Daniel SmithBy Dr. Daniel Smith, WCI Columnist

Many of you already use math in your professional practice. Anesthesiologists calculate volumes of anesthetic and their distributions. Pharmacists calculate renal clearances of drugs. Radiation oncologists calculate radiation doses, and so on. However, as sharp as your medical math may be, there are probably a few arithmetical nuances that, if you dig into, might help you squeeze a bit more return (and perhaps lower some risk!) from your portfolio. 

In this article, I’m going to dig into three mathematical concepts and show you some pitfalls with each and how to avoid them!


#1 Sequence of Returns Risk

The first concept is called sequence of returns risk, which is essentially the risk that poor returns just before or during early retirement tank your portfolio’s value. Jim has written more about this in other posts.

Investing in Retirement

Investing in Retirement Part 3

The Most Important Factor in Retirement Withdrawal Plans

This risk is most evident with a portfolio of volatile assets like stocks. Bonds and other fixed-income tend to weather bear markets and recessions in a more stalwart fashion. Let’s take a look at the fundamentals, how the math actually works, and how to mitigate it. If you’re in your 20s or 30s, either bookmark this page or read it for the benefit of your more elderly relatives.  

Say you start with $500k in assets with 10 years before retirement and a contribution rate of $20K yearly. Your returns are -5%, -4%, -3%, -2%, -1%, 1%, 2%, 3%, 4%, 5% annually. Your final portfolio valuation would be $716,773.43. If you reverse the returns sequence so that the lower returns come last, you end instead with $678,761.30. That’s $38,012.13 difference; nearly 2 years' worth of contributions.

Without context, this doesn’t make a lot of sense. Why would reversing the sequence of returns make a difference? The answer is that with early low returns, you’re putting money into investments as they become cheaper! Later, as the returns increase, you reap the benefits of having purchased assets inexpensively. To the converse, if you buy (invest) while returns are higher, you’re putting money into a portfolio of expensive assets which drop in value more precipitously as the market sours. As you can imagine, the larger the annual contributions to the portfolio, the larger the disparity between the sequences becomes.  

The math is even more telling when you start making withdrawals. You’ve decided to rest on the efforts of your life’s work and decide to draw down a comfortable $160,000/year from a 4 million dollar portfolio. You made this decision after reading a convincing article from your buddy about the “safe” 4% withdrawal rate. Fate, being what it is, has dealt you the returns previously illustrated, starting lowest returns first. At the end of just 5 years, your 4 million dollar nest egg has boiled and is now a paltry $2,687,027.40 despite withdrawing only $800,000 from it! At the end of the next 5 years (with the positive returns), your retirement is looking even less like steak and more like Alpo at $2,221,894.34, and you begin to reconsider donating plasma. While the bleeding has definitely slowed, it hasn’t stopped.  

The magic 8 ball of life gives you a second chance, and you wind up with the good returns as a tailwind early in retirement. Though still not rosy, it’s less grim than what was previously played out. At the end of your tenth year playing senior-league bocce ball, your portfolio’s value is $2,525,991.35. For those playing at home, that’s a difference of $304,097.01.  

The takeaway here is that low or negative returns in close proximity to retirement can put a strain on even a well-funded portfolio.

So, how does one mitigate this pitfall in retirement? Well, there are basically three ways:

  1. Push back your retirement age.
  2. Invest in fixed income assets that have a predictable income stream.
  3. Lower your withdrawals.


Retiring Later

The first choice, retiring later, is probably a fairly simple matter for most of the readers of this blog. I’m willing to bet you all have a fairly marketable set of professional skills which, even if practiced on a part-time basis, could generate a reasonable income. By pushing back retirement (or semi-retiring), you can likely face down the fiercest bear and perhaps even continue to invest, which is usually a good choice in a down market anyway. Bonus points for being able to defer required minimum distributions from your employer-sponsored retirement plan if you’re still working. This might also be a particularly good time to do some Roth conversions!


Invest in Fixed Income

Let’s examine the second option. If perhaps you’re burned out and want to finally make good on that promised trip to Barbados, you might be better suited by a bond or CD ladder. By changing a portion of your investments to those with predictable returns, you essentially mitigate the short-term market volatility that’s at the heart of this particular problem. 

Let’s say you take $500,000 of your portfolio and purchase five high-quality bonds (or bond funds), each with a principal value of $100,000 and maturing in 1, 3, 5, 7, and 9 years. Over the course of those 9 years, you’ll receive the coupon payments for each of those respective bonds and their principal at the end of their respective maturities. 

investing mathIf you’re not wild about purchasing individual bonds and would like to diversify somewhat within that asset class, consider a bond fund with an average maturity that meets your needs. Keep in mind that bond funds can carry bonds of varying investment grades, underlying investment bases (mortgages, treasuries, corporates, municipals, etc.), and somewhat varying maturities. These all somewhat alter the risk composition of the fund.

Since this is supposed to be the safe part of your portfolio, don’t mess around with any fund containing concentrated risks or many if any junk grade bonds. Jim recommends Treasuries or TIPS, which are the safest (with TIPS rising and falling somewhat with inflation), but remember that CDs and their accrued interest are also FDIC insured up to $250,000 per depositor per bank.

Remember that bonds bear interest rate risk and that the rise in interest rates can lower the bond’s market value unless held to maturity. Similarly, inflation will diminish the purchasing power of your dollars, which is the scourge of fixed income. Hedge against this with your existing stock and real estate allocation or by purchasing TIPS. 

For Single Premium Immediate Annuities (SPIAs), understand that you’re paying for a fixed rate of income but are completely giving up your principal. Also, you are dependent upon that company’s solvency to fund your retirement. Variations on annuities—for example, where your heir can receive a lump sum at death—usually have a lower rate of return (after all, they’re hoping to make money on you). The devil is in the details with these products.  


Lower Your Withdrawal Rate

Let’s say you’re disappointed by dismal yields on bonds and can’t abide the idea of working another day, then the last option may be for you. Simply lowering your withdrawal rate can vastly extend the projected longevity of your retirement portfolio. Going back to our $4 million portfolio, let’s say you decide that a fixed 2.5% withdrawal rate doesn’t sound so bad. Even assuming a miserly 2% annualized return on your portfolio (maybe your spouse went to all 20-year treasuries while you were watching Yankees vs Mets), your 2.5% withdrawal rate will extend your portfolio to a Methuselahn 81 years. However, at a 4% withdrawal rate, your portfolio will last around 35 years, which isn’t necessarily a bad thing, provided you’re willing to expire on schedule to avoid penury.  


#2 Arithmetic Return vs Annualized (Geometric) Return

With the lion’s share of this article devoted to the more practical sequence of returns, I’d like to pull back the curtain just a bit on the more esoteric concept of how returns are actually calculated. Let’s perform a thought experiment.

What is the return of a $100,000 portfolio with the returns we so thoroughly investigated above (-5%, -4%…4%, 5%)? Did you think it was $100,000? Actually, it’s $99,451.02. (By the way, it doesn’t matter in what order those returns are placed, i.e. the commutative principle of multiplication means the total return will be the same regardless.) How can that be, you ask, that the total return of a portfolio whose yearly returns average 0% are actually lower than the starting amount? You’re seeing the difference in arithmetic and geometric returns!

Arithmetic returns are those which are simply an average of the yearly returns of a portfolio. Let’s say for simplicity’s sake that your portfolio returned -50% in the first year and +50% in the second year. Your arithmetic return is 0%, but you still have a negative geometric return over the first two years. How is that? Well, if you start with $100 and lose 50%, you have $50 after the first year. If you then earn a positive return of 50%, you’ve gained $25 (50% of $50) for a total return over the two years of -25%; annualized, that number is -13.4%. That annualized return is also known as the geometric return or compounded annual return. Annualized return is the return you’d have to earn each year for an investment to grow (or shrink) from its starting value to its ending value, assuming no cash flow into or out of the investment. The algebra is devastatingly boring (no offense, quants); it’s much easier to use a financial calculator.

The most important point regarding the difference between the two kinds of returns is one of marketing.  

Funds love to market (and magazines publish) arithmetic returns. Don’t be misled by the promise of princely returns only to later find them viridescent and warty. Remember, annualized returns are always lower than arithmetic unless the yearly, periodic returns are exactly the same every year. 


#3 Volatility

I think volatility may be the most underappreciated finance word in the American investor’s lexicon, and it has far more implications than the day-to-day swings of the market. Volatility is difficult to appreciate until you know its derivation but is most commonly defined as standard deviation. Math aside, standard deviation relies on three things:

  1. Number of periods you’re evaluating
  2. Mean of those periods’ returns
  3. Difference between the periodic return (yearly return, monthly return, daily return, etc.) and the mean of the returns 

As the number of periods increases, volatility diminishes. As the periodic returns stray further from their mean, the volatility increases. If you think in algebra (perish the thought), look up the formula for “standard deviation of a sample” and plug in some numbers yourself to see how it works.  

Why is this even important? Put simply, volatility is understanding to what degree returns may differ from their average. If your time horizon for investing is 30 years, then broad market returns over months or even years are just volatility. Spoiler alert, just about every bit of data that crosses the financial news is just volatility. Remember, while historical returns aren’t perfect, they’re likely one of the best estimates retail investors have of future returns. If a stock zooms to the sky in a short period of time, it’s statistically much more likely to crash right back down to earth than to maintain its lofty perch.

Another detriment of high volatility is that it can reduce returns; this is known as volatility drag. More volatile assets, like emerging markets stocks, tend to underperform less volatile securities with the same arithmetic return. This is, in reality, the difference between geometric returns and arithmetic returns made manifest. How does this help you? When it comes to selecting assets for your portfolio, if two assets have similar arithmetic returns and long-term risk profiles, picking the less volatile asset puts money into your pocket. For bonus points, it’s also part of the reason that leveraged stock funds tend to underperform their forecasted returns.  

The last thing I’d like to point out about volatility is its psychological effect on investors. The same folks who put money into the market while it’s going up are also typically the same investors who pull money out near the bottom. The crash and rapid rebound in spring of 2020 was probably a bellwether regarding what kind of investor you are. If you looked that nasty drop in the eye and bought another hundred shares of VT, you’re likely the kind of person whom volatility doesn’t bother. The other end of the spectrum is populated by those who responded to last spring’s stock drubbing with a fast exit to cash and a bottle of Mylanta. If your portfolio’s volatility causes you to sell low just to sleep at night, your portfolio is likely too volatile or too risky.  

In short, volatility is like a region’s weather: variable day to day but consistent when viewed over the long term. The ugly side of volatility is the slow attrition of your returns when compared to similar but less volatile assets. Last, volatility is like a rollercoaster; if you can’t tolerate the ups and downs, don’t get on that particular ride. Full disclosure, I have motion sickness, and you’ll find me planted firmly next to the funnel cake stand.  


In Search of the Conclusion

In summary, hedge your sequence of returns risk with safe fixed income, beware the gleam of the top fund manager’s shiny returns (he statistically won’t be there next year anyway), and treat volatility like static on the radio—tune it out the best you can and ignore the rest until the reception gets better.

What do you think? Do you think of investing in terms of mathematics? Why or why not? Comment below!