Measuring Interest Payments on the U.S. Federal Debt

Measuring Interest Payments on the U.S. Federal Debt George Hall Brandeis University Thomas Sargent New York University May 19, 29 The U.S. government routinely misstates interest costs on government debt. These misstatements occur partly because the government fails to account properly for the real capital losses government creditors experience during inflationary periods. 1 In addition, the government imperfectly measures the fluctuations in returns to its creditors from changes in interest rates and the maturity composition of the debt. When investors compute the real return on an equity or debt investment, they take into account dividend and coupon payments, the change in price of the stock or bond, and the effect of inflation on the general price level. So should the government in accounting for its interest costs to the public. In this paper we update and extend calculations in Hall and Sargent (1997 that estimate the government s true real interest costs. 2 By manipulating the government s period-by-period budget constraint, we identify the interest cost component from the stock of outstanding government obligations for the post-world War II period. We then decompose these costs to uncover the anticipated and unanticipated interest expenses. These updates are of interest in part because of the introduction of inflation-protected Treasury securities (TIPS in late 199s. While all types of debt allow the government to smooth taxes through time, debt denominated in nominal dollars We thank Francisco Barillas, Christian Grewell, and Leandro Nascimento for careful and diligent research assistance. Department of Economics, Brandeis University, 415 South Street, Waltham, MA 2454-911; phone: (781 736-2242; e-mail: ghall@brandeis.edu Department of Economics, New York University, 19 W. Fourth Street, New York, NY 112-1119; phone: (212 998-3548; e-mail: Thomas.Sargent@nyu.edu 1 See, for example, Olivier Blanchard and Jeffrey Sachs, There is No Significant Budget Deficit, New York Times, March 6, 1981, p. A26. 2 Also see Sargent(1993, Hall and Krieger (2, and Sims (21 for similar calculations. 1

allows the government to hedge against fiscal shocks through unexpected changes in inflation. Since the coupon and principal payments of TIPS are indexed by the CPI, the introduction of these securities limits the government s ability to use unexpected changes in inflation to manipulate the return to (and thus share fiscal risks with its creditors. We find that for the period from 1941 to 28 the true cost of funds for the U.S. Government is considerable lower on average and more volatile than the interest cost figures reported by the U.S. Treasury. In particular, for the four years 23-26, the government s return to its creditors was essentially zero. So far, the introduction of inflation protected securities does not appear to have dampened the volatility of the government s cost of funds. 1 A flawed measure of the government s cost of funds Every year the government repays its debt holders in two ways: explicitly in the form of coupons and principal repayments, and implicitly in the form of real capital gains on outstanding debt stemming from the diminished term to maturity of the debt, interest rate changes, and inflation. To measure the government s cost of funds, one must account for the capital gains and losses on outstanding Treasury securities. The federal government reports an incorrect measure of its cost of funds. It records an imperfect measure of its explicit interest costs and ignores its implicit interest costs. The government computes its cost of funds by forming the sum of current coupons on long term coupon bonds and the appreciation on short term discount bonds. 3 Figure 1 reports these official interest expenses as a percent of federal outlays over the period 1941 to 28. 4 The figure displays the spike in interest payments as fraction of expenditures right after World War II and the 198s growth in interest payments, a hallmark of Reaganomics. As the debt-to-gdp ratio fell in the 199s and then rose in the early 2s, so did reported interest costs. This measure of the government s cost of funds is a remarkably smooth series, and it is always positive. The following example illustrates how the government s methodology mismeasures its cost of funds. Consider selling two bonds that would raise the same value for the government at time t =, assuming no uncertainty and a constant real interest rate, r. One is a pure discount (zero 3 The Department of the Treasury calculates the net interest as the sum of coupon payments, accrued interest on bills and zero coupon bonds, and interest on non marketable debt. 4 The series plotted is net interest paid by the Federal Government as a percent of total expenditures reported by the U.S. Treasury. 2

16 14 12 1 8 6 4 2 194 195 196 197 198 199 2 21 Figure 1: Net interest costs computed by the Treasury Department, as a percent of total government outlays. coupon bond with ten periods to maturity, paying off P at time 1; the second is a coupon bond with coupon c, paying off P 1 at time 1. From the net one-period interest rate we can compute the discount factor, 1/(1 + r. The value of the pure discount bond p (t satisfies p (t = (1 + r 1 p (t + 1, for t =, 1,..., 9, where p (1 = P. Evidently, for the pure discount bond, interest accrues through the gradual appreciation in the value of the bond from p ( = (1 + r 1 P at time to p (1 = P at time 1. The rate of appreciation equals the gross interest rate: The value p 1 (t of the coupon bond satisfies 1 + r = p (t + 1. p (t p 1 (t = c + (1 + r 1 p 1 (t + 1, t = 1,..., 9 and p 1 ( = (1 + r 1 p 1 (1, 3

where p 1 (1 = P 1. The interest rate satisfies 1 + r = c p 1 (t + p 1(t + 1 p 1 (t (1 for t = 1,..., 9. For coupon bonds of finite maturity with a principal payment at the end (really a last big coupon, interest payments (that is, the left hand side of equation 1 include more than the coupon. It is not appropriate to measure the interest costs associated with coupon bonds by simply adding up the coupons due this period. Indeed, coupon payments do not represent pure interest in an economic sense; they are partly a repayment of principal. Furthermore, part of the return to investors, and of the cost to the issuer, is in the form of capital gains or losses on bonds as time passes. Any bond with a large final payment is partly a pure discount bond with a significant portion of its return coming in the form of capital gains or losses over time. Our example indicates some but not all of the corrections that we want to make in the government s accounting for its interest costs. The other adjustments have to do with the treatment of inflation, time variation in interest rates, and the existence at any moment of a variety of bonds with various coupon schedules and maturities. Next we expand our example to incorporate all of these features and show how to do the accounting properly. 2 Doing the Accounting Right We build a system for properly counting the government s real interest costs by carefully rearranging the government s period-by-period budget constraint. We then decompose these interest costs in two different ways. First, we manipulate the budget constraint algebraically to isolate explicit and implicit interest expenses. Explicit interest expenses are the real capital gains on one-period discount bonds; implicit interest expenses are the capital gains to the public from holding longer bonds. Second, we decompose the costs into the one-step-ahead anticipated costs and the unanticipated costs. To perform this accounting, we compute at each date t the number of dollars the government has promised for each date t + j in the future. One can regard a coupon bond as a bundle of pure discount bonds of different maturities. We want to treat coupon and principal payments in the same ways. We can price a coupon bond by unbundling it into a set of pure discount bonds, valuing each component individually, and adding up the value of the components. In other words, 4

we strip the coupons from the bond and price the bond as a weighted sum of pure discount bonds of maturities j = 1, 2,..., n. The market and the government already do this. Prestripped coupon bonds are routinely traded. We treat nominal bonds and inflation-indexed bonds separately. For nominal bonds, let s t t+j be the number of time t + j dollars that the government has promised to deliver, as of time t. To compute s t t+j from historical data, we add up all of the dollar principal-plus-coupon payments that the government has promised to deliver at date t + j as of date t. Since zero-coupon bond prices were not directly observable until prestripped coupon bonds were introduced in 1985, we extract the nominal implicit forward rates from government bond price data. We then convert these nominal forward rates on government debt into prices of claims on future dollars. Let qt+j t be the number of time t dollars that it takes to buy a dollar at time t + j: q t t+j = 1 (1 + ρ jt j exp( jρ jt where ρ jt is the time t yield to maturity on bonds with j periods to maturity. Thus, q t t+j nominal pricing kernel at t. To convert t dollars to goods we use is the v t = 1 p t where p t is the price level in base year 1983 dollars, and v t is the value of currency measured in goods per dollar. For inflation-protected bonds, let s t t+j be the number of time t + j goods that the government has promised to deliver as of time t. For indexed debt, when we add up the principal and coupon payments that the government has promised to deliver at date t + j as of date t, we must make adjustments for the past realizations of inflation consistent with the rules governing TIPS. To compute the real price of a promise, sold at time t, of goods at time t + j, ρ jt, we use the real yield to maturity ρ jt = ρ jt π t where as before ρ jt is the time t yield to maturity on bonds with j periods to maturity, and π t is the inflation rate from t 1 to t realized at t. We appeal to a random walk model for inflation 5

to justify how we construct real yield to maturities. 5 time t goods that it takes to purchase a time t + j good, by: We can then compute q t+j t, the number of q t t+j = 1 (1 + ρ jt j exp( j ρ jt. Let def t be the government s real net-of-interest budget deficit, measured in units of time t goods. We can write the government s time t budget constraint as: n n v t qt+js t t t+j + q t+j t s t n t+j = v t q t s t 1 + n q t s t 1 + def t (2 where it is understood that q t t = 1 and q t t = v t. The left hand side of equation (2 is the real value of the interest bearing debt at the end of period t. The first term on this side is the real value of the nominal debt; it is determined by multiplying the number of time t + j dollars that the government has sold in the form of j period pure discount bonds, s t t+j, by their price in terms of time t dollars, qt+j t, summing this product (or value over all such outstanding bonds, j = 1,..., n, and then converting from dollars to goods by multiplying by v t. The second term is the value of the inflation-protected debt, computed by multiplying the number of time t + j goods that the government has promised, s t t+j, by their price in terms of time t goods, qt t+j, and then summing this product over j = 1,..., n, The right side of equation (2 is the sum of the real value of the primary deficit def t and the real value of the outstanding debt that the government owes at the beginning of the period, which in turn is simply the real value this period of outstanding promises to deliver future dollars s t 1 t 1+j and goods s t 1 t 1+j that the government issued last period. The government budget constraint, equation (2, can be rearranged to take the following form: n n v t qt+js t t t+j + q t+j t s t t+j = + v t q t v t 1 q t 1 n s t 1 + v t 1 q t 1 st 1 n q t q t 1 s t 1 + q st 1 t 1 + def t (3 These two forms of the budget constraint are algebraically equivalent. Equation (3 breaks each of the first two terms on the right side of (2 into an interest component and a previous value component. Again, the left hand side of the budget constraint (3 is the real value of government 5 See Atkeson and Ohanian (21 and Stock and Watson (26. 6

debt that the government has outstanding at the end of period t. Thus, the true cost of funds for the government is given by: v t q t v t 1 q t 1 s t 1 + n ( q t q t 1 s t 1. (4 In Figure 2 we contrast the Treasury s official interest cost series with our computed costs of funds series using annual end of the year data from 1941 to 28. The price and quantity data for nominal bonds are from the CRSP Monthly Government Bond File. 6 The quantity outstanding of the Treasury inflation-protected securities (TIPS were obtained from December issues of the U.S Treasury s Monthly Statement of the Public Debt. For the pre-197 period we fit a zero-coupon forward curve from the coupon bond price data via Daniel Waggoner s (1997 cubic spline method. Waggoner fits the zero-coupon one-period forward-rate curve with a cubic spline employing a set of roughness criteria to reduce oscillations in the approximated curve. For 197 to 28 we use the nominal and real zero-coupon yield curves computed by Gurkaynak, Sack and Wright (26, 28. The value of currency v t is the inverse of the December observation of the consumer price index. As can be seen in Figure 2, our computed costs of funds are lower on average and considerably more volatile than what the Treasury reports. 7 Perhaps the most striking event in our series is the large capital loss imposed on bondholders right after World War II. Bondholders received a -12 percent return on government securities in 1946. The government s real borrowing costs were negative during other periods as well: in 195 with the outbreak of the Korean War, during the high inflation episodes of the 197s, and in 1994 and 1999 with the steep fall in bond prices. 8 Overall, the real return on government debt has averaged only 8/1 of one percent since 1941. If we drop the first six years of our sample, the Treasury s real interest costs still averaged only 1.2 percent. When measured correctly, the Treasury s borrowing costs have historically been very low. The first term of equation (4 represents interest on the nominal portion of the government 6 In the CRSP data set the quantity of publicly held marketable debt only goes back to 196. We extended this series using data from the Treasury Bulletin. 7 The Treasury s calculations include some assets (e.g., savings bonds and some securities issued to state and local governments that are not included in our analysis, so these two series are not strictly comparable. Nevertheless we expect that adding these assets to our analysis would not meaningfully change the results. 8 It is interesting to compare these outcomes with predictions of Lucas and Stokey s (1983 model of tax smoothing, according to which government debt pays low returns when there are high government expenditure shocks. 7

3 2 1 1 2 3 4 5 194 195 196 197 198 199 2 21 Figure 2: Real cost of funds computed using equation (4 (solid blue line and the interest costs reported by the Treasury (dashed green line as a percent of real total government outlays. debt, and can be decomposed as ( v t v t 1 q t 1 t t + j=2 s t 1 v t q t v t 1 q t 1 s t 1. (5 The first term in (5 is explicit interest and the second term is implicit interest in the form of real capital gains to the public on its claims on the government. Thus, the term v t v t 1 q t 1 t is the per dollar real capital gain accruing to one-period discount bonds issued at time t 1. The term v t q t v t 1q t 1 is the increase (or the decrease, if negative in the price in terms of goods between t 1 and t of a claim to one dollar in time t 1 + j; multiplying this change in price by the dollar value of time t 1 + j claims outstanding, s t 1 at time t 1, and summing over j gives the capital gain to the public. Similarly, the second term in (4, which represents the interest on the real portion of the government debt, can be decomposed as ( 1 q t 1 t s t 1 t + j=2 q t q t 1 s t 1. (6 In figure 3 we report our breakdown of the total interest costs on the marketable federal debt between explicit and implicit real interest costs from 1941 to 28. In general, the explicit 8

2 1 1 2 3 4 194 195 196 197 198 199 2 21 Figure 3: Explicit (solid blue line and implicit (dashed green line real interest costs as a percent of real total government outlays. interest costs were relatively small and relatively constant over this period. In contrast, the implicit interest costs were substantial, variable, and often negative. Since the real value of the outstanding debt was growing over this period, the s t t+js were growing over time. So the per dollar capital gains are being multiplied by increasingly large numbers. Thus the implicit interest cost became more volatile during the sample period. Alternatively, we can decompose the cost of funds term on nominal debt (i.e. the first term in equation (4 as: n ( E t 1 + v t q t v t 1 q t 1 v t q t v t 1 q t 1 s t 1 s t 1 E t 1 v t q t v t 1 q t 1 s t 1. (7 The first term in (7 is the one-period-ahead anticipated cost of funds and the second term is the unanticipated cost of funds. Since v t 1, q t 1, and st 1 are known at time t 1, we need only to compute the one-step ahead expectation of v t q t for j = 1,..., n. We compute the expected value of v t assuming inflation follows a random walk and expected prices q t from the implied the forward rates; we compute the covariance of v t and q t using ten year rolling windows. 9

Likewise, the cost of funds term for inflation-protected debt (the second term in equation (4 can be decomposed as: n ( E t 1 + q t q t 1 s t 1 q t q t 1 s t 1 E t 1 q t q t 1 s t 1 (8 where the first term is the anticipated component and the second term is the unanticipated component. The one-step-ahead anticipated real cost of funds (i.e. the sum of the first terms in (7 and (8 is plotted in figure 4 along with the calculated cost of funds and the Treasury s reported interest costs. 9 These anticipated interest costs generally go through the middle of the calculated cost of funds. In figure 5 we plot both the anticipated and the unanticipated real returns on U.S. Treasury debt on the same graph. Our results are consistent with similar calculations done by Christopher Sims (21. The unanticipated returns are quite volatile. Although they are roughly zero on average, they regularly exceed 5 percent (in absolute value of total expenditures. In particular, much of the capital loss that occurred in 195 with the outbreak of the Korean War appears to have been unanticipated. Furthermore, during the late 196s, government creditors effectively helped pay for the Great Society and the Vietnam War as the government unexpectedly inflated away part of the real value of the outstanding debt. For the most part, these unanticipated returns continued to be negative in the 197 s as the government offset the fiscal stress of the two oil crises. During the 198s and 199s, as inflation subsided, bondholders usually received positive unanticipated returns. There were several exceptions. In 1994 and 1999 the Fed began raising short-term interest rates causing bond prices to fall. In the next three figures we examine the recent experience in more detail. Figure 6 reports both the official and our calculated cost of funds as a percentage of government expenditures from 1989 to 28. As has been true since World War II, the official numbers continue to overstate the true cost of borrowing. Calculated correctly, the cost has been on average about 5 percent of total expenditures, not the 1 to 15 percent reported by the Treasury. As the debt-to-gdp ratio 9 Our one-year ahead forecasts of inflation are unreliable during the late-194s. In particular, our random walk assumption for inflation has trouble explaining the steep rise in prices that occurred at the end of World War II. Since the expected inflation series seems sensible for the remainder of the sample, we truncated the decomposed series in 1952. 1

25 2 15 1 5 5 1 15 2 194 195 196 197 198 199 2 21 Figure 4: The interest costs reported by the Treasury (dot-dashed blue line, the real cost of funds (dashed green line, and the expected cost of fund(solid red line as a percent of real total government outlays. 2 1 1 2 3 4 194 195 196 197 198 199 2 21 Figure 5: Expected (solid blue line and unexpected (solid green line components of the interest costs as a percent of government outlays. 11

25 2 15 1 5 5 1 1988 199 1992 1994 1996 1998 2 22 24 26 28 Figure 6: Real cost of funds (solid green line and the interest costs reported by the Treasury (solid blue line from 1989 to 28 as a percent of real total government outlays. fell throughout the late 199s, so did interest costs as percentage of government expenditures. It is interesting to note that from 23 and 26, the Federal government borrowed at essentially zero cost. However, in the last two years, borrowing costs have grown significantly. In figure 7, we report the anticipated and unanticipated components of returns since 1989. The anticipated component is always positive and relatively smooth. The unanticipated component is centered around zero and quite a bit more volatile. The variance of the unanticipated component is almost 5 times that on the anticipated component. Even with the introduction of TIPS, much of the volatility in the government s cost of funds have come from unanticipated returns. Figure 8 reports the one-step-ahead anticipated and unanticipated components of the cost of funds associated with the inflation-protected securities (TIPS. Since their introduction in 1997, TIPS have become a larger share of the outstanding stock of debt. Currently about 1 percent of the market value of the Federal debt is indexed. Since the coupon and principal payments of TIPS are indexed by the CPI, the introduction of these securities changes how the government shares inflation risk, since the government cannot induce real unexpected capital gains and losses through changes in unexpected inflation. Nevertheless, although the anticipated cost of the bonds is, with the exception of 22, positive and between 1/2 and 1 percent of outlays, the unanticipated costs 12

2 15 1 5 5 1 15 2 1988 199 1992 1994 1996 1998 2 22 24 26 28 Figure 7: Expected (dashed green line and unexpected (solid red line components of the interest costs and the interest costs reported by the Treasury (solid blue line from 1989 to 28 as a percent of real total government outlays. have been variable and often negative. Since TIPS are typically long-term bonds, unexpected changes in interest rates have a large impact on their values. 3 Conclusion The federal government reports a flawed measured of its own cost of funds. By ignoring the effects of inflation and changes in interest rates on the value of the outstanding federal obligations, the official interest payment calculations make it difficult to accurate assess the true cost of government borrowing. In this note, we have described how to compute a more accurate estimate of the federal government cost of funds. Compared to the numbers from the U.S. Treasury, our measure of these costs is much more volatile and lower on average. 13

1.5 1.5.5 1 1.5 2 2.5 1998 1999 2 21 22 23 24 25 26 27 28 Figure 8: Expected (solid blue line and unexpected (dashed green line components of the interest costs for the TIPS as a percent of government outlays. References [1] Andrew Atkeson and Lee E. Ohanian (21 Are Phillips Curves Useful for Forecasting Inflation? Quarterly Review, Federal Reserve Bank of Minneapolis, issue Win, pages 2-11. [2] Blanchard, Olivier Jean and Jeffrey Sachs (1981 There is No Significant Budget Deficit, New York Times, March 6, page 26. [3] Gurkaynak, Refet S., Brian Sack, and Jonathan Wright (26 The U.S. Treasury Yield Curve: 1961 to the Present. Board of Governors of the Federal Reserve System Working Paper 26-28. http://www.federalreserve.gov/pubs/feds/26/2628/2628abs.html [4] Gurkaynak, Refet S., Brian Sack, and Jonathan Wright (28 The TIPS Yield Curve and Inflation Compensation. Board of Governors of the Federal Reserve System Working Paper 28-5. http://www.federalreserve.gov/pubs/feds/28/285/285abs.html [5] Hall, George J. and Stefan Krieger (2 The Tax Smoothing Implications of the Federal Debt Paydown. Brookings Papers on Economic Activity Vol 2. pp. 253-284. 14

[6] Hall, George J. and Thomas J. Sargent (1997 Accounting for the Federal Government s Cost of Funds. Federal Reserve Bank of Chicago Economic Perspectives Vol 21. No. 4. pp 18-28. [7] Lucas, Robert Jr. and Nancy L. Stokey (1983 Optimal Fiscal and Monetary Policy in an Economy Without Capital. Journal of Monetary Economics, Elsevier, vol. 12(1, pages 55-93. [8] Sargent, Thomas J. (1993 Fact or Fiction: Shortening Debt Maturity Lowers Interest Costs. Catalyst Institute Research Project, December. [9] Sims, Christopher A. (21 Fiscal Consequences for Mexico of Adopting the Dollar. Journal of Money, Credit and Banking Vol. 33, No. 2, Part 2. (May, pp. 597-616. [1] Stock, James and Mark W. Watson (26 Why Has U.S. Inflation Become Harder to Forecast? NBER Working Paper 12324, National Bureau of Economic Research, Inc. [11] Waggoner, Daniel (1997 Spline Methods for Extracting Interest Rate Curves from Coupon Bond Prices. Working paper 97-1. Atlanta, Ga.: Federal Reserve Bank of Atlanta. 15