This post will discuss the idea of the geometric Calmar ratio — a way to modify the Calmar ratio to account for compounding returns.

So, one thing that recently had me sort of annoyed in terms of my interpretation of the Calmar ratio is this: essentially, the way I interpret it is that it’s a back of the envelope measure of how many years it takes you to recover from the worst loss. That is, if a strategy makes 10% a year (on average), and has a loss of 10%, well, intuition serves that from that point on, on average, it’ll take about a year to make up that loss–that is, a Calmar ratio of 1. Put another way, it means that on average, a strategy will make money at the end of 252 trading days.

But, that isn’t really the case in all circumstances. If an investment manager is looking to create a small, meager return for their clients, and is looking to make somewhere between 5-10%, then sure, the Calmar ratio approximation and interpretation makes sense in that context. Or, it makes sense in the context of “every year, we withdraw all profits and deposit to make up for any losses”. But in the context of a hedge fund trying to create large, market-beating returns for its investors, those hedge funds can have fairly substantial drawdowns.

Citadel–one of the gold standards of the hedge fund industry, had a drawdown of more than 50% during the financial crisis, and of course, there was https://www.reuters.com/article/us-usa-fund-volatility/exclusive-ljm-partners-shutting-its-doors-after-vol-mageddon-losses-in-u-s-stocks-idUSKCN1GC29Hat least one fund that blew up in the storm-in-a-teacup volatility spike on Feb. 5 (in other words, if those guys were professionals, what does that make me? Or if I’m an amateur, what does that make them?).

In any case, in order to recover from such losses, it’s clear that a strategy would need to make back a lot more than what it lost. Lose 25%? 33% is the high water mark. Lose 33%? 50% to get back to even. Lose 50%? 100%. Beyond that? You get the idea.

In order to capture this dynamic, we should write a new Calmar ratio to express this idea.

So here’s a function to compute the geometric calmar ratio:

require(PerformanceAnalytics) geomCalmar <- function(r) { rAnn <- Return.annualized(r) maxDD <- maxDrawdown(r) toHighwater <- 1/(1-maxDD) - 1 out <- rAnn/toHighwater return(out) }

So, let's compare how some symbols stack up. We'll take a high-volatility name (AMZN), the good old S&P 500 (SPY), and a very low volatility instrument (SHY).

` `

```
```getSymbols(c('AMZN', 'SPY', 'SHY'), from = '1990-01-01')
rets <- na.omit(cbind(Return.calculate(Ad(AMZN)), Return.calculate(Ad(SPY)), Return.calculate(Ad(SHY))))
compare <- rbind(table.AnnualizedReturns(rets), maxDrawdown(rets), CalmarRatio(rets), geomCalmar(rets))
rownames(compare)[6] <- "Geometric Calmar"
compare

The returns start from July 31, 2002. Here are the statistics.

AMZN.Adjusted SPY.Adjusted SHY.Adjusted Annualized Return 0.3450000 0.09110000 0.01940000 Annualized Std Dev 0.4046000 0.18630000 0.01420000 Annualized Sharpe (Rf=0%) 0.8528000 0.48860000 1.36040000 Worst Drawdown 0.6525491 0.55189461 0.02231459 Calmar Ratio 0.5287649 0.16498652 0.86861760 Geometric Calmar 0.1837198 0.07393135 0.84923475

For my own proprietary volatility trading strategy, a strategy which has a Calmar above 2 (interpretation: finger in the air means that you make a new equity high every six months in the worst case scenario), here are the statistics:

> CalmarRatio(stratRetsAggressive[[2]]['2011::']) Close Calmar Ratio 3.448497 > geomCalmar(stratRetsAggressive[[2]]['2011::']) Close Annualized Return 2.588094

Essentially, because of the nature of losses compounding, the geometric Calmar ratio will always be lower than the standard Calmar ratio, which is to be expected when dealing with the geometric nature of compounding returns.

Essentially, I hope that this gives individuals some thought about re-evaluating the Calmar Ratio.

Thanks for reading.

NOTES: registration for R/Finance 2018 is open. As usual, I’ll be giving a lightning talk, this time on volatility trading.

I am currently contracting and seek network opportunities, along with information about prospective full time roles starting in July. Those interested in my skill set can feel free to reach out to me on LinkedIn here.

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Ilya,

I do not understand what makes geomCalmar geometric.

What makes geomCalmar geometric?

Is not CalmarRatio already geometric?

library(PerformanceAnalytics)

> CalmarRatio

function (R, scale = NA)

uses

> Return.annualized

function (R, scale = NA, geometric = TRUE)

and

> maxDrawdown

function (R, weights = NULL, geometric = TRUE, invert = TRUE,

…)

The near-last lines of CalmarRatio are the following:

drawdown = abs(maxDrawdown(R))

result = annualized_return/drawdown

rownames(result) = “Calmar Ratio”

geomCalmar near last lines are the following:

toHighwater <- 1/(1-maxDD) – 1

out <- rAnn/toHighwater

That is

1/(1-maxDD) -1 = 1/(1-maxDD) – (1-maxDD)/(1-maxDD) = (1 – 1 + maxDD)/(1-maxDD)

= maxDD/(1-maxDD)

The returns may be geometric (compounded on themselves), and the drawdown comes from that, but the ratio itself is *not*. That is, given a 50% CAGR and a 50% max drawdown, your Calmar Ratio will be equal to 1, which generally means you make your money back in a year, correct? Well, if you had a 50% drawdown and made 50% after that, you’d still lose 25%, which I don’t think the regular Calmar ratio captures.

O.K. Thanks.

Why not just use the ratio ytr= -log(1-maxDD)/log(1+annRet) if you really want it to mean the number of years required to recover from the loss with the effect of compounding taken into account? This will lead to (1+annRet)^ytr (1-maxDD)=1, which is what you are after (ytr=’Years to recover’).

For 1998-2018, ytr for SPY is 10.5, and for AMZN it is 9.5 based on the monthly data.

Obviously empirical data suggests that ytr does not come anywhere close to be what it purports to be at least for AMZN (actual number of years to recover from the loss is much less that ytr.

So the good old Calamar Ratio may be enough to give you some idea of the magnitude of the drawdown relative to the returns without any pretension that it gives you some indication of how much time it to takes to recover from the DD.

That is a good calculation as well, though AMZN does seem to be exceptional.

I like it. I’ve sometimes calculated the ‘years to recover’ to consider this effect. I have noticed that using the average compound growth and the max drawdown to calculate recovery time tends to be very conservative. It ignores mean reversion effects – expected returns tend to increase after a major drawdown. Reality is probably somewhere between the geometric Calmar and the standard Calmar.

I do think you have a point. Of course, calling the end of the drawdown is also a very risky proposition!