# Valuing complex share incentives using the Monte Carlo Method

No one can predict the future, even though we try our utmost to secure the outcomes we want – we eat healthier, save for retirement, make investments, and plan our steps. Our decisions today take into consideration several underlying factors in our attempt to create certainty tomorrow.

Some decisions are based on chance, while others are based on historical statistics – for example, betting on the outcome of two consecutive rolls of the dice, or the expected share price at the end of your investment term. One statistical method used in calculating the probability of such an event, is known as the “Monte Carlo” methodology.

Although predicting future events is inherently impossible, mathematical techniques like the Monte Carlo simulation, offer an estimate based on historical patterns and the probability of uncertain occurrences, using random variables to predict where something will move next. In fact, the method derives its name from the renowned gambling destination of Monaco, as it possesses comparable random characteristics to a roulette game.

The model can be utilised to simulate path-dependent outcomes, finding application across a wide range of disciplines, from Astronomy to Finance. The technique used can be specific to, and differ for, each kind of predictive modelling.

## The workings

The formula used in the Monte Carlo Simulation to estimate future share prices was developed based on an observed process called Geometric Brownian Motion. To illustrate this process, imagine you’re on a playground, and you have a friend who’s blindfolded. They want to reach a certain point but can only take steps based on your instructions. You decide to give them a mix of instructions. Sometimes you tell them to take a big step forward, sometimes a small step forward, and sometimes a step backwards. However, you make it more likely for them to take small steps forward and less likely for them to take big steps forward or any steps backwards.

Geometric Brownian Motion is like studying how your blindfolded friend moves based on these instructions. They might take a few steps closer to their goal, but every now and then, they might step back a bit. Over time, they tend to make progress, but there’s still some randomness in the steps they take.

This idea is often used to understand how things change in value over time, like how the prices of stocks in the stock market go up and down with a general upward trend. In the share incentive plan world, Monte Carlo simulations (and Geometric Brownian Motion) play a crucial role in accurately evaluating share incentives, referenced against the performance of other publicly traded companies using their share prices. For these incentives, performance conditions must be met for the incentives to eventually vest, and the valuation of these plans is imperative so the company understands the value of what it is awarding to its employees.

## Valuing share plans using the Monte Carlo technique

Although the methodology is based on very complex principles that can take months to study, the application of the model is what’s of importance when valuing a share incentive plan. There are three main steps to running a Monte Carlo simulation, namely, collecting relevant historical market information to determine the ‘patterns’, simulating expected share price paths using the mathematical formulas and historical distributions, and evaluating the expected payoff of the plan by assessing the performance relative to the targets set.

#### i.          Collecting historic market-related data

The first step is to analyse the current market conditions at the valuation date. This snapshot of the market will be used to estimate the potential expected payoff of the incentive when it becomes available.

To gather the necessary information, we look at historical data on share price movements, expected company distributions (including the total shareholder returns), and current interest rate expectations. It is also important to consider the correlation between a company’s share prices and its industry peers, as similar companies tend to move in sync with each other.

By analysing these factors, we can set up a comprehensive Monte Carlo simulation that evaluates the potential outcomes of the incentive based on current market conditions.

#### ii.          Simulating expected share price paths

Geometric Brownian Motion estimates each daily price into the future using a drift representing a constant forward motion, like a breeze, that influences share prices. The risk-free rate determines the drift, as we expect money to grow at this rate over time.

Using random daily share price movements based on acquired data points, we can estimate tomorrow’s share price as well as the next day… and so on. This process continues at each iteration using the risk-free rate that provides constant upward or downward pressure until reaching a simulated price at the point of vesting.

This method is repeated many times and averaged to obtain an expected terminal share price and payoff.

#### iii.          Evaluating the results

The general principle in evaluating the results is determining the present value of the expected future payout of the share plan. The expected payout means assessing each simulated share price against any pre-defined performance targets set, and averaging this over the number of simulations performed.

There are numerous other complexities in assessing these performance conditions, such as peer group comparisons on a total shareholder return basis (“TSR”), and this will be addressed and tackled in another one of ShareForce’s blogs.

Once the expected payout has been determined, it is converted to a present-day amount known as the fair value.

## Do not leave it to chance

Monte Carlo simulations are often only marginally understood. Valuating and understanding the impact of changing market variables, and interpreting fair values can be complicated and complex, and is best left to the experts.

With ShareForce’s automated valuation engine that applies real-time market data, you can run Monte Carlo valuations and generate audit-ready IFRS2 or GAAP-compliant reports at the click of a button, and get results within seconds. To view a demo of the ShareForce Incentive Plan management tool, fill in the form below or visit www.shareforce.net/contact