file. It shows 2 ways to get the volatility of a return series: (1) a manual approach, and (2) an excel formula STDEV() approach. For practical application, you can just use the excel formula. The manual approach is just to guide you through the equation here.
Note: You can also substitute Rt with log returns. It is actually more mathematically correct to use log returns. But some practitioners prefer working with simple returns because it is more intuitive. And empirically, you will find that both approaches yield very close volatility numbers anyway (but do note the sum of simple and log returns, however, can be drastically different). The more important point is that whichever case you choose, apply the same methodology across all the funds you are comparing against for consistency.
Volatility — Visual Interpretation
This diagram is a simple illustration on the concept of volatility with 3 funds A, B and C. To accentuate the idea, I deliberately made the swings appear even and non-random. But do note that this does not resemble anything in the real world.
Fund A has zero volatility. It appreciates in a straight line at a constant rate.
Fund B has medium volatility. It swings up and down gently.
Fund C has high volatility. It makes big swings both ways.
If all 3 funds deliver about the same annualized returns over the long term, which fund are you going to choose? You will likely go for A because you can be absolutely sure how much the Fund’s value would be at any point in time. With such a fund, you would have no shortfall risk meaning you will never find yourself liquidating at a bad time.
We would wish all funds’ performance profile can look like A. But as always, reality is harsh, such a product unfortunately does not exist.
You might have noticed by now that volatility does not care whether the fund makes or loses money. It penalizes both upside and downside as long as they deviate from the mean. But upside volatility is in fact good for us. So why are we punishing the fund for performing above its expected average return? This raises a question. If our concern is losing money, then shouldn’t we look only at deviations below the expected return or downside volatility?
Mathematically, we can express downside volatility as
n is the total number of months
Rt is the return on investment in month t
MAR is the Minimum Acceptable Return.
The Minimum Acceptable Return is a user-defined parameter. I have set this to zero. This effectively means that I am flushing all positive returns to zero, while keeping the negative returns to compute the downside volatility.
You can refer to the Downside Volatility sheet in the Excel file. Again, there are 2 ways which I did it. The first is the step by step method. The second is to write a VBA function script.
Maximum drawdown is the simplest among the 3 risk measures here. It is the largest historical loss you can experience based on the peak-to-trough NAV of the fund. This occurs when you buy and sell at the worst possible time. Well, shit does happen.
Using the returns, we can construct the NAV of the fund assuming it begins at a value of 1. Basically, there is a drawdown only when the fund is below its previous high. When the fund rallies and make new high, there will be no drawdown i.e. drawdown is 0%.
You can reference the sheet Maximum Drawdown in the excel file. I provided 2 approaches. In the first one, you need to create a new column to track the drawdown each month and then find the minimum in the series. The second made use of vba to create a user defined function to calculate the maximum drawdown directly from the returns series. Personally, I prefer the first method, since it not only just give me information on the maximum drawdown, but also let me see drawdowns in other periods and how long each drawdown lasts.
The diagram below shows the deepest drawdown or peak to trough in the NAV of the fund. So in the worst scenario, an investor who had invested in the fund will lose at most 11.96%. But of course, this is based on historical data and does not mean that the fund will never exceed this drawdown in the future.
The second diagram is a drawdown plot. It clearly shows you where the different drawdowns are and how long the fund took to get out of these drawdowns (i.e. when the drawdown goes back to zero).
The 3 Risk Adjusted Returns — Sharpe, Sortino, Calmar
We are now ready to compute the 3 ratios.
Sharpe ratio was first conceived by Nobel Laureate William F. Sharpe in 1966 and further revised for use by the fund industry. While there are other measures around, Sharpe Ratio is now the de facto standard when the industry talks about risk-adjusted returns. It uses volatility as the common denominator.
Many funds may set the risk free rate to zero in their performance reports. This simplifies matters and apply a common yardstick across all funds. Intuitively, if Fund A has a higher Sharpe Ratio than Fund B, then Fund A is better at extracting returns according to this measure. This is somewhat similar to measuring a worker’s productivity. If Worker A produces more output in an hour than Worker B, then Worker A is more productive than Worker B.
Harry Markowitz is the first to suggest the idea of using only downsides to measure risk, but it was Dr Frank Sortino who came up with the formula for the Sortino ratio. It is essentially a modification of the Sharpe ratio. The difference are the use of downside volatility as the denominator and a Minimum Acceptable Return as the target return.
Calmar ratio was developed by Terry W. Young. Yes, that is right. It is not by someone called Calmar. It is based on the name of his company newsletter instead. This ratio is computed using 36 months of data i.e. the annualized return and the maximum drawdown over the past 36 months.
Because Calmar is based on maximum drawdown, this ratio tends to be more stable than Sharpe or Sortino. Most of the time, it is the numerator that changes.
You can reference the calculations of the ratios in the sheet — Risk Adjusted Returns in the excel file. On the sheet, I calculated the various parameters and ratios for 2 funds A and B. Things are pretty straightforward once you sort out the various risk measures.
When it comes to applying the ratios to assist in performance evaluation, there is no hard and fast rule as long as you enforce a fair and consistent principle. For example, while Calmar is defined to be calculated over 36 months, there is no rule stopping you to extend the period as far back as inception. Although of course, you might end up with a maximum drawdown figure that has not changed for the past 2 decades. This might draw question on how relevant the figure is in today’s context. And even though there is no risk free rate in the equation, adding it in does not do you any harm either. The situation is similar for Sharpe and Sortino, you are free to create a rolling 36-month Sharpe and Sortino ratios for the funds as well. Feel free to experiment as long as the reasoning is sound.
However, you might want to note that assessing a fund’s performance goes much more beyond just looking at a few ratios. There are many other quantitative metrics that the academics and industry have come up with. Statistical measures, while useful, are also only one part of the equation when analyzing a fund or any other investments. The qualitative aspects such as the fund’s philosophy, strategy, mandate, portfolio managers, risk management methodologies etc. are equally important and should be given serious focus when doing any due diligence.
Originally published at investmentcache.com on September 23, 2018.