Risk Adjusted Returns

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Sir G
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Risk Adjusted Returns

Post by Sir G » Thu May 01, 2003 10:50 pm

Here are two links that begin to tell the story of Risk adjusted returns.

c.f.'s g.c. Ratio Paper
docs/g.c.Ratio.pdf

MAR Discussion
viewtopic.php?t=36

Let's run with this very important topic.

Gordon

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Post by Forum Mgmnt » Mon May 05, 2003 4:22 am

I'd like to thank Mark Johnson for his prior posts that got me thinking about this stuff. I think I ended up with a pretty important improvement to my own measures as a result.
In a private message Mark Johnson wrote:The different ways we picked to compute the S.R., sent me on a web hunt. I found Professor Sharpe's website

http://www.stanford.edu/~wfsharpe/art/sr/sr.htm

These paragraphs were helpful to me. They appear right below his equation (10).

In practice, the situation is likely to be more complex. Multiperiod returns are usually computed taking compounding into account, which makes the relationship more complicated. Moreover, underlying differential returns may be serially correlated. Even if the underlying process does not involve serial correlation, a specific ex post sample may.

It is common practice to "annualize" data that apply to periods other than one year, using equations (7) and ( 8 ). Doing so before computing a Sharpe Ratio can provide at least reasonably meaningful comparisons among strategies, even if predictions are initially stated in terms of different measurement periods.

To maximize information content, it is usually desirable to measure risks and returns using fairly short (e.g. monthly) periods. For purposes of standardization it is then desirable to annualize the results.
I have a long-time and well-earned reputation for disagreeing with experts. Let this occasion not pass me up.

I believe that the Sharpe Ratio for monthly data is damn close to useless for measuring what I care about. I really don't care whether or not the monthy returns are smooth.

I'd much rather have returns of 10%, 25%, 10%, 25%, 10%, 25%, than 10%, 12%, 10%, 12%, 10%, 12%, guess which one has the better Sharpe Ratio? Annualizing the monthly numbers, as Sharpe suggests, doesn't help this situation since the basis is still the standard deviation of the monthly returns.

Now, all's not entirely bad with the Sharpe Ratio. The issue is that one should really be using timeframes for the Sharp Ratio that represent the time horizon the investor cares about.

After thinking about it for a bit, I realized that a one year Sharpe Ratio works pretty well for me. I would much prefer a system that returns 50%, 50%, 50%, 50%,... over one that returns 200%, -25%, 200%, -25%, ... even though they compound to the same return.

I'm even okay with using monthly datapoints, one can simply use the trailing 12 month geometric average return each month. This takes out the rather arbitrary western calendar year effect. A 12 month trailing Sharpe Ratio is much better for me.

One could even combine everything into a formula like:

Return X 2.5
----------------------------------------------------------------------------------
(Avg worst 5 DD + 2 Std. Dev. ) X Std. Dev. of trailing 12 month Return

which comes pretty close to incorporating all I care about in one number.

If anyone is curious about the 2.5 in the numerator that comes from:
  • 1 for the MAR Ratio like Return / Drawdown
  • 1 for the Sharpe Ratio like Differential Return / Drawdown
  • 0.5 because contrary to what the various ratios would have you believe higher return is better, so it gets a little extra weight.
So we really have:

1/2 return X MAR Ratio X Sharpe-like Ratio using trailing 12 months

I'm calling it Sharpe-like since I don't use differential returns. I'm not sure that the risk-free rate reduction used by Sharpe is a big factor for the kinds of returns I demand from my trading systems, so I prefer the simplification.

MAR-like accounts for using the average of the five worst drawdowns plus two standard deviations to get a better estimate of the potential worst case than what a single drawdown would suggest. This is a modification to Mark's improvement to the MAR ratio.

He uses the average of the five worst drawdowns. I thought that adding a couple of standard deviations would improve it as a measure of worst-case scenarios.

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Post by Forum Mgmnt » Mon May 05, 2003 5:52 pm

Uh, well, Mark just pointed out that I didn't have a dimensionless measure. Oh, well that's what happens when you write a post at 4 AM. This made me realize that my formula wasn't what I was thinking when I wrote it.

Basic math: R X R is not 2R, duh.

I should have said:

Return X Return X 0.5 Return
----------------------------------------------------------------------------------
(Avg worst 5 DD + 2 Std. Dev. ) X Std. Dev. of trailing 12 month Return

Which doesn't solve the problem mark outlined but does represent what I intended.

Perhaps

Return X Return X 0.5 Return
----------------------------------------------------------------------------------
(Avg worst 5 DD + 2 Std. Dev. ) X Std. Dev. of trailing 12 month Return X Risk Free Rate

would do what I want?

If anyone can figure out a better way to weight the return more without the units/dimension problem let me know.

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Post by damian » Mon Jul 21, 2003 6:48 am

Forum Mgmnt wrote: Return X Return X 0.5 Return
----------------------------------------------------------------------------------
(Avg worst 5 DD + 2 Std. Dev. ) X Std. Dev. of trailing 12 month Return
Hi Forum Mgmnt,

Just above the Sharpe Ratio, I want a metric on my performance table that says "Faith Ratio", thus I have been applying this and other suggestions of yours. I have devised a few ways to approach the above equation, could I please check if I am going about it the right way:

Code: Select all

CAGR	                           89 
Average worst 5 DD	            17 
Std Dev all DD's	               4 
Std Dev rolling 12 mth returns	 44 

(89*89*44.5)/((17+4+4)*44) = 320

I know it seems simple, but is my calculation what you intended?

many thanks
damian

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Post by Mark Johnson » Mon Jul 21, 2003 8:59 am

It isn't a dimensionless quantity: (percent cubed) in the numerator and (percent squared) in the denominator means that the ratio has the dimensions (percent to the 1st power). This may or may not be acceptable to you. I prefer figures of merit that are dimensionless, like the Sharpe Ratio or the MAR Ratio or the Lake Ratio, but that's just me.

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Post by damian » Mon Jul 21, 2003 9:28 am

Dimesnsions in terms of first powers is beyond my understanding :oops: . None the less, in terms of sharp ratios, I have been calculating the following, i think is very much like an equation I have seen elsewhere here:

Sharpe = average of the rolling 12 month returns/St Dev of the rolling 12 month returns.

Using the above system's equity cuve again, i get 92.12%/44.98% = 2.09

Liek wise a MAR ratio that I learnt from this site:

MAR = Average of the rolling 12 mthly returns / average max DD over rolling 12 mths

92.12%/12.93% = 7.02

This needs to be related to the standard MAR calculation to put in more reasonable terms:

CAGR/max DD = 89%/19% = 4.68

I do not think i am revealing anything new here, however I am seeing some nice results. It should be noted that these measures us month end equity curve data. Daily equity curve data points would yield a greater max DD, I think it was 29% from memory.

BTW: The value "average max DD over rolling 12 mths" is found by looking at each 12 month window over the equity curve and for each window, finding the maximim DD that existed.

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Post by Kiwi » Mon Jul 21, 2003 6:21 pm

c.f. and then Damian wrote:
Return X Return X 0.5 Return
----------------------------------------------------------------------------------
(Avg worst 5 DD + 2 Std. Dev. ) X Std. Dev. of trailing 12 month Return
and Mark said that he preferred dimensionless quantities for figures of merit. Now to be picky (always wanted to do that to Mark so I expect a firm response if I get it wrong) this is a dimensionless quantity. For a given time period, its dimensions are:

Return = $/$ = no dimension
DD = $/$ = no dimension
StdDev of Return = no dimension

so its NoD cubed over NoD squared - still NoD. I have this ugly feeling that I shouldnt have excluded time --- and will be hoist by it!

But enough fun. I think Mark has a point and that the discussion of dimensions was hiding it a bit. The point is that when you compare the above figure of merit with a figure of merit like MAR or Sharpe which are respectively similar to the components of the above equation:

Code: Select all

            Return                        and                 Return 
-------------------------------------             --------------------------------
(Avg worst 5 DD + 2 Std. Dev. )                   Std. Dev. of trailing 12 month

You find that this figure of merit is multiplied by the return one extra time. The problem with this is that it biases the ratio to favour systems with a higher return even if they have a higher deviation (as measured by the denominators of the above equations). This is because you have (Return) cubed divided by (Deviation of Return) squared.

OK Mark, I'm ready to take a civilized beating :-)

John

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Post by Mark Johnson » Tue Jul 22, 2003 7:23 pm

There's a reason why so many ratios have been invented: the human beings who use the ratios give different answers to the question "What is the definition of good?" Essentially these ratios try to quantify the goodness of a trading system. But different traders have different aims and thus define "good" differently.

Each of the following statistics ("ratios") could be viewed as a measurement of goodness
  1. CAGR, the compound annual growth rate in percent per year
  2. CAGR^n, the compound annual growth rate raised to the n-th power
  3. (CAGR/MaxDD), the ratio of CAGR to the worst % drawdown ever seen
  4. Lake Ratio, a geometric construct that measures equity drawups vs drawdowns
  5. (CAGR-RFR)/StDev, the Sharpe Ratio
  6. CAGR/TotalPortfolioHeat
  7. Rsquared, a measurement of how perfectly the equity curve resembles a straight line with no jiggles (on semilog axes)
  8. CAGR/AvgDD
  9. CAGR*PctOfDaysMakingNewEquityHighs / NumberDaysLongestDrawdown
Some traders try to quantify Reward vs. Risk. Others try to quantify Reward alone. Still others try to quantify Reward vs. Discomfort. For each, a different measure of "goodness" is probably appropriate.

Much has been written about the Sharpe Ratio and its definition of Discomfort as "standard deviation of returns". Many people don't like this because it penalizes an equity curve with sharp jumps upward. Equity volatility is only bad if I lose money, they say, and so a new ratio, the Sortino Ratio, was born. It only uses the StdDev of equity drawdowns. Ooooohhhhh.

Creating a new measurement and naming a new ratio is harmless enough fun, so I encourage everyone to do it. Heaven knows I've written code to calculate the average of the 5 worst drawdowns, the 99th percentile of the cumulative distribution of drawdowns, the Rsquared of the linear regression of the equity curve, the RootMeanSquare drawdown (even got them to put that into Athena), and countless others, so I can't really criticize other people for tinkering. :lol: But after you do, I promise that when you're studying the output of an optimization run, sorted on YourNewRatio, you'll find yourself looking at a lot more than YourNewRatio when deciding what's "best".

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