In this project, I use monte-carlo simulations to select the optimal portfolio weights in a basket of securities that yields the highest risk-adjusted return. There are two key classes: Security and Portfolio. Securities are individual assets for which we can gather historic prices using the Alphavantage API. Portfolios consist of baskets of securities with their associated weights. The appendix contains details on the attributes and methods used in each class
In a monte-carlo simulation, we can infer the most likely return and risk of a basket of securities based on the returns of each asset (mu), risk (sigma), and covariance among securities in the portfolio. We generate random normal returns for each security in the portfolio, which is correlated with the other randomly generated security returns. We can then determine the weighted portfolio return at each time step over a fixed period. This allows us to determine the ending portfolio value. If we generate many of these paths, we can determine the mean portfolio return and risk for a given portfolio.
The Alphavantage API was used to collect monthly prices for 8 securities. From the prices, we can calculate the mean return, volatility (standard deviation of returns), and the correlations of returns between each security in the form of a covariance matrix.
We start with a portfolio of 8 equally weighted securities and generate 100 possible paths. For our simulation to be realistic, it's important that our randomly generated security returns are correlated with each other.
For example, if two securities returns have a +0.80 correlation, we need to ensure that our randomly generated returns have a similar correlation. The plot below shows the 100 possible paths that a portfolio with a starting value of 100k followed over a 10-year period. From the simulation, we can calculate the mean annualized portfolio return (2.0%) and volatility (9.68%).
Modern portfolio theory suggests that we should be compensated for investing in risky assets. A standard measure of the risk/reward payoff is the sharpe-ratio. It is a ratio of the expected portfolio return minus the return we can expect from a risk-free asset (such as a T-Bill), divided by the expected volatility of the portfolio. The higher the ratio, the better the risk-adjusted return.
sharpe-ratio = (Expected Return - Risk-Free-Rate )/ Expected Volatility
Our starting portfolio of equally weighted securities has a sharpe-ratio of -0.20*. This means we are not being adequately rewarded for investing in risky assets on a risk-adjusted basis. Said another way, we would be better off investing in T-Bills which have a guaranteed return that will likely result in higher portfolio value.
*assuming a risk-free rate of 4.0%, which is what US T-bills were yielding at the time of writing
We can use monte-carlo simulations to find the optimal portfolio from the basket of securities by systematically adjusting each security weight, running a simulation, and tracking the expected return and risk. We can then plot each portfolio by return/risk. The plot below shows the results of the following simulation
- Adjust the 8 security weights by increments of 10% (ie, 0%,10%, 20% 30%)
- No individual security should have a weight greater than 30% of the portfolio
- We don't need to hold all eight securities
- Simulate each possible weight combination & calculate the expected return, risk, and sharpe-ratio
We can see from the plot that we would prefer the portfolios on the top edge as these have greater expected returns per unit of expected risk. This plot is known as the efficient frontier. The colors represent the sharpe-ratio
The table below shows the weights of the top five portfolios sorted by sharpe-ratio. As you can see, the simulation revealed that the best risk-adjusted portfolios hold between 4-6 of the 8 securities.
We can now select one of the top portfolios and re-run the monte-carlo simulation to see how our optimal portfolio compares to our starting portfolio. In this case, we ran the simulation 2,500 times.
The first plot shows the 2,500 possible paths our portfolio took in the simulation.
The box plots show the distribution of returns and volatility of the portfolio. The conditional value-at-risk (CVaR) and maximum drawdown distributions provide some insight into the portfolios potential downside.
The optimal portfolio has an expected return of 5.51%, a risk of 13.08%, and a sharpe-ratio of 0.12. The portfolio is most likely to have an ending value between $103,268 and $267,331. This is a large improvement over the starting portfolio of equal-weighted securities.
Holdings
- SPDR S&P 500 ETF Trust:30.0%
- iShares MSCI EAFE:10.0%
- Mackenzie Canadian Equity Index:30.0%
- BMO Aggregate Bond Index:10.0%
- Gold ETF:20.0%
You could also use this method to evaluate portfolios with different securities and weights
- name = security name
- identifier (used to query alphavantage for prices)
- mu (mean return)
- sigma (mean standard deviation of returns)
- generate_returns(self.mu, self.sigma, self.n). This function simulates returns from the normal distribution based on mu and sigma
A portfolio holds securities at specific weights
- name = portfolio name
- securities = a list of securities
- target_weights = a list of weights associated with each security
- portfolio_value = starting value of the portfolio
- rf = risk free rate (decimal format)
- cov = covariance matrix of the returns of each security in the portfolio
- simulation_results = A dictionary of various outputs from the monte carlo simulation
- mean_return = Mean return of the portfolio. Calculated by the simulation
- mean_volatility = Mean standard deviation of returns of the portfolio - calculated by the simulation
- sharpe_ratio = sharpe-ratio of the portfolio - calculated by the simulation
- expected_values = expected values of the portfolio - calculated by the simulation
- calc_sharpe_ratio(): Calculate the sharpe-ratio of the portfolio
- calc_covariance(): Calculate the covariance between security returns
- convert_from_annual(metric,freq,measure): convert from annual to another frequency (return, risk)
- convert_to_annual(ret,vol,freq,periods): convert return/risk to annual from another frequency
- get_security_prices(yr, key, plot=False,requests_per_min=5): get security prices from alpha vantage
- run_simulation(simulations, years, frequency,robust=False): run the monte carlo simulation
- get_expected_values(): get the expected values from the simulation
- weight_combinations(, total_portfolio_wt, min_security_wt, max_security_wt, weight_increment): get all possible portfolio weight combinations
- build_efficient_frontier(total_portfolio_wt, min_security_wt, max_security_wt, weight_increment, num_sims, years, frequency, verbose=0): Build the efficient frontier
- get_optimal_portfolios(top_n): get the optimal portfolio from the efficient frontier
- plot_boxplots(): plot return/risk in boxplots
- plot_histograms(): plot return/risk in histograms
- plot_portfolio_simulations(): plot the simulation results
- plot_efficient_frontier(): plot the efficient frontier