# Reinforcement Learning Portfolio Management

Mean-variance portfolio problem is a classical problem in Finance. Given the a universe of $N$ stocks with expected returns of $\mu$ and covariance matrix $\Sigma$, mean-variance portfolio problem can be defined as follows:

where $w’\boldsymbol{1}=1$ and $w’\mu =z$. Here $w$ is weights of the portfolio and $z$ is the required return on the portfolio. This problem has a well-known solution.

Despite the analytical solution to the problem, implementing optimal mean-variance portfolio in real-world is difficult mainly because, contrary to the problem statement, true expected returns and covariance matrix is hard to measure. In this post, I will explore using reinforcement learning to tackle the mean-variance portfolio problem.

The reinforcement learning algorithm used in this post is called Deep Q-Network (DQN). DQN is the first deep reinforcement learning algorithm proposed by DeepMind in 2015. Perhaps due to its success in wide range of domains, example codes for DQN can be found in numerous blogs. The DQN code used in this post is based from pythonprogramming.net

#### Training Data

In spirit of the mean-variance portfolio problem, I look at at mean returns and variance / covariances among different stocks as features. For this analysis, I randomly choose 3 stocks from Korea’s KOSPI index and look at daily prices 2010-01-01 ~ 2019-12-31. The three randomly picked stocks are 089470.KS - HDC Hyundai Engineering Plastic Co, 031440.KS - Shinsegae Food Co, and 012030.KS - DB Inc. The stock returns data can be found here. P1,P2, and P3 are the adjusted closing prices of 089470.KS, 031440.KS, and 012030.KS respectively. The stock price data is sourced from FN Guide. As the first step, I need to build the relevant features.

```
import pandas as pd
df = pd.read_csv("stock_prices.csv")
df[['R1','R2','R3']] = df[['P1','P2','P3']]/df[['P1','P2','P3']].shift(1)-1
```

The mean-variance portfolio problem assumes that the stock return distribution is stationary and that $\mu$ and $\Sigma$ do not change. This is unlikely to be true in the real world where stock return distribution is likely to be time-varying. In order to capture the short-term to long-term changes in stock return distrbution, I look at 3 different windows in computing the past mean and variance/covariance.

```
window1 = 20
window2 = 50
window3 = 100
#Mean Returns
df[['M11','M12','M13']] = df[['R1','R2','R3']].rolling(window1).mean()
df[['M21','M22','M23']] = df[['R1','R2','R3']].rolling(window2).mean()
df[['M31','M32','M33']] = df[['R1','R2','R3']].rolling(window3).mean()
#Variance / Covariance of Returns
for i in range(1,4):
for j in range(i,4):
df[f'V1{i}{j}'] = df[f'R{i}'].rolling(window1).cov(df[f'R{j}'])
df[f'V2{i}{j}'] = df[f'R{i}'].rolling(window2).cov(df[f'R{j}'])
df[f'V3{i}{j}'] = df[f'R{i}'].rolling(window3).cov(df[f'R{j}'])
```

Below is the `ModifiedTensorBoard`

class taken from pythonprogramming.net, which allows custom documentation of different training parameters across different epochs.

```
from keras.callbacks import TensorBoard
import tensorflow as tf
class ModifiedTensorBoard(TensorBoard):
# Overriding init to set initial step and writer (we want one log file for all .fit() calls)
def __init__(self, **kwargs):
super().__init__(**kwargs)
self.step = 1
self.writer = tf.summary.FileWriter(self.log_dir)
def set_model(self, model):
pass
# Overrided, saves logs with our step number
# (otherwise every .fit() will start writing from 0th step)
def on_epoch_end(self, epoch, logs=None):
self.update_stats(**logs)
# Overrided
# We train for one batch only, n
# Overriding this method to stop creating default log writero need to save anything at epoch end
def on_batch_end(self, batch, logs=None):
pass
# Overrided, so won't close writer
def on_train_end(self, _):
pass
# Custom method for saving own metrics
# Creates writer, writes custom metrics and closes writer
def update_stats(self, **stats):
self._write_logs(stats, self.step)
```

Below is the class function for DQNAgent. The core of DQN is Q-learning, which is a model-free reinforcement learning algorithm. It roots from state-action value function, also known as Q-function. The Q-function is defined as follows

Intuitively, Q-function defines an intertemporal value of taking a policy $\pi$. If the policy is to always take the optimal action, the Q-function can be recursively written down as

The goal of Q-learning is to find a stable Q-function. This is done through value iteration method

Here $Q_i$ is $i$th iteration estimation of Q-function and $Q^*$ is the best estimate of Q-function. DQN uses deep learning to estimate the Q-function. To effectively train the deep learning model, DQN contains two main features

- Experience Replay
- Target Network

Instead of fitting deep learning model on the entire past data, Experience Replay trains the model in random sample of past data, which reduces chances of overfitting and allows faster training. In my exercise, I used 10,000 replay memory size, 100 minimum replay memory size, and training size( minibatch size) of 64.

Target Network is the concept of updating $Q^*$ less frequently. By keeping $Q^*$ more stable, $Q_i$ can be learned in a more stable manner. In my exercise, I update $Q^*$ every 5 iterations.

```
from keras.models import Sequential
from keras.layers import Dense
from keras.optimizers import Adam
from keras import backend as K
from keras.callbacks import TensorBoard
from collections import deque
import numpy as np
import random
class DQNAgent:
def __init__(self, state_size, replay_memory_size = 10000, min_replay_memory_size = 100, minibatch_size=64, discount = 0.99, update_every = 5):
#Default Params
self.state_size = state_size
self.replay_memory_size = replay_memory_size
self.min_replay_memory_size = min_replay_memory_size
self.minibatch_size = minibatch_size
self.discount = discount
self.update_every = update_every
# Main model
self.model = self.create_model()
# Target network
self.target_model = self.create_model()
self.target_model.set_weights(self.model.get_weights())
#tensorboard
self.tensorboard = ModifiedTensorBoard(log_dir="logs/mean_variance")
#Replay Memory
self.replay_memory = deque(maxlen=self.replay_memory_size)
#Counter to check when to update target network
self.target_update_counter = 0
def create_model(self):
model = Sequential()
model.add(Dense(units=100, input_dim=self.state_size, activation="relu"))
model.add(Dense(units=100, activation="relu"))
model.add(Dense(units=100, activation="relu"))
model.add(Dense(15, activation="linear"))
model.compile(loss=huber_loss, optimizer=Adam(lr=0.001))
return model
def act(self, state):
state = np.array(state)
predicted_qs = self.model.predict(state.reshape(-1,self.state_size))[0]
return np.argmax(predicted_qs)
def update_replay_memory(self, transition):
self.replay_memory.append(transition)
def train(self, terminal_state):
if len(self.replay_memory)<self.min_replay_memory_size:
return
#Sample from replay memory to create training data
minibatch = random.sample(self.replay_memory, self.minibatch_size)
current_states = np.array([transition[0] for transition in minibatch])
current_qs_list = self.target_model.predict(current_states)
new_states = np.array([transition[3] for transition in minibatch])
future_qs_list = self.target_model.predict(new_states)
X,y = [],[]
for index, (current_state, action, reward, new_state, done) in enumerate(minibatch):
max_future_q = np.max(future_qs_list[index])
if done:
new_q = reward
else:
new_q = reward + self.discount * max_future_q
current_qs = current_qs_list[index]
current_qs[action] = new_q
X.append(current_state)
y.append(current_qs)
#Train the model
self.model.fit(np.array(X), np.array(y), batch_size = self.minibatch_size, verbose = 0, epochs = 0, shuffle = False, callbacks = None)
if terminal_state:
self.target_update_counter +=1
if self.target_update_counter>self.update_every:
self.target_model.set_weights(self.model.get_weights())
self.target_update_counter = 0
```

One of the main design choices of DQN is the deep learning model architecture. While many studies in using Reinforcement Learning for trading uses more complicated form of networks such as Convolutional Neural Network (CNN) and Recurrent Neural Network (RNN), since the analytical solution to mean-variance portfolio problem is a simple function of $\Sigma$ and $\mu$, I focus on simple neural network consisting of dense layers as in `create_model`

function. For the loss function, I decided to use Huber Loss function, which I found to be better at fitting in this setup. The code from loss function is derived from here

```
def huber_loss(a, b, in_keras=True):
error = a - b
quadratic_term = error*error / 2
linear_term = abs(error) - 1/2
use_linear_term = (abs(error) > 1.0)
if in_keras:
# Keras won't let us multiply floats by booleans, so we explicitly cast the booleans to floats
use_linear_term = K.cast(use_linear_term, 'float32')
return use_linear_term * linear_term + (1-use_linear_term) * quadratic_term
```

Now that `DQNAgent`

class is done, instance of it can be created. As described previously, this exercise will use $\mu$ and $\Sigma$ estimated at different frequencies as featuers.

```
#Remove observations with not enough past data
dt = df.iloc[window3:].reset_index(drop = True)
train_dt = dt.loc[dt['tradedate']<"2019-01-01"].reset_index(drop = True)
test_dt = dt.loc[dt['tradedate']>="2019-01-01"].reset_index(drop = True)
#Feature column names
features = [c for c in df.columns if c[0] in ['M','V']]
#Variance until now and Proportion of Dataset completed will be added to state variables
agent = DQNAgent(len(features)+2)
```

The next part is training the `DQNAgent`

. One limitation of DQN algorithm is that the action space has to be discrete and finite. For this, I discretize the action space (weights to be placed on each stocks) into a space of 15 actions. Namely, I place a restriction that each stock can only have weights in the multiple of 25%.

Although the original mean-variance portfolio problem attempts to solve minimization problem with the constraint that the mean return be equal to a pre-defined number, it isn’t clear how this constraint can be enforced in DQN context. Even in Ledoit, Wolf (2003), an influential work on empirically estimating $\Sigma$ through shrinkage, mean return is ignored when testing for out of sample performance. This exercise enforces reward function based on portfolio variance.

To prevent overfitting, I randomly pick a sub-period of 1000 trading days on each episode of training.

```
from tqdm import tqdm
#Training Variables
epsilon = 0.8
N_EPISODES = 1000
MIN_EPSILON = 0.05
EPSILON_DECAY = 0.995
EPISODE_LENGTH = 1000
AGGREGATE_STATS_EVERY=5 #Update tensorboard every 5 episodes
weight_map = [[0,0,4],[1,0,3],[2,0,2],
[3,0,1],[4,0,0],[0,1,3],[1,1,2],[2,1,1],
[3,1,0],[0,2,2],[1,2,1],[2,2,0],[0,3,1],
[1,3,0],[0,4,0]]
ep_rewards = [] #History of episode rewards
for episode in tqdm(range(1, N_EPISODES + 1), ascii=True, unit = "episode"):
agent.tensorboard.step = episode
episode_st_idx = np.random.randint(len(train_dt) - EPISODE_LENGTH)
episode_dt = train_dt.iloc[episode_st_idx:episode_st_idx+EPISODE_LENGTH].reset_index(drop=True)
ret_history = [] #History of portfolio returns
current_state = [0]*(len(features)+2) #initialize state variable with 0s
episode_reward = 0
for idx,d in episode_dt.iterrows():
#Epsilon greedy algorithm. Choose random action if random number less than epsilon
if np.random.random()<=epsilon:
action = np.random.randint(0,15)
else:
action = agent.act(current_state)
#Map action to portfolio weights
weights = np.array(weight_map[action])/4
#Process action
ret_history.append(sum(weights*d[['R1','R2','R3']]))
reward = 0
if idx == EPISODE_LENGTH-1:
done = True
reward = -np.var(ret_history)
else:
done = False
episode_reward += reward
new_state = [np.var(ret_history), idx/EPISODE_LENGTH] + list(d[features])
# Every step we update replay memory and train main network
agent.update_replay_memory((current_state, action, reward, new_state, done))
agent.train(done)
#Update current state with new state
current_state = new_state
#Update Tensorboard
ep_rewards.append(episode_reward)
if not episode % AGGREGATE_STATS_EVERY or episode == 1:
average_reward = sum(ep_rewards[-AGGREGATE_STATS_EVERY:])/len(ep_rewards[-AGGREGATE_STATS_EVERY:])
min_reward = min(ep_rewards[-AGGREGATE_STATS_EVERY:])
max_reward = max(ep_rewards[-AGGREGATE_STATS_EVERY:])
agent.tensorboard.update_stats(reward_avg=average_reward, reward_min=min_reward, reward_max=max_reward, epsilon=epsilon)
#Decay epsilon every episode
if epsilon > MIN_EPSILON:
epsilon *= EPSILON_DECAY
epsilon = max(MIN_EPSILON, epsilon)
agent.model.save_weights("trained_model")
```

The trained model weights can be found here. Looking at the train progress, DQN average reward (negative variance) appears to flatten around 0.0003 after 500 episodes.

Now let’s try deploying the model to out of sample data.

```
#load Trained Model
agent.model.load_weights("trained_model")
#initialize variables
current_state = [0]*(len(features)+2)
weight_history = []
ret_history = []
EPISODE_LENGTH = len(test_dt)
for idx,d in test_dt.iterrows():
action = agent.act(current_state)
#Map action to portfolio weights
weights = np.array(weight_map[action])/4
weight_history.append(weights)
#Process action
ret_history.append(sum(weights*d[['R1','R2','R3']]))
reward = 0
if idx == EPISODE_LENGTH-1:
done = True
reward = -np.var(ret_history)
else:
done = False
episode_reward += reward
new_state = [np.var(ret_history), idx/EPISODE_LENGTH] + list(d[features])
current_state = new_state
print("Portfolio Variance:", -reward)
print("S1 Variance:", np.var(test_dt['R1']))
print("S2 Variance:", np.var(test_dt['R2']))
print("S3 Variance:", np.var(test_dt['R3']))
```

```
Portfolio Variance: 0.00016888536339225218
S1 Variance: 0.00039278878062088676
S2 Variance: 0.0002475167239713626
S3 Variance: 0.00031164781447791775
```

It appears portfolio variance by DQN agent has lower variance than any of the singular stock variance.

```
import matplotlib.pyplot as plt
weight_df = pd.DataFrame(weight_history, columns = ["S1","S2","S3"])
weight_df.plot.area()
```

Looking at weights chosen by the DQN agent, 50%, 25%, 25% is chosen for each stocks. This is close to the portfolio weights using shrinkage estimator when the lagrangian to the mean return is approximately 2.

```
from sklearn.covariance import LedoitWolf
mu = train_dt[['R1','R2','R3']].apply(lambda x: np.mean(x), axis = 0)
Sigma = LedoitWolf().fit(train_dt[['R1','R2','R3']]).covariance_
Sigma_inv = np.linalg.inv(Sigma)
A = np.matmul(np.matmul(np.transpose(np.ones(3)), Sigma_inv), np.ones(3))
B = np.matmul(np.matmul(np.transpose(mu), Sigma_inv), np.ones(3))
C = np.matmul(np.matmul(np.transpose(mu), Sigma_inv), mu)
gamma = 2
z = (gamma*(A*C-B**2)+B)/A
optimal_weight = (C - z*B)/(A*C-B**2)*np.matmul(Sigma_inv,np.ones(3)) + (z*A - B)/(A*C-B**2)*np.matmul(Sigma_inv, mu)
print(optimal_weight)
```

```
[0.55581637 0.24408406 0.20009957]
```

reinforcement_learning

python

keras

## Discussion and feedback