Modern portfolio theory

Gestione quantitativa del rischio in Python

Jamsheed Shorish

Computational Economist

The risk-return trade-off

Risk factors: sources of uncertainty affecting return
Intuitively: greater uncertainty (more risk) compensated by greater return
Cannot guarantee return: need some measure of expected return
- average (mean) historical return: proxy for expected future return

Investor risk appetite

Investor survey: minimum return required for given level of risk?
Survey response creates (risk, return) risk profile "data point"
Vary risk level => set of (risk, return) points
Investor risk appetite: defines one quantified relationship between risk and return

Choosing portfolio weights

Vary portfolio weights of given portfolio => creates set of (risk, return) pairs
Changing weights = beginning risk management!
Goal: change weights to maximize expected return, given risk level
- Equivalently: minimize risk, given expected return level
Changing weights = adjusting investor's risk exposure

Modern portfolio theory

Efficient portfolio: portfolio with weights generating highest expected return for given level of risk
Modern Portfolio Theory (MPT), 1952
- H. M. Markowitz (Nobel Laureate 1990)
Efficient portfolio weight vector $w^\star$ solves:

Optimization problem equations for finding the efficient portfolio

The efficient frontier

Compute many efficient portfolios for different levels of risk
Efficient frontier: locus of (risk, return) pairs created by efficient portfolios
PyPortfolioOpt library: optimized tools for MPT
- EfficientFrontier class: generates one optimal portfolio at a time
- Constrained Line Algorithm (CLA) class: generates the entire efficient frontier
  - Requires covariance matrix of returns
  - Requires proxy for expected future returns: mean historical returns

Investment bank portfolio 2005 - 2010

Expected returns: historical data
Covariance matrix: Covariance Shrinkage improves efficiency of estimate
Constrained Line Algorithm object CLA
Minimum variance portfolio: cla.min_volatility()
Efficient frontier: cla.efficient_frontier()

expected_returns = mean_historical_return(prices)

efficient_cov = CovarianceShrinkage(prices).ledoit_wolf()

cla = CLA(expected_returns, efficient_cov)

minimum_variance = cla.min_volatility()

(ret, vol, weights) = cla.efficient_frontier()

Visualizing the efficient frontier

Scatter plot of (vol, ret) pairs

The efficient frontier graph plotted as a series of (risk, return) pairs

Visualizing the efficient frontier

Scatter plot of (vol, ret) pairs
Minimum variance portfolio: smallest volatility of all possible efficient portfolios

The minimum variance portfolio highlighted on the efficient frontier graph

Visualizing the efficient frontier

Scatter plot of (vol, ret) pairs
Minimum variance portfolio: smallest volatility of all possible efficient portfolios
Increasing risk appetite: move along the frontier

Movement along the efficient frontier graph depicted as an arrow moving in the direction of higher risk and higher reward

Let's practice!

Gestione quantitativa del rischio in Python