关于优化目标 --- Sharpe ratio vs. Average daily return vs. Total return

流动的建筑

多谢Diffusion老大，这个贴和前一阵那个哲学炒股的帖子，都很有意思，受教了。

我也说说我的理解。市场（或个股）不管你是否交易，总会在那里上下震荡，volatility大致表达了震荡范围大小的信息。如果你交易，long 的话，入点以下是risk，以上是reward，short的话就反过来。因此可以说Sharpe ratio基本表达了交易者对股价震荡范围的利用率。我个人认为这是一个很有用的指标，不仅是给投资人看的，也是对比不同交易系统的一个有效指标。

在一定市场里，可以提高系统的leverage，Sharpe ratio保持不变，从而取得更大回报。从这个意义上说，return不是表达系统性能的最佳指标。

我个人觉得，可以在限制max drawdown的条件下，最优化Sharpe ratio，也就是说，用经典的Lagrange方法来优化交易系统。

Diffusion

哈哈，等待你功成名就那天提拔小钠一把。
氢氧化钠 发表于 2010-7-11 07:15

    借你吉言。也希望老大也早日功成名就，提拔我一把。

Diffusion

多谢Diffusion老大，这个贴和前一阵那个哲学炒股的帖子，都很有意思，受教了。

我也说说我的理解。市场（ ...
流动的建筑 发表于 2010-7-11 10:57

    谢谢老大发言。

所以在评测系统的时候，应该不考虑leverage。

Lagrange方法，是说这个吗？http://en.wikipedia.org/wiki/Lagrange_multipliers 方法很经典，只是要求f (object) and g (constraint)都可导。可是trading strategy参数稍有变化， equity curve就会完全改变，似乎连续都谈不上...

还有一问，以老大的经验，Max drawdown多少比较合适？Shape ratio多大算是可以有信心交易的系统。

流动的建筑

不一定要全局连续，金融问题里很少有全局连续的。不过我建议你从$SPX做起，股票的跳变过程很困难，需要单独处理。

Sharpe ratio大于2就不错了，要求越高能够交易的机会就越少，max drawdown 当然越小越好，这两个指标会出现矛盾的。

Diffusion

不一定要全局连续，金融问题里很少有全局连续的。不过我建议你从$SPX做起，股票的跳变过程很困难，需要单独 ...
流动的建筑 发表于 2010-7-11 20:38

    还好我现在参数很少，用brute force优化就可以了。等有需要再考虑more advanced method。到时候再来请教。

老大说过用max drawdown做constraint，最大化Sharpe ratio。那么一般这个constraint设成多大？

流动的建筑

还好我现在参数很少，用brute force优化就可以了。等有需要再考虑more advanced method。到时候再 ...
Diffusion 发表于 2010-7-11 19:42

呵呵，既然是Lagrange方法，max drawdown的限制来自于使用者，也就是交易员的心理承受损失，你说了算。

Diffusion

呵呵，既然是Lagrange方法，max drawdown的限制来自于使用者，也就是交易员的心理承受损失，你说了算。
流动的建筑 发表于 2010-7-12 01:04

    我看我可以考虑优化这个 total return / max drawdown.

再次谢谢建筑兄。

Maxie

You might want to look at Sharpe over Sortino.
Sharpe too high = not enough risk taking. No risk = no reward.
Sharpe does not capture tail events it has nothing to do with tail events.
Return / Max DD is a very good measure.

In essence, you are trying to optimize risk adjusted returns where volatility captures risk in general terms and max dd (kinda) captures the fat tails. So, return/sharpe, return/sortino, and return/max dd .....

Diffusion

You might want to look at Sharpe over Sortino.
Sharpe too high = not enough risk taking. No risk = ...
Maxie 发表于 2010-7-12 15:31

    老大也是这行的高手。

那个total return / max drawdown我随口瞎诹的，原来真的有这种用法。 不过我对max drawdown还有有点保留的，因为这个比Sharpe ratio更容易manipulate，有时候微调一下参数就能避免一个类似9/11的crash...

Sortino ratio还是头一次听说，等我研究一下，有问题再问。

粗看了一下，Sortino ratio应该就是传说中的biased Sharpe ratio，就是只penalize downside risk。等我在仔细研究一下。

Diffusion

Denote i as the input parameter. Our objective is to find the optimal i that has maximal positive impact on total return.

Denote avg as the arithmetic average of daily returns, std as the standard deviation of daily returns, and geoavg as the geometric average of daily returns. Suppose avg is fixed, and daily returns follows a normal distribution. Numerical simulation found that the correlation between geoavg and std is -0.83, implying a highly probable linear relation between geoavg and std. Further denote days as the number of trading days, and total as the total return. Because

        total = (1 + geoavg) ^ days - 1,

the -0.83 correlation means that a lower std will lead to a higher total (thus a higher geoavg), if avg is fixed. This observation suggests that the gain of geoavg can be modeled by a sum of two components,

        gain_std, the gain induced by a lower std,
        gain_intrinsic, the excessive gain beside that brought by a lower std. This intrinsic gain is the effect of the variation of the input parameters we want to capture.

The highly probable linear relation can be expressed by

        geoavg = alpha x std + beta.

Linear regression will find alpha and beta. Then

        gain_intrinsic(i) = geoavg(i) - alpha x std(i) - beta.

The maximum gain_intrinsic is the highest excessive gain we are looking for, and the corresponding i is the optimal parameter that achieves the maximum.

Diffusion

Diffusion

Another approach is to maximize the "risk-free daily return", which is defined as the geometric average of daily return over the days that we hold any stocks. For example, if the total return is 100% over 100 days, the geometric average of daily return would be (1+1)^(1/100) - 1 = 0.6956%. If during the 100 days, there are 20 days we didn't hold any stocks, then the "risk-free daily return" is (1+1)^(1/80) - 1= 0.8702%. We exclude the 20 days from the calculation because during those 20 days we are holding cash, and there was no risk (baring any risks of exchange rate, which is out of the scope of this thread). So if several strategies have similar total returns, we would like to chose the one that maximizes risk-free daily return, because it supposed to minimize the risk we were exposed to, while generating maximal return on each day exposed to risk. Below is the equity curve of the Momo strategy that reaches maximal risk-free daily return.



By the way, although the Sharpe ratio of Momo is only 1.6108, the Sortino ratio is 2.1494, meaning, desirably, the downside risk is less than the upside one.