Even Dimensional Fund Advisors Struggles with Inflation Statistics

Inflation is a risk we have to face in financial planning, particularly in retirement.  We need to measure inflation risk correctly to be able to make reasonable financial plans.  The best guide we have to the future takes into account past inflation statistics.  But the field of statistics is full of subtleties, and even Dimensional Fund Advisors (DFA) can make mistakes.

DFA creates good funds, and their advisors tend to do good work for their clients.  I’d prefer to find errors in the work of a less investor-friendly investment firm, but they provided a clear example to learn from.  They misapplied a statistical rule, and as a result, they misinterpreted the history of inflation over the past century.

I discussed this issue with Larry Swedroe in posts on X.  I respect Larry and have followed his work as he tirelessly explains evidence based investing to the masses.

A Simple Example

To explain the problem, let’s first begin with a simpler example.  Someone might offer the following as a rule: when you add two numbers, you get a total that is larger than either of the numbers you added.  This seems sensible.  After all, when two people put their money together, they end up with more together than either of them had individually.

But what if one of the numbers you add is negative?  Then the rule doesn’t work.  If one person is in debt, merging finances with another person leaves that other person with less money than they started with.  The real rule: for positive numbers, when you add them, you get a total larger than either of the numbers you added.  The qualifier “for positive numbers” is important.

Monthly Inflation Data

The amount of inflation we will get next month is a type of random variable.  This random variable has what is called a standard deviation, which is a measure of how widely the inflation outcome can vary.  The larger the standard deviation, the wider the range of possible values we could get.

Over the past century, monthly U.S. inflation has averaged 0.24% with a standard deviation of 0.52%.  For 77% of the months, inflation was within one standard deviation of its average value, i.e., between 0.24% - 0.52% = -0.28% and 0.24% + 0.52% = 0.76%.

Annual Inflation

We tend to be more interested in annual figures than monthly figures.  It’s not hard to find the average annual inflation over the past century; we just multiply the monthly average of 0.24% by 12 to get 2.9%.  But what about the standard deviation?  It doesn’t get 12 times bigger, because sometimes low inflation months tend to cancel high inflation months.

There is a rule in probability and statistics that when you add independent and identically distributed (IID) random variables, the standard deviation grows as the square root of the number of random variables you added together.  So, if you add 100 months of inflation together, this rule says that the uncertainty (standard deviation) of the total would grow by a factor of 10.  Average total inflation for 100 months would be 0.24% x 100 = 24%, and the standard deviation would be 0.52% x 10 = 5.2%.

DFA started with this rule and the fact that there are 12 months in a year.  Then they multiplied the monthly inflation standard deviation of 0.52% by the square root of 12 (roughly 3.46) to get 1.82% as the annual standard deviation.  This is a fairly low figure, and it indicates that inflation has been fairly predictable.  Unfortunately, this isn’t correct, and inflation has been substantially less predictable than this.

The Error

Recall that the standard deviation rule required that the random variables be “independent.”  This means that knowing the inflation in previous months tells you nothing about next month’s inflation.  However, this isn’t true.

There is a measure of the degree to which one month’s data predicts the next month's data.  It’s called autocorrelation or serial correlation, and is related to the idea of momentum.  The autocorrelation coefficient is a number between -1 and +1.  When random variables are independent, the autocorrelation coefficient is zero.

Monthly U.S. inflation over the past century has had an autocorrelation coefficient of 0.47, which is far too high to treat them as independent.  Knowing last month’s inflation narrows down the uncertainty of next month’s inflation.

When we calculate all the actual annual inflation figures for each of the 100 years and find the standard deviation directly instead of calculating it from monthly figures, the result is 3.7%.  This is more than double the 1.82% figure that DFA calculated.

The author of the DFA report seemed to recognize the potential for a problem.  The following note was attached to the reported annual standard deviation of 1.82%.

“Annualized number is presented as an approximation by multiplying the monthly number by the square root of the number of periods in a year.  Please note that the number computed from annual data may differ materially from this estimate.”

Unfortunately, 1.82% is a terrible “estimate” for 3.7%.  This note didn’t help Larry Swedroe who said in our exchange on X, “Fact is though inflation volatility [is] historically low.”  The DFA report led him to believe that inflation is much more predictable than it really is.

Treating financial numbers over time as independent of each other is pervasive in the financial industry.  
They often do this for stock and bond returns, inflation, and other types of returns.  An exception is the work on momentum in asset prices.  We usually see investment models take into account correlations between asset types, but within an asset type over time, independence is often assumed.  However, we should always check autocorrelation to see whether the independence assumption is justified.  

As Larry observed, inflation is “regime related, and regimes last,” and private equity has high autocorrelation due to “stale prices.”  No doubt there are many other causes of high autocorrelation that make the independence assumption inappropriate.

It Gets Worse

What really matters to a financial plan is how inflation builds over decades, not just one year of inflation.  The autocorrelation coefficient for annual inflation is 0.67, which is even higher than it was for monthly inflation.  This means that uncertainty about inflation over a decade is much higher than we might predict knowing the standard deviation of annual inflation.

There are 120 months in a decade.  If we misapply the standard deviation rule and multiply the monthly standard deviation of 0.52% by the square root of 120, we get 5.7%.  However, when we find the standard deviation of inflation over decades directly, it is 25%, a far cry from only 5.7%.

Putting Decade Standard Deviation of Inflation into Context

Suppose we buy a basket of goods for $100, and we want to know what it will cost in a decade.  If we misapply the standard deviation rule and assume what is called a lognormal distribution of inflation outcomes, we think there is only one chance in a thousand that the basket will cost more than $160 a decade later.  

Within the past century, there are a total of 1081 decade-long periods with one decade starting each month.  When we directly calculate how the price of $100 worth of goods grows during each of these 1081 rolling decades in the past century, we find that 23% of the time it exceeds $160.  That’s not a typo.  Somehow 1 in a thousand in theory became 230 in a thousand in reality.

How Relevant is Old Inflation Data?

Some might argue that old inflation data from 50 to 100 years ago isn’t relevant in today’s world.  There may be some truth to this.  However, if we’re going to discount the standard deviation of inflation to reflect reality in the modern world, we have to start  from the correct figure.  The standard deviation of inflation over decades has been 25%, not 5.7%.  Whatever discounting we do, it should start from 25%.

However, we can’t discount inflation uncertainty too far.  As the recent bout of rising prices we had starting in 2021 showed us, central banks cannot fully control inflation.  Future spikes of inflation are possible, and the models we use to account for inflation in financial planning have to reflect this reality.

Annuities

An annuity is a contract where someone hands a lump sum of money to a bank or insurance company in return for guaranteed payments for life, which provides protection against longevity risk and return risk.  Typically, annuity payments are not inflation protected, but as Larry pointed out, inflation-protected annuities now exist.

Decumulation experts know that investors don’t like annuities, even though simulations show that annuities improve the odds that a financial plan will meet client goals.  They call this the annuity puzzle.  

However, if the problem of not modelling inflation properly is pervasive, then maybe annuities (that don’t have inflation protection) aren’t as helpful in lowering portfolio risk as they appear.  Annuities have not fared well in the simulations I do for my own portfolio, but that is because I model the full “wildness” of inflation.

Implication for Financial Planning

Financial planners often use a tool to run what are called Monte Carlo simulations of a financial plan.  The idea is that the simulator creates many plausible histories of investment returns, and checks how often your plan meets your goals.  You might be told that your plan has, say, a 95% chance of success.

However, if the inflation model the simulator uses is wrong, this success rate will be wrong too.  The main error simulators with bad inflation parameters make is to underestimate the uncertainty in the long-term buying power of fixed income securities, such as medium to long-term bonds and annuities and pensions that don’t have inflation indexing.

I pointed out to Larry that the annual inflation standard deviation has been 3.7% and not 1.8%.  He replied “IMO [this] makes little difference because even if [it] was 1.8 not 3.7 one should still consider left tail risks making the portfolio resilient to that, especially given our fiscal situation and the risk of inflation it creates.”

So Larry is saying that even if the Monte Carlo simulations are wrong, financial planners should be checking financial plans for large shocks known as “left tail risks,” such as high inflation, and these checks will uncover any problems.

I agree that we should check financial plans for left tail risks, but I’m not comfortable with this answer.  Suppose we leave simulators unchanged and continue treating inflation as tame.  Suppose further that a financial planner tells a client that their plan has a 95% chance of meeting their goals and has been checked against left tail risks.  Will the planner go on to say that the 95% figure is not really correct, but not to worry because the left tail check will cover things?  Not likely.  If the planner did bring it up, the client would likely ask how far wrong the success probability is.  The planner wouldn’t know.  The client would wonder what the point of the simulation is if the results are known to be wrong.  If the planner doesn’t tell the client that their probability of success isn’t really 95%, the main purpose of the simulations would be to impress clients rather than inform them; Monte Carlo simulations would be marketing rather than substance.

I’m not sure how much of a difference it will make in Monte Carlo simulations to fix the inflation modelling.  I’ve never run simulations with “tame” inflation, and I’ve never seen others run simulations with “wilder” inflation.  But those who have never modelled inflation properly can’t be sure what difference it will make either.

Conclusion

A well known fact about how to calculate the standard deviation of sums of random variables, such as investment returns or inflation, is widely misused in contexts where it does not apply.  Dimensional Fund Advisors did this with inflation data, and this calls into question the accuracy of financial planners’ simulations of their clients’ financial plans.