大盘及策略收益率的公式推导与Python代码

素颜马尾好姑娘i 2023-08-17 16:02 113阅读 0赞

# 一、模型前提与假设 #

设策略总天数为\\(n\\)、第\\(t\\)日大盘的收盘价为\\(P\_t\\)、第\\(t\\)日的单日收益率为\\(r\_t\\)、\\(n\\)天的累积收益率为\\(r\_\{cum\}\\)

假设策略仅买卖大盘指数，第\\(t\\)日的头寸是根据第\\((t-1)\\)日收盘价计算出的\\(s\_\{t-1\}\\)，因此第1天的收益率\\(r\_\{1\}=0\\)

特别注意：为避免未来函数，不能使用\\(s\_\{t\}\\)计算第\\(t\\)日的头寸。

# 二、大盘单日收益率 #

## 1. 离散型 ##

\\\[ r\_t=\\frac\{P\_t\}\{P\_\{t-1\}\}-1 \\\]

对应的Python代码为：

df['market_dis'] = df['close']/df['close'].shift()-1 或 df['market_dis'] = df['close'].pct_change()

## 2. 连续型 ##

\\\[ r\_t=ln\\frac\{P\_t\}\{P\_\{t-1\}\} \\\]

对应的Python代码为：

df['market_con'] = np.log(df['close'] / df['close'].shift())

# 三、大盘累积收益率 #

## 1. 离散型 ##

\\\[ \\begin\{align\} 1+r\_\{cum\}&=(1+r\_\{1\})(1+r\_\{2\})\\cdots(1+r\_\{n\})\\\\\[1.5ex\] &=\\frac\{P\_\{1\}\}\{P\_\{0\}\}\\cdot \\frac\{P\_\{2\}\}\{P\_\{1\}\}\\cdots\\frac\{P\_\{n\}\}\{P\_\{n-1\}\}\\\\\[1.5ex\] &=\\frac\{P\_\{n\}\}\{P\_\{0\}\} \\end\{align\} \\\]

对应的Python代码为：

# 注意：这里的累积收益率是以净值形式体现的，在实际应用中可能需要在此结果基础上-1 df['market_dis_cum'] = (1+df['market_dis']).cumprod()

## 2. 连续型 ##

\\\[ \\begin\{align\} \\text\{exp\}(r\_\{cum\})& = \\text\{exp\}(r\_\{1\}+r\_\{2\}+\\cdots+r\_\{n\}) \\\\\[1.5ex\] & = \\text\{exp\}\\left(\{ln\\frac\{P\_\{1\}\}\{P\_\{0\}\}+ln\\frac\{P\_\{2\}\}\{P\_\{1\}\}+\\cdots+ln\\frac\{P\_\{n\}\}\{P\_\{n-1\}\}\}\\right)\\\\\[1.5ex\] & =\\frac\{P\_\{1\}\}\{P\_\{0\}\}\\cdot\\frac\{P\_\{2\}\}\{P\_\{1\}\}\\cdots\\frac\{P\_\{n\}\}\{P\_\{n-1\}\}\\\\\[1.5ex\] & =\\frac\{P\_\{n\}\}\{P\_\{0\}\}\\\\\[2ex\] \\end\{align\} \\\]

对应的Python代码为：

# 注意：这里的累积收益率是以净值形式体现的，在实际应用中可能需要在此结果基础上取np.log() df['market_con_cum'] = df['market_con'].cumsum().apply(np.exp)

# 四、策略单日收益率 #

## 1. 离散型 ##

\\\[ r\_t= \\begin\{cases\} 0&,t=1\\\\\[2ex\] s\_\{t-1\}\\left(\\cfrac\{P\_t\}\{P\_\{t-1\}\}-1\\right)&,t=2,3,\\cdots,n\\\\\[2ex\] \\end\{cases\} \\\]

对应的Python代码为：

df['strategy_dis'] = df['position'].shift()*df['market_dis']

## 2. 连续型 ##

\\\[ r\_t= \\begin\{cases\} 0&,t=1\\\\\[2ex\] s\_\{t-1\}ln\\cfrac\{P\_t\}\{P\_\{t-1\}\}&,t=2,3,\\cdots,n\\\\\[2ex\] \\end\{cases\} \\\]

对应的Python代码为：

df['strategy_con'] = df['position'].shift()*df['market_con']

# 五、策略累积收益率 #

## 1. 离散型 ##

\\\[ \\begin\{align\} 1+r\_\{cum\}&=(1+r\_\{2\})(1+r\_\{3\})\\cdots(1+r\_\{n\})\\\\\[1.5ex\] &=\\left\[1+s\_\{1\}\\left(\\frac\{P\_\{2\}\}\{P\_\{1\}\}-1\\right)\\right\]\\left\[1+s\_\{2\}\\left(\\frac\{P\_\{3\}\}\{P\_\{2\}\}-1\\right)\\right\]\\cdots\\left\[1+s\_\{n-1\}\\left(\\frac\{P\_\{n\}\}\{P\_\{n-1\}\}-1\\right)\\right\]\\\\\[1.5ex\] \\end\{align\}\\\\ \\\]

对应的Python代码为：

# 注意：这里的累积收益率是以净值形式体现的，在实际应用中可能需要在此结果基础上-1 df['strategy_dis_cum'] = (1+df['strategy_dis']).cumprod()

## 2. 连续型 ##

\\\[ \\begin\{align\} \\text\{exp\}(r\_\{cum\})& = \\text\{exp\}(r\_\{2\}+r\_\{3\}\\cdots+r\_\{n\}) \\\\\[1.5ex\] & = \\text\{exp\}\\left(\{s\_1ln\\frac\{P\_\{2\}\}\{P\_\{1\}\}+s\_2ln\\frac\{P\_\{3\}\}\{P\_\{2\}\}+\\cdots+s\_\{n-1\}ln\\frac\{P\_\{n\}\}\{P\_\{n-1\}\}\}\\right)\\\\\[1.5ex\] & =\\left(\\frac\{P\_\{2\}\}\{P\_\{1\}\}\\right)^\{s\_1\}\\left(\\frac\{P\_\{3\}\}\{P\_\{2\}\}\\right)^\{s\_2\}\\cdots\\left(\\frac\{P\_\{n\}\}\{P\_\{n-1\}\}\\right)^\{s\_\{n-1\}\}\\\\\[1.5ex\] \\end\{align\} \\\]

对应的Python代码为：

# 注意：这里的累积收益率是以净值形式体现的，在实际应用中可能需要在此结果基础上取np.log() df['strategy_con_cum'] = df['strategy_con'].cumsum().apply(np.exp)

转载于:https://www.cnblogs.com/oddgod/p/11516169.html