Why does maximizing returns perform worse than maximizing (price - price*returns)

I don't understand why maximizing returns yields much worse performance than maximizing (price - price*returns).

I've attached my returns maximization algorithm here.

1
Total Returns
--
Alpha
--
Beta
--
Sharpe
--
Sortino
--
Max Drawdown
--
Benchmark Returns
--
Volatility
--
 Returns 1 Month 3 Month 6 Month 12 Month
 Alpha 1 Month 3 Month 6 Month 12 Month
 Beta 1 Month 3 Month 6 Month 12 Month
 Sharpe 1 Month 3 Month 6 Month 12 Month
 Sortino 1 Month 3 Month 6 Month 12 Month
 Volatility 1 Month 3 Month 6 Month 12 Month
 Max Drawdown 1 Month 3 Month 6 Month 12 Month
import quantopian.algorithm as algo
from quantopian.pipeline import Pipeline
from quantopian.pipeline.data import Fundamentals
import quantopian.optimize as opt
from quantopian.pipeline.factors import SimpleBeta,Returns
from quantopian.pipeline.data.builtin import USEquityPricing
from quantopian.pipeline.data import EquityPricing
import numpy as np
import pandas as pd

def initialize(context):
beta = SimpleBeta(
target=sid(8554),
regression_length=260,
)
pipe_beta = Pipeline(
columns={
'beta': beta,
},
screen=beta.notnull(),
)
algo.attach_pipeline(pipe_beta, 'pipe_beta')
algo.attach_pipeline(returns_pipeline(), 'returns')

algo.schedule_function(
rebalance,
algo.date_rules.every_day(),
algo.time_rules.market_open(hours=1),
)

def returns_pipeline():
pipe = Pipeline(
columns={
'Returns': Returns(window_length=2),
},
screen=base_universe)
return pipe

context.returns = algo.pipeline_output('returns')
context.pipeline_beta_data = algo.pipeline_output('pipe_beta')

def constraints(context):
constraints = []
pipe_beta_data = context.pipeline_beta_data.dropna()
beta_neutral = opt.FactorExposure(
pipe_beta_data[['beta']],
min_exposures={'beta': -0.05},
max_exposures={'beta': 0.05},
)
constraints.append(beta_neutral)
MAX_GROSS_LEVERAGE = 1.05
constrain_gross_leverage = opt.MaxGrossExposure(MAX_GROSS_LEVERAGE)
constraints.append(constrain_gross_leverage)
dollar_neutral = opt.DollarNeutral()
constraints.append(dollar_neutral)
MAX_SHORT_POSITION_SIZE = 0.01  # 1%
MAX_LONG_POSITION_SIZE = 0.01   # 1%
# Define the position concentration constraint.
constrain_pos_size = opt.PositionConcentration.with_equal_bounds(
-MAX_SHORT_POSITION_SIZE,
MAX_LONG_POSITION_SIZE,
)
constraints.append(constrain_pos_size)
constrain_sector_style_risk = opt.experimental.RiskModelExposure(
#min_momentum=-0.1,
#max_momentum=0.1,
)
constraints.append(constrain_sector_style_risk)
return constraints
def rebalance(context, data):
returns = context.returns.dropna()
objective = opt.MaximizeAlpha(returns['Returns'])
algo.order_optimal_portfolio(objective = objective,
constraints=constraints(context)
,)

There was a runtime error.
7 responses

And here's the maximizing (price - price*returns)

3
Total Returns
--
Alpha
--
Beta
--
Sharpe
--
Sortino
--
Max Drawdown
--
Benchmark Returns
--
Volatility
--
 Returns 1 Month 3 Month 6 Month 12 Month
 Alpha 1 Month 3 Month 6 Month 12 Month
 Beta 1 Month 3 Month 6 Month 12 Month
 Sharpe 1 Month 3 Month 6 Month 12 Month
 Sortino 1 Month 3 Month 6 Month 12 Month
 Volatility 1 Month 3 Month 6 Month 12 Month
 Max Drawdown 1 Month 3 Month 6 Month 12 Month
import quantopian.algorithm as algo
from quantopian.pipeline import Pipeline
from quantopian.pipeline.data import Fundamentals
import quantopian.optimize as opt
from quantopian.pipeline.factors import SimpleBeta,Returns
from quantopian.pipeline.data.builtin import USEquityPricing
from quantopian.pipeline.data import EquityPricing
import numpy as np
import pandas as pd

def initialize(context):
beta = SimpleBeta(
target=sid(8554),
regression_length=260,
)
pipe_beta = Pipeline(
columns={
'beta': beta,
},
screen=beta.notnull(),
)
algo.attach_pipeline(pipe_beta, 'pipe_beta')
algo.attach_pipeline(returns_pipeline(), 'returns')

algo.schedule_function(
rebalance,
algo.date_rules.every_day(),
algo.time_rules.market_open(hours=1),
)

def returns_pipeline():
pipe = Pipeline(
columns={
'Returns': Returns(window_length=2),
},
screen=base_universe)
return pipe

context.returns = algo.pipeline_output('returns')
context.pipeline_beta_data = algo.pipeline_output('pipe_beta')

def constraints(context):
constraints = []
pipe_beta_data = context.pipeline_beta_data.dropna()
beta_neutral = opt.FactorExposure(
pipe_beta_data[['beta']],
min_exposures={'beta': -0.05},
max_exposures={'beta': 0.05},
)
constraints.append(beta_neutral)
MAX_GROSS_LEVERAGE = 1.05
constrain_gross_leverage = opt.MaxGrossExposure(MAX_GROSS_LEVERAGE)
constraints.append(constrain_gross_leverage)
dollar_neutral = opt.DollarNeutral()
constraints.append(dollar_neutral)
MAX_SHORT_POSITION_SIZE = 0.01  # 1%
MAX_LONG_POSITION_SIZE = 0.01   # 1%
# Define the position concentration constraint.
constrain_pos_size = opt.PositionConcentration.with_equal_bounds(
-MAX_SHORT_POSITION_SIZE,
MAX_LONG_POSITION_SIZE,
)
constraints.append(constrain_pos_size)
constrain_sector_style_risk = opt.experimental.RiskModelExposure(
#min_momentum=-0.1,
#max_momentum=0.1,
)
constraints.append(constrain_sector_style_risk)
return constraints
def rebalance(context, data):
returns = context.returns.dropna()
- returns['Returns']*data.current(returns.index,'price')
algo.order_optimal_portfolio(objective = objective,
constraints=constraints(context)
,)

There was a runtime error.

You observe that Max(returns) results in worse performance than Max(price - price*returns) = Max(price*(1-returns)).
Honestly i don't know the answer, but i think your returns are 1-day returns, right?
So that seems to me to imply that there is a tendency for higher prices (positive 1 d returns) generally to be followed by lower prices (negative 1d returns), which is consistent with a general tendency toward short-term Mean-Reversion type of behavior. Does that make sense?

It makes sense in theory, but I've extended the time window and am seeing the same thing.

They are one day returns but I get similar results for 20 days (pure returns is only down 20%). For 200 days pure returns has a positive return (about 5%) and (price - price*returns) has returns of ~10% - slightly worse than the shorter time windows.

One question. What is the term price - price*returns or, written another way price*(1-returns), supposed to represent? It's probably obvious but help me out.

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It's mean reversion adjusted price. The price of yesterday

I am very sorry. Still don't get it. Wouldn't yesterday's adjusted price be

price_yesterday = price / (1 + returns)