new Risk Measure suggestion: MAPE from approximations

06May11

We introduce a new performance measure that scores how consistent, how smooth the profit of the strategy is.
In some sense it measures the risk.
We measure it on the $PV (Portfolio Value) chart. On the first day, the Portfolio Value is $1.

Methods for measuring risk:

1.
Usually, a highly variable PV has a high SD (standard deviation). That is one measure of the inconsistent profit.
The drawbacks of the SD as a measure of risk:
it measures volatility around the Mean. Blah. That mean is static. In a 23 years backtest, measuring the variance around a static mean is silly. The mean should move. (as with a SMA, or EMA)
it is an absolute value and it is not a percentage of the PV. So, if a strategy takes 20 years, it is naturally achieve higher PVs at the end than a strategy backtested for only 1 year.
This high PV at the end result that the SD will be high in all case.
– Symmetric around the mean. SD considers the PV above the average equally bad as under the average. (Usually investors are happy if there are above the mean, and very unhappy under the mean curve.)

2.
Another widely used risk measure is the maximum drawdown.
It measures risk only under the mean; in theory it measures only the bad risk(, not the good risk (the good risk is risk to the upside)). That is good.
It measures the deviation as a percentage (not the value of the underlying). That is good too.
However some drawbacks:
– it says nothing about how long the maximum drawdown lasted. We call the Maximum Drawdown Suffering days the number of days until we are in the MaxDD valley. It does matter greatly that the suffering days are 10 days, or 10 years.
– also, what about the second, third, etc. MaxDD? What if the MaxDD was -50% and the MaxDD suffering Days was 10 days, but the second largest DD was -45% and its suffering days was 10 years? MaxDD doesn’t tell us about this behaviour along the PV curve.

3.
We need something new.
First we realise, we need to work on percentages.
We suggest using 2 measures here:
-The average percentage deviation from the linear approximation and
-The average percentage deviation from the multiplicative approximation.

The MAPE stands for Mean Absolute Percentage Error: see wiki here.

The Linear PV approximation should be a simple linear function connecting the first point to the last point.
(note: it is not the linear regression line. It could be, but this linear PV is much simpler to compute)

However, there is one caveat here:
LinearApprox is not a perfect choice in long term backtests. In a 23 years backtest, the daily ‘deltaInc’ (see source code) is $0.05. That is 5% daily profit on the first day, but the same $0.05 is only 0.0001% profit on the last day. The problem is that it is the additive profit factor.

So, let’s introduce a multiplicative approximation. That calculates the geometric mean of the daily profits, and uses that to estimate a smooth PV
that could have been generated by our strategy, if our strategy was perfectly consistent and generating the same profit% every day for 23 years.

Here is the code of the two error metrics:

and here is how the linear and multiplicative approximation look like (in a linear scaled chart!!!)

Note, it would look very differently in a log scaled chart. In a log chart, the multiplicative (green) would look linear.
It is surprising to see that the best, most consistent approximation of the PV line is not the linear, but the multiplicative one.

In the future, we would like to use the MAPE_from_MulAppr. metric to express the riskiness (the volatility) of the strategy‘s return.

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