Volatility- and drawdown-adjusted returns are the most commonly used values to judge the performance of a trader or a backtest. However, neither of those truly measures the consistency of the returns. Long periods of low-volatility, sideways movement in an equity curve are obviously undesirable, but do are not shown in the Sharpe or MAR ratios. Instead, we need to look at specialized consistency (or “straightness”) metrics.
There are of course some “standard” straightness metrics. R-squared is the most popular, and it works pretty well. I like to raise it to the 4th power or so in order to magnify small differences and make it a bit more “readable”. Another popular metric is the K-Ratio, of which there are at least 3 different versions floating around. The K-Ratio also takes returns into account, so it’s not purely a straightness measure. I prefer the Zephyr version which is calculated as the slope of the equity curve divided by its standard error.
Let’s see if we can construct some alternatives. To start out, we need a benchmark to measure straightness against. That is the “ideal line”. It is the straight line that connects the first and last points of the equity curve. The further away the equity curve is from the ideal, the less desirable it is.
Equity curve, ideal line, and the difference between them.
There are certain obvious principles we can derive from this simple analysis:
- We want to minimize the area of deviation from the ideal line.
- The further away we are from the ideal, the worse (non-linearly).
- Being below the ideal is worse than being above it.
- Being below the ideal for long periods of time is undesirable.
We can easily quantify these ideas into a useful measure of equity curve straightness by using numbers such as the total area of deviation from the ideal, the volatility of the deviation, the length of time spent below the ideal, etc.
An interesting heuristic to look at is the number of times the equity curve crosses the ideal line. The closer the equity tracks the ideal line, the more times it will cross it. This metric fails in idealized tests, but works well in real-world scenarios. It also tends to fail when there are few trades in the sample. Divide the number by the total number of observations in the sample to standardize it.
A similar metric is the average drawdown length. Perhaps it is even more useful because presumably long deviations below the ideal are more important than deviations above it, and the number of crosses does not differentiate between the two.
Some other numbers I think may be interesting: the ratio between the area of difference above and below the ideal, the volatility of the difference, the volatility of the difference below the ideal, average absolute deviation, and average absolute deviation below the ideal (both standardized to the magnitude of the curve).
I created a metric that arbitrarily and haphazardly combines some of the above concepts, and I’m calling it the QUSMA Equity Curve Straightness, Downward Deviation, and Stability Measure (QECSDDSM, pronounced /keɪks-du-sʌm/). It is intended purely as a straightness measure, and does not take into account returns or the slope of the equity curve. It is calculated as follows (see the excel file at the bottom to make sense of it):
Let’s take a look at some extreme examples:
First of all, note that both the Sharpe ratio and the MAR ratio would select the “wrong” strategy if they were used naively: they both prefer Series 3 & 4 over Series 2. Both the K-Ratio and QECSDDSM correctly prefer the first two. Note that the number of crosses is a useless metric here because the most perfect line has very few of them, simply due to being “too straight”. The ratio of the areas above and below the idealized line is not very useful in these scenarios because they are so extreme.
In general most of the numbers roughly agree with each other in terms of ordering the curves from best to worst, so the actual formulation of QECSDDSM doesn’t really matter all that much.
Let’s look at a slightly more realistic assortment of equity curves:
In this case the intuition behind the number of crosses metric becomes obvious. Interestingly QECSDDSM is the only metric to prefer Series 4 to Series 3, which I think is undesirable. Series 4 highlights a problem with the metrics that measure volatility or focus on the area below the ideal: simply having very few trades “gets around” them and produces an overly-high score. Again the Sharpe and MAR ratios produce an “incorrect” ranking by preferring Series 3 to Series 2. The difference mainly comes from the fact that the curve is not very volatile and does not spend a lot of time below the ideal. Some fine tuning of the parameters should smooth things out pretty easily, though.
Another potentially interesting approach to the issue would be to do some sort of regime change detection on the returns (here’s one simple approach). A straight curve will obviously have fewer changes in the average of the returns.
Finally, here’s an excel file that you can play around with.
gregorsam says:
The best way to do this is to construct the notion of difference between two equity curves ab initio. It should be defined such that:
1. The difference diff(A,B) between the equity curves A and B should be negative of diff(B,A), the difference between B and A.
2. If diff(A,C)>0, and diff(C, B)>0, then one must have diff(A,B)>0.
3. Obviously, diff(A,A) must be zero.
If you can come with such a measure that conforms to your expectations for some typical curves, you are there.
Peter Wang says:
Great post!
A quick question about R^6 computation, In your spreadsheet, you used power(R^2,4) to comput R^6. I am wondering why not using power(R^2,3)? I am not good at statistics, so please forgive me if this is a stupid question. Thanks!
qusma says:
Hi Peter,
You’re absolutely right, of course.
David Ellis says:
I don’t understand why a linear equity curve is ideal. Shouldn’t it ideally be exponential (e.g. of the form: principal*(1 + rate_of_return)^t )? Maybe I don’t really understand what an equity curve is.
qusma says:
Hi David,
Of course you want exponential growth. This whole discussion implicitly assumes that you take a logarithm of the actual, compounded equity curve (alternatively it can be conceptualized as the equity curve if the capital used per trade were fixed). Steady exponential growth will lead to a straight log of the equity curve.
David Ellis says:
Ahh! I follow you now. Thanks!
Jacob M. Morley says:
Very interesting analysis. I’m curious on your thoughts on a few things:
Metrics like R^2 (or R^2 to some power) are inherently standardized on [0,1], which lends itself to ready comparability. R^2 (or another power) is also less computationally intensive. Do you think QECSDDSM is a better linearity metric?
qusma says:
Even if QECSDDSM is a better linearity metric (how do you measure quality? It’s a pretty subjective thing), I think that in practice, R^2 and QECSDDSM are going to result in the same ranking in ~99% of cases. So given the obvious advantages of R^2 that you mentioned, it’s probably the best thing to use.
Jacob M. Morley says:
Okay, thanks for the response–I was just wondering about your motivation for the QECSDDSM metric.