Modern Portfolio Theory: Basics, Efficient Frontier, and Sharpe Calculation

中文 📅 2023-12-13 👁 views this month —

Modern Portfolio Theory (MPT) was proposed by Harry Markowitz in 1952 and stands as one of the seven foundational theories of modern finance. It uses mathematical terminology to describe concepts such as diversification and risk management, providing investors with a toolkit for constructing diversified portfolios. Specifically, it assumes that an investor holds multiple assets and seeks to optimize the portfolio to minimize risk while satisfying a given expected return constraint.

All these portfolios form a curve (with the portfolio’s standard deviation on the x-axis and expected return on the y-axis), known as the efficient frontier curve. The upper portion of this curve is referred to as the efficient frontier.

This diagram is vertically symmetric. For every point on the lower half, there is a corresponding point on the upper half with the same risk but a higher expected return. Therefore, portfolios on the lower half are not held by any rational investor.

Each investor selects their specific portfolio along the efficient frontier based on their risk tolerance (utility function).

Markowitz was awarded the 1990 Nobel Memorial Prize in Economic Sciences for his contributions to MPT. He strongly advocated the idea of diversifying investment risk through portfolio construction, famously stating that “diversification is the only free lunch in investing.”

Basic Concepts

We will explain the efficient frontier using the simplest case: a portfolio of two risky assets.

$$ R_p = w_1R_1 + (1-w_1)R_2 $$

Where:

$R_p$ is the portfolio return $W_1$ is the weight of Asset 1 $R_1$ is the return of Asset 1.

Portfolio risk is measured by variance:

$$ \sigma_p = \sqrt{w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2Cov(R_1, R_2)}\newline =\sqrt{w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\rho_{1,2}\sigma_1\sigma_2} $$

Here, $\rho_{1,2}$ is the correlation coefficient between Asset 1 and Asset 2.

When the correlation coefficient between the two assets is 1, the equation simplifies to:

$$ \sigma_p =\sqrt{w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\sigma_1\sigma_2}\newline =\sqrt{(w_1\sigma_1 + w_2\sigma_2)^2}\newline = w_1\sigma_1 + w_2\sigma_2 $$

When the correlation coefficient between the two assets is -1:

$$ \sigma_p =\sqrt{w_1^2\sigma_1^2 + w_2^2\sigma_2^2 - 2w_1w_2\sigma_1\sigma_2}\newline =\sqrt{(w_1\sigma_1 - w_2\sigma_2)^2}\newline = w_1\sigma_1 - w_2\sigma_2 $$

From these equations, we can see that if two assets are perfectly negatively correlated, an equal-weight allocation results in zero portfolio return, creating a risk-free combination. Since $\sigma_p$ equals zero, this point becomes the leftmost point on the graph.

When one asset comprises 100% of the portfolio, it forms the top and bottom points on the graph.

In this series, we will first use a portfolio of four assets to demonstrate how to find the optimal asset allocation using both Monte Carlo methods and optimization algorithms. These are foundational approaches. Once we understand the principles, we can use third-party libraries to automate this work.

Calculating Returns, Sharpe Ratio, and Volatility

Let’s first assign a random set of weights to see how the portfolio’s return, Sharpe ratio, and other parameters perform. In this process, we aim to understand:

How to fetch data.
How to generate random weight vectors (focusing on the constraint that weights sum to 1).
How to calculate metrics such as the Sharpe ratio and volatility.

Assume our asset portfolio consists of:

1234600519 贵州茅台
300750 宁德时代
300059 东方财富
601398 工商银行

We use the following code to obtain their returns over the past year, saving them in a returns DataFrame:

12345678910111213141516171819202122232425262728293031import arrow
import akshare as ak
import pandas as pd
import numpy as np
from IPython.display import display


stocks = ["600519", "300750", "300059", "601398"]

frames = {}

now = arrow.now()
start = now.shift(years = -1)
end = now.format("YYYYMMDD")
start = start.format("YYYYMMDD")

for code in stocks:
    bars = ak.stock_zh_a_hist(symbol=code, 
                              period="daily", 
                              start_date=start, 
                              end_date=end, 
                              adjust="qfq")
    
    bars.index = pd.to_datetime(bars["日期"])
    frames[code] = bars["收盘"]

prices = pd.DataFrame(frames)
returns = prices.pct_change()

returns.dropna(how='any', inplace=True)
display(returns.head().style.format('{:,.2%}'))

We have already encountered this code in our previous article on CAPM.

Next, we randomly assign a weight vector (the first step of the Monte Carlo method) and calculate its Sharpe ratio:

123456789101112131415161718192021222324252627import numpy as np
from empyrical import sharpe_ratio

weights = np.array(np.random.random(4))
print('Random Weights:')
print(weights)

print('\nRebalance')
weights = weights/np.sum(weights)
print(weights)

# 生成每日每个标的对组合的贡献
weighted_returns = weights * returns
weighted_returns.head()

# 把每一行按列加总，就得到了每日资产收益
port_returns = weighted_returns.sum(axis=1)

# 然后计算组合资产的波动
cov = np.cov(port_returns.T)
port_vol = np.sqrt(np.dot(np.dot(weights, cov), weights.T)) # 0.01

# 使用 sharpe_ratio来计算夏普率
sr = sharpe_ratio(port_returns) # 0.18

print("Sharpe Ratio and Vol")
print(f"{sr:.2f} {port_vol:.2f}")

In the code, we first randomly generate a weight matrix and then normalize it (ensuring the sum of the weight matrix elements equals 1).

Some articles use log returns. However, most practitioners consider geometric variance meaningless when calculating volatility (based on Google search results).

We use the sharpe_ratio method from empyrical to calculate the Sharpe ratio. For simplicity, we set the risk_free rate to 0. empyrical is a tool developed and open-sourced by Quantopian for calculating various strategy metrics, such as Sharpe ratio, Sortino ratio, and max drawdown. Common quantitative libraries like this are introduced in the Monopoly Quant Trading Course.

Ultimately, we obtain the following results:

1234567Random Weights:
[0.48349071 0.32903015 0.85308562 0.64038565]

Rebalance
[0.20966711 0.14268485 0.36994299 0.27770505]
Sharpe Ratio and Vol
0.72 0.01

We have determined that the Sharpe ratio for this portfolio is 0.72. Generally, a Sharpe ratio above 1 is considered acceptable for index or blue-chip investments; for other high-risk equity investments, it typically needs to exceed 1.8, though few assets surpass 3. In the Monopoly Quant Trading Course, we discuss the relationship between the Sharpe ratio and max drawdown using Monte Carlo methods—specifically, what max drawdown is likely to occur when the Sharpe ratio is 1, when it is 2, and so on.

With all preparations complete, we will next introduce how to use the Monte Carlo method to determine the optimal allocation for the aforementioned portfolio.