Scientific Calculations with Julia/Pension Simulator

Scientific Calculations with Julia Pension Simulator Mortgages and Loans Risk Minimization in Finance To calculate the future capital of a supplementary pension assuming a fixed rate of return r, the classic compound annuity formula is used: where (P) = periodic payment (r) = average return (n) = number of periods In reality, various scientific publications assume that rates of return r fluctuate around an average value. For example: | year | return | | ---- | ----- | | 1 | 7% | | 2 | -3% | | 3 | 11% | | 4 | 2% | thumb|left|Normal distributions with mean and standard deviation or volatility This results in an approximately bell-shaped or normal distribution, which is caused by the fact that financial returns are influenced by many factors: economics politics interest rates news investor behavior The combination of these factors often leads to a distribution similar to the normal, but this is an approximation; in fact, many analysts prefer to approximate rates with the Student t-distribution. Monte Carlo simulation can be used to obtain a distribution of final pension assets by randomly varying the annual rate of return according to a normal distribution with only two parameters: mean and standard deviation or volatility , and iterating the process, for example, 10,000 times. Each of the 10,000 simulations produces a different future and a different final net worth. Once this distribution is obtained, percentiles are calculated, allowing us to understand how much net worth could be had in the various scenarios. For example, suppose we run 10,000 Monte Carlo simulations of net worth at 30 years. We obtain 10,000 final net worth values. | Percentile | Net Worth | | ---------- | ----------- | | 10% | 350.000 € | | 50% | 700.000 € | | 90% | 1.300.000 € | Interpretation: 10% → pessimistic scenario (10% of the scenarios have a value lower than €350,000) 50% → typical scenario (the median - 50% of the scenarios have a net worth lower or higher than €700,000) 90% → very favorable scenario (only 10% of the scenarios have values ​​higher than €1,300,000) Implementation in Julia First, I install the Julia libraries I need: using Pkg Pkg.add("Distributions") Pkg.add("Statistics") To calculate the net worth obtainable after 30 years with a fixed rate of return of 5%, I use the compound annuity formula in Julia, obtaining €249,677: function pac(payment, r, n) payment * ((1+r)^n - 1) / r end pac(300, 0.05/12, 12*30) 249677.59060843976 To reach a target pension capital of €500,000 after 30 years with an initial capital of €20,000, an annual payment of €5,000, and a variable rate of return r approximately according to the normal with a mean of 5% and a standard deviation or volatility of 15%, 10,000 Monte Carlo simulations are implemented in Julia and the 10%, 50%, and 90% quantiles are calculated, as well as the probability of reaching the target. target: using Distributions, Statistics # parameters initial_capital = 20000 deposit = 5000 years = 30 target = 500000 mean = 0.05 volatility = 0.15 simulations = 10,000 # return distribution dist = Normal(mean, volatility) function simulate() capital = initial_capital for i in 1:years return = rand(dist) capital = capital*(1 + return) + deposit end return capital end # run simulations results = [simula() for i in 1:simulations] # final capital percentiles p10 = quantile(results, 0.10) p50 = quantile(results, 0.50) p90 = quantile(results, 0.90) # probability of reaching target prob_target = mean(results .>= target) println("Probability of reaching target: ", prob_target) println("10th percentile: ", p10) println("Median: ", p50) println("90th percentile: ", p90) Probability of reaching target: 0.2676 10th percentile: 176089.39588073976 Median: 352329.90761352365 90th percentile: 731757.8309400893 So: 26.76% probability of reaching the €500,000 target in In the worst-case scenario, the figure is approximately €176,000. In the median scenario, it's €352,000. If we want to create a "complete pension simulator" with 40 years of work in which assets are accumulated with a variable rate of return r approximately according to the normal, with a mean of 5% and a standard deviation or volatility of 15%, and then 25 years of pension in which the withdrawal is equal to the difference between expenditure and the pension, minus inflation of 2%, we obtainand that the probability of not exhausting the capital is 78.29%, and then in the median scenario, it can reach €799,000: using Distributions, Statistics initial_capital = 20000 annual_contribution = 6000 work_years = 40 pension_years = 25 average_return = 0.05 volatility = 0.15 inflation = 0.02 annual_expense = 35000 pension = 20000 simulations = 10,000 dist = Normal(average_return, volatility) function simula_vita() capital = initial_capital # accumulation phase for i in 1:work_years r = rand(dist) capital = capital*(1+r) + annual_contribution end # pension phase expenditure = annual_expenditure for i in 1: pension_years r = rand(dist) # pension income withdrawal = max(expenditure - pension, 0) capital = capital*(1+r) - withdrawal # expenditure inflation expenditure *= (1 + inflation) if capital <= 0 return 0 end end return capital end results = [simula_vita() for i in 1: simulations] # final wealth percentiles p10 = quantile(results, 0.10) p50 = quantile(results, 0.50) p90 = quantile(results, 0.90) prob_success = mean(results .> 0) println("10th percentile: ", p10) println("Median: ", p50) println("90th percentile: ", p90) println("Probability of not exhausting the capital: ", prob_success) 10th percentile: 0.0 Median: 799229.9586610713 90th percentile: 4.859875070862252e6 Probability of not exhausting the capital: 0.7829 FIRE Method The FIRE (Financial Independence, Retire Early) method is based on a simple principle: accumulate enough assets to generate passive income to cover living expenses, making work optional. Its fundamental pillars are the 4% rule (you can withdraw 4% of your assets each year without exhausting them) and its inverse (1/0.04 = 25), the 25x rule (you need 25 times your annual expenses). Implementation in Julia Assuming €30,000 in annual expenses, applying the 25x rule, the FIRE goal becomes €30,000 x 25 = €750,000, which must be reached to cover living expenses, making work optional. It takes us a long time to reach the FIRE goal and the accumulated wealth history : # FIRE goal with 25x rule annual_expenses = 30,000.0 fire_goal = annual_expenses * 25 # €750,000 # Savings rate (primary leverage) net_income = 4,000.0 # monthly monthly_expenses = 2,500.0 monthly_savings = net_income - monthly_expenses savings_rate = monthly_savings / net_income # 37.5% # Year-by-year simulation function fire_simulation( initial_assets::Float64, annual_savings::Float64, return::Float64, goal::Float64 ) assets = initial_assets history = [assets] year = 0 while assets < goal && year < 60 assets = assets * (1 + return) + annual_savings push!(history, assets) year += 1 end return year, history end years, history = simulation_fire(50,000.0, monthly_savings * 12, 0.07, fire_goal) println("Years to FIRE: $years") Years to FIRE: 18 for a in 1: years println("Year $a: €$(round(history[a], digits=2))") end Year 1: €50,000.0 Year 2: €71,500.0 Year 3: €94,505.0 Year 4: €119,120.35 Year 5: €145,458.77 Year 6: €173,640.89 Year 7: €203,795.75 Year 8: €236,061.45 Year 9: €270,585.76 Year 10: €307,526.76 Year 11: €347,053.63 Year 12: €389,347.39 Year 13: €434,601.7 Year 14: €483,023.82 Year 15: €534,835.49 Year 16: €590,273.97 Year 17: €649,593.15 Year 18: €713,064.67 With the 4% rule, you can withdraw the 4% of assets each year without depleting them: # Decumulation phase: function decumulation(assets, annual_expenses, return, years) p = assets for a in 1: years p = p * (1 + return) - annual_expenses println("Year $a: €$(round(p, digits=0))") p < 0 && (println("Assets depleted!"); break) end end decumulation(fire_goal, annual_expenses, 0.04, 40) Year 1: €750,000.0 Year 2: €750,000.0 Year 3: €750,000.0 Year 4: €750,000.0 Year 5: €750,000.0 Year 6: €750,000.0 Year 7: €750,000.0 Year 8: €750,000.0 Year 9: €750,000.0 Year 10: €750,000.0 Year 11: €750,000.0 Year 12: €750,000.0 Year 13: €750,000.0 Year 14: €750,000.0 Year 15: €750,000.0 Year 16: €750,000.0 Year 17: €750,000.0 Year 18: €750,000.0 Year 19: €750,000.0 Year 20: €750,000.0 Year 21: €750,000.0 Year 22: €750,000.0 Year 23: €750,000.0 Year 24: €750,000.0 Year 25: €750,000.0 Year 26: €750,000.0 Year 27: €750,000.0 Year 28: €750,000.0 Year 29: €750,000.0 Year 30: €750,000.0 Year 31: €750,000.0 Year 32: €750,000.0 Year 33: €750,000.0 Year 34: €750,000.0 Year 35: €750,000.0 Year 36: €750,000.0 Year 37: €750,000.0 Year 38: €750,000.0 Year 39: €750,000.0 Year 40: €750,000.0 By withdrawing 3% annually, the assets decrease; with 5% annually, the assets grow beyond the fire target: decumulation(fire target, annual expenses, 0.03, 40) decumulation(fire target, annual expenses, 0.05, 40)