# Beyond the Mean: Exploring Tail Risks in Inflation Expectations

A crucial issue for monetary policy decision-making is projecting future inflation rates and assessing their associated risks. This blog post explores the inflation expectations held by participants in financial markets. Specifically, it leverages financial market data to estimate the probability distribution of expected consumer price index (CPI) inflation over the next five years.

Data from the Federal Reserve Bank of Minneapolis indicate that the distribution’s mean, standard deviation and skewness have all risen relative to prepandemic levels, which is not surprising given the recent uptick in inflation and the greater uncertainty surrounding future inflation.

The skewness of the market’s expected inflation distribution speaks more directly to tail risk. In 2018, the market-implied skewness was negative, indicating higher left-tail risks—meaning a greater risk of inflation being lower than expected. However, the current market-implied skewness is positive, signaling right-tail risk—a greater risk of inflation being higher than expected.

While the estimates of the mean, standard deviation and skewness provide useful information, it is challenging to draw quantitative conclusions from them. Instead, I utilize these three data moments to estimate a probability distribution that accounts for tail risks. As I describe in this post, my analysis based on these market-implied moments suggests a 22% probability that the expected annual CPI inflation rate over the next five years will average between 2.5% and 3.5%, and a 28% probability that inflation will exceed 3.5% during this future period.

Although the mean inflation expectation only rose from about 2% before the pandemic to 2.5% in May 2024, the analysis in this post suggests there is a substantial right-tail risk that should be taken into account when considering the inflation outlook.

## Estimation of Distributions with Tail Risks

To calculate distributions that account for tail events, I follow the methodology laid out in my 2020 paper with co-authors Laura Veldkamp and Venky Venkateswaran.See “The Tail That Wags the Economy: Beliefs and Persistent Stagnation,” *Journal of Political Economy*, August 2020, Vol. 128, No. 8. This approach requires estimating the mean, µ; the standard deviation, σ; and the skewness, ɣ. Let *x* be the random variable of interest (e.g., CPI over the next five years), and w be the standardized variable defined as ɷ = (x-µ)/σomega equals left parenthesis x minus mu right parenthesis divided by sigma. The proposed estimate for the distribution of ɷ is:

The estimate for the probability distribution is equal to phi left parenthesis omega right parenthesis left parenthesis 1 minus gamma left parenthesis left parenthesis 3 omega minus omega cubed right parenthesis right parenthesis divided by 6 right parenthesis

where ɸ(ɷ) is the standard normal density function. Note that if ɣ = 0, then it is a standard normal distribution. However, when the skewness, ɣ, is nonzero, it allows for a fatter tail in the distribution.

The estimation process requires three data moments: the mean, the standard deviation and the skewness. Fortunately, the Federal Reserve Bank of Minneapolis reports weekly estimates of the market-implied mean, standard deviation and skewness for expected CPI inflation over the next five years. These estimates are derived from the prices of inflation caps and floors traded on the Bloomberg platform, which reflect financial market participants’ expectations for future inflation.Caps and floors are option-like derivative contracts often used to hedge against high or low inflation, respectively. For a given maturity period and prespecified inflation rate (the strike rate), the buyer of a cap pays an upfront premium. If actual inflation over that maturity exceeds the strike rate, the seller compensates the buyer. Floors work inversely—if inflation falls below the strike, the seller pays out. By comparing premiums of caps or floors with different strike rates for the same maturity, it becomes possible to estimate market-based probabilities for inflation over that period. See, for example, Michael McCracken and Aaron Amburgey’s 2021 blog post. There are concerns about the liquidity of this market; see, for example, New York Fed President John C. Williams’ 2023 speech.

## Market-Implied Mean, Standard Deviation and Skewness

The first figure shows the market-implied mean for the distribution of expected CPI inflation over the next five years, starting in January 2018. While the mean was around 2% in 2018, it peaked at 3.5% in 2022 and then fell to 2.3% by January 2024. However, the mean has since risen to about 2.5% by May 2024, reflecting the recent uptick in inflation in the U.S. economy. The second figure shows the market-implied standard deviation, which has also seen a substantial increase, from 0.8% in January 2018 to 1.7% in May 2024. This increase in the standard deviation captures the recent heightened volatility in inflation as well as greater uncertainty surrounding future inflation levels.

The third figure presents the skewness of the expected inflation distribution. Skewness measures the asymmetry of a probability distribution. A negative skew indicates that the left side is the larger of the distribution’s two tails, while a positive skew means the right side is the larger one. In 2018, the market-implied skew was around -0.20, implying left-tail risks—a higher risk of lower-than-expected inflation. However, the market-implied skew as of May 2024 was 1.18, signifying right-tail risk—a higher risk of higher-than-expected inflation.

**Inflation Expectations over the Next Five Years**

**Standard Deviation of Inflation Expectations over the Next Five Years**

**Skewness in the Distribution of Inflation Expectations over the Next Five Years**

SOURCE FOR THE THREE FIGURES: Federal Reserve Bank of Minneapolis and author’s calculation.

NOTE FOR THE THREE FIGURES: The market-based inflation expectations are for CPI inflation over the next five years.

## Probability Distributions

While the three moments (mean, standard deviation and skewness) provide useful information, drawing quantitative conclusions from them alone may be challenging. However, one can apply the estimation methodology described earlier to map these moments onto probability distributions. I estimate the distribution of future inflation by dividing it into four groups: (i) below 1.5%, (ii) between 1.5% and 2.5%, (iii) between 2.5% and 3.5%, and (iv) more than 3.5%. Note that, by construction, the sum of the probabilities across these four groups must equal 100%.

The next figure shows the time series of probabilities for each of the four inflation groups. Three distinct regimes are evident: pre-COVID-19, early COVID-19, and late COVID-19 (starting around mid-2022).

**Probabilities for Expected Inflation Ranges over the Next Five Years**

SOURCE: Federal Reserve Bank of Minneapolis and author’s calculations.

NOTE: The figure shows the probability that average annual inflation for the next five years belongs to one of four possible intervals: 1) less than or equal to 1.5%, 2) greater than 1.5% and less than or equal to 2.5%, 3) greater than 2.5% and less than or equal to 3.5%, or 4) greater than 3.5%.

Before the COVID-19 pandemic, the second group (expected inflation of 1.5% to 2.5%) had the highest market-based probability. The negative skewness, as noted earlier, implies the first group (below 1.5%) had a positive probability, but the chances of the third group (2.5% to 3.5%) were relatively small, while the probability of the fourth group (above 3.5%) was almost zero. These estimates are consistent with the view that inflation was roughly around the Federal Reserve’s 2% target,The Fed’s inflation target is stated in terms of the annual percent change in the personal consumption expenditures (PCE) price index, which is a broader price index than the CPI. with risk tilted toward lower-than-desired inflation levels.

The second episode, from the start of the COVID-19 pandemic until the second quarter of 2022, showed increased volatility in the estimates, reflecting the heightened volatility in markets and the overall economy due to the ongoing pandemic disruptions.

The most interesting results emerge after the second quarter of 2022, when the estimates seem to settle at a more stable level. The probabilities assigned to the third and fourth groups remain relatively high: There is a 22% probability that CPI inflation over the next five years hovers between 2.5% and 3.5%, and 28% probability that it exceeds 3.5%. Hence, the market implies there is a 50% probability that inflation will be above 2.5% over the next five years, according to my analysis.

To conclude, the mean inflation expectation only rose from about 2% before the pandemic to 2.5% in May 2024. However, the standard deviation also increased substantially, and the skewness switched from negative to positive, signaling right-tail risks on inflation expectations. The analysis in this post suggests that both second- and third-order moments should be taken into account to make assessments about future inflation and their risks.

**Notes**

- See “The Tail That Wags the Economy: Beliefs and Persistent Stagnation,”
*Journal of Political Economy*, August 2020, Vol. 128, No. 8. - Caps and floors are option-like derivative contracts often used to hedge against high or low inflation, respectively. For a given maturity period and prespecified inflation rate (the strike rate), the buyer of a cap pays an upfront premium. If actual inflation over that maturity exceeds the strike rate, the seller compensates the buyer. Floors work inversely—if inflation falls below the strike, the seller pays out. By comparing premiums of caps or floors with different strike rates for the same maturity, it becomes possible to estimate market-based probabilities for inflation over that period. See, for example, Michael McCracken and Aaron Amburgey’s 2021 blog post. There are concerns about the liquidity of this market; see, for example, New York Fed President John C. Williams’ 2023 speech.
- The Fed’s inflation target is stated in terms of the annual percent change in the personal consumption expenditures (PCE) price index, which is a broader price index than the CPI.

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