> For the complete documentation index, see [llms.txt](https://laurence-wilse-samson.gitbook.io/textbooks/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://laurence-wilse-samson.gitbook.io/textbooks/financial-economics-claims-prices-holders/part-iii-asset-markets/chapter_09_fixed_income.md). # Chapter 9: Fixed Income, the Term Structure, and Safe Assets *Part III: Asset Markets — Financial Economics: Claims, Prices, and Holders* *** ## Opening Episode: The Conundrum, February 2005 On June 30, 2004, after a year at one percent, the Federal Open Market Committee began raising the federal funds target. It raised it again in August, September, November, December, and on February 2, 2005 — six moves of twenty-five basis points each, a hundred and fifty basis points of tightening announced in advance and delivered on schedule. Every textbook says what should follow. Short rates rise; expected future short rates rise with them; long rates, being averages of expected short rates, rise too. They fell. The ten-year Treasury yield was about 4.6 percent on the day the tightening began and about 4.2 percent when Alan Greenspan appeared before the Senate Banking Committee on February 16, 2005 to deliver the semiannual monetary policy report. The two-year yield had risen, measured from the start of 2004, by roughly the full amount of the funds rate move; the ten-year had gone the other way; the spread between them had compressed by more than a hundred basis points. Greenspan told the committee that "the broadly unanticipated behavior of world bond markets remains a conundrum." The word stuck, and the puzzle it named ran for another eighteen months: the Fed took the funds rate to 5.25 percent by June 2006 and the ten-year yield ended that cycle roughly where it had started. The explanations offered at the time have a common shape, and it is worth naming before we get to any of them. None of them is about the payoff of a Treasury note, which is fixed and known. Every one of them is about who was buying. **Foreign official demand.** Ben Bernanke, then a governor, gave the argument its name in a lecture in Richmond on March 10, 2005: a *global saving glut*. Asian and oil-exporting economies were running large current account surpluses and recycling them into dollar reserves, and reserve managers buy Treasury securities. Japan's Ministry of Finance had intervened on a scale of roughly $320 billion in the fifteen months to March 2004; China's foreign exchange reserves roughly doubled between the end of 2003 and the end of 2005. Warnock and Warnock's later estimate put the effect of foreign official flows on the ten-year yield at something like eighty basis points. Reserve managers do not trade the yield curve. They buy what their currency policy leaves them holding, at whatever price the market shows them. **Pension duration demand.** At the same moment, European and British pension regulation was moving toward marking liabilities at market discount rates, which turns a long-dated benefit promise into a long-dated bond position that a plan is now measured against. Plans responded by buying duration at the very long end. The UK Debt Management Office issued its first fifty-year conventional gilt in May 2005 and its first fifty-year index-linked gilt that September, and the fifty-year sterling swap rate traded *below* the thirty-year and, at moments, below the ten-year — a downward-sloping long end, which no expectations story generates and a demand story generates immediately. In the United States, the Treasury announced in August 2005 that it would bring back the thirty-year bond it had discontinued in 2001, with the first auction in February 2006. Governments were being asked for duration and were manufacturing it. Both stories, and every serious competitor to them, say that the price of a long-dated claim was set by the identity and the constraints of the people who had to hold it. That is this chapter's thesis, and the yield curve is where it is easiest to see, because the yield curve is two things at once. It is a pricing object — the term structure of $$E\[m]$$, exactly as Chapter 3 §3.6 promised, with the machinery of duration, convexity, and no-arbitrage forward rates hanging off it. And it is a record of who needs duration and who is willing to supply it. Sections 9.1 through 9.4 build the first object. Section 9.5 reads the second, and returns to February 2005 with tools rather than adjectives. *** ## 9.1 Bond Prices, Yields, and Duration ### Prices and yields A bond is a dated stream of payoffs. Chapter 3's notation already covers it: a claim paying $$x\_t$$ at each date $$t = 1, \dots, n$$ has price $$ p = \sum\_{t=1}^{n} \frac{x\_t}{(1+y)^t} $$ when every payment is discounted at the same rate $$y$$. That rate is the **yield to maturity**: not a forecast, not a return the holder will earn, but the single internal rate of return that reproduces the observed price. It is a price quoted in units of a rate, which is why it is convenient and why it misleads. Two bonds with the same yield and different coupons are being discounted by different implicit term structures. The honest object is the **zero curve**: the schedule $$y^{(n)}$$ of yields on zero-coupon claims, one per horizon, so that the price of a claim to one dollar at date $$n$$ is $$ p^{(n)} = \frac{1}{\left(1+y^{(n)}\right)^{n}} $$ and any coupon bond is priced by discounting each payment at its own horizon's rate. Zero prices are the term-structure version of Chapter 3's state prices: primitives, from which coupon bonds are portfolios. From the zero curve two derived curves follow. The **forward rate** $$f^{(n)}$$ is the rate for borrowing between $$n-1$$ and $$n$$ contracted today, pinned by no arbitrage at $$f^{(n)} = p^{(n-1)}/p^{(n)} - 1$$; the cash-and-carry argument of Chapter 3 §3.3 is the enforcement. The **par curve** gives, at each maturity, the coupon rate that makes a bond trade at face value — the coupon a new issue would carry — and it is the curve most often plotted as "the yield curve." **Table 9.1: One zero curve and the two curves implied by it (annual compounding)** | Maturity $$n$$ | Zero yield $$y^{(n)}$$ | Discount factor $$p^{(n)}$$ | Forward $$f^{(n)}$$ | Par yield | | -------------- | ---------------------- | --------------------------- | ------------------- | --------- | | 1 | 4.00% | 0.961538 | 4.000% | 4.000% | | 2 | 4.40% | 0.917485 | 4.802% | 4.391% | | 3 | 4.70% | 0.871284 | 5.303% | 4.680% | | 4 | 4.90% | 0.825844 | 5.502% | 4.870% | | 5 | 5.00% | 0.783526 | 5.401% | 4.965% | *Source: Author's calculation from the assumed zero curve in column two.* Three things in the table are general. Forwards lie above zeros when the zero curve slopes up, and they exaggerate its slope — the five-year forward turns down while the zero curve is still rising, because a flattening zero curve requires the marginal one-year rate to fall. Par yields lie below zeros for an upward-sloping curve, because a par bond front-loads some of its value into earlier, more cheaply discounted payments. And the whole apparatus is arithmetic: nothing about expectations, preferences, or risk has entered yet. Figure 9.10 draws the three curves the table tabulates, and separates out the par-zero gap the common axis cannot resolve. ![Figure 9.10: The yield curve, decomposed](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-804c8e6b437bbd3c064cfe99a74720894eb9698e%2Ffig_09_10_zero_forward_par.png?alt=media) **Figure 9.10: The yield curve, decomposed.** Table 9.1's assumed zero curve and the two curves that follow from it by arithmetic alone — nothing about expectations, preferences or risk has entered. Panel (a): the zero yields $$y^{(n)}$$, the one-year forward rates $$f^{(n)} = p^{(n-1)}/p^{(n)} - 1$$ they imply, and the par coupons that make a bond of each maturity trade at face. Forwards lie above zeros wherever the zero curve slopes up and exaggerate its slope, and the five-year forward turns down while the zero curve is still rising, because a flattening zero curve requires the marginal one-year rate to fall; the shaded wedge is that exaggeration. Panel (b): par yields lie below zeros, by 0.9 basis points at two years and 3.5 at five. That ordering is one of the three general facts Section 9.1 reads off the table, and it is drawn separately because on the common axis of panel (a) it is smaller than the width of the line. *Source: Author's construction from the assumed zero curve of Section 9.1, Table 9.1; discount factors, forward rates and par coupons are computed from that curve.* ### Duration Chapter 3's Table 3.1 made the point that a discount-rate move costs a thirty-year claim far more than a one-year claim, and promised this chapter would name it. The name is **duration**. Differentiate the price with respect to the yield and divide by the price: $$ -\frac{1}{p}\frac{dp}{dy} = \frac{1}{1+y}\sum\_{t=1}^{n} t \cdot \frac{x\_t/(1+y)^t}{p} \equiv \frac{D\_{\mathrm{Mac}}}{1+y} \equiv D\_{\mathrm{mod}} $$ **Macaulay duration** $$D\_{\mathrm{Mac}}$$ is the present-value-weighted average time to payment — a number measured in years, and the horizon at which the bond's price risk and reinvestment risk offset. **Modified duration** $$D\_{\mathrm{mod}}$$ is that quantity converted into a price elasticity: the percentage price change per unit change in yield. The first is a fact about the claim's timing; the second is the risk measure, and it is the unit in which every institution in Part IV measures the interest-rate exposure of its balance sheet. Take a ten-year bond with a 4 percent annual coupon and face value 100, priced to a yield of 5 percent. Its price is 92.2783. Its Macaulay duration is 8.3596 years — less than ten, because the coupons pay before maturity — and its modified duration is $$8.3596/1.05 = 7.9615$$. A one-basis-point rise in yield therefore costs about 0.0796 percent of value, or about 7.3 cents per 100 of face. ### Convexity and the price-yield asymmetry Duration is a first derivative, and the price-yield relation is curved. **Convexity** is the second: $$ C = \frac{1}{p}\frac{d^2p}{dy^2} = \frac{1}{(1+y)^2}\sum\_{t=1}^{n} t(t+1)\frac{x\_t/(1+y)^t}{p} $$ and the second-order expansion of the price change is $$ \frac{\Delta p}{p} \approx -D\_{\mathrm{mod}}\Delta y + \tfrac{1}{2}C(\Delta y)^2 $$ The bond above has $$C = 78.29$$. Table 9.2 shows what that buys. **Table 9.2: Exact and approximate price changes, ten-year 4% bond priced to yield 5%** | Yield change | New price | Exact % change | Duration only | Duration + convexity | | ------------ | --------- | -------------- | ------------- | -------------------- | | −200 bp | 108.5302 | +17.612% | +15.923% | +17.489% | | −100 bp | 100.0000 | +8.368% | +7.962% | +8.353% | | −25 bp | 94.1377 | +2.015% | +1.990% | +2.015% | | +25 bp | 90.4639 | −1.966% | −1.990% | −1.966% | | +100 bp | 85.2798 | −7.584% | −7.962% | −7.570% | | +200 bp | 78.9293 | −14.466% | −15.923% | −14.357% | *Source: Author's calculation. The bond prices at exactly 100 when the yield equals its 4% coupon, which is the arithmetic check on the table.* Read the ±100 basis point rows against each other. A hundred basis points down is worth 8.37 percent; a hundred basis points up costs 7.58 percent. The gain exceeds the loss by nearly eighty basis points of price, and duration alone — which is symmetric by construction — misses the asymmetry entirely. Figure 9.2 draws the object the table samples. ![Figure 9.2: Duration and convexity](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-c7fc4494c8d80197db2dc6499c6df7a0a395f648%2Ffig_09_02_duration_and_convexity.png?alt=media) **Figure 9.2: Duration and convexity.** Panel (a) is the price-yield relation for the ten-year 4 percent bond, drawn over a much wider range of yields than Table 9.2 tabulates, because curvature is the whole subject and over 200 basis points it is a line width. The dashed line is duration alone — the tangent at a 5 percent yield, with slope minus 7.9615 times the price — and it lies below the curve on both sides, which is the geometric content of positive convexity: a bond's price falls by less than the tangent says when yields rise, and rises by more when they fall. The dotted line adds the second-order term and tracks the curve closely for several hundred basis points before it too pulls away, this time from the other side. The shaded column is the window Table 9.2 reports. Panel (b) is the error each approximation makes at the table's six yield moves. Duration alone is wrong by 1.69 percent of price at minus 200 basis points and by 1.46 percent at plus 200, and — the point — it is wrong in the same direction both times, understating what the holder gains and overstating what the holder loses. Adding convexity cuts the error to about a tenth of that. A holder measuring a portfolio in duration alone is not making a small error symmetrically; they are systematically understating the value of owning an ordinary bond. That asymmetry is what positive convexity *is*, and it is a real economic good: an ordinary bond's holder is long a small option on rate volatility, for which the market charges in the level of the yield. It is also why the sign of convexity organizes so much of fixed income. Long-dated zeros have the most of it, so barbells beat bullets of the same duration when rates move a lot. Callable bonds and mortgage-backed securities have *negative* convexity, because the borrower's option to prepay shortens the claim exactly when rates fall — which turns their holders into forced sellers of duration into a rally and forced buyers into a selloff. That feedback loop is Chapter 13's subject — §13.3 for the sign of convexity, §13.5 for the hedging flows it generates — and it is one of the three mechanisms Section 9.5 needs. *** ## 9.2 The Term Structure ### The expectations hypothesis and its failure ![Figure 9.1: Four yield curves](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-1154b83d00bffa31913ca4043f193e7981d13efd%2Ffig_09_01_four_yield_curves.png?alt=media) **Figure 9.1: Four yield curves.** Constant-maturity Treasury yields on four dates, on a logarithmic maturity axis so that the money-market end of the curve is visible at all. Two of the dates are the opening episode's: 30 June 2004, the day the tightening began, when the curve was steep — 117 basis points at three months against 462 at ten years — and 30 June 2006, the day it ended, when the whole curve sat within a quarter of a point of five percent. Between those two dates the funds target rose by more than four points and the ten-year yield moved from 4.62 to 5.15. The curve did not shift upward; it rotated about its long end, which is the fact §9.5 has to explain. The other two dates are the shapes a reader needs for the rest of the chapter. On 19 October 2023 the curve was inverted — three-month yields above ten-year ones — the configuration the expectations hypothesis reads as an expected easing and §9.2 reads as a term premium. And on 9 March 2020 the entire curve, out to thirty years, sat below one percent, which is the level from which Chapter 19's opening episode begins. *Source: Board of Governors of the Federal Reserve System, H.15 constant-maturity Treasury yields, via FRED.* Why is the curve shaped as it is? The oldest answer is the **expectations hypothesis**: long rates are averages of expected future short rates, so that $$ y^{(n)}\_t \approx \frac{1}{n}\sum \_{i=0}^{n-1} E\_t\left\[y^{(1)} \_{t+i}\right] $$ and, equivalently, today's forward rate is the market's forecast of the future spot rate. In Chapter 3's language this is the claim that $$m$$ and future short rates are uncorrelated, so that $$E\_t\[m\_{t\to t+n}]$$ factors into a product of expectations. It has a clean testable content: an upward-sloping curve should forecast rising short rates, and every bond should earn the same expected return over any holding period. It fails, and the way it fails is instructive. Campbell and Shiller (1991) ran the two regressions the hypothesis implies. In the first, the change in the long yield over the life of the short bond is regressed on the current spread between long and short yields, scaled so that the expectations hypothesis predicts a coefficient of one. The estimated coefficients are negative, and they grow more negative with maturity. When the curve is steep, long yields subsequently do not rise as the hypothesis requires; on average they *fall*. In the second regression, the change in the short rate over the next period is regressed on the appropriately scaled spread; the coefficient is positive but well below one. Together the pair says that the spread contains real information about the near-term path of the short rate, and that essentially all of the rest of it is something other than a forecast. That something is a risk premium, and the same evidence read in return space says so directly: excess returns on long bonds over short bonds are *predictable* from the shape of the curve. A steep curve forecasts high realized excess returns on duration, not rising long yields. This is the fixed-income twin of Chapter 7's return predictability from valuation ratios, and it has the same interpretation. Prices move because discount rates move, and the discount rates are time-varying. ### Term premia and the decomposition problem Define the **term premium** as the residual: $$ \mathrm{tp}^{(n)}\_t = y^{(n)}\_t - \frac{1}{n}\sum \_{i=0}^{n-1} E\_t\left\[y^{(1)} \_{t+i}\right] $$ the compensation, in yield terms, for holding a long claim rather than rolling a short one. Three properties matter. It is not a constant, so the older "liquidity premium" version of the expectations hypothesis — long rates equal expected short rates plus a fixed markup — is also rejected. It is *countercyclical* in the same way Chapter 5's habit story requires: the compensation for bearing duration risk is high when the marginal holder's risk-bearing capacity is low, which in a Campbell-Cochrane economy is when the surplus consumption ratio is low, and in this book's other register is when the holder's balance sheet is under strain (Chapter 16 §16.5). And it can be negative: if long bonds pay off in bad states — if a recession means falling short rates and rising bond prices — duration is a hedge, and a hedge earns a negative premium. Whether the term premium is positive is therefore an empirical question about the correlation between bonds and consumption, and the answer has changed sign across regimes. It was positive in the inflationary 1970s and 1980s, when bad news was inflationary and bonds fell with stocks; it was frequently negative in the 2010s, when bad news was deflationary and bonds rallied when stocks fell. ![Figure 9.3: Expectations and the term premium](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-3ea19bcd64aa82f3268fa7768cb519ad7537100b%2Ffig_09_03_expectations_and_the_term_premium.png?alt=media) **Figure 9.3: Expectations and the term premium.** Panel (a) splits the ten-year yield into the expected path of short rates and the residual, monthly since 1961, on the Adrian-Crump-Moench estimates the New York Fed publishes. The shaded gap between the two lines is the term premium, drawn red where it is negative. Panel (b) is that residual alone. Read the three properties off it. It is not a constant: it runs from +5.2 points in May 1984 to −1.4 in July 2020, which disposes of the fixed-markup version of the expectations hypothesis as well as the pure one. It is large relative to the yield it sits inside — through the early 1980s more than a third of a fourteen-percent yield was compensation for duration rather than a forecast of anything. And it has spent forty-eight consecutive months below zero, with decade means of +3.4 in the 1980s, +0.4 in the 2010s and −0.2 in the 2020s, a decline whose candidates are the same ones this chapter's §9.5 will name. One caution the panel cannot draw. The split is the output of an affine model with an assumed short-rate process, not a measurement, and the next paragraph is about exactly how much that assumption does. *Source: Adrian, Crump and Moench term premium estimates, Federal Reserve Bank of New York.* The hard part is that the definition contains an expectation nobody observes. Splitting an observed yield into an expected-short-rate path and a residual requires a model of expectations, and small differences in that model produce large differences in the split. The reason is the persistence of the short rate. If the short-rate process is very persistent, today's low short rate implies low expected short rates far out, and the model attributes little of a low long yield to a term premium; if it is less persistent, expectations mean-revert quickly and the premium absorbs the difference. Persistence is estimated with substantial small-sample bias — the standard autoregressive downward bias, which Bauer, Rudebusch, and Wu (2012) showed materially shifts term-premium estimates once corrected. This is the **decomposition problem**, and it is why every term premium series in circulation is a model output rather than a measurement, and why serious work reports more than one. ### ★ Affine term structure models The workhorse framework makes the term structure **affine**. Posit a low-dimensional state vector $$X\_t$$ — in practice the first few principal components of the yield curve, which are recognizably a level, a slope, and a curvature factor and together explain nearly all of the variation in yields — following a Gaussian vector autoregression. Let the one-period rate be affine in the state, $$y^{(1)}\_t = \delta\_0 + \delta\_1' X\_t$$, and let the SDF be exponentially affine, $$ m\_{t+1} = \exp\left(-y^{(1)}\_t - \tfrac{1}{2}\lambda\_t'\lambda\_t - \lambda\_t'\varepsilon \_{t+1}\right), \qquad \lambda\_t = \lambda\_0 + \lambda\_1 X\_t $$ where $$\varepsilon\_{t+1}$$ are the state innovations and $$\lambda\_t$$ is the vector of market prices of risk, written in the notation of Chapter 3's price of risk because that is exactly what it is. Then bond prices inherit the form $$p^{(n)}\_t = \exp(A\_n + B\_n' X\_t)$$, with $$A\_n$$ and $$B\_n$$ generated by a recursion obtained from $$p^{(n+1)}\_t = E\_t\[m \_{t+1}p^{(n)} \_{t+1}]$$, and yields are affine in the state: $$y^{(n)}\_t = -(A\_n + B\_n'X\_t)/n$$. Two features of the construction are the whole point. First, the entire curve at every maturity is generated by a handful of factors and one pricing kernel, so a model fitted to some maturities must price the others — this is **pricing-kernel discipline**, the term-structure form of Chapter 3's law of one price, and it is what stops a term-premium decomposition from being an arbitrary curve-fitting exercise. Second, setting $$\lambda\_t \equiv 0$$ collapses the model to the expectations hypothesis up to a Jensen term, so the market-price-of-risk parameters *are* the term premium, and their dependence on $$X\_t$$ through $$\lambda\_1$$ is what makes premia time-varying. Problem 6 works the smallest case in full. The practical outputs are two decompositions in wide use. Adrian, Crump, and Moench (2013) estimate a five-factor Gaussian affine model by a sequence of ordinary least squares regressions rather than by likelihood, which makes it fast, transparent, and reproducible; the Federal Reserve Bank of New York publishes the resulting term premium series daily and free. Kim and Wright's model, maintained at the Federal Reserve Board, uses a three-factor specification estimated with survey forecasts of future short rates as additional observations, precisely to discipline the persistence problem. The two agree on the broad movements and disagree on levels by tens of basis points, which is an honest statement of what the decomposition problem costs. Use them as this chapter uses them: to date and sign the movements in the premium, not to assert its level to a basis point. *** ## 9.3 Real Rates and TIPS Every yield so far has been nominal. Since January 1997 the United States has issued **Treasury Inflation-Protected Securities**, whose principal accretes with the consumer price index, so their yields are real. The difference between the nominal yield and the TIPS yield at the same maturity is the **breakeven inflation rate**: the inflation rate at which an investor would be indifferent between the two claims. The breakeven is not expected inflation. It is $$ \text{breakeven} = E\_t\[\text{inflation}] + \text{inflation risk premium} - \text{TIPS liquidity premium} $$ and the last term is the reason to be careful. The TIPS market is an order of magnitude smaller than the nominal Treasury market and is far less liquid, so TIPS trade cheap — their yields are too high, and breakevens are therefore biased down, by an amount that widens exactly when liquidity is scarce. In the fall of 2008 the ten-year breakeven collapsed toward zero, which no forecaster read as a forecast of a decade of price stability; it was a liquidity event, and it was visible directly in the fact that a TIPS bond combined with an inflation swap — a synthetic nominal Treasury with identical cash flows — was persistently cheaper than the actual nominal Treasury. Fleckenstein, Longstaff, and Lustig (2014) documented that gap, found it averaged tens of basis points and reached far more in the crisis, and named it the TIPS-Treasury puzzle. It is a violation of the law of one price on the safest pair of claims in the world, and Chapter 3 §3.2's framework says what to conclude: the arbitrage requires balance sheet, and balance sheet was the scarce thing. With that caveat, TIPS deliver something no other market does — a directly observed history of long-horizon *real* interest rates, rather than one inferred by subtracting an inflation forecast. What it shows is a long decline. The ten-year TIPS yield was above 4 percent shortly after the market's founding, fell through the 2000s, spent much of 2012 to 2021 negative, and rose back to around 2 percent by the mid-2020s. That series is the market's version of the decline in the natural rate of interest that the macroeconomics literature estimates from the real side, and it is the single most important input into any claim about long-run valuations, because it is the level from which every risk premium in this book is measured. *** ## 9.4 Safe Assets and the Treasury Market ### The convenience yield Chapter 3 §3.1 flagged an institutional wedge and pointed here: some claims are discounted at rates that no risk-based model can produce, because the claim does something for its holder beyond paying its cash flows. A Treasury security is the leading case. It can be turned into cash the same day in size, it is the standard collateral in repo and in derivatives margin, it is what a money market fund can hold under Rule 2a-7, it carries a zero risk weight in bank capital rules and no charge in insurers' risk-based capital (Chapter 16, Table 16.3), and it is what a central bank holds as a reserve. Those services are worth something, and their value shows up as a price above the discounted value of the cash flows — equivalently as a yield below what the SDF alone would justify. That gap is the **convenience yield**, $$\mathrm{cy}\_t$$. (Chapter 8 §8.2 uses the same phrase for a different object: the benefit of holding a physical commodity rather than a claim to it, which enters the cost of carry. The shared name reflects a shared logic — a holding service the cash flows do not capture — and the two never appear together.) Krishnamurthy and Vissing-Jorgensen (2012) made this measurable. Their central instrument is the spread between the yield on Aaa-rated corporate bonds and the yield on long Treasuries. Aaa credit risk is small and slow-moving; if the spread nonetheless varies systematically with the *quantity* of Treasury debt outstanding, the variation is not about credit. It does, on their sample, and the sign is unambiguous: when Treasury debt is small relative to GDP, the spread is wide, and when debt is large the spread is narrow. Treasuries are a good with a downward-sloping demand curve, and the government is its monopoly supplier. Their estimate of the average value of the liquidity and safety services over 1926 to 2008 is on the order of seventy basis points a year. At a marketable stock of roughly $27 trillion, seventy basis points is something near $200 billion a year of interest the Treasury does not pay — a seigniorage-like transfer earned by issuing a claim people want to hold for reasons other than its return. ![Figure 9.5: The convenience yield](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-f27ce6bd21fab6474230d432f298ccb18338f332%2Ffig_09_05_the_convenience_yield.png?alt=media) **Figure 9.5: The convenience yield.** Krishnamurthy and Vissing-Jorgensen's instrument, on the free series that carry it: Moody's Aaa corporate yield less the long Treasury, against federal debt held by the public as a share of GDP, annually from 1939. Panel (a) is the pair over time; panel (b) is the relation, split at 2008. On the window their paper works — 1939 to 2008, which contains the wartime debt peak and its long postwar run-down — the fit is negative, about nine tenths of a basis point of spread per point of debt to GDP, correlation −0.33. That is the finding: Aaa credit risk is small and slow-moving, so a spread that moves with the *quantity* of Treasuries outstanding is not a statement about credit. It is the price of a holding service, and the Treasury is its monopoly supplier. Add the years since 2008 and the fit turns over. Two reasons, and neither of them rescues a credit story. The Federal Reserve took a large share of the new stock onto its own balance sheet, so debt held by the public overstates the free float actually available to private holders; and 2008, 2020 and 2022 put genuine credit risk into an Aaa spread that had carried very little of it before. The point of drawing both fits is that the convenience yield is identified off the *supply* of Treasuries, and the post-crisis period is one in which the supply available to private holders and the supply outstanding stopped being the same thing. *Source: Moody's Aaa corporate bond yield via FRED, less the long Treasury yield (FRED's ten-year constant maturity spliced to Shiller's long rate before 1953); federal debt held by the public as a percent of GDP via FRED. The construction follows Krishnamurthy and Vissing-Jorgensen (2012). Author's calculations.* Two implications follow immediately and both are used later in this book. Debt management is not neutral: how much the Treasury issues, and at which maturities, changes the price of safety and therefore of everything priced against it. And the private sector has an incentive to manufacture substitutes — which is what the pre-crisis machine for turning mortgages into Aaa-rated tranches was, and why Chapter 10 treats safe-asset manufacture as a credit-market phenomenon rather than a technical one. ### Who holds Treasuries, and the four great shifts Chapter 2's Table 2.5 gives the Treasury row of the master holdings table. Table 9.3 opens it out over time, because the composition has been rebuilt twice in twenty years. **Table 9.3: Holders of US Treasury securities ($ trillions)** | Holder | 2007:Q4 | 2014:Q4 | 2026:Q1 | | ------------------------------------------ | ------- | -------- | -------- | | Federal Reserve (SOMA) | 0.7 | 2.7 | 4.0 | | Foreign official | 1.7 | 4.1 | 3.9 | | Foreign private | 0.6 | 2.0 | 5.4 | | Money market funds | 0.2 | 0.5 | 3.4 | | Other funds (mutual, ETF, state and local) | 0.6 | 1.3 | 4.0 | | Banks and depositories | 0.1 | 0.5 | 2.2 | | Insurers and pensions | 0.5 | 0.8 | 1.8 | | Households and hedge funds (residual) | -0.3 | 0.2 | 3.0 | | **Total** | **4.4** | **12.6** | **29.0** | *Source: Financial Accounts of the United States (Z.1), table L.210, holder rows through their FRED mirrors; data through 2026:Q1, retrieved 25 August 2026. Rows do not exhaust the table: brokers and dealers, GSEs, issuers of asset-backed securities and nonfinancial business hold the remaining $0.2, $0.4 and $1.2 trillion in the three columns. The residual row follows the Z.1 convention in which the household sector absorbs unallocated positions — including hedge funds, whose Treasury holdings the Z.1's separate memo puts at $0.3 trillion in 2025:Q4 — and a residual can be negative, as it is in 2007:Q4. Cross-checks: the Treasury International Capital data give $3.9 trillion official and $5.5 trillion non-official against the Z.1's $3.9 and $5.4; the Federal Reserve's H.4.1 carries $4.4 trillion of Treasuries held outright against the Z.1's $4.0, the difference being unamortized premiums and inflation compensation, and it puts the SOMA peak at $5.8 trillion in June 2022.* Four shifts account for nearly all of the change between the columns, and Box 9.1 takes them in turn. ![Figure 9.6: Who holds Treasuries](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-8c1e2cf38ec6131b6ef39ef1dc0741e929a4ed04%2Ffig_09_06_who_holds_treasuries.png?alt=media) **Figure 9.6: Who holds Treasuries.** Table 9.3's column, run backwards through sixty-five years. Every one of the four shifts Box 9.1 names is visible as a line crossing another: banks, the dominant holder in the 1960s, fall from thirty percent to under five; the rest of the world rises from four percent to a peak of fifty-one in 2008 and has been falling since; the Federal Reserve's share, an operating balance before 2008, goes to a quarter of the market under quantitative easing and back; and money market funds appear from nothing after the 2016 reform. What the picture is for is the observation §9.5 builds on. Look at the list of holders and ask which of them buys Treasuries because it expects a good return. A reserve manager accumulating dollars, a central bank executing a purchase programme, a bank meeting a liquidity requirement, a money fund restricted to government paper — none of them. The marginal buyer of duration in this market is usually somebody with a mandate, and that is why the term premium is a quantity story rather than a forecast. *Source: Financial Accounts of the United States (Z.1), table L.210, Treasury securities by holder, via the FRED mirror; author's calculations.* > **Box 9.1 — The four great shifts in Treasury ownership** > > **The Fed became the largest single holder, then stopped.** Pre-crisis the System Open Market Account held Treasuries as an operating balance. Three rounds of quantitative easing and the March 2020 expansion took the holding to a peak of roughly $5.8 trillion by mid-2022; quantitative tightening has run it down since. The Fed is the purest price-insensitive buyer in the table: the quantity is announced by a committee. > > **Foreign officials rose and then plateaued.** Reserve accumulation carried official holdings from well under a trillion at the turn of the century to around $4 trillion by 2014 — the flow the opening episode is about — after which the stock stopped growing. Because the total kept growing, the official *share* has fallen by roughly half. Private foreign holdings have taken up the slack, and private foreign holders behave quite differently: they hedge, they respond to yields, and they leave. > > **Money market funds became a major holder after 2016.** The SEC's money market reform, effective October 2016, imposed a floating net asset value and liquidity gates on institutional prime funds and left government funds alone; roughly a trillion dollars moved from prime to government funds within months. A second wave followed the 2022-23 tightening, when money fund assets rose above $6 trillion and the Fed's overnight reverse repo facility — an alternative to bills for exactly these holders — peaked near $2.5 trillion at the end of 2022. Money fund demand is concentrated at the very short end and is close to perfectly elastic to a few basis points of yield, which makes it the most price-sensitive column in the table. > > **Hedge funds moved into the residual row.** The cash-futures **basis trade** — long the cash Treasury, short the corresponding futures contract, financing the cash leg in repo at haircuts that permit leverage of many tens to one — exists because asset managers want duration in futures form and somebody must take the other side. It has grown to something on the order of a trillion dollars of gross exposure by the mid-2020s, on estimates from the Office of Financial Research and CFTC position data. Much of the growth in Table 9.3's residual row is this. It is genuinely useful: it is the mechanism by which cash and futures prices stay tied together. It is also the most fragile row in the table, because it is the only one whose holder is levered against the position itself. ### March 2020 from the claim's side Between March 9 and March 18, 2020, the ten-year Treasury yield *rose* while equities fell by a third, off-the-run Treasuries traded at large discounts to their on-the-run twins, and the cash-futures basis blew out. Chapter 19 opens with that episode and narrates it from the dealer's side, where the binding constraint sat; the point to take here is what it did to the claim. For a few days the convenience yield went the wrong way. The instrument that is supposed to be worth *more* than its cash flows because it can always be sold became the instrument that was hardest to sell, and by the measures used above it traded cheap rather than rich — He, Nagel, and Song's (2022) study of the episode calls the result an "inconvenience yield." The reason is that convenience is not a physical property of a Treasury security. It is a service produced jointly by the security and by the dealer balance sheets that stand ready to turn it into cash, and when the second input is rationed the service is not produced. Safety, in this book's sense, is a statement about who can hold the claim, not only about who issued it. ### The debt ceiling as a convenience-yield experiment The debt ceiling episodes of 2011, 2013, and 2023 are as close to a controlled experiment as this literature gets, because they change the probability of a technical missed payment on a *narrow window* of bills while leaving everything else — the issuer, the tax base, the collateral rules — untouched. Each time, the yields on bills maturing just after the projected exhaustion date rose sharply relative to bills maturing just before and just after that window, by tens of basis points in 2013 and by more in 2023, while longer Treasury yields barely moved and in some episodes fell. Two readings follow. Investors price a genuine, if small, probability of a delayed payment, and they price it in exactly the maturities where a delay would bite. And the *money-like* portion of the convenience yield is destroyed by even a remote timing risk: a bill that might not pay on the day is no longer usable as a cash equivalent, whatever its recovery value, which is why the effect appears in the specific instrument rather than in the sovereign's credit spread. *** ## 9.5 Who Holds the Bond, and What Their Constraints Do to Its Price Duration is a risk that someone has to bear, so for a bond the question is really who the marginal buyer of duration is. The Treasury issues it, the private sector holds it, and the term premium is the price at which the market clears. The mandated question of this book's Part III — who holds this claim, and what their constraints do to its price — has an unusually sharp answer here, because bond holders announce their constraints in public. ### Preferred habitat, made modern Modigliani and Sutch proposed in 1966 that investors have **preferred habitats**: maturity segments they occupy for reasons of their own, from which they move only for compensation. The idea was descriptively appealing and analytically inert, because a market in which everyone is segmented has no arbitrage and therefore no discipline — nothing stops the two-year and the two-year-and-a-day from having unrelated yields. Vayanos and Vila (2021) supplied the missing half. **Habitat investors** demand specific maturities inelastically, for reasons exogenous to the model — a liability schedule, a currency policy, a mandate. **Arbitrageurs** are risk-averse optimizers who trade the whole curve and are the only agents enforcing no-arbitrage. Arbitrageurs absorb whatever habitat demand leaves unfilled, and they charge for it, because absorbing it puts interest-rate risk on a finite amount of capital. Write $$Q^{(n)}$$ for the supply of the maturity-$$n$$ claim the private market must hold, $$Q^{(n)}\_h$$ for inelastic habitat demand, $$\gamma$$ for arbitrageurs' risk aversion, and $$\sigma\_n$$ for the claim's return volatility. The reduced form is one equation: $$ \mathrm{tp}^{(n)} = \gamma\sigma\_n^2\left(Q^{(n)} - Q^{(n)}\_h\right) $$ Everything the model is for is visible in it. Setting $$\gamma \to 0$$ restores the expectations hypothesis exactly, and habitat demand moves nothing; $$\gamma \to \infty$$ gives complete segmentation. Real markets sit between, so a demand shock at one maturity moves *that* maturity most and its neighbors in proportion to return correlation — the local-supply prediction that distinguishes preferred habitat from every expectations story, and is testable. It is also the theory of how quantitative easing works: the central bank buys, $$Q^{(n)}$$ falls, arbitrageurs hold less duration, the premium falls, and no signal about future policy is required. Krishnamurthy and Vissing-Jorgensen's decomposition of the QE announcements and D'Amico and King's finding that yields fell most in the maturity sectors actually purchased are the empirical counterparts. The same equation says any program's effect depends on how much capital arbitrageurs have when it runs, which is why the March 2020 purchases moved prices further per dollar than a calm-market program does. ### The liability-driven bid The largest habitat investors in the world are pension plans and life insurers, and Chapter 16 §16.2 says why. A defined benefit promise is a long-dated, rate-sensitive liability valued by discounting a fixed schedule, so a plan measured on its funding ratio is short duration by construction and closes the gap by buying long bonds regardless of their expected return; risk-based capital pushes insurers the same way from a different direction. This is demand for duration as a *hedge*, and it is the book's cleanest example of a demand curve that is nearly vertical in price. Two consequences. The demand is largest where supply is thinnest — beyond twenty years, where few issuers naturally borrow — which is why the sterling and euro long ends inverted in the mid-2000s and why the opening episode's fifty-year gilts were issued at all. And the hedge is usually levered, through swaps and repo, because a plan wants the duration without surrendering the return-seeking assets that close its funding gap. Leverage converts an inelastic buyer into a forced seller when prices move against it fast enough, which is what happened to UK LDI mandates in September 2022 — Chapter 16 §16.2 carries that episode as its case study. The most price-insensitive holder of duration in ordinary times can become the most price-sensitive seller of it in a week. > **Box 9.2 — Silicon Valley Bank, March 2023** > > Duration is a property of a claim; insolvency is a property of a holder. At the end of 2022 Silicon Valley Bank held roughly $117 billion of securities against $212 billion of assets, $91 billion of it designated held to maturity and overwhelmingly long-duration agency mortgage-backed securities and Treasuries bought near the yield lows. Section 9.1's arithmetic is the whole analysis: at a modified duration of six, three hundred basis points of yield costs about eighteen percent of value before convexity, and yields rose by more than that through 2022. The unrealized loss reached roughly $15 billion against tangible common equity near $16 billion. > > None of it reached reported capital. Securities designated **held to maturity** are carried at amortized cost, so the mark sat in a footnote while the bank reported itself well capitalized — **Appendix C** §C.4.2 owns that accounting, and the footnote was public throughout. > > The loss was not distinctive. Jiang, Matvos, Piskorski and Seru (2024) put the banking system's mark-to-market securities loss near two trillion dollars in early 2023; almost every bank was carrying one. What was distinctive was who held the liabilities. Roughly nine-tenths of SVB's deposits sat above the insurance limit, in a concentrated clientele of venture-backed firms — Chapter 19 §19.2's coordination problem with the guarantee removed above a threshold. Some $42 billion of withdrawal requests arrived on 9 March 2023 and the bank was closed the next day; the Federal Reserve's post-mortem (Barr, 2023) records both the unhedged rate exposure and the funding concentration. Chapter 13 §13.5 runs the episode from the mortgage side, where the missing convexity hedge is the point. The claim was one every holder in Table 9.3 owns; the run was a fact about one holder's funding. ### Price-insensitive official demand Foreign official reserve managers are habitat investors with an unusual objective. A central bank accumulating reserves is not choosing a portfolio against a return target: the quantity is a byproduct of exchange-rate policy and the maturity is chosen for safety and depth. Table 9.3's official row is a $$Q^{(n)}\_h$$ that moves for reasons entirely outside the bond market. Why the dollar holds that role belongs to the companion volume; what belongs here is the pricing consequence, which is that a large, growing, price-insensitive bid concentrated in intermediate and long maturities compresses the term premium without any change in the expected path of policy. The domestic central bank is the same kind of holder acting from a different motive — Del Negro, Eggertsson, Ferrero and Kiyotaki (2017) price the balance sheet that official liquidity provision creates — and the equation above does not care which official institution the inelastic demand comes from. ### The conundrum, resolved Now return to February 2005 with the equation. The Federal Reserve raised the short rate, which moves the expectations component and did in fact carry the two-year yield up by nearly the full amount. Over the same window $$Q^{(n)}\_h$$ rose on two fronts: reserve managers were absorbing a historically large flow into intermediate and long Treasuries, and newly liability-measured pension plans were bidding for duration at the very long end. Habitat demand rose, supply did not, arbitrageurs were left holding less duration, and $$\mathrm{tp}^{(n)}$$ fell. A falling premium against a rising expectations component nets to a long yield that goes nowhere and a curve that flattens from the long end rather than the short. That is not a puzzle in this framework; it is what the framework predicts. The evidence since agrees: affine decompositions put most of the mid-2000s decline in long yields in the term premium, with the ACM ten-year estimate falling by well over a percentage point between mid-2004 and 2006 and approaching zero. The exact magnitudes are model-dependent — the decomposition problem doing its work — but no serious decomposition assigns the move to expectations, and the Data Exercise reproduces the picture. ![Figure 9.4: The conundrum window](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-d6a5b2031a21cbfbc6c1650d2cce225e3b420641%2Ffig_09_04_the_conundrum_window.png?alt=media) **Figure 9.4: The conundrum window.** The federal funds target against the two- and ten-year Treasury yields, 2003 through 2007, with the start of the tightening marked and the window to Greenspan's testimony shaded. Read the three lines against the equation. The target is a step function the Committee announced in advance and delivered on schedule, from one percent to 5.25. The two-year yield tracks it, which is the expectations component doing what the expectations hypothesis says it should. The ten-year does not: over the shaded window it falls from 4.62 to 4.16 while the target rises 1.25 points, and the two-to-ten spread compresses by 117 basis points. By the end of the cycle the target was 5.25 and the ten-year was back within a few basis points of where it had started two years earlier. One caution about the two-year line. Measured from the start of 2004, before the market had priced the cycle, the two-year rose by essentially the full amount of the target's move; measured from 30 June, when much of the tightening was already in the price, it rose by about two thirds of it. Either baseline supports the section's point, which is about the ten-year. *Source: Board of Governors of the Federal Reserve System, via FRED. Author's calculations.* The general lesson is the one Chapter 3 §3.7 planted. When the marginal holder of a claim buys it for a reason unrelated to its expected return, the claim's price stops being a forecast and becomes a demand curve read at a point. Greenspan's conundrum was a conundrum only for a model in which the yield curve is a forecast. Read as a record of who needed duration, it was a story about reserve managers and pension regulators, and the tools to tell it existed in 2005. In Chapter 1 §1.2's terms, a term premium that falls while the short rate rises is a flow with a constraint behind it and not news: duration arriving in the hands of holders who did not choose it for its expected return, absorbed by arbitrageurs whose capacity to hold it is finite. *** ## Elsewhere in the Series * **International bond markets institutionally: issuance, sovereign hierarchy, currency of denomination, and cross-border investor bases** — *International Finance*, Chapter 12. That chapter also discusses credit-spread components conceptually. All term-structure and bond-pricing theory is this chapter's; the institutional geography is that one's. * **The dollar's international role and reserve accumulation** — *International Finance*, especially Chapters 2 and 11. Section 9.5 uses foreign official demand as a holder fact and does not argue the monetary economics behind it. * **Quantitative easing as policy implementation: operating frameworks, announcement design, balance sheet policy** — *International Finance*, Chapters 15 and 16. Section 9.5 keeps only the pricing channel. * **The Treasury row of the master holdings table** — this book, Chapter 2, Table 2.5, from which Table 9.3 is opened out. * **Countercyclical risk premia and the habit mechanism** — Chapter 5 §5.5. **Constrained capital and margin spirals, stated canonically** — Chapter 16 §16.5. **LDI, risk-based capital, and the 2022 UK gilt episode** — Chapter 16 §16.2. **March 2020 narrated from the dealer's side** — Chapter 19's opening episode. **Negative convexity and mortgage hedging flows** — Chapter 13 §§13.3 and 13.5, which answers this chapter's forward pointer and sets Table 13.2 against §9.1's Table 9.2. **Demand systems that estimate the curves this chapter assumes** — Chapter 20. **The corporate issuer on the other side of §9.4's convenience yield and §9.5's habitat demand** — Chapter 23 §23.7, whose gap-filling and safe-asset-supply arguments make corporate maturity choice a response to the quantities this chapter takes as given. *** ## Summary 1. **A bond is a dated payoff stream and the zero curve is its primitive.** Yield to maturity is a price quoted as a rate, not a forecast and not a return; forwards and par yields are arithmetic consequences of the zero curve, and Table 9.1 shows that forwards exaggerate its slope while par yields understate it. 2. **Duration is the price elasticity, convexity is its curvature.** A ten-year 4 percent bond at a 5 percent yield has a modified duration of 7.96 and convexity of 78.3. Duration alone predicts a symmetric response to yield moves; the actual response is asymmetric, with a 100 basis point rally worth 8.37 percent against a 7.58 percent loss on a 100 basis point selloff (Table 9.2). 3. **Negative convexity turns holders into procyclical traders.** Callable bonds and mortgage securities shorten when rates fall, forcing their holders to buy duration into a rally and sell into a selloff — the mechanism Chapter 13 develops and Section 9.5 uses. 4. **The expectations hypothesis fails in a specific direction.** Campbell and Shiller's regressions produce coefficients that are negative and grow more negative with maturity: when the curve is steep, long yields subsequently fall rather than rise. Read in return space, the same fact says excess returns on duration are predictable from the shape of the curve. 5. **Term premia are time-varying, countercyclical, and can be negative.** They are the failure of the expectations hypothesis measured in yield units, they rise when the marginal holder's risk-bearing capacity falls, and their sign depends on whether bonds hedge or amplify bad states — which has changed across regimes. 6. **The decomposition problem is the central measurement obstacle.** Splitting a yield into expectations and premium requires a model of short-rate persistence that samples of the available length estimate poorly. Affine models impose pricing-kernel discipline on the split; ACM and Kim-Wright are the practical outputs, they agree on movements, and they disagree on levels by tens of basis points. 7. **TIPS give a direct history of real rates, with a liquidity wedge attached.** Breakevens are expected inflation plus an inflation risk premium minus a TIPS liquidity premium, and the last term widens exactly when it matters, as the 2008 collapse in breakevens and the TIPS-Treasury puzzle both show. 8. **Treasuries price above their SDF value because they provide money-like services.** The convenience yield, measured by Krishnamurthy and Vissing-Jorgensen through the Aaa-Treasury spread's dependence on the quantity of debt outstanding, has averaged on the order of seventy basis points — near $200 billion a year at present debt levels — and makes debt management a pricing instrument. 9. **The Treasury holder base has been rebuilt twice in twenty years** (Table 9.3): the Fed from an operating balance to the largest single holder and partway back, foreign officials from rising to flat, money funds from negligible to $2.5 trillion, and hedge funds into the residual row via the basis trade. March 2020 showed that convenience is produced jointly by the security and by dealer balance sheets, and can go into reverse; the debt ceiling episodes show it destroyed by timing risk in the specific bills exposed. 10. **The marginal buyer of duration is identifiable, and that is the chapter's answer.** Vayanos and Vila's preferred habitat gives the pricing equation $$\mathrm{tp}^{(n)} = \gamma\sigma\_n^2(Q^{(n)} - Q^{(n)}\_h)$$, nesting the expectations hypothesis and full segmentation as limits and delivering quantitative easing as a corollary. Greenspan's conundrum is what that equation predicts when reserve managers and liability-driven pension plans raise habitat demand while the central bank raises the short rate. *** ## Key Terms * **Yield to maturity** $$y$$: The single discount rate that reproduces a bond's observed price; a price quoted in units of a rate, not a forecast and not an expected return * **Zero curve**: The schedule of yields $$y^{(n)}$$ on zero-coupon claims by horizon; the primitive from which coupon bonds are portfolios * **Par curve**: The coupon rate at each maturity that prices a bond at face value; the curve usually plotted as "the yield curve" * **Forward rate** $$f^{(n)}$$: The rate for borrowing between $$n-1$$ and $$n$$ contracted today, pinned by no arbitrage at $$p^{(n-1)}/p^{(n)} - 1$$ * **Macaulay duration** $$D\_{\mathrm{Mac}}$$: The present-value-weighted average time to a claim's payments, in years * **Modified duration** $$D\_{\mathrm{mod}}$$: $$D\_{\mathrm{Mac}}/(1+y)$$; the percentage price change per unit change in yield, and the unit in which institutions measure rate exposure * **Convexity** $$C$$: The second derivative of price with respect to yield, scaled by price; positive convexity makes rallies worth more than selloffs cost * **Negative convexity**: The property of callable bonds and mortgage securities whose duration shortens as rates fall, turning holders into procyclical buyers and sellers of duration * **Expectations hypothesis**: The claim that long yields are averages of expected future short yields; equivalently, that $$m$$ and future short rates are uncorrelated * **Term premium** $$\mathrm{tp}^{(n)}\_t$$: The gap between a long yield and the average expected short rate over its life; compensation for bearing duration risk * **Decomposition problem**: The dependence of any expectations/term-premium split on an unobserved model of short-rate persistence, which samples of available length identify weakly * **Affine term structure model**: A model in which the short rate and the market prices of risk are affine in a low-dimensional state, so that yields are affine in that state and one kernel prices the whole curve * **Breakeven inflation**: Nominal yield minus TIPS yield; expected inflation plus an inflation risk premium minus a TIPS liquidity premium * **Convenience yield** $$\mathrm{cy}\_t$$: The price premium a claim commands for the money-like services it provides — settlement, collateral, regulatory eligibility — over and above the discounted value of its cash flows * **Safe asset**: A claim whose value is expected to be preserved precisely when other values are not, and which is therefore usable as collateral and as a store of value in stress; a joint product of the issuer and of the intermediaries who make it tradable * **Preferred habitat**: A maturity segment an investor occupies for reasons exogenous to bond returns, leaving it only for compensation * **Basis trade**: Long cash Treasuries against short futures, financed in repo at high leverage; the mechanism tying cash and futures prices together and the most fragile holder position in the market *** ## Readings ### Required * Krishnamurthy, A. and A. Vissing-Jorgensen (2012). "The Aggregate Demand for Treasury Debt." *Journal of Political Economy* 120(2): 233-267. *Establishes that Treasuries carry a convenience yield by showing the Aaa-Treasury spread falls systematically as the quantity of debt rises — a demand curve for safety, estimated, and the empirical foundation of §9.4.* * Vayanos, D. and J.-L. Vila (2021). "A Preferred-Habitat Model of the Term Structure of Interest Rates." *Econometrica* 89(1): 77-112. *Turns Modigliani and Sutch's descriptive idea into a general equilibrium model with risk-averse arbitrageurs, delivering the pricing equation of §9.5 and, as a corollary, the transmission channel of quantitative easing.* ### Recommended * Lucas, R. E. (1978). "Asset Prices in an Exchange Economy." *Econometrica* 46(6): 1429-1445. *The pricing kernel that §9.2's affine models presuppose and never restate. Read it to see where the risk-neutral measure of the term-structure literature comes from, and why a term premium is a covariance before it is a regression residual.* * Campbell, J. Y. and R. Shiller (1991). "Yield Spreads and Interest Rate Movements: A Bird's Eye View." *Review of Economic Studies* 58(3): 495-514. *The canonical rejection of the expectations hypothesis; read it for the pattern of coefficients across maturities rather than for any single number.* * Adrian, T., R. Crump and E. Moench (2013). "Pricing the Term Structure with Linear Regressions." *Journal of Financial Economics* 110(1): 110-138. *The affine model estimated by OLS instead of likelihood, which is why its term premium series is published daily and can be reproduced by a student; the practical output §9.2 uses.* * Del Negro, M., G. Eggertsson, A. Ferrero and N. Kiyotaki (2017). "The Great Escape? A Quantitative Evaluation of the Fed's Liquidity Facilities." *American Economic Review* 107(3): 824-857. *The central bank's balance sheet as a price, not as an operating procedure: what happens to the return on liquid assets when the official sector supplies liquidity against illiquid collateral. The pricing channel §9.5 keeps while routing the operating framework to the companion volume on international finance.* * Gürkaynak, R., B. Sack and J. Wright (2007). "The U.S. Treasury Yield Curve: 1961 to the Present." *Journal of Monetary Economics* 54(8): 2291-2304. *Documents the fitted zero-coupon curve the Federal Reserve Board publishes and updates; the data behind most of the empirical work in this chapter, and free.* * Dimson, E., P. Marsh and M. Staunton. *Credit Suisse Global Investment Returns Yearbook 2014*. Credit Suisse Research Institute, 2014. *Long-run real returns on bonds and bills across more than a century and a score of markets. The natural check on §9.3: the real rate is not a constant of nature, and the yearbook's cross-country tables show how wide the historical range has been.* * Fabozzi, F. J., ed. *The Handbook of Fixed Income Securities*, 7th ed. McGraw-Hill, 2005. *The manual counterpart to the mortgage handbook Chapter 13 requires, and the reference for everything §9.1 states without deriving: settlement and accrual conventions, the full duration and convexity apparatus, and the sector-by-sector institutional detail this chapter compresses into a paragraph each.* * Thau, A. *The Bond Book*, 2nd ed. McGraw-Hill, 2001. *The practitioner's register, aimed at the individual investor. Useful for §9.1 precisely because it explains the price-yield asymmetry without a derivative in sight, which is a good test of whether the reader has understood convexity or only computed it.* *** ## Discussion Questions 1. **Was the conundrum a conundrum?** Section 9.5 resolves February 2005 with a preferred-habitat argument. State precisely what an observer in 2005 would have needed to measure to distinguish that explanation from two rivals: that the market expected the tightening cycle to be short and to be followed by cuts, and that long-run inflation expectations had fallen because the Fed's tightening was credible. Which of the three does the behavior of the *fifty-year sterling* curve help discriminate, and why is a foreign long end informative about a US puzzle at all? 2. **Did March 2020 dethrone the safe asset?** For nine trading days the world's benchmark safe claim sold off in a flight to safety, and its convenience yield turned negative. Argue both sides. Against: this was a temporary failure of an input (dealer balance sheet) rather than of the claim, and it was repaired in days by a buyer with no balance sheet constraint. For: a safe asset that requires a central bank to remain safe is a different object from the one the theory describes, and the theory should say so. What evidence from the following five years would settle it, and has any of it arrived? 3. **A term premium you cannot measure.** Section 9.2 argues that the expectations/premium split is model-dependent to the tune of tens of basis points. A central bank official cites a decline in the term premium as evidence that policy has not become restrictive. What are you entitled to conclude, and what would you ask her for? Is there a policy question for which the *level* of the premium matters, as opposed to its change? 4. **Whose convenience?** The convenience yield is measured in aggregate, but the services are not valued equally by every holder — a money market fund values Rule 2a-7 eligibility, a dealer values repo specialness, an insurer values a zero capital charge, a reserve manager values depth. Sketch how you would estimate a *holder-specific* convenience yield, and say what the dispersion across holders would tell you that the aggregate number does not. (Chapter 20's demand system is the machinery; the question is what you would want it to estimate.) 5. **The habitat that levers.** Section 9.5 argues that liability-driven investors are the most price-insensitive buyers of duration in normal times and can become the most price-sensitive sellers in a crisis. Does that make LDI a stabilizing or a destabilizing institution for the long end of the curve? Would a rule forbidding leverage in LDI mandates improve matters, and what would it do to the demand for duration and hence to its price in ordinary times? *** ## Problems **Problem 1 — Price, duration, convexity.** A bond pays an annual coupon of 6 percent on face value 100 and matures in seven years. The yield is 4.5 percent. (a) Compute the price. (b) Compute Macaulay duration and modified duration. (c) Compute convexity. (d) Using duration alone, then duration and convexity, estimate the price change for a 150 basis point rise in yield. Compare each with the exact repricing and report the two errors in cents per 100 of face. (e) Repeat (d) for a 150 basis point fall. Explain in one sentence why the duration-only errors have the same sign in both directions. **Problem 2 — Zeros, forwards, and par.** The one-, two-, and three-year zero yields are 3.00, 3.60, and 4.00 percent (annual compounding). (a) Compute the three discount factors. (b) Compute the one-year forward rates $$f^{(2)}$$ and $$f^{(3)}$$. (c) Compute the two-year and three-year par yields. (d) A three-year bond with a 5 percent annual coupon trades at a price of 103.20. Is it cheap or dear relative to the zero curve? State the arbitrage that would be available if you could trade zeros freely, and name the market frictions that would stop you (Chapter 3 §3.2). **Problem 3 — The expectations hypothesis and the forward rate.** Using the curve of Problem 2, an investor buys the two-year zero and plans to sell it after one year. (a) Compute the one-year forward rate implied by the curve. (b) Show that if the one-year spot rate one year from now equals that forward rate, the investor's realized one-year holding return equals today's one-year spot rate exactly. Verify numerically. (c) Suppose instead the realized one-year rate is 3.30 percent. Compute the realized holding return and the excess return over the one-year rate. (d) Now suppose the two-year yield carries a term premium of 40 basis points over its expectations-hypothesis level. What one-year rate does the market actually *expect* one year from now, given the same observed curve? Compare it with the forward rate from (a), and explain why an econometrician who assumed the expectations hypothesis would report the market as having been systematically wrong in one direction. **Problem 4 — Reading the convenience yield.** Moody's Aaa corporate bond yield is 5.15 percent and the twenty-year Treasury constant maturity yield is 4.60 percent. (a) Compute the Aaa-Treasury spread in basis points. (b) An analyst estimates that 20 basis points of the spread compensate for Aaa default risk and corporate-bond illiquidity. What is the implied convenience yield? (c) Marketable Treasury debt outstanding is $27 trillion. What is the annual interest saving the convenience yield represents? (d) A fitted relation from the Krishnamurthy-Vissing-Jorgensen regression gives the spread, in percentage points, as $$1.00 - 0.90\times(\text{Debt}/\text{GDP})$$. With GDP of $29 trillion, the Treasury issues an additional $2 trillion. By how much does the spread narrow, and what happens to the *total* dollar value of convenience services supplied? Interpret the sign of the change in total value. **Problem 5 — Preferred habitat comparative statics.** In the reduced form of §9.5, $$\mathrm{tp}^{(n)} = \gamma\sigma\_n^2(Q^{(n)} - Q^{(n)}\_h)$$. Take $$\gamma = 0.25$$ per trillion dollars, $$\sigma\_n = 0.10$$, $$Q^{(n)} = 14$$ (trillions of long-maturity Treasuries the private market must hold), and $$Q^{(n)}\_h = 8$$ (trillions of inelastic habitat demand). (a) Compute the baseline term premium in basis points, and the sensitivity $$d\thinspace\mathrm{tp}/dQ^{(n)}$$ in basis points per trillion dollars. (b) Foreign official demand rises by $2 trillion. Compute the new term premium. (c) Starting from the baseline, the central bank instead buys $2 trillion of long bonds. Compute the new term premium, and explain why (b) and (c) give the same answer while being economically different events. (d) A funding shock doubles arbitrageurs' risk aversion. Recompute the baseline premium. If the same shock simultaneously forces levered habitat investors to sell $1 trillion, what is the premium then? (e) The empirical literature on quantitative easing finds effects on the order of tens of basis points per trillion dollars of purchases. Is the calibration in (a) consistent with that? What would have to be true of $$\sigma\_n$$ or $$\gamma$$ for the model to match a much larger estimate? **Problem 6 ★ — A one-factor affine model.** The short rate follows $$y^{(1)}\_{t} = X\_t$$ with $$X \_{t+1} = (1-\phi)\bar{X} + \phi X\_t + \sigma\varepsilon \_{t+1}$$, $$\varepsilon \_{t+1}\sim N(0,1)$$ i.i.d., and the SDF is $$m \_{t+1} = \exp(-X\_t - \tfrac{1}{2}\lambda^2 - \lambda\varepsilon \_{t+1})$$ with $$\lambda$$ constant. Use $$\bar{X} = 0.04$$, $$\phi = 0.9$$, $$\sigma = 0.01$$, $$\lambda = -0.20$$, and a current short rate of $$X\_t = 0.03$$. (a) Verify that $$p^{(1)}\_t = E\_t\[m \_{t+1}] = e^{-X\_t}$$, and state what that verification is checking. (b) Derive $$p^{(2)}\_t = E\_t\[m \_{t+1}p^{(1)} \_{t+1}]$$ in closed form, and show that it takes the affine form $$\exp(A\_2 + B\_2 X\_t)$$. Identify $$A\_2$$ and $$B\_2$$. (c) Compute the two-period yield $$y^{(2)}\_t$$, the expectations-hypothesis component $$\tfrac{1}{2}(X\_t + E\_t\[X \_{t+1}])$$, and the term premium in basis points. Separate the term premium into its Jensen and risk-compensation parts. (d) What sign must $$\lambda$$ have for the term premium to be positive, and what does that sign mean about the covariance between the SDF and bond returns? Relate the answer to Chapter 5's countercyclical risk premia. (e) Write the recursion that generates $$A \_{n+1}, B \_{n+1}$$ from $$A\_n, B\_n$$, and describe without computing it what happens to the term premium per period as $$n$$ grows when $$|\phi| < 1$$. *** ## Selected Solutions *Solutions to Problems 1 and 5 follow. Solutions to the remainder are in the instructor materials.* **Problem 1.** (a) Discounting six coupons of 6 and a final 106 at 4.5 percent: $$ p = \sum\_{t=1}^{6}\frac{6}{1.045^t} + \frac{106}{1.045^7} = 108.8391 $$ (b) The present-value-weighted average maturity is $$D\_{\mathrm{Mac}} = 5.9684$$ years, so $$D\_{\mathrm{mod}} = 5.9684/1.045 = 5.7113$$. Duration is well short of the seven-year maturity because a 6 percent coupon returns a substantial share of value early. (c) $$C = 41.2946$$. (d) For $$\Delta y = +0.015$$: duration alone gives $$-5.7113\times0.015 = -8.567$$ percent; adding convexity gives $$-8.567 + \tfrac12(41.2946)(0.015)^2 = -8.102$$ percent. Exact repricing at 6.0 percent gives **100.0000** — the yield now equals the coupon, so the bond is at par, which is the arithmetic check — a change of $$-8.121$$ percent. The duration-only error is 0.446 percent of 108.8391, or **48.5 cents per 100 of face**; the duration-plus-convexity error is -0.019 percent, or **2.0 cents** in the other direction. (e) For $$\Delta y = -0.015$$: duration alone gives $$+8.567$$ percent; with convexity, $$+9.032$$ percent; exact repricing at 3.0 percent gives 118.6908, a change of $$+9.052$$ percent. The duration-only error is 0.485 percent, or **52.8 cents**; the duration-plus-convexity error is 0.020 percent, or 2.2 cents. Both duration-only errors are positive because the true price-yield curve lies above its tangent line everywhere: duration understates the gain and overstates the loss, which is positive convexity restated. (The residual after the convexity term changes sign because it is third-order, not second.) **Problem 5.** (a) $$\gamma\sigma\_n^2 = 0.25\times0.01 = 0.0025$$ per trillion, so the sensitivity is **25 basis points per trillion dollars**. The residual arbitrageurs must hold is $$14 - 8 = 6$$, so $$\mathrm{tp} = 0.0025\times6 = 0.015$$, or **150 basis points**. (b) $$Q\_h^{(n)} = 10$$, residual 4, $$\mathrm{tp} = 0.010$$, or **100 basis points**. A fifty basis point compression from a demand shift alone, with no change in the expected path of policy — the opening episode in one line. (c) $$Q^{(n)} = 12$$, residual 4, $$\mathrm{tp} = 0.010$$, or **100 basis points**. The equation only sees the residual, so a purchase and a habitat-demand increase of the same size are identical in their price effect. They differ in everything else: the central bank's holding is reversible by announcement and is financed by reserves that stay in the banking system, while a reserve manager's is driven by an exchange-rate policy and unwinds when that policy does. The pricing model is deliberately silent on the difference, which is a limitation to state rather than to hide. (d) With $$\gamma = 0.5$$ the sensitivity doubles to 50 basis points per trillion; the baseline residual of 6 gives $$\mathrm{tp} = 0.030$$, or **300 basis points**. If levered habitat investors are also forced to sell 1 trillion, the residual becomes 7 and $$\mathrm{tp} = 0.035$$, or **350 basis points**. The two effects compound, which is the model's version of Chapter 16 §16.5: the shock that impairs the arbitrageur's risk-bearing capacity is the same shock that hands him more inventory. (e) Yes, to an order of magnitude: 25 basis points per trillion is inside the range of published QE estimates, which cluster in the tens of basis points per trillion for purchases in calm markets. To generate a much larger effect the model needs either a higher $$\sigma\_n$$ — a more volatile long end, as in a crisis — or a higher $$\gamma$$, which is exactly what a funding shock delivers. That the same parameters that make QE powerful are the ones that appear in a crisis is not a coincidence in this model; it is the mechanism. *** ## Data Exercise: The Conundrum in Data All four parts run on free sources. **Part A — Fit a curve and compute a duration (FRED).** Download the constant-maturity Treasury series `DGS1`, `DGS2`, `DGS3`, `DGS5`, `DGS7`, `DGS10`, `DGS20`, `DGS30` for the most recent business day. 1. These are par-equivalent yields, not zero yields. Bootstrap an approximate zero curve from them, interpolating linearly in yield between the quoted maturities to fill the missing years. Plot the par curve and the bootstrapped zero curve on the same axes and confirm the ordering §9.1 predicts. 2. Compute the implied one-year forward rates out to thirty years. Plot them against the zero curve, and identify the maturities at which the forward curve turns down. 3. Choose one on-the-run note — a ten-year, say — and compute its price, Macaulay duration, modified duration, and convexity on a semiannual basis at the current yield. Reprice it exactly at ±100 and ±200 basis points, and reproduce the structure of Table 9.2 for your bond. Report the convexity gain in cents per 100 of face at ±200 basis points. 4. As a robustness check on step 1, download the Gürkaynak-Sack-Wright fitted zero-coupon parameters published by the Federal Reserve Board and compare their zero curve with your bootstrap at five, ten, and twenty years. Where do you differ most, and why? **Part B — The conundrum, plotted.** Download the Adrian-Crump-Moench term premium estimates from the Federal Reserve Bank of New York's website (free, daily, ten-year `ACMTP10` and the fitted yield and expected-short-rate components), and `DFF` or `FEDFUNDS` from FRED. 5. Plot the effective federal funds rate and the ten-year yield from January 2004 through December 2006 on one panel. Mark June 30, 2004 and February 16, 2005. Report the cumulative change in each over the tightening cycle. 6. On a second panel, plot the ACM ten-year term premium and the ACM expected-average-short-rate component over the same window. Report how much of the change in the ten-year yield each component accounts for. 7. Extend both panels through the 2015-2019 and 2022-2024 tightening cycles. In which cycles did the long yield fall while the funds rate rose, and in which did the term premium do the work? Write one paragraph on whether "conundrum" describes a single episode or a recurring feature. 8. State one thing your decomposition cannot establish, referring to §9.2's decomposition problem. **Part C — The convenience yield.** Download `AAA` (Moody's seasoned Aaa corporate bond yield) and `DGS20` from FRED, monthly, over the longest common sample, together with `GFDEBTN` (federal debt held by the public) and `GDP`. 9. Construct the Aaa-Treasury spread and plot it. Mark 2008-09, March 2020, and the debt-ceiling episodes of 2011, 2013, and 2023. 10. Regress the spread on debt-to-GDP. Report the coefficient and compare its sign and rough magnitude with §9.4's account of Krishnamurthy and Vissing-Jorgensen. State clearly what the regression does *not* control for and why the maturity mismatch between `AAA` and `DGS20` matters. 11. Using your fitted relation, compute the implied dollar value of convenience services at the start and end of your sample. Explain why the total can move in the opposite direction to the per-dollar convenience yield. **Part D — Who holds it.** From the Financial Accounts (Z.1) table L.210 and the Treasury International Capital release, rebuild the columns of Table 9.3 with exact figures for the three dates, splitting foreign holdings into official and private. 12. Plot each holder's share of the total from 2000 to the present. Identify the two composition breaks §9.4 names, and date them. 13. The household row includes hedge fund positions the accounts cannot separate. Using the Office of Financial Research's Hedge Fund Monitor and the CFTC's Traders in Financial Futures data on leveraged-fund Treasury futures positions, construct an independent estimate of the basis trade's size and compare it with the growth of the residual row. (Chapter 2's Data Exercise sets up the same reconciliation from the accounting side.) **Part E ★ (if you have licensed data).** Using CRSP US Treasury issue-level data, construct the on-the-run/off-the-run yield spread for the ten-year sector daily from 1990. Plot it against your Part C convenience yield measure, and against a dealer balance sheet proxy from the Federal Reserve's primary dealer statistics. Test whether the two safety measures separate during March 2020 in the way §9.4 claims: a liquidity-based measure should invert while a longer-horizon safety measure does not.