Explaining the Monte Carlo Simulation methodology for valuing equity

It has not been unusual for chairman and remco chairs to be flummoxed when questioned on the fair value basis sometimes used for granting employee equity, and appearing in statutory remuneration tables and notes to the accounts.

This article aims to provide the directors with an explanation that has a bit of history.

During a summer’s evening, back in 1913 at the tables within the Monte Carlo casino, a roulette wheel landed on black 26 times in a row.

If you had started with an initial bet of 5 francs, and doubled up every time – until a particularly well-timed drink break at the 27th spin, you could have ended up with 167,772,160 francs (provided there were no table limits and you weren’t dragged out the back for a walloping beforehand). On the other hand, if you had tried to apply the same tactic on red, you would have handed over the keys to your fresh out of the factory Bugatti Type 13 (see HERE) many spins before the 26th.

Let us think of these 26 spins on the 18th of August 1913 collectively as a single outcome within millions of possible combinations of red, black and the dreaded green zero. For example, when we simulate the path of a stock price over 26 months, we may observe an outcome whereby the price rises continuously each month, or an outcome whereby the price falls continuously each month. So too, the price may remain flat, or slightly up, or slightly down – there are many millions of possible outcomes of a stock price along a path. The point of Monte Carlo simulation is to try and replicate these 26 spins, or months, millions of times in order to achieve the expected outcome. Those who have sat at a roulette table long enough will know that the expected outcome of landing on black once is 18/37 (the number of black numbers divided by the total number of numbers which includes the depressing green zero). The chance of landing on black 26 times in a row is 18/37 to the power of 26, or 73 in 10 billion times.

Unfortunately for stock prices, modelling the possible outcomes is not as straightforward. Several smart individuals throughout history helped derive the mathematics behind the possible outcomes of stock prices.

We start with Robert Brown, a Scottish botanist and palaeobotanist born in 1773. He was examining grains of pollen of the pinkfairies plant under a microscope, when he observed that the minute particles ejected from the pollen grains executed a continuous random motion. With this powerful observation, Brownian motion was discovered.

Brownian motion was eventually applied to stock price movement (known formally as Geometric Brownian motion). Don’t worry, we will not be diving into the mathematics behind that transformation. However, one who did was Norbert Wiener, an American mathematician prodigy born in 1894.

Norbert Wiener graduated from high school at the age of 11, and by the age of 14, was awarded his Bachelor of Arts in mathematics. At 17 years old, Wiener received a Ph.D. from Harvard for a dissertation on mathematical logic. The Wiener process, named in his honour, is the mathematical process commonly used today to simulate a stock’s return. In a simple example, think of plugging in a stock’s price, its volatility and the risk-free rate into the Wiener process – and out pops another random stock price. Do this 26 times (for example) whilst continuously updating the formula with your new stock price, and you have a single stock price outcome within countless different possible outcomes.

What we aim to achieve by utilising a Monte Carlo simulation, is the expected outcome. We simulate a stock’s returns over a specified time period, and record this as a single outcome. Then, with help from powerful computer programs, the process is repeated up to millions of times. Finally, we can take the average across all of the recorded outcomes to arrive at the expected value of the stock.

In relation to equity grants, there are specific instances that require a simulation (or similar technique) to determine the fair value. These include grants that vest based on share price appreciation or total shareholder returns (the most commonly used LTI measure in Australia). These are referred to as market measures. In addition to the requirement to use simulations, there are distinct advantages to using them. Because the simulation runs over a defined time period, the inputs can be varied over the that time. This allows a more sophisticated approach to valuing equity as, for example, volatility can be varied (i.e. decay) from the current level to a long-term level over the simulation period.

By applying a performance test and recording the vesting outcome and stock price within each simulation, we could arrive at a fair (expected) value by taking the average across all simulations. This is the methodology used to determining the fair value of equity grants that incorporate a market measure.

© Guerdon Associates 2024
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