What Is Monte Carlo Simulation?

Written By: Carly Sanfilipo

Performance Share plans that are contingent on Relative Total Shareholder Return (“RTSR”) continue to gain in prevalence. However, the accounting rules under ASC 718 create financial reporting needs on the date of grant, as it is considered a market condition, and will generally require a Monte Carlo simulation. (For a full primer on performance award accounting, please see Accounting for Performance Awards).

A Monte Carlo simulation is a forecasting model comprised of mathematical algorithms that project future stock prices using a random number generator. Using widely accepted financial theorems to simulate all probable future stock price paths, we are able to estimate the value of a market condition. The underlying financial theory behind the simulation is under a risk-neutral framework (therefore consistent with other models used for stock-based compensation like the Black-Scholes formula). The intent of this article is to provide a basic understanding of Monte Carlo simulation, along with the required assumptions.

The first step is determining the assumptions and data that will be required in a Monte Carlo simulation, summarized in the chart below.

Data / AssumptionConsiderations
Effect on Fair Value
Current Stock Price
(for Company and each peer)
• Can be determined through virtually any financial data provider, and should be based on the accounting measurement date
• No effect on the Fair Value as a percentage of the Current Stock Price
Risk-Free Rate of Return• Based on U.S. zero coupon Treasuries, and should be based on the accounting measurement date
• Minimal, but fair value increases as the risk-free rate increases
Expected Dividend Yield
(for Company and each peer)
• Can be calculated based on historical or market expectations for future dividend yield
• Some companies will develop a spot dividend yield (not recommended due to volatility), or some over a period of historical time commensurate with the performance period, and others will be based on market analyst expectations
• If a company re-invests dividends in the TSR calculation, and pays dividend equivalents at Target, then a practical expedient can be applied of a 0% dividend yield
• None for companies that pay dividend equivalents
• If a company does not pay dividend equivalents during the performance period, there is a slight reduction in the fair value depending on the size of the dividend yield
Expected Volatility
(for Company and each peer)
• Should be selected consistent with process for other equity instruments (stock options, etc.) if possible
• Typically, historical volatility is calculated over a period of time commensurate with the performance period
• Implied volatility can be challenging when determining the validity for all other peers
• If your volatility goes up, the impact of higher possible stock prices is largely offset by an increased likelihood of underperformance
• As peer volatility goes up, fair value increases because outlier simulations become more frequent
Correlation Coefficient between peers• Typically, correlation coefficients are calculated over a historical period of time commensurate with the performance period and consistent with the prices used to calculate expected volatility
• The higher the correlation coefficients, then the lower the valuation due to fewer outlier simulations
Stochastic modeling via Monte Carlo simulation• Higher volatility companies will require more simulations to achieve lower levels of "simulation error"
• Random simulation should be compared against a closed-form projected stock price formula to determine magnitude of simulation error
• Typically 1% or lower error is desirable through the random simulation process

Once you’ve determined the assumptions required for the model, the basic process of a Monte Carlo simulation for a RTSR valuation is as follows:

STEP 1: Project the stock prices over the performance period for the company and each peer company (or Index). Prices fluctuate based on the random number generator and the risk-free rate, with the magnitude of change based on each company’s volatility and correlation to the peers or Index.

STEP 2: Determine the TSR of each company or Index based on the simulated ending average prices compared to the beginning average prices.

STEP 3: Determine the rank of the company compared to the peers, or determine the outperformance compared to the Index. Compare the rank or outperformance to the payout schedule to determine the payout.

STEP 4: The present value for one simulation of the model is equal to the product of the payout percentage, the company’s ending price, and the present value factor based on the length of time and the risk-free rate.

STEP 5: The process is repeated hundreds of thousands of times to minimize the error (and should try to ensure that the simulation error is less than 1%).  The average value of all the simulations is the accounting fair value.

The valuation of a RTSR performance share is complex with many variables that ultimately determine the fair value. Additionally, there can be a wide variance in valuation outcomes that can become troublesome for both the financial reporting under ASC718, as well as how you develop your target grant sizes.


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