Personal Finance and Simulation Modeling
CFA studying can be tough, and sometimes you need to take a break; During one of my study breaks, I came up with a Monte Carlo model to estimate my (or your) future net worth. In this posting I explore that model, and take you on a journey of personal finance wonderment.
The Model
A Monte Carlo simulation (at least in my model) works by generating random numbers that act as inputs into a predefined model (with appropriate assumptions). Each time the model is taken through an iteration, a different result will occur, driven by these random inputs. When the model is run multiple times, you are able to determine the probabilities that certain outcomes will occur.
Below you can see the results of several of the iterations in my model[kml_flashembed movie="http://simran.crownpac.net/blog/wp-content/uploads/2006/08/iterationgraph1.swf" height="230" width="450" wmode="transparent" /]
As mentioned above, each iteration is driven by both inputs and assumptions. The inputs for my model were the return on equity for each of the 40+ years, while the assumptions used are listed below (and are fairly realistic):
- a relatively aggressive but realistic savings level of ~20% of personal income (decreases as a proportion of personal income over time)
- a real return on equity of ~7%
- inflation of ~2%
- borrowing rate at Prime plus 1.5%
- annual return standard deviation of ~10% with returns normally distributed
- implementation of a properly balanced portfolio (approximating “the market” with an overall beta of 1)
Insights
As you can see, the results from each iteration appear unique. Although you can anecdotally get a sense of what you might expect to be worth by watching each iteration, the real insight comes when you start to produce a histogram (all values are shown in today’s dollars):
Probability of Net Worth at Age Seventy
Interpreting this graph, you can see that you will have a 1% likelihood of being worth $0-$2M, and a 19% chance of being worth $5M-10M. Notice that despite our assumption of a normal distribution of annual returns, the expected value of the portfolio is positively skewed: There are a greater number of very high results, and the most expected result is lower than the average.
What I find more useful than the histogram, however, are the cumulative probabilities as shown below. Interpreting the following graph, you can see that at age 70 (and with no debt in the portfolio), there is an 80% chance you’ll be worth at least $2M-5M, and a 12% chance of being worth at least $12M-15M.
[kml_flashembed movie="http://simran.crownpac.net/blog/wp-content/uploads/2006/08/leverage1.swf" height="250" width="450" wmode="transparent" /]
As with any model, the fun part comes when you play with the assumptions. I’ve included some buttons on the above graph to facilitate your play: you can choose the leverage scenario for the portfolio, and see the impact it has on the cumulative probabilities.
Notice that as you increase the debt level, your likelihood of being very rich increases, while the likelihood of being worth less remains the same? Debt in your portfolio, over the long term, is a good thing.
Downside of Debt
The graph above shows only the upside of increased leverage. There are, of course, downsides that will ultimately determine the optimal balance of debt and equity for your portfolio. What are the constraints that should limit debt exposure and determine your optimal capital structure? read away:
- Portfolio value variability - Your personal risk aversion level determines how much you can take. My personal feeling is that many investors are overly cautious when it comes to use of leverage, but not careful enough when it comes to individual stock selection vs asset allocation. Using a monte carlo model can help you quantify the risks, and determine what level you really should be at.
- Probability of Bankruptcy (and associated costs) – Because the costs of personal bankruptcy are so high, any chance of total portfolio loss should be avoided. Fortunately, bankruptcy only becomes an issue at extremely high levels of leverage; well beyond the 65% leverage scenario I included above (I ran a test on the simulation, but have not included it in this posting)
- Greater demands on cashflow management – The impact on the portfolio resulting from margin calls, and portfolio rebalancing adds extra cost and care to managing the portfolio
- Greater Need to rebalance portfolio - Changes to asset class values will be magnified by the use of leverage. Depending on your rebalancing approach, this could add significant costs
- Increased borrowing costs at higher debt levels – This depends on how big you shoot. If you are pushing 8 figures, you might want to read up on some Miller and Modigliani
Extensions
There are many more variations we can build off of this model. For instance, if you where interested how soon you could expect to be worth $5,000,000, you could use the following graph:
With a little bit of work, we could also examine:
- How often periods of low cashflow would occur
- How often Portfolio rebalancing would be necessary
- Likelihood of bankruptcy
- Optimal portfolio construction
Whatever you are interested in, the model can be built to examine your needs.
Conclusions
Despite the long post, this is a relatively brief look at what these kinds of models can do for you. Some takeaways:
- Modeling brings Clarity – Despite the uncertainty around future market returns, you can develop rational expectations on where you will be in the future by using tools like simulation modeling. Combine a knowledgeable financial modeler with powerful computing, and a model can be adapted to address any issue that you might be concerned with.
- Time is your Friend – When you no longer have the ability ski moguls because your knees are titanium, you can take comfort in the fact that you can cruise the Mediterranean, be a philanthropist, or buy a fancy Skoda. You already know that it is good to save, but it is nice to be able to quantify it.
- Debt is your Friend – Assuming you’ve got time (measured in decades), it probably makes sense to bite off some debt. You’ve seen above the result of adding leverage to my model, and the resulting net worth values are large. Always remember however that there is a big difference between consumer debt, and leveraged investments; make sure your debt is working for, not against you.
I hope you found this long and flashy post interesting. Let me know if you have any comments or questions.
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Excellent use of modeling to demonstrate the variability of future scenarios. Instead of monte carlo, have you explored agent based models or system dynamics models? We’ve found them to be the next step in the evolution of computer modeling for enhanced learning and improved decision making. I would love to speak with you some more about your modeling experience as your writing here is clear and compelling. The animated graphs are also interesting. I assume you did these with Flash?
Two comments:
1. Is the networth value adjusted for the inflation assumption? Otherwise, you can put a high inflation value, and get to be “rich” easily.
2. Debt as a leverage only works when your return on the leveraged money exceeds the interest rate that you pay. More often than not, this difference is quite close to 0%. I think a real return of 7% on equity is too high, especially after a super-bull market in 2000. Furthermore, inflation averages to be about 3.3%, rather than 2%.
Thanks Justin,
I have some experience with system dynamics models, although they have been for business process design, and workflow management – Haven’t applied the technique to finance before.
I have not used agent based models, and I don’t have a high level of understanding – I’ve always imagined agent based models as a direct extension of monte carlo models – is this a correct assumption?
The graphs are indeed flash, but i used a couple programs to put them together
Hi Frugal, thanks for you comment – your points are addressed below:
1. Is the networth value adjusted for the inflation assumption? Otherwise, you can put a high inflation value, and get to be “rich” easily.
As I mentioned in the posting (perhaps too subtlety) all the values mentioned throughout are real values, and are shown in todays dollars. The values are therefore adjusted for inflation.
2. Debt as a leverage only works when your return on the leveraged money exceeds the interest rate that you pay. More often than not, this difference is quite close to 0%.
I can currently borrow at 5.5% – With an expected nominal ROE of 9.5% (2.5% inflation plus 7% real return), that is a 4% differential (9.5%-5.5%)
I think a real return of 7% on equity is too high, especially after a super-bull market in 2000.
Firstly, due to the time horizon of the model (40+ years), I feel a 7% real return on equity is reasonable. If the time horizon were 10 years, I would agree that it could be too high, but over 40 years it is entirely reasonable to assume that the ROE will approximate the historical average. For more on why 7% real return works, check out the following article from the Social Security Advisory Board (PDF).
Furthermore, inflation averages to be about 3.3%, rather than 2%.
There are couple points regarding inflation:
Hi Simran Gill,
I’m just using your assumption here and recalculate all the numbers since 1980. Simply using S&P500 index, since 1980 Jan 1st to 2000 Jan 1st, S&P 500 returned 14% annually compounded. I used the inflation calculator from here:
http://minneapolisfed.org/Research/data/us/calc/
And I got an annual compouned inflation of about 3.8% for the same period. Therefore from 1980 to 2000, the real return from equity S&P500 is 10.2%.
Since your data indicates that equity real return approaches 7% in a 40 year timeframe, if I simply take a timeframe from 1980 to 2020, S&P500 should return (1.07)^40, but since S&P500 already returned (1.102)^20 for the first 20 years, the next 20 years, it should probably return about ( (1.07^40) / (1.102^20) ) ^ (1/20) = 1.039, or 3.9% real return.
I don’t know whether how accurate 7% is, but I do know that after an extended bull market, I really expect a lackluster market performance. 3.9% may be just what you will be looking at. And I bet that putting in a 3.9% to your model, it won’t give you a very comfortable result.
Just my two cents.
Another possible error in your modelling is that you are applying 7% equity return on the entire networth, while most of the time, a person may have mortgage debt (fixed negative return) + stocks + house (which has a lower return compared to stocks). More often than not, portfolio size of the house is much bigger than size of the stocks at least in the initial stages of probably 10 to 20 years. Using 7% real return for both stocks & house will give you an exaggerated return.
You must adjust that to get any accurate result.
By the way, you didn’t include taxes on equity returns either.
I have a java calculator at
http://www.1stMillionAt33.com/java_codes/retire.html
It includes everything, debt, tax, stock return. I also didn’t separate the return of house from stocks, but I let you enter the average asset return.
I don’t do much Monte Carlo because I believe that investment returns are less than random, but more of a cyclical nature. With a cycle view, if you’re at the downhill, it’s downhill. Throwing dice won’t increase your probability of getting richer. I think trying to catch cycles probably help more.
What software did you use to do the modeling in?
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Like Ben, I too am curious to know what software you used for the modeling. The analysis is intriguing.
Hi Duane and Ben,
Glad you found the analysis of use -
For the modelling I used Excel, and to present it I used a combination of Macromedia Flash, and Visio.
Frugal,
Because the portfolio return, initial size, Std Dev., and contributions are all inputs – its not a modelling error if the values I chose are different from what you would choose. It’s up to the individual application to determine an appropriate return level, risk, and whether or not real estate (home) is included in the portfolio.
The benefit of monte carlo is that it doesn’t just give ‘point’ estimates for future worth – it’s a way of evaluating risk. The model you’ve linked to is excellent, but it’s hard to determine the likihood of achieving a specific return.
Tax was excluded from the model because it depends on your holding period, use of tax-free investment plans (RRSP’s in canada), tax-free bonds (muni’s), country, tax bracket, tax changes, etc. That would be an extensive extension of the model, but a valuable one.
Simon,
Definitely I am not saying that you have any modeling errors, and your model is certainly better than mine.
I am merely saying that your inputs are probably not correct.
Excellent post. I did something similar with an eye more towards answering what the right level of volatility to target is given your saving and spending goals. I.e., assuming that accepting more market risk provides increasing returns, when does it makes sense to hold the market portfolio vs. skewing towards bonds (lower risk) or levering via LEAPS, margin, or high beta portfolios.
My analysis is here: http://kurtjohnson.net/Interests/InvestmentRiskOptimization.htm
As a sidenote, your return trajectories are a bit of a conundrum since there is some mean reversion; returns are not random from year to year and I expect that using the mathematical conversion of annual return volatility to convert to volatility per decade would really overstate volatility observed in past decades (multiply the annual return volatility by 10^.5).
As a sidenote, I believe 10% annual stdev of returns is probably low; I expected a standard deviation closer to 20-25% for US equities. Using this larger number you may need to convert the model to a lognormal distribution to avoid drawing trials where the return from a normal distribution is
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Indeed an intriguing post. I dunno if it will be asking for too much but i was wondering if you would like to share the spreadsheet as well to give an insight of the model….
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That was a very interesting idea, I planned to make similar graphics for my credit card debt. That was actually one of the major spending issues in my finance, and credit card expert site said it is crucial to make your debt visible to reduce spendings… Anyway, you gave me inspiration)))
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[...] Simran Gill: Personal Finance and Simulation Modelling. Using Monte Carlo simulation to project a future [...]