Margin of safety vs odds from Monte Carlo simulation

Fundamentals 29. This article summarizes the insights gained from using Monte Carlo simulation for stock valuation. 

Are you tired of guessing when it comes to valuing companies? What if I told you that a simple tweak in your valuation approach could unlock a treasure trove of insights, giving you the edge over other investors? 

In the fast-paced world of finance, traditional point estimates may lead you astray, leaving you vulnerable to market surprises. Imagine confidently predicting a company’s true worth, not just with a single number, but through a dynamic range of scenarios that account for real-world uncertainties. 

Welcome to the world of Monte Carlo simulations! This powerful technique transforms your understanding of intrinsic value, allowing you to navigate investment opportunities precisely.

Some of the key insights from my Monte Carlo simulations are:

• If the margin of safety is between 0 % and 30 % from a point estimate, a Monte Carlo simulation can provide a better picture of whether the company is an investment opportunity.

• A Monte Carlo enables you to better handle situations where there are asymmetric distributions of the key variables.

Curious to discover how this innovative method can supercharge your investment strategy? Let’s dive into the game-changing approach that’s set to revolutionize how you view company valuations!

Contents

• Beyond point estimates

• Monte Carlo simulation

• Identifying the range and distributions

• My Montel Carlo Simulation approach

• Margin of Safety vs Monte Carlo simulation odds

• Conclusion

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Beyond point estimates 

I currently value companies using a discounted cash flow model with the following key variables – revenue, growth rate, margin, capital efficiency, and discount rate. The steps taken to determine the intrinsic value can be summarized as follows:

• I first carry out a detailed fundamental analysis covering both qualitative and historical trends. I also looked at competitors’ performance and industry analysis.

• I then determine the point estimates for each of the variables based on the results of the fundamental analysis. I consider this my base case.

• I feed the base case point estimates into my valuation model to get a point estimate of the intrinsic value. 

• Sometimes, in addition to the base case, I determine the optimistic and pessimistic intrinsic values. These are based on estimates of the appropriate optimistic and pessimistic values of the key variables.

Finally, I compare the intrinsic values with the market price to determine the margin of safety. I defined the margin of safety as = (intrinsic value / market price) – 1. The goal is to invest in those companies with a margin of safety greater than 30%.

The challenge with the above approach is that there is always some uncertainty about the point estimates of the various variables. Even the best-case and worse-case scenarios assume that all the variables have the best or worst values at the same time.

In reality, some variables are strongly correlated. Also, having the best or worst values at the same time are probably extreme cases that do not have the same likelihood of occurring as the base case.

A better approach is to do an unlimited number of scenarios and then see how the intrinsic values are distributed. This is the idea behind the Monte Carlo simulation.

Monte Carlo simulation

Imagine you want to determine the intrinsic value of a company, but you are not sure about the exact values of key variables like revenue, growth rate, and margin. Instead of picking just one number for each variable, Monte Carlo simulation allows you to explore many possible scenarios by randomly selecting values for each variable.

How Does It Work?

• Identify the variables. First, I list the key variables I use in the discounted cash flow model - revenue, growth rate, margin, capital efficiency, and discount rate.

• Define the ranges. For each variable, I define a range of possible values. For example, if I think revenue could realistically be between RM 1.1 million and RM 1.2 million, I set that range.

• Random sampling. The simulation randomly picks values for each variable from the defined ranges. It does this thousands of times to create different scenarios.

• Calculate values. For each scenario, the valuation model calculates the intrinsic value of the company.

• Analyse results. After running all these scenarios, I get a distribution of intrinsic values instead of a single-point estimate. This distribution helps me see a range of possible outcomes, rather than just the best-case or worst-case scenarios. 

The chart below shows the results of such a Monte Carlo distribution. Refer to “Is Hap Seng Consolidated an investment opportunity?”

• The point estimate resulted in an intrinsic value of RM 4.13 per share compared to the market price of RM 4.03 per share. This is only a 2% margin of safety.

• The Monte Carlo simulation shows the range of possible intrinsic values. There is a 0.7 to 1 odds that the intrinsic value is greater than the market price.

Chart 1: Monte Carlo simulation of intrinsic values.

Why Use the Monte Carlo simulation?

• Captures uncertainty. Unlike my original point estimate method, which assumes certain values for the variables, the Monte Carlo simulation reflects the uncertainty and variability of real life. It shows how likely different outcomes are based on random combinations of the variables.

• Identifies risk. I can see not just the average intrinsic value, but also the likelihood of various values occurring. This helps me understand the risk involved in investing in the company.

• Margin of safety. After running the simulation, I can determine how often the intrinsic value exceeds the market price. I then compare this with how often the intrinsic value is lower than the market price. This gives the odds of having the intrinsic value being higher than the market price. It provides a clearer picture of the risk.

In essence, a Monte Carlo simulation allows me to have a more comprehensive view of the potential value of a company by considering a wide range of possibilities and their probabilities. It is like taking a more realistic snapshot of the investment landscape instead of relying on a few fixed estimates. This method helps me better prepare for the uncertainties when valuing a company.

Identifying the range and distributions

You can see that there are 2 challenging steps in carrying out a Monte Carlo simulation:

• Establishing the range of each variable.

• Assessing how the values of a variable are spread out over a range. It indicates the likelihood of each possible outcome.

If you are able to assess the best and worst-case scenarios based on the fundamental analysis, you have already established the range. I must admit that it involves a mix of analysis and judgment. But this is the easier part of the 2 challenges.

It is likely that for a particular variable, not all the values within the range have a similar likelihood of it occurring. Some values may be more likely than say the extreme values. This is where you have to next determine the likelihood of each value. 

In layman's terms, this is called determining the probability distribution of the variable. A probability distribution describes how the values of a variable are spread out over a range. It indicates the likelihood of each possible outcome. 

In statistics, there are several common types of probability distributions.

• Normal distribution. This is often used for variables that are expected to cluster around a mean (like revenue growth rates in stable industries).

• Uniform distribution. This is used when all outcomes within a range are equally likely.

• Triangular distribution. This is useful when you have a most likely value along with minimum and maximum values. It visually represents the most likely scenario between the extremes.

• Log-Normal distribution. This is common for financial metrics, where values cannot be negative and are skewed to the right.

Challenges in identifying the distribution

How do you identify the probability distribution? An appropriate statistical approach is to look at the historical data and try to assess how they are distributed. 

Unfortunately, this can be challenging when looking at a company’s performance. For example, I based my valuation on the annual performance of companies. While in theory you can go back and get 20 to 30 years of annual data, the profile of the company would have changed over the years. 

Valuation is about projecting the future. As such even if I rely on historical data, I want to ensure that it reflects the future business profile. Once I take this into account, I will be lucky to be able to find 10 annual data points.

With 10 data points, it is meaningless to try to find the best-fit distribution. There are suggestions that I could have rolling-4 quarters performance and hence increase the number of data points. 

I have tried this and while better than 10 data points, there are separate issues with quarterly data. There may be seasonal patterns. You will have year end audit adjustments. Even if I can adjust for these, there are questions on whether the best-fit distribution is meaningful. 

Imagine finding the histogram with 15 bins with 40 data points and you will understand what I mean. In practice, I have even less than 10 annual data points. 

The chart below is from my analysis of the past 7 years' operating margin of Bursa FoundPac. I have chosen 10 bins for each case.

• The left part of the chart shows the histogram of the past 7 years' annual operating margin.

• The right part of the chart shows the histogram of the rolling 4 quarters' operating margin

Is this a left-skewed distribution? If skewed, does the skew come from the oldest data points and hence may not reflect the current business profile?

You can appreciate why I think trying to get the best-fit distribution with 10 annual data points or 40 quarterly annual points may not be meaningful.

Chart 2: Comparative histograms – Annual data vs rolling 4-quarters

Damodaran has suggested that you look at cross-section distributions. In other words, look at the industry distribution. Again, there are practical problems as many times, I already find it hard to identify 10 competitors.

My Monte Carlo simulation approach

The way I use the Monte Carlo simulation is shaped by the following.

• The outcomes are based on my estimates of the range of the various variables. I am not simulating the range of historical outcomes.

• There will always be limited data points. As such I focus on the triangular distribution as this is the best distribution when you have limited data points.

• The simulation is carried out in EXCEL using its random number generator function. The shape of the triangular distribution generated by my 2,000 runs is not what you imagine.

• I use a DCF model to estimate the intrinsic value based on the following key variables – revenue, growth rate, margin, capital efficiency, and discount rate. The outcome is some non-linear combination of these variables. 

Estimating the range

When carrying out the point estimates, the base values for the various variables are the expected values based on my analysis. 

There is also some uncertainty about the expected values. The range in my Monte Carlo simulation reflects my view of the uncertainty. 

I set the range in the context of this uncertainty. In most cases, I set the range and – 10 % to + 10% of the expected values.

• If I gauge that the range could be positively skewed, I set the range as – 5% to + 10 %. 

• If the range could be negatively skewed, I set the range as – 10% to + 5%.

In a sense, I am setting boundaries to the range. It is a judgment call based on the fundamental analysis.

I seldom look at the actual historical ranges as I am not simulating the historical outcome. I am catering to the uncertainty about the expected value. 

EXCEL’s Triangular distribution 

While there are several probability distributions to consider, I resorted to using the triangular distribution from a practical perspective. The triangular distribution is the best choice if you have limited data.

This is not so bad because when I use the EXCEL function to simulate the triangular function, I get the picture below. I suspect that I do not get a triangle shape with the EXCEL simulation because I only have 2,000 runs. It is possible that we 100,000 simulations, I could get a more triangular shape.

Chart 3: Simulated triangular distribution 

You can see that the picture is one with a uniform middle section with tapering ends. Depending on where I set the maximum and minimum values relative to the mode, I can get a positively or negatively skewed distribution.

I think this type of distribution can serve my Monte Carlo simulation.

There are inherent challenges in using historical data for future projections in company valuation. I believe that my approach of using a triangular distribution is a sensible compromise given the practical limitations. 

It allows me to incorporate assumptions about future performance while still acknowledging the uncertainty. Ultimately, the goal is to create a model that is useful for decision-making, and I would like to think that my strategy aligns with that objective.

Non-linear relationships

I carry out a Monte Carlo simulation of the value of a company based on the following DCF equation. 

Value = Earnings X (1-tax rate) X (1- Reinvestment rate) X (1 + growth rate) / (WACC - growth rate). 

The DCF formula involves multiplication and division, which introduces non-linear interactions among the variables. Non-linear functions can amplify or dampen the effects of input variations, leading to skewed output distributions.

• The variables interact in complex ways. For example, if the growth rate is high, it can disproportionately affect the value compared to when it is low.

• The WACC serves as a denominator in the equation. If WACC is near the growth rate, small changes in either can lead to large changes in the computed value, causing a concentration of outputs around certain values.

• Some parameters may have a more significant impact on the outcome than others. This sensitivity can create a distribution that heavily favours certain ranges of outputs.

Overall, the non-linear nature of the DCF equation and the varying sensitivity of the outputs to changes in those inputs lead to a non-uniform distribution of outcomes in the Monte Carlo simulation.

The Central Limit Theorem suggests that the sum of many independent random variables tends toward a normal distribution. But the model’s output is a complex function of those inputs rather than a simple sum. The resulting distribution may not resemble a normal distribution, depending on how those inputs interact.

Chart 4 illustrates this where I carried out a Monte Carlo simulation of a company with similar expected values and ranges. But they differ in the probability distribution.

• The left chart shows the results assuming a triangular distribution of – 10 % to + 10% of the various variables. 

• The right chart shows the same simulation but based on a uniform distribution for all the variables.

You can see that the shapes of the intrinsic values are similar. A uniform distribution of the inputs does not result in a uniform distribution of the output. 

This is another reason why I use the triangular distribution. It is simple and I can easily set the range to reflect the appropriate skewness. 

Chart 4: Comparative simulation - Triangular vs Uniform distribution

Overall, I would like to think that my approach reflects a practical way to overcome the various challenges in using Monte Carlo simulation. Remember that the Monte Carlo simulation is only a small part of my fundamental analysis. It is meant to provide additional insights and not replace the fundamental analysis. 

Interpreting the results

The goal of the Monte Carlo simulation is to understand the range of possible outcomes. As such I plot the histogram of the various intrinsic values and then estimate the odds that the intrinsic values are greater than the market price compared to the case when it is less than the market price.

This is estimated by dividing the frequency the values are greater than the market price by the frequency the values are less than the market price

The Charts below illustrate them.

Chart 5: Equal odds

Chart 6: 24 to 1 odds

Chart 7: < 1 odds

Margin of Safety vs Monte Carlo simulation odds

I carried out a DCF valuation of companies using both a point estimate and a Monte Carlo simulation. I then compared the results from the margin of safety with the odds from a Monte Carlo simulation. 

• The margin of safety was obtained by comparing the point estimate of the intrinsic value with the market price of the company. 

• The odds were computed by comparing the % of the values greater than the market price with the % less than the market price. 

The results are shown in Table 1 where I have grouped them into 3 based on the margins of safety. 

Table 1: Margin of safety vs Monte Carlo odds

There appears to be a positive correlation for many companies - higher margins of safety often correspond to higher odds. However, this is not universally true, as some companies with moderate margins exhibit low odds.

Here are some key points to consider:

• Negative margins of safety. Companies with negative margins of safety consistently have odds of 0. In these cases, the market prices are higher than the intrinsic values. The Monte Carlo simulations confirm this by showing less than 0.5 to 1 odds of intrinsic values exceeding market prices.

• Positive margins of safety of less than 27%. Companies with positive margins of safety of less than 27 % can have varying odds. 

• FoundPac has a margin of safety of 13% with odds of 10 to 1. This indicate a strong likelihood that its intrinsic value exceeds the market price in many simulations. 

• Stanley Black & Decker has a margin of safety of 11% but odds of 0. This suggests that even with a small margin, the probability of intrinsic value exceeding market price is low.

• High margins of safety. Companies like Can One and Star Bulk show both high margins of safety and high odds, which indicates a robust investment case.

I would interpret the results as follows.

• If there is no margin of safety, the odds of having intrinsic values greater than the market price are almost nil. 

• If the margin of safety is greater than 30 %, you are likely to get 3 to 1 odds that the intrinsic values are greater than the market price.

• If the margin of safety is between 0 to 30 %, a Monte Carlo simulation may weed out those that are poor investment opportunities. But you need to do this on a case-to-case basis.

If nothing else, it suggests that a 30% margin of safety from a point estimate of the intrinsic value is a reasonable one. 

Conclusion

In the investment world where uncertainty reigns, relying solely on traditional point estimates for company valuations can leave you exposed to unforeseen risks. By integrating Monte Carlo simulations into your valuation toolkit, you gain a powerful ally.  

The Monte Carlo simulation approach allows for a more nuanced understanding of potential outcomes. It can accommodate the complexities and variabilities inherent in financial markets. 

Through the insights gained from Monte Carlo simulations, you can identify a range of possible intrinsic values. You are not fixated on a single estimate, thereby enhancing your ability to gauge the margin of safety. It also provides you with a clearer picture of the risks involved.

As we have seen, the relationship between the margin of safety and the odds derived from simulations provide valuable guidance, helping you make more informed investment decisions.

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If the above article was useful, you can find more insights on how to make money in my e-book. The e-book is now available from Amazon, Kobo and Google Play.

PS: If you are in Malaysia or Singapore, the e-book can only be download from Kobo and Google Play. 

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I am not an investment adviser, security analyst, or stockbroker.  The contents are meant for educational purposes and should not be taken as any recommendation to purchase or dispose of shares in the featured companies.   Investments or strategies mentioned on this website may not be suitable for you and you should have your own independent decision regarding them. 

The opinions expressed here are based on information I consider reliable but I do not warrant its completeness or accuracy and should not be relied on as such. 

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