“Sure, it’s just investment advice…”

Monte Carlo Methods for Properly Arguing with Your Father-in-Law

Finances are tough. Every day a new app pops up, claiming to assist millennials in our endeavors to turn acorns, pennies or whatever else we’ve got either into cash or appreciating assets.

What’s the wary consumer to do? How should one evaluate whether or not to invest with your broker or keep it simple and dump my monthly allotment into a nice, stable, ETF?

Unfortunately, such a problem came to a head for friend we’ll call Charlie. See Charlie is in the unfortunate position that his father-in-law, John, is, well, both his father-in-law and would-be mutual fund manager. “Alarm bells are ringing, Willie”.

“Check, mate.”

John, being the sound financial adviser that he is wants the best for his daughter, which means wanting what’s best for Charlie’s money even if that 5.75% sales load sounds a bit daunting. With sound logic, John immediately jumps to the handy compounding interest formula and a bit of knowledge about the past performance of the fund he manages and the S&P 500.

John’s initial numbers to sell Charlie on his fund.

John crunches the numbers and his conclusion? Invest $10,000 per year with me at the fund’s sales load over the next 30 years and we’ll beat your ETF investment by nearly half-a-million dollars.

Sounds great, right?

I stare at Charlie over the slowly warming pint he’s interrupted to tell me this story and I can see him hoping, begging for some alternative solution. Fighting back the intense urge to stare back blankly until the conversation returns to the playoffs, an image of their intense financial standoff flashes across my mind. Charlie is outmatched at the tables of the Monte Carlo Casino, or is he?

A Puncher’s Chance

“You see”, I slowly ease out, “sure your father-in-law is great at his job and promises you above market average returns, but how certain is he about those rates?” Charlie’s last hope in winning his pivotal struggle was to go out guns blazing in a flash of numerical methods.

Howard Marks’ classic graph explaining the value added by a fund manager.

You see, our intuition, however well-intentioned, often doesn’t correspond with reality, especially in regards to random processes. A key linchpin in John’s argument is that rates of return remain constant between years, which is decidedly not true and since John has neglected to mention anything about the variance or risk in his fund’s performance Charlie really doesn’t have enough information to make a good decision. What Charlie really wants to know is: Is this guy worth the 5.75% per year?

Being the good friend that I am, I offer to look into this problem and extend it with the help of a few random processes.

“It’s okay, Charlie, we’ll show John our smarts together.”

Hello, Darkness, my old friend

Now there’s the easy way and the hard way to do everything, I’m sure you’ve figured out, which route I took. I started out by defining two classes: one object to represent an individual investment strategy and another to represent the comparison of two investment strategies.

The key thing for the investment strategy is to define a method that calculates the return for a single year given the rate of return, a method that completes a full investment cycle (e.g., thirty years) and finally a method to simulate a number of random investment cycles. Together these methods correspond to the Monte Carlo method, which is a fancy way of saying, “Let’s take a lot random samples and see what happens.” First, let’s check out the dependencies:

Import the random module for lean access to a number of random generators.

The investment strategy object:

Methods of the object:

The key “Monte Carlo”ness of these methods comes from the use of random.gauss() in longTermReturn.

Finally, the class to compare strategies has some useful helper methods and standard experimental setup assumptions:

Note that each strategy must be of the AnnualCompoundingInvestment class.

Are we there yet?

I like to make sure my code’s not wrong before writing blog posts about it (*normally, cough, cough*). As a sanity check, our Monte Carlo simulation result should come to the same answer as the constant rate of return that John calculated for a near zero risk level.

Investing with John is on top, going with an ETF is the bottom.

Thankfully, this sanity check passes: If John is able to maintain above market returns at an equivalent near-zero level of risk, then John should be able to make Charlie $500,000 more over the thirty year investment period. The next question is:

• If we assume both investment strategies have the same expected return (i.e., mean) and risk (i.e., standard deviation) of long-term annual returns, then which strategy is better (I assume a normal distribution of the annual return data, which should be valid for data sets longer than 30 years)?

Cumulative distribution function shows that both strategies result in the same final account value at about the same probability. ETF is slightly better (Note: y-axis corresponds to the probability that Charlie’s return is less than or equal to the matching x-axis value).

In this scenario I assumed that both the market and John’s fund have the same distribution on rate of return with a roughly 17% annual standard deviation. Even with the 5.75% commission going back to John, the ETF strategy is not markedly different, which supports the typical intuition that Charlie shouldn’t pay for returns he can get by just tracking the market (now having to pay your father-in-law is probably a different question). Glancing at the cumulative distribution function highlights the ever-so-slight advantage of the ETF.

Finally, I asked:

• If we look back at our original scenario and add uncertainty to the annual rate of return, which investment strategy is the better option?

Whether Charlie invests with John or an ETF the expected return is about equal, but John offers him the chance to luck up and get rich.

In this scenario we assume that John provides a higher rate of return, but a higher level of risk, therefore to justify investing with John requires that the risk-adjusted return be worthwhile.

It’s finished…

Staring duly into the abyss of my IDLE shell, a sense of relief floods my body; Charlie may still be in the dark, but now we have a bit more information: Even if John is worse at keeping his fund’s annual returns centered near the long-term average (i.e., more risk/higher standard deviation) and charges Charlie to participate in this managed goodness, Charlie won’t do worse than the market (i.e., same expected value).

To advise Charlie on a decision, however, I turned back to what I “learned” listening to The Most Important Thing by Howard Marks on audiobook, risk may be a theoretical construct useful for simulations, but when it comes down to where you put your money judge your adviser on this:

1. Does the adviser provide higher returns than an investment with the same level of risk (i.e., equivalent standard deviation)?
2. Or does the adviser provide equivalent returns as a similar investment at a lower level of risk (i.e., equivalent mean, smaller standard deviation)?

In this case, until John can fork up more data about the underlying variance of the fund, we can only speculate that the alpha provided by John’s services don’t outweigh the security of an ETF.

Crisis averted? Potentially.

Did we learn some Python? Definitely.