Monte Carlo simulation is a computerized mathematical technique to generate random sample data based on some known distribution for numerical experiments. This method is applied to risk quantitative analysis and decision making problems. This method is used by the professionals of various profiles such as finance, project management, energy, manufacturing, engineering, research & development, insurance, oil & gas, transportation, etc.
This method was first used by scientists working on the atom bomb in 1940. This method can be used in those situations where we need to make an estimate and uncertain decisions such as weather forecast predictions.
Monte Carlo Simulation ─ Important Characteristics
Following are the three important characteristics of Monte-Carlo method −
- Its output must generate random samples.
- Its input distribution must be known.
- Its result must be known while performing an experiment.
Monte Carlo Simulation ─ Advantages
- Easy to implement.
- Provides statistical sampling for numerical experiments using the computer.
- Provides approximate solution to mathematical problems.
- Can be used for both stochastic and deterministic problems.
Monte Carlo Simulation ─ Disadvantages
- Time consuming as there is a need to generate large number of sampling to get the desired output.
- The results of this method are only the approximation of true values, not the exact.
Monte Carlo Simulation Method ─ Flow Diagram
The following illustration shows a generalized flowchart of Monte Carlo simulation.
If you are near or in retirement, you must have consulted with at least one would-be financial advisor. If not, you may have run a retirement planning program on one of the brokerage sites. Likely you saw curves from 10,000 Monte Carlo simulations of some recommended plan, with a conclusion that your funds would have a 90% chance of surviving for 30 or 40 years. What does this mean? How sensible was this?
You also may have read article or books by Wade Pfau or others, or seen Seeking Alpha articles by me or others, employing Monte Carlo simulations and showing their results. How should you think about these?
This article is devoted to explaining what Monte Carlo simulations are and how they can be useful in financial planning. It also strongly criticizes the way Financial Advising firms often use them. My goal is to leave you better prepared to understand and apply their results.
What Are Monte Carlo Simulations?
Monte Carlo simulations are used in many fields of human activity. Here we are concerned only with their application to financial planning. Monte Carlo methods work for cases where one cannot know what the outcome will be but does understand something about the average return (the mean) and its variations.
Before one can do a simulation, one needs, at minimum, the mean return and the standard deviation corresponding to the history of each investment. These represent their statistical behavior. If one has a sequence of returns, Excel has functions for each of these.
One also needs to know or to assume the shape of the likely outcomes. This is generally taken to be the classic bell-shaped curve, whose fancy name is the 'Normal Distribution'. One can find discussions of alternative distributions. The daily behavior of the stock market is not accurately described by the Normal Distribution, but the average annual behavior is described well enough that way, as we will see.
A single simulation works as follows, marching forward in time. We will here assume steps of one year. One starts with an initial portfolio, with some percentage in each investment. For the first year, one takes a random result from the Normal Distribution to find the annual return of a given investment, based on its mean and standard deviation. Doing this for each investment in a portfolio produces the end-of-year values of the investments. Then one makes the planned adjustments. These typically would involve withdrawals and rebalancing. One then repeats this process for the next year, then the next, and so on to the end of the period being modeled.
A single simulation like that just described has little value by itself. It shows one possible path through the future. What has value is collecting the statistics that result from doing many such simulations. Then one can see the average long-term gain for the portfolio, and the distribution of anticipated outcomes. This prediction of the future will not be accurate if the statistical behavior of the investments changes significantly. We will return to this point later.
An alternative approach, producing similar results, is to evaluate the probabilities by sampling the actual historical record. This is discussed in detail in Pfau's book.
Application to U.S. Stocks
Let's work through this process for U.S. stocks. We are fortunate that real annual returns and much other data can be found on Professor Shiller's website. Figure 1 shows the annual (real) returns, binned into intervals of 5% in return. This plot emphasizes the actual variability, even compared to the statistical model. The histogram looks much prettier if binned into intervals of 10%, which is what you tend to find on websites. This might satisfy PR departments and comfort frightened investors. The binning shown here emphasizes the potential for your experience to differ from the model.
The Normal Distribution (the blue curve) is a good representation of the data. If you use it in a Monte Carlo simulation and draw 147 years of returns, the histogram usually shows deviations from the blue curve that are of the same magnitude as those you see in the actual data. Where they appear differs from one simulation to the next.
Figure 1. Distribution of real returns of U.S. Stocks, from 1871 through 2018. Data from Shiller Plot by author.
What one actually knows about any investment is that the return will be within one standard deviation of the mean return about two-thirds of the time. In the case of U.S. stocks, this is between -9.3% and + 26.7%. The region beyond one standard deviation from the mean is known as the 'tails' of the distribution. One does not have enough data to know much about the relative likelihood of annual returns in the tail. It would not matter if you did, because you only get to personally experience actual returns for a few decades. There is no way to predict what the actual returns of your investment will be. One can say that most of the time it will fall in a certain range, if the future statistical behavior of that investment does not change.
This is a good place to make an additional point. Sampling about 100 cases, as the historical record does, is enough to see the main trends in the response. In modeling though, one has a choice. One can do 100 cases, 1,000, 10,000, or some other number. As one increases the number, the histograms get smoother. Their prettier appearance can make them seem very precise and scientific. Do not let their pretty smoothness make you think that things are actually so well known. The plots shown below are from sets of 10,000 simulations.
Performance of a Portfolio of U.S. Stocks
Above we looked at the distribution of one-year returns from U.S. stocks. Now we consider supporting retirement from a portfolio of U.S. stocks. We pull out some funds each year, according to some rule, to support spending in retirement. A single simulation will sample the distribution of returns 40 times in sequence, getting one possible result for how things would work out. Then one repeats this many times and looks at the statistics of the results.
Figure 2 shows a histogram of results, assuming that the real rate of withdrawal is 4% of the initial portfolio balance each year. The leftmost bar shows the fraction of cases for which the portfolio will have run out of money. The next bar shows the chance that the final value of the portfolio will be positive, but no greater than its initial value. And so on. One and see that the retiree is very likely to end retirement with a net wealth that far exceeds the value at the start of retirement. Yet there is also some chance that the portfolio will become exhausted, if the withdrawal rule is slavishly followed.
Figure 2. Histogram of outcomes for a portfolio invested only in U.S. stocks. The annual withdrawal, in constant dollars, is 4% of the initial portfolio value. Model and plot by author.
It is difficult to judge from Figure 2 just what is the fractional likelihood of various outcomes. The standard way to see this is with what is called a Cumulative Distribution Function (CDF). The CDF starts at the left of the histogram and moves rightward, keeping track of the total probability of a result further to the left. Figure 3 shows the CDF for the histogram of Figure 2. One can see that one has a 50% chance of ending retirement with more than about five times the initial portfolio value. One also has just under a 20% chance of running out of funds, if one blindly keeps spending in the event that bad luck makes the portfolio steadily shrink.
Figure 3. Results for a portfolio of U.S. Common Stocks, whose statistical behavior is that of a Normal Distribution based upon the historical returns. The real fraction of the initial portfolio withdrawn each year is 4%. The left plot shows the probability that the portfolio value is below the number shown on the abscissa. The right plot shows, for those cases for which funds are exhausted, the year in which this occurs. Models and plots by author.
My first reaction to this plot is sign me up! In the small likelihood that my portfolio is shrinking, I will have to adjust. But in most cases I will get to live better and also leave a nice legacy. But for the most part, humans fear loss far more than they embrace gain.
Normal human reactions have a lot to do with what advisors can sell. Many people cannot tolerate, psychologically, the sequences of down years that stocks sometimes produce. Many people cannot accept a model that has more than a ten percent chance of 'failure', if that. This is true even though the failure is artificial. On commenter on Seeking Alpha was unwilling to pursue a path that had more than a million-to-one chance of failure.
Beyond that, there are no guarantees that the statistics that create Figures 2 and 3 will be those that apply in the future. This is, in my view a far bigger risk than the risk of a sequence of very bad years. Fortunately, one can structure one's portfolio to reduce the statistical likelihood of failure while also reducing the impact of the limitations of the statistics. We will explore this in a next article.
Figure 3 also shows, on the right, the probability that the funds will have run out by any given year. I find this second plot helpful, though I've never seen it in material from a financial advisor. My personal point of view is that I don't care so much about sustaining my spending level in my 3rd and perhaps 4th decade of retirement, if I even make it that far. These are the classic 'no-go' years.
The upshot is this. If 1) a 4% withdrawal rate would support the spending I want to do, and 2) I was sure that the future returns of the stock market would resemble those of the past, then I would cheerfully throw my money into index funds and think about something other than investments to stave off Alzheimer's.
I also did a test in which I forced the annual return to lie within the body of the distribution, not permitting it to lie on the tails. There were almost no failures. The lesson is this: the cases that require a change of spending plans are those where the returns repeatedly are much less than anticipated from the historical record. If that happens to you, you will be unlucky. The question for you is whether to live your life in fear of being unlucky or in the expectation that you will probably experience returns within the top 80% of the likely range.
Figure 4 shows one aspect of what can go wrong beyond the results of the standard model above. Many commentators believe that stock returns in coming decades will not resemble the historic rates. There are a number of arguments for this. One of them is that the need to refinance the Federal debt in the U.S. crowds out investment that can raise productivity, limiting future growth. Another one is that easing by the Federal Reserve has driven stock prices to unreasonable heights. There are others. Some of them seem sensible to me.
So let us suppose that the average real returns from U.S. stocks in the next few decades is 6%. Figure 4 shows what that does to the outcomes of the Monte Carlo simulations. Now roughly 1/3 of the time the portfolio runs low on funds. Yet more than half the time one's legacy will have more money than one had upon retirement. Even so, one would prefer to reduce the risk of running low on funds, even though this will mean reducing the maximum potential legacy. I will discuss some options for this in Part II.
Figure 4. Results for a portfolio of U.S. Common Stocks, whose statistical behavior is that of a Normal Distribution with an average return of 6%. The left plot shows the probability that the portfolio value is below the number shown on the abscissa. The right plot shows, for those cases for which funds are exhausted, the year in which this occurs. Models and plots by author.
Conclusions
This article was stimulated by an exchange with a commenter on Seeking Alpha. That particular individual wanted absolute certainty that his portfolio would never be exhausted. He missed the point. One uses a withdrawal rule in modeling a portfolio in order to assess the range of likely outcomes. One can prevent the exhaustion of one's funds by limiting withdrawals to some maximum fraction of the remaining portfolio, and making lifestyle adjustments if necessary. Only an idiot would let a fixed withdrawal rule propel him into poverty. (I say him deliberately, as this level of dumb behavior is far easier to imagine from a male than a female.)
In my view, Monte Carlo simulations are useful for seeing the average behavior that will result from a specific approach to working with some collection of investments. They are also useful for seeing the spread of likely outcomes. Your own experience is most likely to fall in the range of 20% to 80% cumulative probability.
We don't know the tails of the probability distributions, and especially not the distributions that you will experience, well enough to say much about the differences between 5%, 10%, and 15% probability. So when a financial advisor tells you about 90% chances, just nod and smile. That is unless you feel like asking what probability distribution was used and what accuracy they assign to its tails.
To my own mind, it makes sense to design your portfolio so that it will meet your needs if the investment outcomes lie within the 'one-sigma' range of negative statistical outcomes, roughly above 20% on the cumulative distribution. One can hope to do much better. One can hope to beat the average statistics, by intelligent selection of investments and by monitoring them so one can get out ahead of problems. Alas, one also might encounter investment performance that is worse than the lowest one's plans can accommodate. If and as this happens, one will have to change one's plans. But living in fear of very poor investment performance seems a to me a wasteful way to spend one's retirement.
Even so, one can reduce the odds that one will have to change course, by diversification. That will be the subject of Part II.
Disclosure:I/we have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours.I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.
Additional disclosure: I am not a financial adviser or a tax advisor, but am an independent investor. Any securities or classes of securities mentioned are not recommendations.
(Last Updated On: 26 March, 2019)Monte Carlo analysis is a statistical way to analyze a circuit. This simulation allows us to test the process variation and mismatching between devices in a single chip or wafer.
Contents
- Example of Monte Carlo simulation in Cadence with ADE-XL
What is the difference with the CORNER ANALYSIS?
The Corner analysis simulates your design with the minimum and maximum value of each parameter. But it does not reproduce the mismatching between devices! The corner analysis makes the problem much harder than it really is. It simulates extreme cases, that in real fabrication process will never occur, because of the correlation between parameters are not taken in account.
A basic corner analysis can be around 65 simulations, taking the maximum and minimum of the process variables:
- CMOS thickness: wp, ws, wo, wz.
- Resistor value: wp, ws.
- Capacitor value: wp, ws.
- Temperatures: (typ.)-20 to 85ºC
- Voltage supply: depend on your supply source, etc.
Typical number of simulations: tm+ 4CMOS*2RES*2CAP*2TEMP*2Vsupply = 65
*Where:
- ws = worse speed
- wp = worse power
- wo = worse one
- wz = worse zero
For example, if the CMOS gate oxide thickness is big for NMOS, it will be also big for the NMOS. It is not realistic to simulate the case when the thickness of the NMOS is minimum and the PMOS is maximum.
Corner analysis guarantees that the circuit will work, of course, under all possible consideration, but it overdesigns your circuit.
Monte Carlo Simulation in Circuit Design
Monte Carlo analysis is based on statistical distributions. It simulates mismatching and process variation in a realistic way. On eachsimulationrun, it calculates every parameter randomly according to a statistical distribution model. With this analysis, you will see in which region your circuit will work most of the time.
The drawback of Monte Carlo is the large amount of simulations required to have acceptable results. It should be at least 250 to have a significant sample, but the minimum amount of simulations is not trivial (it follows the statistical model), but as a thumb rule, the more simulation the more significant the test is.
The amount of simulations to run with Monte Carlo, is much higher comparison than the Corner analysis.
Coming back to the previous example with the gate oxide thickness:
In the corner analysis it is simulated the worst case of the minimum and maximum values of the thickness parameter resulting in a variation of the threshold voltage:
But in real fabrication cases, if the NMOS is thin, the PMOS transistor will have also similar thickness. This area is shown in the yellow region.
Another type of corner analysis is known as “statistical corner models”, here thousands of real produced wafers are measured. Now the statistical corner analysis considers correlations of map parameters. This analysis improves the Monte Carlo pure statistical method with the feedback of the real wafer measurements.
Actualization: Example of Monte Carlo simulation in Cadence
In this example, a clock is going to be simulated. This clock has a configurable frequency output from 0.84MHz to 1.88MHz depending on a digital input of 4 bits (16 steps).
First, we make sure that the simulation is working fine in nominal conditions and try to shorten the simulation time as much as possible. If you want to run hundreds or even thousands of simulations, you have to optimize your computational resources.
- Reducing the amount of time (in the ~microseconds range normally) to be simulated for each run.
- Adjusting carefully the resolution steps.
- Selecting only the minimal amount of nets to be saved. But they have to be the minimum necessary to interpret the results and detect where the problem may come from.
In the following picture, the nominal results for the clock are shown. The frequency f_0 and f_15 are the values to be tested on the simulation. The nominal values are 836kHz and1.89MHz. Let’s see later how much do they change with the process variations…
Before Monte Carlo, normally a corner simulation is performed, in order to have a preview of the worse case and see the bounded of the Monte Carlo simulation. For this example, the Process Corners simulations looked like:
To proceed with the Monte Carlo:
First, select “Monte Carlo Sampling” (as the picture below shows) and choose the number of simulations on the configuration window:
Later go to the Corners set-up, as shown in the picture below, and choose the parameters you want to vary, Usually, the temperature and other parameters. In my case, I want variations on the temperature (-20-to+85°C) and in VDD (the power supply from 1.1V to 1.3V). Then depending on the technology used, you will have another Monte Carlo model files.
Deactivate the nominal simulation because we already run and saved it. Then, run the Monte Carlo pressing the green play button.
The results “Yield” window will appears similar to the picture below. In this window, the average, standard deviation, and other statistical parameters are shown.
In Monte Carlo, a histogram plot is used to see graphically the results. For this, click on the historiogram icon and then select “histogram” as shown in the screen shot below.
A new window will prompt, and there you can play with the parameters or selecting which data you want to plot. If you want to combine all the cases you plotted (in my case 4 cases because of 2 temperatures and 2 VDD). You can press “Combine” to plot all together as shown in the next picture:
Then you can export the graph choosing a white background. This is very useful and will give you way better appearance when you insert it in your documentation or reports:
Example of Monte Carlo simulation in Cadence with ADE-XL
In this other example, a cascade circuit is going to be used to make the Monte Carlo simulation.
First, add the Dc- simulation to save the operation point of the circuit.
To have a good performance, you can put many jobs in parallel, depending on how many cores does your machine has. For this example I was using a 8 core machine, so I set 8 simulations in parallel.
Go to Options>> Job setup.
Make a DC-simulation a plot the currents.
Add this values from the calculator into the output window in the ADE XL.
Copy this expression and paste them on the ADE XL.
You can add them by Outputs>>Setup
You should get something like. Delete every corner and leave one, that we are going to use as a reference.
We want to make the Monte Carlo Simulation for the nmos transistor, we chose the model cmosmc (Monte Carlo). If we wanted the Monte Carlo for the resistor you should chose “resmc”.
Select it on the Data View.
Go to options for the Monte Carlo Simulation
Set the configuration parameters. Set to process if you want to check the changes on the process in the electrical parameters. If you want to test the changes by matching on the transistors, select then Mismatch. And the number of points, start with something low, for testing, then you can increase it to a larger amount of simulations.
After the simulation, you can right-click and select “historiogram” to plot the results.
For a larger number of simulations:
Related Articles:
What is a Monte Carlo Simulation?
Monte Carlo simulations are used to model the probability of different outcomes in a process that cannot easily be predicted due to the intervention of random variables. It is a technique used to understand the impact of risk and uncertainty in prediction and forecasting models.
Monte Carlo simulation can be used to tackle a range of problems in virtually every field such as finance, engineering, supply chain, and science.
Monte Carlo simulation is also referred to as multiple probability simulation.
Monte Carlo Simulation
Explaining Monte Carlo Simulations
When faced with significant uncertainty in the process of making a forecast or estimation, rather than just replacing the uncertain variable with a single average number, the Monte Carlo Simulation might prove to be a better solution. Since business and finance are plagued by random variables, Monte Carlo simulations have a vast array of potential applications in these fields. They are used to estimate the probability of cost overruns in large projects and the likelihood that an asset price will move in a certain way. Telecoms use them to assess network performance in different scenarios, helping them to optimize the network. Analysts use them to assess the risk that an entity will default and to analyze derivatives such as options. Insurers and oil well drillers also use them. Monte Carlo simulations have countless applications outside of business and finance, such as in meteorology, astronomy and particle physics.
Monte Carlo simulations are named after the gambling hot spot in Monaco, since chance and random outcomes are central to the modeling technique, much as they are to games like roulette, dice, and slot machines. The technique was first developed by Stanislaw Ulam, a mathematician who worked on the Manhattan Project. After the war, while recovering from brain surgery, Ulam entertained himself by playing countless games of solitaire. He became interested in plotting the outcome of each of these games in order to observe their distribution and determine the probability of winning. After he shared his idea with John Von Neumann, the two collaborated to develop the Monte Carlo simulation.
Example of Monte Carlo Simulations: The Asset Price Modeling
One way to employ a Monte Carlo simulation is to model possible movements of asset prices using Excel or a similar program. There are two components to an asset's price movements: drift, which is a constant directional movement, and a random input, which represents market volatility. By analyzing historical price data, you can determine the drift, standard deviation, variance, and average price movement for a security. These are the building blocks of a Monte Carlo simulation.
To project one possible price trajectory, use the historical price data of the asset to generate a series of periodic daily returns using the natural logarithm (note that this equation differs from the usual percentage change formula):
Periodic Daily Return=ln(Previous Day’s PriceDay’s Price)
Next use the AVERAGE, STDEV.P, and VAR.P functions on the entire resulting series to obtain the average daily return, standard deviation, and variance inputs, respectively. The drift is equal to:
Drift=Average Daily Return−2Variancewhere:Average Daily Return=Produced from Excel’sAVERAGE function from periodic daily returns seriesVariance=Produced from Excel’sVAR.P function from periodic daily returns series
Alternatively, drift can be set to 0; this choice reflects a certain theoretical orientation, but the difference will not be huge, at least for shorter time frames.
Next obtain a random input:
Random Value=σ×NORMSINV(RAND())where:σ=Standard deviation, produced from Excel’sSTDEV.P function from periodic daily returns seriesNORMSINV and RAND=Excel functions
The equation for the following day's price is:
Next Day’s Price=Today’s Price×e(Drift+Random Value)
To take e to a given power x in Excel, use the EXP function: EXP(x). Repeat this calculation the desired number of times (each repetition represents one day) to obtain a simulation of future price movement. By generating an arbitrary number of simulations, you can assess the probability that a security's price will follow given trajectory. Here is an example, showing around 30 projections for the Time Warner Inc's (TWX) stock for the remainder of November 2015:
The frequencies of different outcomes generated by this simulation will form a normal distribution, that is, a bell curve. The most likely return is at the middle of the curve, meaning there is an equal chance that the actual return will be higher or lower than that value. The probability that the actual return will be within one standard deviation of the most probable ('expected') rate is 68%; that it will be within two standard deviations is 95%; and that it will be within three standard deviations is 99.7%. Still, there is no guarantee that the most expected outcome will occur, or that actual movements will not exceed the wildest projections.
Crucially, Monte Carlo simulations ignore everything that is not built into the price movement (macro trends, company leadership, hype, cyclical factors); in other words, they assume perfectly efficient markets. For example, the fact that Time Warner lowered its guidance for the year on November 4 is not reflected here, except in the price movement for that day, the last value in the data; if that fact were accounted for, the bulk of simulations would probably not predict a modest rise in price.
If you are near or in retirement, you must have consulted with at least one would-be financial advisor. If not, you may have run a retirement planning program on one of the brokerage sites. Likely you saw curves from 10,000 Monte Carlo simulations of some recommended plan, with a conclusion that your funds would have a 90% chance of surviving for 30 or 40 years. What does this mean? How sensible was this?
You also may have read article or books by Wade Pfau or others, or seen Seeking Alpha articles by me or others, employing Monte Carlo simulations and showing their results. How should you think about these?
This article is devoted to explaining what Monte Carlo simulations are and how they can be useful in financial planning. It also strongly criticizes the way Financial Advising firms often use them. My goal is to leave you better prepared to understand and apply their results.
What Are Monte Carlo Simulations?
Monte Carlo simulations are used in many fields of human activity. Here we are concerned only with their application to financial planning. Monte Carlo methods work for cases where one cannot know what the outcome will be but does understand something about the average return (the mean) and its variations.
Before one can do a simulation, one needs, at minimum, the mean return and the standard deviation corresponding to the history of each investment. These represent their statistical behavior. If one has a sequence of returns, Excel has functions for each of these.
One also needs to know or to assume the shape of the likely outcomes. This is generally taken to be the classic bell-shaped curve, whose fancy name is the 'Normal Distribution'. One can find discussions of alternative distributions. The daily behavior of the stock market is not accurately described by the Normal Distribution, but the average annual behavior is described well enough that way, as we will see.
A single simulation works as follows, marching forward in time. We will here assume steps of one year. One starts with an initial portfolio, with some percentage in each investment. For the first year, one takes a random result from the Normal Distribution to find the annual return of a given investment, based on its mean and standard deviation. Doing this for each investment in a portfolio produces the end-of-year values of the investments. Then one makes the planned adjustments. These typically would involve withdrawals and rebalancing. One then repeats this process for the next year, then the next, and so on to the end of the period being modeled.
A single simulation like that just described has little value by itself. It shows one possible path through the future. What has value is collecting the statistics that result from doing many such simulations. Then one can see the average long-term gain for the portfolio, and the distribution of anticipated outcomes. This prediction of the future will not be accurate if the statistical behavior of the investments changes significantly. We will return to this point later.
An alternative approach, producing similar results, is to evaluate the probabilities by sampling the actual historical record. This is discussed in detail in Pfau's book.
Application to U.S. Stocks
Let's work through this process for U.S. stocks. We are fortunate that real annual returns and much other data can be found on Professor Shiller's website. Figure 1 shows the annual (real) returns, binned into intervals of 5% in return. This plot emphasizes the actual variability, even compared to the statistical model. The histogram looks much prettier if binned into intervals of 10%, which is what you tend to find on websites. This might satisfy PR departments and comfort frightened investors. The binning shown here emphasizes the potential for your experience to differ from the model.
The Normal Distribution (the blue curve) is a good representation of the data. If you use it in a Monte Carlo simulation and draw 147 years of returns, the histogram usually shows deviations from the blue curve that are of the same magnitude as those you see in the actual data. Where they appear differs from one simulation to the next.
Figure 1. Distribution of real returns of U.S. Stocks, from 1871 through 2018. Data from Shiller Plot by author.
What one actually knows about any investment is that the return will be within one standard deviation of the mean return about two-thirds of the time. In the case of U.S. Download preset smoth transition 2018 smuggler. stocks, this is between -9.3% and + 26.7%. The region beyond one standard deviation from the mean is known as the 'tails' of the distribution. One does not have enough data to know much about the relative likelihood of annual returns in the tail. It would not matter if you did, because you only get to personally experience actual returns for a few decades. There is no way to predict what the actual returns of your investment will be. One can say that most of the time it will fall in a certain range, if the future statistical behavior of that investment does not change.
This is a good place to make an additional point. Sampling about 100 cases, as the historical record does, is enough to see the main trends in the response. In modeling though, one has a choice. One can do 100 cases, 1,000, 10,000, or some other number. As one increases the number, the histograms get smoother. Their prettier appearance can make them seem very precise and scientific. Do not let their pretty smoothness make you think that things are actually so well known. The plots shown below are from sets of 10,000 simulations.
Performance of a Portfolio of U.S. Stocks
Above we looked at the distribution of one-year returns from U.S. stocks. Now we consider supporting retirement from a portfolio of U.S. stocks. We pull out some funds each year, according to some rule, to support spending in retirement. A single simulation will sample the distribution of returns 40 times in sequence, getting one possible result for how things would work out. Then one repeats this many times and looks at the statistics of the results.
Figure 2 shows a histogram of results, assuming that the real rate of withdrawal is 4% of the initial portfolio balance each year. The leftmost bar shows the fraction of cases for which the portfolio will have run out of money. The next bar shows the chance that the final value of the portfolio will be positive, but no greater than its initial value. And so on. One and see that the retiree is very likely to end retirement with a net wealth that far exceeds the value at the start of retirement. Yet there is also some chance that the portfolio will become exhausted, if the withdrawal rule is slavishly followed.
Figure 2. Histogram of outcomes for a portfolio invested only in U.S. stocks. The annual withdrawal, in constant dollars, is 4% of the initial portfolio value. Model and plot by author.
It is difficult to judge from Figure 2 just what is the fractional likelihood of various outcomes. The standard way to see this is with what is called a Cumulative Distribution Function (CDF). The CDF starts at the left of the histogram and moves rightward, keeping track of the total probability of a result further to the left. Figure 3 shows the CDF for the histogram of Figure 2. One can see that one has a 50% chance of ending retirement with more than about five times the initial portfolio value. One also has just under a 20% chance of running out of funds, if one blindly keeps spending in the event that bad luck makes the portfolio steadily shrink.
Figure 3. Results for a portfolio of U.S. Common Stocks, whose statistical behavior is that of a Normal Distribution based upon the historical returns. The real fraction of the initial portfolio withdrawn each year is 4%. The left plot shows the probability that the portfolio value is below the number shown on the abscissa. The right plot shows, for those cases for which funds are exhausted, the year in which this occurs. Models and plots by author.
My first reaction to this plot is sign me up! In the small likelihood that my portfolio is shrinking, I will have to adjust. But in most cases I will get to live better and also leave a nice legacy. But for the most part, humans fear loss far more than they embrace gain.
Explain What A Monte Carlo Simulation Is
Normal human reactions have a lot to do with what advisors can sell. Many people cannot tolerate, psychologically, the sequences of down years that stocks sometimes produce. Many people cannot accept a model that has more than a ten percent chance of 'failure', if that. This is true even though the failure is artificial. On commenter on Seeking Alpha was unwilling to pursue a path that had more than a million-to-one chance of failure.
Beyond that, there are no guarantees that the statistics that create Figures 2 and 3 will be those that apply in the future. This is, in my view a far bigger risk than the risk of a sequence of very bad years. Fortunately, one can structure one's portfolio to reduce the statistical likelihood of failure while also reducing the impact of the limitations of the statistics. We will explore this in a next article.
Figure 3 also shows, on the right, the probability that the funds will have run out by any given year. I find this second plot helpful, though I've never seen it in material from a financial advisor. My personal point of view is that I don't care so much about sustaining my spending level in my 3rd and perhaps 4th decade of retirement, if I even make it that far. These are the classic 'no-go' years.
The upshot is this. If 1) a 4% withdrawal rate would support the spending I want to do, and 2) I was sure that the future returns of the stock market would resemble those of the past, then I would cheerfully throw my money into index funds and think about something other than investments to stave off Alzheimer's.
I also did a test in which I forced the annual return to lie within the body of the distribution, not permitting it to lie on the tails. There were almost no failures. The lesson is this: the cases that require a change of spending plans are those where the returns repeatedly are much less than anticipated from the historical record. If that happens to you, you will be unlucky. The question for you is whether to live your life in fear of being unlucky or in the expectation that you will probably experience returns within the top 80% of the likely range.
Figure 4 shows one aspect of what can go wrong beyond the results of the standard model above. Many commentators believe that stock returns in coming decades will not resemble the historic rates. There are a number of arguments for this. One of them is that the need to refinance the Federal debt in the U.S. crowds out investment that can raise productivity, limiting future growth. Another one is that easing by the Federal Reserve has driven stock prices to unreasonable heights. There are others. Some of them seem sensible to me.
So let us suppose that the average real returns from U.S. stocks in the next few decades is 6%. Figure 4 shows what that does to the outcomes of the Monte Carlo simulations. Now roughly 1/3 of the time the portfolio runs low on funds. Yet more than half the time one's legacy will have more money than one had upon retirement. Even so, one would prefer to reduce the risk of running low on funds, even though this will mean reducing the maximum potential legacy. I will discuss some options for this in Part II.
Figure 4. Results for a portfolio of U.S. Common Stocks, whose statistical behavior is that of a Normal Distribution with an average return of 6%. The left plot shows the probability that the portfolio value is below the number shown on the abscissa. The right plot shows, for those cases for which funds are exhausted, the year in which this occurs. Models and plots by author.
Conclusions
This article was stimulated by an exchange with a commenter on Seeking Alpha. That particular individual wanted absolute certainty that his portfolio would never be exhausted. He missed the point. One uses a withdrawal rule in modeling a portfolio in order to assess the range of likely outcomes. One can prevent the exhaustion of one's funds by limiting withdrawals to some maximum fraction of the remaining portfolio, and making lifestyle adjustments if necessary. Only an idiot would let a fixed withdrawal rule propel him into poverty. (I say him deliberately, as this level of dumb behavior is far easier to imagine from a male than a female.)
In my view, Monte Carlo simulations are useful for seeing the average behavior that will result from a specific approach to working with some collection of investments. They are also useful for seeing the spread of likely outcomes. Your own experience is most likely to fall in the range of 20% to 80% cumulative probability.
We don't know the tails of the probability distributions, and especially not the distributions that you will experience, well enough to say much about the differences between 5%, 10%, and 15% probability. So when a financial advisor tells you about 90% chances, just nod and smile. That is unless you feel like asking what probability distribution was used and what accuracy they assign to its tails.
To my own mind, it makes sense to design your portfolio so that it will meet your needs if the investment outcomes lie within the 'one-sigma' range of negative statistical outcomes, roughly above 20% on the cumulative distribution. One can hope to do much better. One can hope to beat the average statistics, by intelligent selection of investments and by monitoring them so one can get out ahead of problems. Alas, one also might encounter investment performance that is worse than the lowest one's plans can accommodate. If and as this happens, one will have to change one's plans. But living in fear of very poor investment performance seems a to me a wasteful way to spend one's retirement.
Even so, one can reduce the odds that one will have to change course, by diversification. That will be the subject of Part II.
Disclosure:I/we have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours.I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.
Additional disclosure: I am not a financial adviser or a tax advisor, but am an independent investor. Any securities or classes of securities mentioned are not recommendations.