Sharpe ratio

[raymond008375]

Quote from MTE:

Exactly, a normal distribution can be described by its mean and standard deviation. So if the distribution is not normal then those two are not enough.

By the way, Markowitz portfolio theory assumes normal distribtion, because it states that a portfolio can be described by its mean and standard deviation.

IC......but I still have some problems.....

Other than Normal, there will be some other dists can have the same properties too, won't it? e.g. t-dist......it also has zero third and fourth moments too.....why must we use Normal instead of other candidates? Do you know if there're any researches talking about this?

And......do prop traders really can beat the "Normality fate"? Do they really can maintain a return with negative skewness? I'm now working in Swift Trade......yet I don't really understand about it

 

[MTE]

Quote from raymond008375:

IC......but I still have some problems.....

Other than Normal, there will be some other dists can have the same properties too, won't it? e.g. t-dist......it also has zero third and fourth moments too.....why must we use Normal instead of other candidates? Do you know if there're any researches talking about this?

And......do prop traders really can beat the "Normality fate"? Do they really can maintain a return with negative skewness? I'm now working in Swift Trade......yet I don't really understand about it

T-dist. is used as well. The point is that Sharpe ratio assumes normal or t-dist, and most strategies are not, so blindly using Sharpe ratio can be misleading.

Why would you want to have a negative skew!? Negative skew means signficant losses. You want a positive skew distribution!

Can't comment on the prop traders.

 

[raymond008375]

Quote from MTE:

T-dist. is used as well. The point is that Sharpe ratio assumes normal or t-dist, and most strategies are not, so blindly using Sharpe ratio can be misleading.

Why would you want to have a negative skew!? Negative skew means signficant losses. You want a positive skew distribution!

Can't comment on the prop traders.

Negative skew means the opposite.......
The tail is on the left.....meaning that the probability of loss is low. You may check wikipedia or other sources for confirmation.

And.....do you mean that comments on prop traders is prohibited in this forum?

 

[MTE]

Quote from raymond008375:

Negative skew means the opposite.......
The tail is on the left.....meaning that the probability of loss is low. You may check wikipedia or other sources for confirmation.

And.....do you mean that comments on prop traders is prohibited in this forum?

I know that the tail is to the left. This means that there's some probability of an extreme loss, i.e. getting wiped out by a "black swan" event. By the way, by saying that you are actually making an assumption that the majority of returns are positive, which may or may not be true. You can have a distribution where the bulk is around zero and then a very negative tail. Or to put it more technically, you are assuming that the mode of the distribution is in the positive.

It's not prohibited, I just have nothing to say on the topic.

 

[raymond008375]

Quote from MTE:

I know that the tail is to the left. This means that there's some probability of an extreme loss, i.e. getting wiped out by a "black swan" event.

It's not prohibited, I just have nothing to say on the topic.

But positive skewness would mean that the "camel hump" is on the left......This means loss is more frequent, isn't?

Referring to my graph just before, both of them are negatively skewed since they both have "camel hump" on the right. The only difference is that you would calculate a smaller third moment (i.e. E(r^3)) about the right one than the left one. This is because the right graph has a tiny "camel hump"on the left where the left graph doesn't have. It offsets the skew negativity a certain little bit.

That's why I like to exactly measure the third moment instead of just saying "positive" or "negative" skewed. When any black swan event happens, all negatively skewes strategies would certainly have some probability of extreme loss. But what I mean is that the less "negative" one (just like the right graph) would be "relatively" more probable to incur extreme loss.

 

[MTE]

Quote from raymond008375:

But positive skewness would mean that the "camel hump" is on the left......This means loss is more frequent, isn't?

Referring to my graph just before, both of them are negatively skewed since they both have "camel hump" on the right. The only difference is that you would calculate a smaller third moment (i.e. E(r^3)) about the right one than the left one. This is because the right graph has a tiny "camel hump"on the left where the left graph doesn't have. It offsets the skew negativity a certain little bit.

That's why I like to exactly measure the third moment instead of just saying "positive" or "negative" skewed. When any black swan event happens, all negatively skewes strategies would certainly have some probability of extreme loss. But what I mean is that the less "negative" one (just like the right graph) would be "relatively" more probable to incur extreme loss.

It obviously depends on the location of the distribution.

 

[vita]

Quote from raymond008375:

Is it really necessary to let the histrgram resemble Normality? What if it isn't Normal? I didn't see any assumption about Normality in Port. Theory by Harry Markowitz......

Raymond, you notice that the Sharpe Ratio calculation is based on the assumption that the mean of the return histogram grows with "time" while the risk i.e. stddev grows with "sqrt(time)". This assumption only holds for histogram that is unimodal and "looks" Normal.

If your return distribution does not look Normal, you need to measure your performance for longer period. For example, if your daily histogram is multi-modal or highly skewed, you need to look at multi-day or weekly returns instead. If you plot your weekly returns, the resulting histogram should looks more Normal, and gives you a more accurate Sharpe Ratio.

I hope this clarifies why we can't simply report Sharpe Ratios for highly skewed, multimodal histograms.

 

[raymond008375]

Quote from vita:

Raymond, you notice that the Sharpe Ratio calculation is based on the assumption that the mean of the return histogram grows with "time" while the risk i.e. stddev grows with "sqrt(time)". This assumption only holds for histogram that is unimodal and "looks" Normal.

If your return distribution does not look Normal, you need to measure your performance for longer period. For example, if your daily histogram is multi-modal or highly skewed, you need to look at multi-day or weekly returns instead. If you plot your weekly returns, the resulting histogram should looks more Normal, and gives you a more accurate Sharpe Ratio.

I hope this clarifies why we can't simply report Sharpe Ratios for highly skewed, multimodal histograms.

Well....can I interpret what you said in this way?

When we talk about Port. Theory, we're going to estimate E(r) and ƒÐ.
Since our estimation method "usually" use sample returns with a certain period (i.e. across a time period), actually we implicitly assume that the probability distribution (and so as E(r) and ƒÐ) don't change over time.......or in terms of time series analysis, the return process should be assumed as stationary.

Otherwise, if it's non-stationary (i.e. the distribution changes over time), then our sampling method as mentioned above would just be unreliable, is it?

Using an unrealistic example, if we had a time-machine, we could record the return in one moment, then went back to that exact moment, and record the return (perhaps it would be a different number) again, then went back again......and so on until we had recorded plenty of return samples such as a size of 30. Then we just calculate the average of this 30 numbers to be an estimate of E(r) at that exact moment........And we can just continue this procedure for other time moments.........

Then even the distribution and E(r) change over time, we won't worry about it.......we're doing repeated sampling on just one moment at all

 

[vita]

Quote from raymond008375:

Well....can I interpret what you said in this way?

When we talk about Port. Theory, we're going to estimate E(r) and ƒÐ.
Since our estimation method "usually" use sample returns with a certain period (i.e. across a time period), actually we implicitly assume that the probability distribution (and so as E(r) and ƒÐ) don't change over time.......or in terms of time series analysis, the return process should be assumed as stationary.

Otherwise, if it's non-stationary (i.e. the distribution changes over time), then our sampling method as mentioned above would just be unreliable, is it?

Using an unrealistic example, if we had a time-machine, we could record the return in one moment, then went back to that exact moment, and record the return (perhaps it would be a different number) again, then went back again......and so on until we had recorded plenty of return samples such as a size of 30. Then we just calculate the average of this 30 numbers to be an estimate of E(r) at that exact moment........And we can just continue this procedure for other time moments.........

Then even the distribution and E(r) change over time, we won't worry about it.......we're doing repeated sampling on just one moment at all

You nailed it. The problem is that unless our return distribution is stationary (i.e., having the same mean and stddev) over time, all of the theories in quantitative finance including Sharpe Ratio, Portfolio Theory, and Options pricing (e.g. using Black-Scholes) fall apart. To overcome this, we need to first inspect our the returns, and change the time period to calculate them until we get a relatively stationary distribution, or at least a distribution which has a single mode, and least skewness.