Equity Curve R squared

I do not think I can agree with your statements regarding risk. First of all, you are ignoring the risk of ruin. I think we can both agree that if you increase leverage by a lot that your risk of bankruptcy will exponentially increase. Hence, risk is not a linear function of leverage. Hence, also R^2 will not remain unchanged when you increase leverage, which is a statement you made.

But to get back to R^2 when applying to the equity curve, I am still and always have been confused what it exactly measures. Also, the value of R^2 itself is meaningless. It does not tell you anything other than that it is high or low. However a Sharpe ratio tell me exactly the factor of adjusted returns as a function of risk, taken. How is R^2 on an equity curve even computed? I just do not see the relationship why anyone would want to regress portfolio returns over time or vice versa.

 

There are many measures of risk. The risk measure used to calculate Sharpe Ratio is just standard deviation of returns. This doesn't "know" about risk of ruin; unless you use geometric means, which most people don't. It also scales with leverage.

R squared gives you an indication of how consistent the performance is. It's calculated by regressing the log equity curve (cumulated series r_t where r_t is the percentage return) on the time index (1....T)

Consider the following series of returns:

-3.00%
4.00%
6.00%
-2.00%

This has a Sharpe of 0.28 and an R squared of 0.625. Notice the standard deviation is 4.4%. If I double that set of returns, i.e. double the leverage, I get:

-6.00%
8.00%
12.00%
-4.00%

The standard deviation and the mean both double. Hence the Sharpe Ratio is invarient to changes in leverage. The R squared is also exactly the same.

However the geometric sharpe ratio is 0.26 in the first example and 0.23 in the second example. If I increase leverage enough (17 times to be exact) then I will get a negative geometric sharpe ratio because there will be a risk of ruin.

Now consider the following series of returns:

-1.00%
-1.00%
-1.00%
10.00%

This has a higher Sharpe Ratio than the first example: 0.32 (the geometric Sharpe is also higher). But the R squared is just 0.23 - because the returns are much less consistent.

Which series would you rather buy? Perhaps not a straightforward question as it depends on your risk preference (mainly for insurance premia, skew and option gamma) but the inconsistency of returns in the second series should give you pause for thought. Perhaps period 4 was just a fluke. And that is what the R squared is highlighting.

Don't get me wrong. I love Sharpe Ratio. It's my number one favourite individual measure. But to get a view on performance you need to look at several measures. There are many things wrong with R squared, but it adjusts for risk in exactly the same way as Sharpe Ratio. So that alone isn't a reason for R squared being inferior in some way to Sharpe Ratio.

GAT

 

I would clearly prefer the second series, because I care about downside variation not upside variation. Hence some use an alternative sharpe ratio measure that only takes into account downside variation.

But getting back to the topic of leverage. I am afraid you approach this issue from a mathematical perspective, only. That is very dangerous, imho.

Imagine you have an account that is funded with 100k USD. If I trade a 100,000 usd unleveraged position in a currency pair then in most all cases I do not take the risk of bankruptcy even in a string of losses. However if I leveraged the investment and was allowed to trade a 10 million dollar position then an approximate 1% move in the currency pair would completely bankrupt me. So, as you correctly stated with higher leverage the sharpe ratio and R squared are unchanged, however the huge problem here is that I am basically fooled by the just mentioned risk metrics.

Not only has my risk of ruin almost climbed to 100% but there are many other subtle issues at play that magnify risk. Think of the risk of changed margin requirements, think of the fact that your sharpe ratio or r squared only measure realized return volatility. If intraday a move is exhibited of >1% against me with my 100 fold leverage I am basically bankrupt even if the currency pair reverses and closes at a smaller change than 1% from trade entry.

Please see the following for a more in-depth discussion:
http://quant.stackexchange.com/questions/18413/sharpe-ratio-and-leverage


My whole point is 2 fold:

a) Sharpe ratio and r squared are not very good measures of risk in isolation, in fact they can dangerously mislead. I am not saying that you postulated using them in isolation but I feel it is worth making the point anyway.

b) R squared is a very weak metric in that it means nothing when you get a result of 0.63 or 0.3 other than knowing that 0.63 represents a higher return variation than 0.3. But when you use sharpe ratio you know exactly what 0.5 or 1.3 stands for. It is the multiple of the excess return relative to the risk taken. That is very meaningful and makes intuitive sense. R squared values do not make intuitive sense.

c) And a point to OP's firs post: If he combines several strategies and his r squared improves "only" from 0.95 to 0.98xx then that basically means that the additional strategies are very highly correlated with the original strategy in terms of return profile. They hardly add any value to the the first strategy. But that takeaway one would someone most likely also get using sharpe ratio.

In summary, I concur only partly with your statement that "R^2 will account for risk - it is invariant to leverage". It only accounts for a portion of risk. It will not account for the real risk in the trading strategy/ies.

 

A quick simulation with normally distributed returns gives the following results:

SR R2
1.4 0.97
1.1 0.82
1.0 0.70, 0.87,
0.80 0.92
0.40 0.54

"I would clearly prefer the second series, because I care about downside variation not upside variation. Hence some use an alternative sharpe ratio measure that only takes into account downside variation."

I also prefer that .... to a point. But let's take it to the extreme. You have a series with 30 years of small negative returns and one massive positive return. Your Sharpe and Sortino ratio would be amazing. Would you really trade that? Purely from a statistical perspective it's more likely the positive return is a fluke than a meaningful event.

"Sharpe ratio and r squared are not very good measures of risk in isolation"

They aren't measures of risk at all....

"But getting back to the topic of leverage. I am afraid you approach this issue from a mathematical perspective, only. That is very dangerous, imho."

I don't think you've read my post carefully. My mathematical perspective means I understand the difference between arithmetic means and sharpe ratios; and geometric means and sharpe ratios. The latter will get worse with higher leverage. Therefore when using the former it's important to understand this shortcoming. It's clear you understand the difference as well from your example, but knowing the maths as well doesn't make my understanding any worse than yours - I'd argue it makes it better.

"R squared is a very weak metric in that it means nothing when you get a result of 0.63 or 0.3 other than knowing that 0.63 represents a higher return variation than 0.3. "

Actually it's the other way round. The higher the R^2 the more consistent the return series.

Yes I also don't have a good intuition about R^2; if you actually read my posts you'll see I said repeatedly that I prefer SR if I have to pick one measure, precisely for that reason. But I still think there is some value in using other risk measures. Personally I use the p-value of the bootstrapped sharpe ratio of the return series. I find that intutive - you may not - but it has the disadvantage of being much harder to replicate than R^2.

GAT

 

hence, the need for many more statistical results such as drawdowns, recovery velocity and time period of recoveries from drawdowns, avg gain per trade, avg loss per trade ,....

P.S. And sorry, it was a typo, obviously a r square of 0.63 has a lower return variation than a measure of 0.3. And yes, a better mathematical grasp of methodologies is better than a worse one, but still utterly useless when ignoring the real and much more prevalent risks of trading such as foolish position sizing. Not implying anything about your approach of course. I have an advanced degree in quant finance myself but having professionally traded for 14 years I learned that metrics such as the ones we have been discussing mean hardly anything in the context of the bigger picture when evaluating performance, especially when analyzed in isolation.