Scientific Calculations with Julia/Portfolio Optimization in Finance

Scientific Calculations with Julia Portfolio Optimization in Finance Maximizing Returns in Finance Basic Model of Sports Betting In finance, when you want to optimize a portfolio according to Harry Markowitz, consisting, for example, of n stocks, you solve the problem by minimizing the following objective function: subject to: where is the variance of the portfolio you want to minimize with expected return ≥ target. If a stock has an expected return of 7%, but sometimes returns +20%, sometimes returns -15%, then it is risky not because it always loses, but because it is unpredictable. The mathematical measure of this unpredictability is the portfolio's variance, which must be minimized by ensuring that the expected return is greater than or equal to the target, while volatility is the square root of the variance. is the covariance matrix, and w is the weight vector , which represents the percentage of the portfolio invested in each asset. For example: → 40% in asset 1 → 35% in asset 2 → 25% in asset 3 And the following always hold: and the weights are all positive. The covariance matrix is one of the most important mathematical objects in quantitative finance. For a portfolio with n assets, the matrix is: Where is the return series for asset ( i ). On the main diagonal is the variance of the various assets, so: which measures the volatility of the asset (the greater it is, the riskier it is) Off the diagonal is the covariance: and measures whether two assets move together, so if: Cov > 0 → they move in the same direction Cov < 0 → they move in opposite directions Cov ≈ 0 → independent movements If you have two stocks where: one rises when the other falls → negative covariance → excellent diversification they always move together → positive covariance → little diversification independent movements → Covariance near 0 → discrete diversification Implementation in Julia First, I install the Julia libraries I need: using Pkg Pkg.add(["Convex", "SCS", "LinearAlgebra","YFinance", "DataFrames", "Statistics"]) Suppose we consider three stocks: Apple (AAPL), Microsoft (MSFT), and Amazon (AMZN). Using the YFinance package, I download daily prices for the last 5 years and transform the data into a DataFrame: using YFinance, DataFrames, Statistics tickers = ["AAPL", "MSFT", "AMZN"] data = get_prices.(tickers, range="5y", interval="1d") data = data |> DataFrame I create a dictionary and then a DataFrame with the closing prices of the stocks and their respective dates: prices = Dict() for t in tickers prices[t] = data[data.ticker.==t,"adjclose"] # adjusted price column end # Convert to a single DataFrame df = DataFrame(Date = data[data.ticker.=="AMZN","timestamp"]) for t in tickers df[!, t] = prices[t] end df The average returns of the 3 stocks are calculated and we note that Apple makes more: AAPL = reduce(vcat, df.AAPL) MSFT = reduce(vcat, df.MSFT) AMZN = reduce(vcat, df.AMZN) # Calculate daily returns rets = dropmissing(DataFrame( AAPL = diff(log.(AAPL)), MSFT = diff(log.(MSFT)), AMZN = diff(log.(AMZN)) )) μ = mean(Matrix(rets), dims=1)' println("Average returns:") println(μ) Average returns: [0.0006538039701943343; 0.0005005829765010237; 0.00029305866759470864;;] I calculate the covariance matrix: Σ = cov(Matrix(rets)) 3×3 Matrix{Float64}: 0.000299483 0.000171464 0.000210499 0.000171464 0.000272221 0.000233099 0.000210499 0.000233099 0.000493231 Finally, I calculate the minimum variance of the portfolio and its weights: #Target return daily: r_target = 0.0004 #Decision variables: n = length(μ) w = Variable(n) #Optimization issue: # constraints constraints = [ sum(w) == 1, μ' * w >= r_target, w >= 0 ] # objective function: objective = quadform(w, Σ) # construction of the problem problem = minimize(objective, constraints) solve!(problem, SCS.Optimizer) weights = evaluate(w) println("Optimal weights = ", weights) println("Minimum variance = ", evaluate(objective)) ------------------------------------------------------------------ SCS v3.2.11 - Splitting Conic Solver (c) Brendan O'Donoghue, Stanford University, 2012 ------------------------------------------------------------------ problem: variables n: 5, constraints m: 12 cones: z: primal zero / dual free vars: 1 l: linear vars: 4 q: soc vars: 7, qsize: 2 settings: eps_abs: 1.0e-04, eps_rel: 1.0e-04, eps_infeas: 1.0e-07 alpha: 1.50, scale: 1.00e-01, adaptive_scale: 1 max_iters: 100000, normalize: 1, rho_x: 1.00e-06 acceleration_lookback: 10, acceleration_interval: 10 compiled with openmp parallelization enabled lin-sys: sparse-direct-amd-qdldl nnz(A): 22, nnz(P): 0 ------------------------------------------------------------------ iter | pri res | dua res | gap | obj | scale | time (s) ------------------------------------------------------------------ 0| 1.45e+01 1.00e+00 2.01e+01 -9.96e+00 1.00e-01 4.88e-04 75| 1.77e-04 7.28e-05 2.82e-05 2.60e-04 1.00e-01 6.02e-04 ------------------------------------------------------------------ status: solved timings: total: 6.04e-04s = setup: 2.04e-04s + solve: 4.00e-04s lin-sys: 3.97e-05s, cones: 2.52e-05s, accel: 7.34e-06s ------------------------------------------------------------------ objective = 0.000260 ------------------------------------------------------------------ Optimal weights = [0.35076741527347427, 0.37629612818017755, 0.27293645299415875] Minimum variance = 0.00024558676226717267 # Daily return portfolio_return_daily = dot(μ, weights) # Daily volatility portfolio_vol_daily = sqrt(weights' * Σ * weights) # Monthly return (%): A stock market month has approximately 21 days. portfolio_return_month = portfolio_return_daily * 21 * 100 # Annual return (%): A stock market year has approximately 252 days. portfolio_return_year = portfolio_return_daily * 252 * 100 # Monthly volatility (%) portfolio_vol_month = portfolio_vol_daily * sqrt(21) * 100 # Annual volatility (%) portfolio_vol_year = portfolio_vol_daily * sqrt(252) * 100 println("Monthly return (%) = ", portfolio_return_month ) println("Annual return (%) = ", portfolio_return_year ) println("Monthly volatility (%) = ", portfolio_vol_month ) println("Annual volatility (%) = ", portfolio_vol_year ) Monthly return (%) = 0.9913696368340023 Annual return (%) = 11.896435642008028 Monthly volatility (%) = 7.163080305760683 Annual volatility (%) = 24.813638056547024 Conclusion: A minimum variance of 0.00026 is obtained with 35.07% Apple, 37.63% Microsoft, and 27.29% Amazon in the portfolio with an expected return greater than 0.0004. However, from the covariance matrix, we see that the covariances are all positive, so there is little diversification, meaning that if one stock rises, the other also rises, and vice versa, which is normal because they all belong to the US tech sector.