Modeling finances typically involves learning various heuristics and following them. However, when it comes to choosing a risk tolerance, it’s a little more black-and-white than I am personally comfortable with (or at least, so is my understanding of those heuristics).

While various financial planning tools exist, none of them fulfill the je ne sais quoi of writing your own over-engineered particle-based simulations. For instance, you might be in a situation where you are interested in understanding the difference between organic account growth and risky investments, while having a good understanding of what your exposure to the market is at a time where you are least solvent in your risk-taking era. Or you may be interested in understanding the correlations between various different risky strategies, and how these risks may compound.

The goal of this Julia package is to put quantitative numerical rigour to individual (and other) investment accounts.

Methods

The Particles Method

When running a financial simulation, instead of using a typical numeric quantity, for instance a Float64, we instead use a Particle as implemented in MonteCarloMeasurements.jl [1]. The name particle follows from the particle filtering literature. A Particle object contains a set of particles ($O(1000)$ particles) that are used to represent distinct trajectories in a Monte Carlo simulation. These particles sample a probability distribution differently, and as a result, can be combined after applying some computation $f$ on each trajectory (as you would on a float), to get an output probability distribution after performing $f$.

For instance, if $f$ is a linear map that preserves gaussians, say, $f(x)=x+2$, where we know $x\sim \mathscr{N}(0, 1)$ takes from a normal distribution, then we can calculate the PDF of $f(x)$ which should be $\sim \mathscr{N}(2, 1)$. When the map is not linear and/or the probability distribution being used isn’t preserved under $f$, this gets much more complicated. This can be further complicated in the presence of multiple variables where they could be correlated by different amounts.

This calculation is neat since each particle inside a Particle object is disjoint and can be parallelized over (e.g. SIMD), which is what MonteCarloMeasurements.jl does [1].

This type of simulation also lends particular usefulness when considering low probability events and highly discontinuous or non-linear systems – things a person might be interested in capturing when investing their money.

Implementation

A few classes exist in the package that model different types of investment accounts, these accounts can be thought of as our functions $f$. A SavingsAccount, for instance, implements typical simple and compound interest over some period of time. By starting a simulation of placing $1,000 $\pm$ 100 into an account we can see how the amount grows and how the uncertainty grows as a function of the time and yield.

sa = InvestmentAccounts.SavingsAccount(1000. ± 100., 0.03/12)
sa2 = InvestmentAccounts.SavingsAccount(1000. ± 100., 0.07/12)
dates = 1:15*12

ribbonplot(dates, InvestmentAccounts.calculate_time_dependent_total_value(sa, dates), label="APY of 3%")
ribbonplot!(dates, InvestmentAccounts.calculate_time_dependent_total_value(sa2, dates), label="APY of 7%")

xlabel!("Months")
ylabel!("Value of Savings Account")

The ± in the code above indicate a gaussian distribution, but one in principle is interested in distributions beyond that. For instance, one could choose to model a higher-risk accounts return as bimodal, where you are more likely to make or lose more money and the chance the account value is unchanged is small. For the purposes of illustration, let’s model this as a univariate biased coin flip + gaussian: (-0.5 ± 0.1) ⊞ Bernoulli(0.7).

References

1. https://baggepinnen.github.io/MonteCarloMeasurements.jl/stable/