# How does Monte Carlo function use historical data

I am interested in understanding how the MC function uses data and does its calculations. Namely, is the historical range used to establish a range of possible returns for the specific asset class (ie -40% to +30%) and to randomly pick a figure from within the historical range to apply to the asset amount from the prior year, independently of the prior years returns.

Or, does is the range of possible returns for a given year, influenced by the prior years returns?

Similarly, how does the MC engine work across asset classes. Given two highly correlated assets, ie two US Equity funds of different composition, will the MC engine select and utilize a predicted return for each fund that is consistent with the historical correlation of those assets as shown in the historical data for the asset, or will it select returns independently and ignore historical correlations.

Thanks

### Hi, Happy to discuss on the

Hi, Happy to discuss on the phone. Number is 617 83 2148. But we take the historic asset return data and use it to parameterize a multivariate log normal distribution from which we draw returns for each of your assets. So, yes, we fully consider correlations across asset returns. Doing it any other way would be malpractice. best, Larry

### Hi, Happy to discuss on the

Hi, Happy to discuss on the phone. Number is 617 83 2148. But we take the historic asset return data and use it to parameterize a multivariate log normal distribution from which we draw returns for each of your assets. So, yes, we fully consider correlations across asset returns. Doing it any other way would be malpractice. best, Larry

### Hi Larry, thanks for the

Hi Larry, thanks for the reply. Very clear on the cross correlation. Is the distribution applied 'fresh' each year, or, does the parameterizing approach consider prior year applied return as a constraint on the possible returns for a current year?

Thanks again, Greg

### The approach described by

The approach described by Larry seems reasonable for accounting for correlations across assets in any given year, but what are the assumptions about time series correlations? Is there any accounting for persistence of business cycles over several years, for example? Does the MC engine adjust the multivariate log-normal distribution for the current year to condition on recently realized returns? (i.e. something like a multivariate log-normal VAR model) I realize that doing so with any precision would require more historical return data to calibrate the model, but it seems like this could be done for a simple portfolio consisting of some broad stock and bond indices. (p.s. I'm a professional economist and would love to see more technical discussion of how the MC modeling works and what assumptions it incorporates about asset return distributions.)

### The following applies to the

The following applies to the MC over in MaxiFi, though aspects of it may also apply to the MC in ESPlanner. In other words, I've copied this from the application help area in MaxiFi. You can see it there for yourself as a MaxiFi user, but wanted to follow up. Kotlikoff would be happy also to respond to further specific questions you have.

--------

Understanding MaxiFi’s Monte Carlo Risk Analysis
MaxiFi’s risk analysis recognizes that you are investing at risk, i.e., that you can’t tell precisely what real rates of return (returns adjusted for inflation) you will earn each year on your regular and retirement account assets. We call a particular sequence of annual real returns that you may receive a trajectory of returns.

Your investment strategy – the assets you tell MaxiFi you'll invest in through time – determines what trajectories of real returns you may experience. Suppose your investment strategy is to hold stocks for 15 years and then switch to bonds. On average, stocks yield higher annual real returns than do bonds. But their annual real returns are far more variable. Hence, your trajectories of real returns will be generally high, but very variable for the first 15 years and generally low and smooth thereafter.

Or your investment strategy may entail holding half of your assets in stocks and half in bonds on an ongoing basis. In this case, your real return trajectories will generally lie midway between the typical all-stocks or all-bonds return trajectories. This mixed investment strategy would exhibit less real return variability than the all-stocks strategy, but more variability than the all-bond strategy.

For any given investment strategy, the particular trajectory of real returns you end up receiving will differ from the average real returns trajectory because of the random nature of the real returns you earn on your investments. In other words, you can’t expect to earn the average real return every year. Instead, there will be high and low real returns along any trajectory. You can also experience lots of low real returns or lots of high real returns in a row. This is called sequence of return risk.

MaxiFi can’t tell you what trajectory of real returns you’ll receive through time. It can and does show you the range (distribution) of real return trajectories you may face given your investment strategy. Keep in mind that the historical real return data we use to help predict future real returns are no guarantee of future returns. It’s also important to bear in mind that you only go through life once and will only experience one trajectory of real returns.

Translating a Real Returns Trajectory Into a Living Standard Trajectory
The trajectory of real returns you experience will help determine, along with your other economic and demographic inputs, the living standard your household will experience each year. We call a trajectory of annual living standards a living standard trajectory. For each real return trajectory you may experience, there is a corresponding living standard trajectory.

There is another key factor that enters into the determination of your living standard trajectory given your real return trajectory and other inputs. This other factor is your spending behavior, which you can control by setting an annual safe rate of return. This is the return you can count on receiving, on average. We ask you to be cautious in setting your safe rate of return as well as all other inputs.

MaxiFi’s Monte Carlo Living Standard Risk Analysis
Once you’ve specified your investment strategy and safe rate of return, MaxiFi produces hundreds of possible living standard trajectories to show you the range of living standard trajectories you may experience.

MaxiFi repeats the following steps to generate the hundreds of living standard trajectories. In this description, we’ll assume you are 50 years old to make things concrete. But the same process is followed regardless of your current age. We’ll also assume you specified a 3.5 percent safe rate of return together with a 2.5 percent inflation rate. This implies a roughly 1 percent safe annual real return.

MaxiFi draws, at random, a real return trajectory – a trajectory of annual real returns for your regular assets, your retirement account assets, and your spouse/partners’ retirement account assets (if applicable). These draws are based on your specified investment strategy, which can entail holding different portfolios (combinations) of assets over time.

MaxiFi calculates what you should spend this year (at age 50) if you knew for sure you were going to earn, in each future year, the safe real return you specified. MaxiFi records your household’s living standard at age 50. Note that if you set a lower safe real rate of return, MaxiFi will calculate a smaller spending amount. Hence, your choice of the safe real rate of return controls your spending behavior. The lower the safe rate you set, the less you will spend in the present and, therefore, the lower the chances of having very low spending in the future.

MaxiFi advances you by one year to age 51, but in calculating what assets you’ll have at age 51, it uses the actual real returns that it drew for you for age 50 in Step 1.
MaxiFi calculates what you should spend this year (at age 51) if you knew for sure you were going to earn, in each future year, the safe real return you specified. MaxiFi records your household’s living standard at age 51.
MaxiFi advances you by one year to age 52 and so on. In other words, it runs your household forward in this manner, calculating a living standard for your household for each year through your household’s last possible year of life.
MaxiFi collects all your annual living standards and stores them as the first of your living standard trajectories.
MaxiFi draws a new return trajectory and repeats steps 1 through 6.

After generating 500 living standard trajectories, MaxiFi ranks them based on their average annual living standard. It then displays five living standard trajectories (as well as their associated discretionary spending levels) – the trajectory with the 95th highest average annual living standard, the trajectory with the 75th highest average annual living standard, the median trajectory with the 50th highest average annual living standard, the trajectory with the 25th highest average annual living standard, and the trajectory with the 5th highest average annual living standard.
Using MaxiFi’s Monte Carlo Risk Analysis

Once you’ve run MaxiFi’s risk analysis based on your specified investment strategy and spending behavior (determined by your specified safe real rate of return), you can try other investment strategies and spending behaviors. You’ll see that less aggressive investing will lead to living standard trajectories that are lower, but less variable. You’ll also see that less aggressive spending, generated by a lower assumed safe real rate of return, will produce living standard trajectories that start out lower, but have less chance of being low in the future.

Bear in mind that you will experience just one trajectory of returns through time and one living standard trajectory based on that trajectory of returns and your spending strategy. Consequently, you face three risks when it comes to deciding how aggressively to invest and save:

You’ll end up on a living standard trajectory that’s lower, on average, than you’d like.

You’ll end up with more annual variability in your living standard than you’d like.
You’ll experience more downside living standard risk toward the end of your life than you’d like.

MaxiFi doesn’t suggest what investment strategies and safe rate of return you should adopt. This depends on your attitudes toward risk. What MaxiFi does is show you the implications of investing and spending more or less aggressively to assist you in deciding how to invest and spend through time.

### Thanks for the lengthy reply,

Thanks for the lengthy reply, but it doesn't actually answer my relatively simple question. It does help me reframe it, however. So here is the question again, phrased differently: You say (above) that "MaxiFi draws, at random, a real return trajectory – a trajectory of annual real returns." My question is about HOW it "draws a trajectory." One simple method would be to make independent draws for each future year and concatenate them to form a trajectory. Then the returns would be uncorrelated over time. That approach would be consistent with the efficient markets hypothesis that says that future returns follow a random walk and so are completely unpredictable apart from their mean. (Technically, the trajectory is a "martingale.") But nowhere in the documentation do I see that commonly made assumption stated. And there are other possibilities in which returns for a single asset are correlated over time. See for example the discussion at https://www.investopedia.com/articles/07/mean_reversion_martingale.asp

### Hi Scott, We assume that real

Hi Scott, We assume that real returns are joint logarithmically distributed. We used Cholesky decomposition in forming the draws. In short, we are taking full account of the covariation of the returns of the assets you specify you are holding. We form the variance-covariance matrix and mean returns using conforming historical data. Hope this answers your questions. And, yes, this is all consistent with return on one's following random walks. But the returns on separate securities don't move independently of one another.
best, Larry

### Thanks Larry. The random walk

Thanks Larry. The random walk part of your answer is what I was looking for. I couldn't find any mention of that aspect in the documentation or online Q&A. I'm just getting started with the MC features of the product and am looking forward to using it.

We use cookies to deliver the best user experience and improve our site.