If you read much about retirement finance, then you've probably
seen a piece or two on sequence of return (SOR) risk. They don't provide much
clarity, IMHO.

Different writers define SOR risk differently (part of the problem), but I use the
term to describe the observation that two investors who withdraw or invest

*periodically from their retirement portfolio can experience the same average stock market returns and have very different outcomes when the**order*of those returns differs.
Those investors subject to SOR risk would include most who
save in a 401(

*k*) savings plan, investors who dollar cost average, retirees who implement a safe withdrawal rates strategy and investors who save or spend random amounts from their retirement portfolio.
In fact, as I will show, there is only one way to avoid it.

I read a nice post by Dana Anspach entitled, "Sequence of Returns Risk Misunderstood by Many Retirees." It explains the effects well, except that the assertion that sequence of returns risk does not apply in the accumulation phase of retirement funding is incorrect, as I will demonstrate

^{1}.

The article shows that a series of 10 market returns:

· 1996 23.10%

· 1997 33.40%

· 1998 28.60%

· 1999 21.0%

· 2000 -9.10%

· 2001 -11.90%

· 2002 -22.10%

· 2003 28.70%

· 2004 10.90%

· 2005 4.90%

would result in a terminal portfolio value of $164,513

^{2}for an investor who started out with $100,000 and spent $6,000 per year. An investor who did exactly the same but experienced those returns in reverse order (4.9% in year one, 10.9% in year two, etc.) would end up with only $125,691.
One thing that bothers me about this analysis, which
is quite common in the literature, is that the reader may come to think that he
or she is endowed with a pre-ordained set of future market returns and need only worry
about their sequence.

The article compares two of these series, forward and
backward, but there are actually 10!, or 3,628,800 different permutations of
these ten returns (a rare opportunity to use the factorial key on my
decades-old HP 12C calculator. It’s right there under the “3”.)

Retirement plans typically cover 30 years or more. How
many different sequences can you create from 30 numbers? There are
approximately 2.65 x 10

^{32}permutations of 30 years of returns, or 265 followed by 30 zeroes, which rounds off to a gazillion bazillion.
These two aren’t even the best and worst series orders for
the ten years of returns. The largest portfolio value will result when the returns arrive
in descending order ($181,026) and the smallest will occur when they arrive in ascending
order ($80,994).

Of course, we’re not limited to permutations of 30 or so
market returns in retirement, the possibilities are limitless. So there is no
“sequence” of future returns, as if when you don’t get one then you'll get the
other. We have no idea what those 30 annual returns will be, so it seems
pointless to hope that, whatever they might be, they arrive in decreasing order.

Sequence of returns risk isn’t a phenomenon, at all. It’s
algebra.

Let’s say a retiree has a portfolio worth $100,000 and
experiences annual returns of 20% followed by a loss of 7% followed by a gain
of 3%. His terminal portfolio value (TPV) can be calculated as:

We can factor out the $100,000 initial portfolio value and get

The terminal portfolio value after three years is $114,998. The commutative property of multiplication tells us that we will get the same result no matter what order we multiply these four factors. So, the portfolio represented by this equation will always accumulate the same total value regardless of the sequence of the returns.

*If we don’t add new funds or subtract spending from our portfolio each year, there is no SOR risk.*

What happens to the algebra if we do withdraw, say $4,000 at
the end of each year? The portfolio value is then calculated as follows:

There are three things to observe about equation [2]. First, the equation no longer contains only multiplication, so the commutative property does not apply and the order in which the market returns appear affects the solution. Swap the 1.2 and the 0.93 in equation [2], for example, and you change the result to $101,884.

Because we are subtracting amounts each year, we now see
sequence of returns risk that wasn’t present in equation [1].

The second thing to observe is that this would be the case
if we added $4,000 each year instead of subtracting it, so

*SOR risk is present in both accumulation and spending phases*(just change the minus signs to plusses in equation [2]).
The third observation is that sequence of returns would also
matter if we subtracted or added random amounts each year. SOR risk is present
when we withdraw or save fixed dollar amounts periodically, as the safe withdrawal
strategy requires, or if we spend or save a different dollar amount every year.

There was no SOR risk before we introduced periodic buying
and/or selling from the portfolio. That tells us that SOR risk is simply the
uncertainty of the stock prices at those periodic buy and sell points in our
future.

Now, what happens to equation [1] if our investor decides to
withdraw say, 4% each year of his portfolio’s current value instead of some
dollar amount? Portfolio value is now calculated as:

or

which simplifies to:

Equation [3] now looks like equation [1]. The percentages are each smaller by 0.04, but once again, the expression is the product of four factors, so the commutative property of multiplication applies and the sequence doesn’t matter. There is no sequence of returns risk.

Also note that if the percentage isn’t the same every year
(e.g., we spend 4% of remaining portfolio value the first 5 years and 5% the
second 5 years), we re-introduce sequence of returns risk.

What does this high school algebra exercise tell us about
sequence of returns risk in investing for retirement?

- SOR risk is present when you invest your retirement portfolio in volatile assets like stocks. It is not present in fixed annuities and is negligible in low-volatility portfolios like short and intermediate Treasury bond ladders.

- SOR risk can be present in both accumulation and spending phases of retirement funding.

- The only way to avoid SOR risk with a risky portfolio is to withdraw the same percentage of your portfolio’s remaining balance every year.

I was exposed to SOR risk while I
saved for retirement, but

*saving*a set percentage of my increasing portfolio balance wasn’t feasible. There was no practical way to avoid SOR risk in the accumulation phase.
I withdraw about 4% of my current portfolio
value every year, so I’m not exposed to SOR risk in retirement.

Lack of exposure to SOR risk did
not, however, protect me from the 2007-2009 market crash (my portfolio
allocation did). So, it didn't protect me from the risk that a large portfolio
loss early in retirement would threaten my portfolio’s survival if I live a
long life.

That’s another way financial
writer’s define sequence of returns risk, but I believe that is a different
risk.

I’ll tackle that in Part Two.

----------------------------

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^{1}The reader can work through the same three-year example I provide but adding a fixed dollar amount each year instead of subtracting and observe that sequence of risk does, in fact, exist in the accumulation phase.

^{2}The referenced article states the total of $162,548 but I believe this is in error. That number appears to be the portfolio value after 9 years, not 10. The mistake is inconsequential. I note it only to avoid confusion.

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