Random Equity Returns

Overview


The standard quantitative model of equities is the Geometric Brownian Motion. (see quantitative equity modeling).
{% d S(t) = \mu S(t) dt + \sigma S(t) dW(t) %}
  • {% \mu %} - the arithmetic average return
  • {% \sigma %} - the volatility

The analysis assumes that there is an equity index that the retiree has their money invested in. That is, we ignore the fact that the retiree could be investing in multiple different stocks and just approximate it by a single index.

Simulating an Ito Process


The Ito process can be simulated easily using the Ito library. (see Simulating Ito Process)

In order to simulate an Ito process, one must choose a drift and a volatility. Typically, the volatility can just be chosen to be the average equity volatility. The harder challenge is to choose a drift rate. This is typically also chosen to be equal to the average historical value. However, there are issues with this.

  • Small changes to the average return assumption can have big changes to the final value of the portfolio
  • If the drift is set to a constant average value, nothing stops simulated equity values from growing without bound over the given time frame. (see tehtering below) 



let ito = await import('/lib/statistics/simulations/v1.0.0/ito.mjs');

let withdrawal = 40000;
let wealth = 500000;
let data = [];
let log = Math.log(wealth);
for(let i=0;i<100;i++){ 
	let sims=ito.generate(50, S=>0.1*S[S.length-1]-withdrawal, S=>0.15*S[S.length-1], wealth)
	let index = -1;
	for(let j=0;j<sims.length;j++){
		if(sims[j]<=0) {
			index=j;
			break;
		}
	}
	data.push(index);
}
Try it!

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