马克维茨的组合收益率公式如上，这个公式我认为对对数收益率的E(Ri)应该不成立  或者说只是近似成立
不知道我理解得对不对  多谢版主

Forgive my sleeping since it is too late.

I am happy you notice this problem. But remember, if you use geometric return, you should use the corresponding geometric average otherwise you are ignoring the convexity of the exponential function if you treat them in a linear way.

You can show that if log return is normal then arithmetic return is log normal so does the asset price. However, the summation of log normal is not log normal any more. It seems a little wierd, since you may say that you can derive the portfolio's return as Pt+1/Pt -1=Σwi(Ait+1/Ait -1) where Ait is the price of ith asset in the portfolio at time t. Remember, if you accept the fact that the At+1/At is log normally distributed, Pt+1/Pt can not be log normal , so that there is some inconsistency. This is one draw back of arithmetic return. That' why log return is so popular.

For log return, the summation is still normal. It is convenient. This question is a little trivial if you really want to dig out the detailed part. The main reason we use the log return is mainly describe in my explanation above. Anyway, you don't need to be worried about that, you can use both returns.  Actually log return works both correct in mathematical derivation and better in statistical property. So take it easy.

What I said does not mean you're wrong. What you said is also reasonable. I just give some advice to calm you down.

Hope help~

非常感谢版主如此热心！！
您说的我觉得非常有道理！我之前也是这样考虑的（正态分布，并且线性，方便加和，多期后总收益不会偏差等等）
我只是觉得使用日对数收益率也有一定的问题，就是在合成组合的日对数收益率的时候：
如果用各种资产的对数收益率（我没有用收益率）按权重加和，得到组合的对数收益率，这种算法似乎也有点偏差——
正确的算法似乎应该是：把各种资产的对数收益率换算回收益率，再按权重加权，得出的总值再取对数（这样的话，马克维茨模型计算方差的公式就不是原来的公式那样了，会很复杂）



思路好像有点乱。。。不好意思
不知版主的看法如何，多谢多谢

No problem, this is my duty.

Yes,that is the definition of the geometric average and you can do that.

GR works good for small time interval, both in the normal and linear weighted aspects. Actually my JP Morgan professor told us that usually they use log return for portfolio management. So I recommend you to use that although you have pointed out that it has some linear weight summation issue. In your daily return case, it should works good and it has many good properties.

You don't need to worry about that. If the committee challenge you , I have given you sufficient reasons to argue

As far as i know, arithmetic return is more broadly used by practitioner especially when you do not reinvest on daily basis. Also when you need to do linear combination of returns. Yet log return is more favored by academical world because its nice mathematical property(especially when you do time series models and do simulation.). In terms of your problem. I'm assuming you're trying to estimate expected return and volatility by daily data. Both of them works fine. But you need to be careful. Say you need use yearly return and volatility to calculate shape ratio,  arithmetic return makes more sense.
  

agree.

One thing need to be pointed out is that GR and AR are just two ways to calcuate return. It does not mean that if the time is discrete you need to use AR, while if it is continuous you have to use GR.  You can use both but GR is more consistent with continous time finance which does not indicates GR is very bad for discrete time case. In discrete time basis, both works. (just a big diffence between AR and GR). It does not make sense that if you use log return for daily basis but their summation can not be used in the yearly basis. Actually for monthly and yearly basis, log return works better than daily basis since it is more normally distributed with is garanteed by central limit theorem.

So the most practical way to handle with this question is that you derive the model in the log return framework, since it has good mathematical and statistical property so the result is more stable and then finally convert it into the arithmetical one if you want.

Actually on the street, people has a approximation for GR and AR, GR=AR-1/2*variance. This result can be easily get if you are familiar with Geometric Brownian motion.

Different people has different perspective. AR has it's advantage when calculate the linear weighted return. I prefer log return since it has more advantages.

我之前的做法是：
步骤1.求取各ith资产的日对数收益率的均值LRi、标准差Qi以及他们之间的相关系数矩阵
步骤2.使用马克维茨的模型标准公式：组合日对数收益率LRp=sum wi*LRi,wi为权重，以及组合的标准差Qp
步骤3.计算日sharpe ratio
步骤4.比较夏普比例大小确定风险资产的最优组合

我觉得好像不太对，就来发帖子询问。

版主 多谢
我的做法如上，做的时候就是用日对数收益率 套经典的马克维茨 是不是有问题

请问楼主：马克维茨投资组合的收益率是单个资产的对数收益率加权平均还是简单收益率加权平均呢？