The Curve — The Rate, Plate xii

The volume has spoken of the rate as though time had one price, but the market offers a different quotation for each distance into the future. A loan until morning, a claim payable in a year, a bond that returns its principal in thirty: each exchanges present command for a different promise. Arrange comparable yields by maturity and the price of time becomes a line. This is the term structure, or yield curve. It may rise, flatten, bend into a hump or turn downward, with a longer commitment yielding less per year than a shorter one. The shape is often read as a forecast, especially when an inversion is taken to announce a downturn. Yet the line records prices at which people transact today, and a price contains more than an opinion about tomorrow. It also contains the terms on which someone will accept the uncertainty of being wrong. The final plate asks how to read those two things together: what the curve says about the rates people expect, and what it says about the risks they are willing to hold.

A meaningful curve compares claims whose differences have been made explicit. Combining a risky corporate loan with a government bill would mix maturity with credit risk; combining currencies would mix the terms of time with exchange-rate exposure. Even within one government's debt, coupon payments, liquidity and the privileges of particular securities can complicate the comparison. The clean analytical object is a set of zero-coupon bonds, each making a single payment at its own maturity. Each price then gives a discount factor for one date, and each annualized yield expresses that discount over the relevant interval. A coupon bond makes several payments, so its yield to maturity is a single summary rate fitted to a stream, rather than the price of just its final date.¹ The distinction matters whenever the curve is used to value another promise. A payment next year and a payment ten years hence need not be discounted at the same annual rate. Time's many prices are already present in the simplest bond arithmetic; a single number conceals them for convenience.

1 Tobias Adrian, Richard Crump and Emanuel Moench, “Pricing the Term Structure with Linear Regressions”, Federal Reserve Bank of New York Staff Report 340, revised April 2013, §§1–2.4. Their distinction between zero-coupon yields and returns on coupon-bearing securities is preserved here; quoted market yields require attention to their payment and compounding conventions.

The prices at different dates can imply a future rate without telling us what that future rate will actually be. Suppose a one-year zero-coupon yield is two per cent and a two-year yield is three per cent, with annual compounding and otherwise identical certain payments. Investing for two years grows a unit to 1.03 squared. Investing for one year and arranging today to reinvest the proceeds for the second year must produce the same amount in a frictionless market. The implied second-year forward rate is therefore about 4.01 per cent: 1.03 squared divided by 1.02, less one.² This is a relation among prices and commitments available now. It does not promise that a fresh one-year investment, chosen only when next year arrives, will pay 4.01 per cent. That later rate remains unknown. The forward contract fixes terms in advance and allocates the consequences of the difference. A forecast is a belief about the later quotation; a forward rate is a price at which exposure to it can be exchanged. The two are connected, but equating them would remove precisely the uncertainty for which the market may demand compensation.

2 The two-rate example is illustrative: (1.03² ÷ 1.02) − 1 = 0.040098…, or about 4.01 per cent. It assumes identical default-free monetary payoffs, unrestricted borrowing and lending, and no taxes or transaction costs. Ben Bernanke, “Reflections on the Yield Curve and Monetary Policy” (20 March 2006), “The Recent Behavior of Longer-Term Yields,” separates implied forward rates from expected future spot rates.

Expectations become one part of a long yield when the choice is between holding a longer bond and repeatedly buying shorter ones. In the usual decomposition, the long yield is approximately the expected average of future short rates plus a term premium. The approximation keeps compounding and other valuation details from disappearing into an exact-looking slogan. The premium also needs to be distinguished from the visible spread between a long yield and today's short rate: that spread can reflect expected changes in short rates as well as compensation for risk. Nor must the premium be positive. A long fixed payment exposes its holder to changes in interest rates and purchasing power, but a bond whose value rises when other assets fare badly can supply a hedge for which investors accept a lower expected return. The risk depends on the portfolio and the economic states in which the payment is valuable.³ This is why the curve cannot be read by a rule that rising always means future tightening and falling always means future easing. An altered expected path can change its shape, and so can an altered willingness to bear the risks of that path. The observed line joins the two without labeling their separate contributions.

3 Richard Clarida, “Monetary Policy, Price Stability, and Equilibrium Bond Yields: Success and Consequences” (12 November 2019), yield decomposition and the discussion of a negative equilibrium term premium. Adrian, Crump and Moench, §2.4, explains the model-based separation of fitted yields and the expected short-rate component. The premium is inferred, not separately quoted.

⁂

Inversion earns attention because the slope has contained useful information about subsequent economic weakness. In their 1996 study, Estrella and Mishkin tested the spread between the ten-year Treasury note and the three-month Treasury bill against other forecasting indicators. Their out-of-sample comparisons found useful recession-prediction performance at horizons of several quarters, and they expressed the result as a probability rather than a date stamped on the next downturn.⁴ There are intelligible channels behind the association. Restrictive short rates can slow expenditure; investors who expect weaker activity or lower inflation may also expect the central bank to cut later, drawing longer yields downward. But a spread can move because the premium changes, and expectations of lower inflation need not mean an expectation of recession. The relation is therefore an empirical one, conditioned by the period, the maturities compared and the other information available. A signal can deserve attention without deserving obedience. Its value lies in changing the assessment of possibilities and prompting a closer inquiry into their causes, not in relieving the observer of judgment.

4 Arturo Estrella and Frederic Mishkin, “The Yield Curve as a Predictor of U.S. Recessions”, Current Issues in Economics and Finance 2, no. 7 (June 1996), pp. 1–5, including notes 4–8. The plate reports the historical study and its probability framework; it supplies neither a current yield reading nor a current recession forecast.

The central bank participates in the curve it reads. Its present decision affects a short rate, and its account of future decisions can alter the expected sequence from which longer yields are partly formed. Purchases of longer securities can also change the supply of duration that private investors must absorb. These are channels of influence, not a power to select every maturity's yield independently of expectations, competing assets and the response of the economy.⁵ A promise of low rates may be understood as reassurance about policy or as news that conditions are weaker than investors had thought. The bank must interpret a market that is simultaneously interpreting the bank. This does not make the curve false; it makes the explanation of a movement consequential. A lower long yield caused by a smaller premium is not the same event as a lower long yield caused by expectations of a severe slump. Models such as Adrian, Crump and Moench's seek to separate the components, but the separation requires assumptions and can vary with the specification. The line itself is observable. The beliefs and compensation assigned to its parts are an inference made from it.

5 Bernanke, “Some Reasons for the Decline in Far-Forward Rates” and “Long-Term Yields and Monetary Policy”; Clarida, discussion of long-duration asset purchases; Adrian, Crump and Moench, §3.5, on comparisons across model specifications. These accounts support the channels and the problem of interpretation, without implying that any announced purchase or promise moves all yields by a fixed amount.

The curve closes the volume by giving the price of time a form adequate to its complexity. There is a quotation for each horizon, an expected path beneath the quotations, and compensation whose meaning depends on what risks the holder already bears. None is the calendar's automatic reward. The earlier plates have found grounds for a payment in waiting, productive opportunity and the surrender of liquidity; they have also found monetary institutions, changing purchasing power, ethical choices and divided intentions within what looked like one rate. The curve gathers these questions across dates. To ask whether its return is earned is to ask what service or exposure has been supplied, on what terms, and with what alternatives. Policy's presence does not settle that judgment, any more than a market quotation does. The borrower and the lender still commit resources before they know how the world will turn. That uncertainty carries the inquiry into The Premium, the next volume: the price demanded for bearing what time has not yet disclosed.