Decoding the Bond Market

What are the expected future interest rates implied by these par yields? The conversion involves three steps:

1. Convert from par yields to zero coupon bond yields, then to forward yields
2. Adjust for the convexity in forward yields that arises from profit on forward contracts being extra nice, since they’ll get discounted to the present at a lower interest rate while losses are discounted at higher rates⁴
3. Estimate and adjust for the risk premium investors are normally offered to take interest rate risk.⁵ There are different plausible ways to represent the interest rate risk premium. We like and use the approach proposed by Andy Morton (2022) in which the risk premium takes the form λ σ T, where λ is the risk premium parameter and σ is the variability of interest rates for term T (see footnote for more detail).⁶ λ can be interpreted as a Sharpe ratio, which is assumed to be constant for bonds of all different maturities and coupons. Fitting the risk premium requires the further assumption that far enough into the future, the market will not have sufficient information to meaningfully differentiate between the expected short-term interest rate on different dates.

In the chart below, we show the result of these adjustments for secured US nominal and real interest rates.⁷ We can see that far into the future, the market expects about 3.85% for the nominal short-term interest rate, and 1.75% for the real interest rate. The expectation for the difference between the two — the inflation rate — is 2.1%.⁸

While the Fed would surely be delighted to see the bond market’s confidence in the Fed’s ability to maintain inflation right around 2% in the long term,⁹ we suspect there are many investors who would feel this expectation is perhaps a bit too optimistic. For example, over the past 125 years, US inflation has averaged 3%, and in the 54 years since the US went off the gold standard, the average inflation rate has been 3.9%.

In contrast to the prediction embedded in bond prices, the Fed’s March 2025 Fed “dot plot” predicts a long-term level for the short rate of 3%, and a real rate of about 1%. Both of these rates are about 1% lower than those implied by the market.

Interest rate risk premia

Built into the nominal and real yield curves are the risk premia that investors are requiring to bear interest rate risk. For nominal bonds, the implied risk premium is 10% of the risk they are bearing, meaning the excess-return-to-risk ratio (i.e. the Sharpe ratio) of bonds of all maturities is 0.1. For example, if you own a 10-year bond (with no risk of default) and that bond has an annual price risk of 6%, then you’d expect to earn 0.6% above the short-term risk-free rate.¹⁰

For real interest rates, the risk premium implied by the term structure of real rates is 8%, only a tad lower than for nominal interest rates. This is somewhat surprising, as many financial economists consider long-term inflation-protected bonds to be the closest thing to a risk-free asset.¹¹ The reason is that if the main purpose of wealth is to support consumption over our lifetimes — and possibly beyond through bequests — then we should measure our wealth in terms of a long-term, inflation-protected annuity, and TIPS come close to delivering those desired cash flows.

Furthermore, the 0.1 Sharpe ratio implicit in the nominal rate term structure seems low, compared to Sharpe ratios offered by other risky asset classes such as equities. This low nominal Sharpe ratio may make sense in a world where interest rate risk is negatively correlated with stock market risk. However, if the recent positive correlation between the returns of stocks and bonds continues, perhaps we will see the bond market start to offer investors a higher risk premium.

We can also infer the risk premium associated with bearing pure inflation risk, which comes in at 7%.¹² This is also puzzling, as it is not only low in absolute terms, but it is also lower than the risk premium for bearing real interest rate risk. High inflation feels like a more significant systematic risk than the risk coming from changes in long-term real rates, but the market doesn’t see it that way.

Estimating risk premia from historical returns

Another approach taken by researchers is to try to estimate the bond market risk premium by measuring the historical return of bonds versus the return of short-term Treasury bills.13 This approach has two shortcomings: 1) the bond market risk premium very likely changes over time, and 2) it takes a lot of historical data to estimate the risk premium accurately even if it didn’t change over time. For example, if we assumed that the bond market risk premium was constant over the past 30 years and we found that bonds delivered a Sharpe ratio of 0.1, all we could say is that it’s 90% likely that the true bond market risk premium was between -0.2 and 0.4.

Estimating the bond market risk premium using historical returns is similar to estimating the expected return of the stock market using historical returns, rather than using a forward-looking metric such as earnings yield. As a general matter, forward-looking metrics based on cash flows, when available, give better expected return estimates than we can get from extrapolating historical returns.

US Treasury sovereign default risk

Table 2 below shows normal US Treasury bond yields to the right of the secured, extra-safe yields from Table 1. Notice that Treasury bonds have higher yields than the rates for secured, extra-safe bonds, and that the spread gets higher with maturity, reaching 0.84% at 30 years.

One way (but not the only way) to interpret these rates is to see the higher yields as representing some combination of expected loss from a default on US Treasury obligations, and a risk premium for bearing that risk of loss. Assuming that a default would result in a 50% hit to promised payments, and assuming that three-quarters of the spread represents an expectation of default and the other quarter is a risk premium,¹⁴ we calculate the cumulative probability of default to 10-year and 30-year horizons at about 10% and 50% respectively.

To the extent one sees prices of US Treasury bonds through the lens of credit risk, the message being sent by long-term Treasury yields being close to 1% higher than corresponding interest rate swaps is a very sobering one indeed.¹⁵ While there are less ominous interpretations of these spreads, such as that they represent bank balance sheet constraints,¹⁶ the fact that US sovereign debt no longer carries the highest credit rating from the main credit rating agencies suggests that investors should not be too dismissive of default risk on this very important asset class. In the end, it may not matter whether in reality the spread is representative of a market micro-structure imbalance rather than signaling a significant probability of default — it is unequivocally a “bad look.” Given the potentially alarming message the spread is sending, perhaps we’ll see the US Treasury or Fed entering the long-maturity interest rate swap market as a fixed rate payer in an effort to narrow the spread between long-term Treasuries and swaps. And it’s not a bad trade either, extending the duration of the debt at an expected long-term short rate of a bit under 4%.

Connecting the dots

Our reading of bond market messaging is that it is more attractive to hold exposure to long-term interest rates in inflation-protected form, as long-term nominal rates are not offering adequate compensation for the inflation risk inherent in them. If you feel that the probability of US sovereign default in the long term is much lower than that implied in market pricing, then US TIPS might be a doubly good way to hold your long-term interest rate exposure.

While the bond market is in general pretty efficient in its pricing, there may be times when it can be significantly out of line with investor expectations. At such moments, investors should be well-rewarded for making the effort to decode what the bond market is saying.

Endnotes

¹ Of course, it’s also useful to be aware of the interest rate and inflation surveys compiled by the Fed and other bodies.

² In this note, we use “expectations” in the mathematical sense of a probability-weighted outcome, not as the most likely or median outcome. As we’ll discuss in more detail shortly, interest rate futures are assumed to reflect a combination of real-world expectations and risk premium. For short-term rates 18 months in the future, it is generally agreed that the risk premium effect is not large.

³ These secure bonds are equivalent to overnight lending secured by Treasury collateral plus SOFR interest rate swaps. We pulled the swap rates used in the table from this link.

⁴ The convexity adjustment for long-dated forwards depends primarily on the volatility of interest rates, and secondarily on the specific process that interest rates follow over time (e.g. normal, log-normal, in between). We can glean much of the necessary information from the interest rate options market. Here, we use an assumption that nominal interest rates are normally distributed with a standard deviation of. 0.75% per annum – a bit lower than where long-term interest rate options are trading currently. For real rates, we assume 0.60% volatility. A good rule-of-thumb estimate for the convexity adjustment is: ½ T2 σ2.

⁵ Though infrequent, there have been times when the bond market offered investors a negative risk premium, meaning investors viewed interest rate risk as desirable and were willing to pay to get that risk.

⁶ Here is Morton’s proposed model of term premium, from Morton (2022) – the references to Libor can be read today as applying to short-term secured rates such as SOFR:“With convexity eliminated, the main drivers of Libor futures rates are expected Libor at maturity and a term premium. I propose that for a sufficiently distant maturity T the market will expect the economy to have passed through any transient cycles and/or that there is insufficient information to distinguish one maturity from another so the expected spot rate will converge, to some r∞. For such T, I suggest Fut(·) should roughly have the form: Fut(T) ≈ r∞ λ T σT (1) where λ is a (possibly negative) risk premium parameter.Why? Suppose you want to go long the market (receive fixed), and are considering at what maturity to do so. For large T, if you trade at rate Fut(T) priced as in (1), your expected profit is λ T σT, since by assumption the expected settlement rate is r∞. You earn that profit at rate λ σT and the option market prices the annualized standard deviation of it at σT: the (Sharpe) ratio of the two is λ, independent of T. Here, I deploy my second assumption: in the mostly one factor world of interest rates, the Sharpe ratio of two very similar trades (realizations in the distant future of the short rate) should be equal, using market provided estimates of each trade’s volatility.”

⁷ Rates are continuously compounded annual rates, which are virtually the same as monthly compounded rates.

⁸ The expectation for inflation as the difference between the expected nominal and real rate is an approximation, which is close for nominal and real rates in the low single digits. The relation between the nominal rate, the real rate and the inflation rate is usually stated as (1 + rn) = (1 + rr) * (1 + rinf). Also, we are relying on the assumption that nominal and real rates are normally distributed and unbounded.

⁹ In fact, as the Fed’s preferred inflation measure is PCE inflation, which tends to run about 0.25% lower than the CPI measure, the market is expecting inflation a bit below the Fed’s target!

¹⁰ As time goes by and the duration of the bond gets shorter, the extra return it needs to deliver declines.

¹¹ Some observers suggest that the real rate risk premium may come from investors worrying about the integrity of the CPI measurement process.

¹² For this calculation we generate an inflation term structure consistent with the nominal and real rate term structures, and assume a volatility of inflation of 0.5% per annum, uncorrelated with changes in the real rate.

¹³ For example, see Kim and Orphanides (2005) and Kim and Wright (2005).

¹⁴ It’s difficult to estimate the split between expected default and a risk premium for taking that risk. We could imagine, using an expected utility analysis, investors with typical degrees of risk-aversion, holding 20% of their portfolios in Treasury bonds with an expected return equal to about one third of the expected loss from default.

¹⁵ A similar message is being signaled by five-year credit default swaps on US Treasury bonds, which are quoted at 0.5% per annum – a level pretty close to where default swaps are trading on Italian and Greek debt!

¹⁶ Though if that were the case, we might expect that the spreads would not start close to zero and expand with maturity so sharply.

Further Reading and References

• Cochrane, J. and Piazzesi, M. (2008). “Decomposing the Yield Curve.” SSRN.
• Fama, E. and Bliss, R. (1987). “The Information in Long-Maturity Forward Rates.” American Economic Review 77, 680-92.
• Kim, D. and Orphanides, A. (2005). “Term Structure Estimation with Survey Data on Interest Rate Forecasts.” Finance and Economics Discussion Series 2005-48. Board of Governors of the Federal Reserve System.
• Kim, D. and Wright, J. (2005). “An Arbitrage-Free Three-Factor Term Structure Model and the Recent Behavior of Long-Term Yields and Distant-Horizon Forward Rates.” Finance and Economics Discussion Series 2005-33. Board of Governors of the Federal Reserve System.
• Morton, A. (2022). “Inferring an Expected Rate and a Term Premium from Long Maturity Swaps and Options.” Unpublished.
• Rebonato, R. (2018). Bond Pricing and Yield Curve Modeling: A Structural Approach. Cambridge University Press.
• Stambaugh, R. (1988). “The information in Forward Rates: Implications for Models of the Term Structure.” Journal of Financial Economics 22, 3-25.
• Tuckman, B. and Serrat, A. (2022) Fixed Income Securities: Tools for Today’s Markets. Fourth Edition. Wiley. See chapters 8 and 9 and related appendices.

This is not an offer or solicitation to invest, nor are we tax experts and nothing herein should be construed as tax advice. Past returns are not indicative of future performance.

The inspiration for this article is an unpublished note by our friend and head of markets at Citigroup, Andy Morton: “Inferring an Expected Rate and a Term Premium from Long Maturity Swaps and Options.” (2022). Thank you also to our friends John Cochrane, Larry Hilibrand, Vladimir Piterbarg, Bill Montgomery, Jeffrey Rosenbluth and Bruce Tuckman for their very helpful discussion and comments.

Victor Haghani is founder & CIO of Elm Wealth, a Philadelphia-based asset manager. James White is Elm Wealth’s CEO.

Learn more at www.elmwealth.com.

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