# Why don't these betas match?

I am sure I am missing something simple, but I would expect my portfolio beta when regressed against the market to match my individual component betas multiplied by the portfolio weights. I have created a simple example below. Any help in explaining where I have gone wrong would be much appreciated.

import pandas as pd
import numpy as np
import statsmodels.api as sm
from statsmodels import regression

def beta(x, y):
model = regression.linear_model.OLS(y, x).fit()
# Remove the constant now that we're done
x = x[:, 1]
return model.params

bond_one = [100, 96, 102, 88, 96, 101, 120, 110, 105, 107, 106]
bond_two = [98, 102, 88, 95, 105, 100, 101, 99, 104, 108, 112]
mkt = [1000, 1004, 1000, 1010, 1020, 1000, 990, 995, 1005, 1025, 1035]

df_mkt = pd.DataFrame(mkt, columns = ['mkt'])
df_mkt = df_mkt.pct_change().dropna()
df = pd.DataFrame(bond_one, columns = ['bond_one'])
df['bond_two'] = bond_two

df_price = df.copy()
df = df.pct_change().dropna()

notionals = {'bond_one': 2500000,
'bond_two': 6500000}

mkt_values = {key: value*(df_price[key].iloc[-1]/100)
for (key, value) in notionals.items()}

#create portfolio market value
tot_port = sum(list(mkt_values.values()))
#generate weights
wts = {key: value/tot_port for (key, value) in mkt_values.items()}

#create portfolio returns
df_port = df.copy()*0
df_port = df.mul(list(wts.values()), axis=1)
df_port['port'] = df_port.sum(axis=1)

#add port and market into original dataframe
df['port'] = df_port['port'].copy()
df['mkt'] = df_mkt['mkt'].copy()

#run OLS on individuals and portfolio
b1_beta = regression.linear_model.OLS(x = df['bond_one'].values, y=df['mkt'].values).fit()
b2_beta = beta(x=df['bond_two'].values, y=df['mkt'].values)
port_beta = beta(x=df['port'].values, y=df['mkt'].values)

calc_beta = wts['bond_one']*b1_beta + wts['bond_two']*b2_beta
###why don't calc_beta and port_beta match?

• But why exactly do you expect that to be the case ? Have you verified by the beta formula that the sum of betas is the beta of the sum ? I haven’t but given the definition of beta, I highly doubt it. This is the place to start. – Ivan Oct 8 '19 at 15:03
• I was basing my understanding on this StackExchange post: quant.stackexchange.com/questions/22002/… which suggests they should be the same. That is why I was asking the question. – dsugasa Oct 8 '19 at 17:47
• how off are they? – Chris Oct 8 '19 at 21:27
• quite different, which makes me think I am doing something wrong or my understanding is incorrect. I tried to provide a complete simple example of what I am doing to illustrate the problem. – dsugasa Oct 8 '19 at 22:51
• Nicely answered at stackoverflow.com/questions/58284469/why-dont-these-betas-match – dsugasa Oct 9 '19 at 13:29