# Background and Research

#### Calculating Inflation Expectations Using Two Kinds of Treasury Securities

Inferring long-term inflation expectations from the data is difficult. One measure, obtained from the University of Michigan's Survey of Consumers, asks participants what they expect inflation to be over the next 5 to 10 years, but it consistently runs above other measures and is probably not very reliable. Another measure of long-term inflation expectations is the Survey of Professional Forecasters (SPF), which asks professional forecasters what they expect inflation to be over the next 10 years. The SPF appears to measure expected inflation well on average but appears to be excessively smooth and does not pick up week-by-week or even month-by-month variation in actual expected inflation.

In theory, the yields on two different kinds of Treasury securities—nominal treasury notes and treasury inflation-protected securities (TIPS)—can be used to calculate a market-based estimate of expected inflation. Nominal treasury notes earn a fixed nominal rate of interest on a fixed amount of principal, whereas the principal of TIPS is adjusted for inflation (and thus so are the coupon payments). Because the return on nominal treasuries is vulnerable to inflation, it most assuredly contains compensation to investors for any losses they expect to incur from inflation if they hold the bond. TIPS therefore protect the bondholder from losses due to inflation, but nominal treasuries do not. Whereas nominal treasury notes earn a fixed nominal rate of interest, TIPS earn a fixed real rate of interest. In principle, one ought to be able to simply subtract the real yield on TIPS from the nominal yield of treasury notes of the same maturity to derive expected inflation.

Yet even that measure is imperfect. The TIPS-derived measure of inflation expectations underestimates SPF expected inflation and on average actual expected inflation by 50 basis points. If we could be assured that the difference between TIPS expected inflation and actual expected inflation was constant, we could easily correct for this bias. But this bias is probably not constant. There are reasons to suspect that the difference between a nominal security and an inflation protected TIPS security may not correctly measure expected inflation.

There are actually 2 different factors that cause TIPS to be a biased predictor of expected inflation: an inflation-risk premium and a liquidity premium. To make matters more difficult, these biases likely go in different directions. We attempt to correct for both of these biases.

The existence of an inflation-risk premium suggests that TIPS expected inflation likely overestimates actual expected inflation. This occurs because variable inflation implies that the real return on nominal treasury securities is uncertain, while, by definition, a TIPS real return is constant. To compensate for this inflation risk, the real return on TIPS will be less than the average return on nominal bonds. Studies suggest that because of inflation risk, TIPS expected inflation will overestimate actual expected inflation by 50 to 100 basis points. Although this bias may not be constant over very long periods of time, monthly movements in the bias are not likely to be important. If the bias is nearly constant, correcting for it is straightforward.

More difficult to correct for is the bias due to liquidity risk because it is likely not constant over time. While the TIPS market is deep, it is, nevertheless, less liquid than the market for nominal treasury securities. Because of this relative liquidity difference, a TIPS real return should be more than the real return on nominal government securities. That is, TIPS-derived expected inflation will underestimate actual expected inflation. This bias, however, does not appear to be constant and can change on a weekly basis.

#### Preparing the Adjusted Series:

Correcting for Illiquidity and Inflation Risk

To correct for the relative illiquidity of TIPS, we need a proxy for liquidity risk and an unbiased (but not necessarily efficient) predictor of expected inflation. The adjustment to the unadjusted TIPS expected inflation series is based on the following assumptions:

- While nominal treasuries are extremely liquid instruments, there is still a small liquidity premium priced into these bonds, and we assume that the liquidity premium in the TIPS market is correlated with the liquidity premium in the nominal treasuries market.
- One measure of the liquidity premium in the nominal treasuries market is the difference in the yields on nominal treasuries in the primary and secondary markets. (The primary market refers to bonds bought directly from the Treasury at auction; bonds bought in this market are said to be bought "on the run." The secondary market refers to bonds bought from other investors—"off the run".) We assume that the liquidity risk for TIPS is larger than the risk of nominal treasuries bought in the secondary market because the TIPS market is less developed.
- We assume that the difference observed between two measures of expected inflation—that reported in the Survey of Professional Forecasters and that derived from unadjusted TIPS yields—is largely driven by the liquidity risk. Furthermore, this liquidity risk does not affect the difference between actual and SPF expected inflation.

The liquidity premium in the nominal treasuries market is calculated as the difference between the yields on 10-year on-the-run and off-the-run treasury notes. (The data are obtained from the Board of Governors of the Federal Reserve System.) Regressing the spread between SPF expected inflation and unadjusted TIPS-derived expected inflation on the liquidity premium results in the following equation:

Spread = 0.948 - 12.71(LP) + 20.9(LP)^{2},

(where LP = liquidity premium).

The constant picks up the bias due to inflation risk, while rest of the equation picks up the bias due to liquidity. This equation demonstrates that if there were no liquidity risk in the nominal treasuries market, and thus no liquidity risk in the TIPS market, expected inflation derived from TIPS would overstate actual expected inflation by 95 basis points (0.95 percent). This overstatement is basically the size of the inflation risk predicted by earlier research, which lends credence to our correction method.

As liquidity risk rises in the nominal treasuries market, liquidity risk in the TIPS market also rises, so that the unadjusted TIPS-derived expected inflation series understates actual expected inflation. We can, therefore, correct unadjusted TIPS-derived expected inflation by subtracting the spread, estimated using the equation above, and the current liquidity premium in the nominal treasuries market:

Adjusted TIIS expected inflation = Unadjusted TIIS expected inflation - 0.9485 + 12.7(LP) - 20.9(LP)^{2}