具有分组边界的 SciPy 优化 [英] SciPy optimization with grouped bounds
问题描述
我正在尝试执行投资组合优化，返回使我的效用函数最大化的权重.我可以很好地完成这部分，包括权重总和为 1 的约束，并且权重也给我一个目标风险.我还包括了 [0 <= weights <= 1] 的边界.这段代码如下所示:
I am trying to perform a portfolio optimization that returns the weights which maximize my utility function. I can do this portion just fine including the constraint that weights sum to one and that the weights also give me a target risk. I have also included bounds for [0 <= weights <= 1]. This code looks as follows:
def rebalance(PortValue, port_rets, risk_tgt):
#convert continuously compounded returns to simple returns
Rt = np.exp(port_rets) - 1
covar = Rt.cov()
def fitness(W):
port_Rt = np.dot(Rt, W)
port_rt = np.log(1 + port_Rt)
q95 = Series(port_rt).quantile(.05)
cVaR = (port_rt[port_rt < q95] * sqrt(20)).mean() * PortValue
mean_cVaR = (PortValue * (port_rt.mean() * 20)) / cVaR
return -1 * mean_cVaR
def solve_weights(W):
import scipy.optimize as opt
b_ = [(0.0, 1.0) for i in Rt.columns]
c_ = ({'type':'eq', 'fun': lambda W: sum(W) - 1},
{'type':'eq', 'fun': lambda W: sqrt(np.dot(W, np.dot(covar, W))
* 252) - risk_tgt})
optimized = opt.minimize(fitness, W, method='SLSQP', constraints=c_, bounds=b_)
if not optimized.success:
raise BaseException(optimized.message)
return optimized.x # Return optimized weights
init_weights = Rt.ix[1].copy()
init_weights.ix[:] = np.ones(len(Rt.columns)) / len(Rt.columns)
return solve_weights(init_weights)
现在我可以深入研究这个问题，我将权重存储在一个 MultIndex 熊猫系列中，这样每个资产都是一个与资产类别相对应的 ETF.当一个同等权重的投资组合被打印出来时看起来像这样:
Now I can delve into the problem, I have my weights stored in a MultIndex pandas Series such that each asset is an ETF corresponding to an asset class. When an equally weights portfolio is printed out looks like this:
equity CZA 0.045455
IWM 0.045455
SPY 0.045455
intl_equity EWA 0.045455
EWO 0.045455
IEV 0.045455
bond IEF 0.045455
SHY 0.045455
TLT 0.045455
intl_bond BWX 0.045455
BWZ 0.045455
IGOV 0.045455
commodity DBA 0.045455
DBB 0.045455
DBE 0.045455
pe ARCC 0.045455
BX 0.045455
PSP 0.045455
hf DXJ 0.045455
SRV 0.045455
cash BIL 0.045455
GSY 0.045455
Name: 2009-05-15 00:00:00, dtype: float64如何包含额外的界限要求，以便当我将这些数据组合在一起时，权重总和落在我为该资产类别预先确定的分配范围之间?
how can I include an additional bounds requirement such that when I group this data together, the sum of the weight falls between the allocation ranges I have predetermined for that asset class?
具体来说，我想包括一个额外的边界，使得
So concretely, I want to include an additional boundary such that
init_weights.groupby(level=0, axis=0).sum()
出[264]:
equity 0.136364
intl_equity 0.136364
bond 0.136364
intl_bond 0.136364
commodity 0.136364
pe 0.136364
hf 0.090909
cash 0.090909
dtype: float64在这些范围内
[(.08,.51), (.05,.21), (.05,.41), (.05,.41), (.2,.66), (0,.16), (0,.76), (0,.11)]
[更新]我想我会用一个我不太满意的笨拙的伪解决方案来展示我的进步.也就是说，因为它没有使用整个数据集来解决权重，而是逐个资产类别来解决权重问题.另一个问题是它返回的是序列而不是权重，但我相信有人比我更合适，可以提供有关 groupby 函数的一些见解.
[UPDATE] I figured I would show my progress with a clumsy psuedo-solution that I am not too happy with. Namely becuase it doesn't solve the weights using the entire data set but rather asset class by asset class. The other issue is that it instead returns the series rather than the weights, But I am sure someone more apt than myself, could offer some insight in regards to the groupby function.
因此，对我的初始代码进行轻微调整后，我得到了:
So with a mild tweak to my initial code, I have:
PortValue = 100000
model = DataFrame(np.array([.08,.12,.05,.05,.65,0,0,.05]), index= port_idx, columns = ['strategic'])
model['tactical'] = [(.08,.51), (.05,.21),(.05,.41),(.05,.41), (.2,.66), (0,.16), (0,.76), (0,.11)]
def fitness(W, Rt):
port_Rt = np.dot(Rt, W)
port_rt = np.log(1 + port_Rt)
q95 = Series(port_rt).quantile(.05)
cVaR = (port_rt[port_rt < q95] * sqrt(20)).mean() * PortValue
mean_cVaR = (PortValue * (port_rt.mean() * 20)) / cVaR
return -1 * mean_cVaR
def solve_weights(Rt, b_= None):
import scipy.optimize as opt
if b_ is None:
b_ = [(0.0, 1.0) for i in Rt.columns]
W = np.ones(len(Rt.columns))/len(Rt.columns)
c_ = ({'type':'eq', 'fun': lambda W: sum(W) - 1})
optimized = opt.minimize(fitness, W, args=[Rt], method='SLSQP', constraints=c_, bounds=b_)
if not optimized.success:
raise ValueError(optimized.message)
return optimized.x # Return optimized weights
以下 one-liner 将返回稍微优化的系列
The following one-liner will return the somewhat optimized series
port = np.dot(port_rets.groupby(level=0, axis=1).agg(lambda x: np.dot(x,solve_weights(x))),
solve_weights(port_rets.groupby(level=0, axis=1).agg(lambda x: np.dot(x,solve_weights(x))),
list(model['tactical'].values)))
Series(port, name='portfolio').cumsum().plot()
[更新 2]
以下更改将返回受约束的权重，但仍然不是最佳的，因为它在组成资产类别上被分解和优化，因此当目标风险的约束被认为只有初始协方差矩阵的折叠版本可用
The following changes will return the constrained weights, though still not optimal as it is broken down and optimized on the constituent asset classes, so when the constraint for target risk is considered only a collapsed version of the initial covariance matrix is avaliable
def solve_weights(Rt, b_ = None):
W = np.ones(len(Rt.columns)) / len(Rt.columns)
if b_ is None:
b_ = [(0.01, 1.0) for i in Rt.columns]
c_ = ({'type':'eq', 'fun': lambda W: sum(W) - 1})
else:
covar = Rt.cov()
c_ = ({'type':'eq', 'fun': lambda W: sum(W) - 1},
{'type':'eq', 'fun': lambda W: sqrt(np.dot(W, np.dot(covar, W)) * 252) - risk_tgt})
optimized = opt.minimize(fitness, W, args = [Rt], method='SLSQP', constraints=c_, bounds=b_)
if not optimized.success:
raise ValueError(optimized.message)
return optimized.x # Return optimized weights
class_cont = Rt.ix[0].copy()
class_cont.ix[:] = np.around(np.hstack(Rt.groupby(axis=1, level=0).apply(solve_weights).values),3)
scalars = class_cont.groupby(level=0).sum()
scalars.ix[:] = np.around(solve_weights((class_cont * port_rets).groupby(level=0, axis=1).sum(), list(model['tactical'].values)),3)
return class_cont.groupby(level=0).transform(lambda x: x * scalars[x.name])
推荐答案
经过很长时间，这似乎是唯一适合的解决方案......
After much time this seems to be the only solution that fits...
def solve_weights(Rt, b_ = None):
W = np.ones(len(Rt.columns)) / len(Rt.columns)
if b_ is None:
b_ = [(0.01, 1.0) for i in Rt.columns]
c_ = ({'type':'eq', 'fun': lambda W: sum(W) - 1})
else:
covar = Rt.cov()
c_ = ({'type':'eq', 'fun': lambda W: sum(W) - 1},
{'type':'eq', 'fun': lambda W: sqrt(np.dot(W, np.dot(covar, W)) * 252) - risk_tgt})
optimized = opt.minimize(fitness, W, args = [Rt], method='SLSQP', constraints=c_, bounds=b_)
if not optimized.success:
raise ValueError(optimized.message)
return optimized.x # Return optimized weights
class_cont = Rt.ix[0].copy()
class_cont.ix[:] = np.around(np.hstack(Rt.groupby(axis=1, level=0).apply(solve_weights).values),3)
scalars = class_cont.groupby(level=0).sum()
scalars.ix[:] = np.around(solve_weights((class_cont * port_rets).groupby(level=0, axis=1).sum(), list(model['tactical'].values)),3)
class_cont.groupby(level=0).transform(lambda x: x * scalars[x.name])
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