FI 1 – The Term Structure and Interest Rate Dynamics

Governments issue coupon bonds, not zeros. The observable curve is therefore usually the par curve: the yields to maturity on coupon-paying government bonds priced at par, across a range of maturities. In practice the most recently issued bonds at each maturity, the on-the-run issues, are used to build it, because they are the most liquid and trade at or near par.

The par curve is valuable precisely because a zero curve can be recovered from it. Treat a coupon bond as a bundle of zeros, one for each dated payment. Then solve for the zero rates one at a time, starting from the shortest maturity and working outward, using each solved rate as an input to the next equation. That forward substitution procedure is called bootstrapping.

Note the pattern: with an upward-sloping curve, each par rate sits below the zero rate of the same maturity, because the par yield is dragged down by the earlier, lower-discounted coupons.

Reading the shape of the curve

In developed markets the curve usually slopes upward, and it usually flattens as maturity lengthens, so equal steps out in maturity buy progressively smaller increases in yield. Two forces produce that shape. Nominal yields embed a premium for expected inflation, so an upward slope is generally read as a market view that inflation will rise or at least hold steady, which in turn tends to accompany reasonably strong growth. Risk premiums add to it, since longer bonds carry more interest rate risk.

An inverted curve is much less common. Because a nominal yield contains an inflation premium, a downward slope can signal an expectation that inflation will decline from a relatively high starting level, and an expected slowdown in activity is one obvious reason to expect that. Inverted curves are frequently seen ahead of recessions. A flat curve typically appears only briefly, during the transition between an upward and a downward slope in either direction. A humped curve, rarer still, occurs when intermediate rates sit above both short and long rates.

Yield to maturity is the most familiar number in the bond market and the most frequently misread. For a coupon bond maturing at T, the yield to maturity is not the T-year spot rate. No arbitrage requires the bond to be worth the sum of its payments discounted at their own spot rates, and the yield to maturity is the single rate that reproduces that same total. It is therefore a weighted average of the spot rates that actually price the bond, with the weights determined by where the cash flows fall.

When the yield to maturity is a poor estimate of return

An investor earns the yield to maturity only under conditions that are seldom met. The yield to maturity is the expected return on a bond held to maturity, assuming every coupon and the principal arrive in full and on time, and assuming every coupon is reinvested at the original yield to maturity. As rates move, that reinvestment assumption fails. Four situations make the yield to maturity an unreliable guide to expected return:

1. Interest rates are volatile, so reinvestment will not occur at the assumed rate.
2. The yield curve slopes, upward or downward, which again means reinvestment happens at rates other than the yield to maturity.
3. Default risk is significant, so actual cash flows may differ from the promised ones used in the calculation.
4. The bond carries one or more embedded options, such as a call, a put or a conversion feature, whose exercise would end the holding period before the stated maturity.

Against that expected figure stands the realised return, which is the return actually earned over the investor holding period. It depends on the reinvestment rates that actually occurred and on the yield curve prevailing when the holding period ends. Only with perfect foresight would expected and realised returns coincide.

Active bond management rests on one proposition: today’s forward curve is a prediction of tomorrow’s spot curve, and a manager who disagrees with it can profit. To make that operational it is necessary to know exactly what happens when the forward curve turns out to be right, so that any deviation can be attributed correctly.

The central result is that the forward contract price does not move at all, so long as spot rates evolve exactly as today’s forward curve implies. Solving the forward pricing model for the price gives the starting point.

Now let t periods pass and suppose the new discount function for a maturity of T is exactly the forward discount function implied today, that is, the new factor equals DF_t+T divided by DF_t. Rewrite the forward price at the new date using the shortened remaining terms, substitute the assumed new discount factors, and the common factor DF_t cancels from numerator and denominator. What survives is DF_B divided by DF_A, which is the original forward price. Nothing has changed. It follows that any change in a forward contract price must come from the spot curve departing from what today’s forward curve predicted.

A numerical demonstration

Take a flat yield curve at 4%. The discount factors, to four decimal places, are DF₁ = 1 ÷ 1.04 = 0.9615, DF₂ = 1 ÷ (1.04)² = 0.9246 and DF₃ = 1 ÷ (1.04)³ = 0.8890. The forward contract that delivers a one-year bond at year two is priced at F_2,1 = 0.8890 ÷ 0.9246 = 0.9615.

Let one year pass and assume the discount function at year one is precisely the forward discount function implied at year zero. The new one-year and two-year factors are 0.9246 ÷ 0.9615 = 0.9615 and 0.8890 ÷ 0.9615 = 0.9246. The forward contract now has one year to expiry and is priced at 0.9246 ÷ 0.9615 = 0.9615, exactly what it was a year ago.

The same figures show what a bond earns when the curve is unchanged. A three-year bond bought at 0.8890 is worth 0.9246 one year later, a return of (0.9246 − 0.8890) ÷ 0.8890 = 4%, equal to the spot rate. After another year it is worth 0.9615, a further (0.9615 − 0.9246) ÷ 0.9246 = 4%, equal to the implied one-year forward rate. Each bond rolls down the curve and collects the current one-period spot rate followed by the successive forward rates.

The one-period return result

Rearranging the forward rate model with the horizon set to one period gives a compact statement of the same idea: if the spot curve one period from now equals today’s forward curve, the total return over that period is the one-period risk-free rate, whatever the maturity of the bond held.

Turning the result into a valuation rule

The comparison in the previous example generalises into a test of cheapness. If any of the future spot rates an investor expects lies below the quoted forward rate for the same period, then, all else equal, the market is discounting the payments at a higher rate than the investor would. The market price sits below the intrinsic value the investor perceives, so the bond looks undervalued. Reverse the inequality and the bond looks overvalued.

The results of the previous section support one of the best-known active trades in fixed income. When the curve slopes upward, the forward curve lies above the spot curve, which means the market is implicitly forecasting that rates will rise. A manager who expects the curve instead to stay where it is can buy a bond longer than the investment horizon and sell it before maturity. As time passes, the bond is repriced at successively shorter maturities and therefore at successively lower yields and higher prices. That is rolling down the yield curve, also called riding the yield curve.

The total return on a fixed-rate bond that neither defaults nor is called breaks into three parts, and the rolldown trade works on the third:

• the promised coupons and principal;
• the interest earned on reinvesting those coupons;
• the capital gain or loss if the bond is sold before maturity.

The conditions the trade depends on

Three conditions are doing all the work. The curve must slope upward, so that shorter maturities carry lower yields. Yields must not change, or at least must not rise enough to erase the rolldown gain. And the bond bought must mature after the investment horizon, so that there is curve left to roll down at the point of sale. The total return depends on the spread between the forward rate and the spot rate, and the longer the maturity of the bond, the more sensitive its total return is to that spread.

The same logic explains a strategy that became widespread after 2008. With many central banks holding short rates near zero, curves turned steeply upward, and active managers were handed an incentive to fund at short maturities and invest at long ones. That is a maturity spread carry trade: borrow short, lend long, in the same currency. It is common whenever the curve is upward sloping, and it carries substantial interest rate risk, because an unexpected jump in future spot rates, from an inflation shock for instance, hits the long position while the funding cost resets upward.

Stated plainly, when the curve slopes upward a bond that approaches maturity is repeatedly valued at lower yields and higher prices. Holding it through part of that appreciation and selling before maturity can raise total return, and as long as rates stay stable and the slope stays positive the strategy can be repeated to add return continuously.

The government spot curve is one measure of the time value of money. The swap curve is the other, and in large parts of the world it is the more relevant one.

An interest rate swap exchanges fixed-rate interest payments for floating-rate payments. Each leg is computed by applying its rate to a notional principal for each interest period over the life of the contract, and only the net amount changes hands. The fixed leg rate is the swap rate. It plays the same role for a swap that the yield to maturity plays for a government bond, with one important difference: the swap rate is built from short-term lending rates rather than default-risk-free rates. Floating legs historically referenced survey-based rates such as three-month or six-month US dollar Libor and are transitioning to transaction-based market reference rates derived from secured overnight funding.

The curve of swap rates across maturities is the swap curve. Because benchmark swaps are par swaps, with the fixed rate chosen so that no money changes hands at inception, the swap curve is itself a type of par curve. When the term par curve is used elsewhere in this reading it refers to the government par yield curve.

Why the swap market is so liquid

Two features explain the depth of the market. First, a swap has no borrower and no lender, only two counterparties exchanging cash flows, which allows terms to be tailored freely to what each side needs. Second, swaps are among the most efficient instruments available for hedging interest rate risk. The Bank for International Settlements put the notional amount outstanding on interest rate swaps at nearly $350 trillion as of June 2020.

Where the swap curve is the natural benchmark depends on the market. Many countries lack a liquid government bond market beyond one year, so the swap curve is the only usable term structure available. Where the private sector dwarfs the public sector, the swap curve is a better measure of the time value of money than the cost at which the government happens to borrow. Swaps are frequently used as the benchmark in Europe, while in Asia the swap and government bond markets developed alongside one another and both are used in credit and loan valuation.

Within a single market the choice can turn on the institution rather than the country. In the United States both a deep Treasury market and a deep swap market exist, and the benchmark chosen tends to follow the interest rate exposure profile of the user. Wholesale banks that hedge their balance sheets with swaps value assets and liabilities off the swap curve. Retail banks with little swap exposure are more likely to use the government spot curve.

Discount factors, forward rates and the swap rate

Every forward date carries a discount factor, the value today of a unit payment received on that date, expressed as a decimal fraction. If a security paying ₩10,000 in one year currently costs ₩9,259.30, the one-year discount factor is 9,259.30 ÷ 10,000 = 0.92593, and the rate implied by it is 1 ÷ 0.92593 − 1 ≈ 8.00%.

Pricing a swap means solving for the constant fixed rate that sets the present value of the fixed leg equal to the present value of the floating leg over the life of the contract. The fixed leg is straightforward once the rate is set. The floating leg is harder, because by construction its payments change with future rates, and the forward rate for each floating payment date has to be taken from the forward curve. At origination the floating leg is worth par, which produces a clean condition on the swap rate s_T.

The swap spread is the spread paid by the fixed-rate payer of an interest rate swap over the yield on the on-the-run, that is the most recently issued, government security of the same maturity. It captures the yield premium the market demands for credit exposure relative to the default-risk-free benchmark. Because swap rates are constructed from market rates on short-term risky debt, the swap spread works as a barometer of perceived credit risk. It typically moves counter to the cycle, widening in recessions and narrowing in expansions.

The arithmetic is simple. If the fixed rate on a five-year fixed-for-floating swap referencing MRR is 2.00% and the five-year Treasury yields 1.70%, the swap spread is 2.00% − 1.70% = 0.30%, or 30 bps. In euro-denominated markets the benchmark is usually Bunds of matching maturity, and in the United Kingdom it is gilts.

The same term is used in a second, narrower sense: a bond quoted at a spread over the interest rate swap curve. To keep the two apart, the spread of a bond yield over the swap rate of the same maturity is called the I-spread, or ISPRD, or interpolated spread. In its simplest form the I-spread is the difference between the bond yield to maturity and the swap rate obtained by straight-line interpolation along the swap curve. A related measure is the zero volatility spread, or Z-spread, which is the constant spread added to every point of a government or swap spot curve that makes the discounted cash flows equal the market price.

Why market participants like the swap benchmark

Historically the Libor swap curve was taken to reflect the default risk of commercial banks rated A1 or A+. The move from Libor to reference rates built on secured overnight funding transactions will increase the influence of supply and demand in government debt markets on swap rates. The swap market is also led by major financial institutions rather than controlled by governments, so swap rates are more comparable from country to country. And it offers more maturity points from which to build a curve than government bond markets do: deposit rates covered the short end, interest rate futures covered maturities out to about a year, and swap rates extend as far as 50 years in US dollars and euro.

Conventions and the treatment of maturity

Treasury curves and swap curves are different benchmarks, so it matters whether a quoted spread is measured against a government yield or against a swap rate. The two curves can differ for several reasons: swap cash flows carry greater default risk than Treasury cash flows; liquidity varies by maturity, and some parts of the term structure trade more actively in swaps than in bonds; and arbitrage between the two markets cannot be executed perfectly.

Defining a swap spread runs into a maturity problem. A 10-year swap matures in exactly 10 years, but a 10-year government bond has exactly 10 years to run only on its issue date. By convention, therefore, the 10-year swap spread is the difference between the 10-year swap rate and the yield on the 10-year on-the-run government bond, and spreads at other maturities are defined the same way.

Short-term spreads as economy-wide risk gauges

Spreads between short-term government and risky rates are widely used to read economy-wide credit and liquidity conditions. The TED spread is the difference between MRR and the yield on a Treasury bill of the same maturity. The name combines the abbreviation for the US T-bill with the ticker for the MRR-based Eurodollar futures contract. A rising TED spread signals greater perceived credit and liquidity risk, as it did in early 2020 during the market turmoil associated with the COVID-19 pandemic.

The MRR–OIS spread, formerly the Libor–OIS spread, is the difference between MRR and the overnight indexed swap rate. An overnight indexed swap is a swap whose floating leg equals the geometric average of a daily unsecured overnight rate, typically the rate at which banks lend to one another overnight, such as the federal funds rate for US dollars or Eonia for euros.

The reference rates themselves are changing. As participants move away from survey-based Libor toward benchmarks built on actual transactions, the secured overnight financing rate, or SOFR, has gained prominence and is expected to replace Libor. SOFR is a daily volume-weighted index of qualifying transactions in the US Treasury repurchase market, and it therefore responds to supply and demand in secured funding. Comparable substitutions are under way elsewhere. In the euro area the recommended successor to Eonia is the secured euro short-term rate, known as ESTR. In Canada the proposal is to retire the survey-based Canadian Dollar Offered Rate, or CDOR, in favour of the Canadian Overnight Repo Rate Average, or CORRA.

Shaping risk is the sensitivity of a bond price to changes in the shape of the yield curve, as distinct from changes in its level. It matters because the shape of the curve changes continually and shifts are rarely parallel. A manager may want to trade a forecast of curve shape, or to hedge the curve risk in a portfolio using swaps. Shaping risk also drives the value of many options, which is significant given how many fixed-income instruments carry embedded options.

The historical record of swap curves in the United States and Europe from March 2006 to March 2020 illustrates how much the curve moves. In both markets the pre-crisis curve of March 2006 carried the highest yields and the March 2020 curve, during the pandemic-related market turmoil, the lowest. The paths in between diverged: in the United States the end of quantitative easing and tighter policy produced a rebound in swap yields before 2020, whereas in Europe continued accommodation kept yields low or negative.

Reducing the movements to a few factors

The practical problem is that the curve can take almost any shape. The response is to find a model that compresses the space of possible movements into a probabilistic combination of a small number of standard ones. A yield curve factor model is a description of yield curve movements that looks realistic when tested against historical data.

The best known such model is the three-factor specification of Litterman and Scheinkman (1991). Their finding was that the history of curve movements can be reconstructed accurately from just three mutually independent movements, to which they attached the labels level, steepness and curvature. The three factors are recoverable from the variance and covariance matrix of historical interest rate movements.

• Level. An upward or downward shift of the whole curve.
• Steepness. A non-parallel shift in which short-term rates change by more than long-term rates, or the reverse.
• Curvature. A movement across three segments at once: the short and long segments rise while the middle segment falls, or the reverse.

The three factors are not equally important. The level factor explains most of the total change in swap and bond yields, and can be read as the parallel component in which rates move in the same direction and by a similar order of magnitude. Steepness comes second, with short-term yields typically moving more than long-term yields; because these changes accumulate over time, this factor explains less of the total variance than level does. Curvature comes third and has the smallest impact of the three, pushing intermediate yields in one direction while pushing short and long yields in the other. It is the factor that captures the twist in the curve.

Quantifying how volatile rates are at each maturity matters for two reasons. Most fixed-income instruments and derivatives contain embedded options, and option values, and therefore the values of the instruments containing them, depend crucially on the level of interest rate volatility. Separately, controlling the effect of rate volatility on price volatility is a core part of any risk management process.

The term structure of interest rate volatilities describes the yield volatility of a zero-coupon bond at every maturity. This volatility curve, often shortened to the vol or the volatility term structure, is a measure of yield curve risk.

Volatility is not uniform across the curve. Under the usual lognormal assumption, the uncertainty attached to an interest rate is measured by the annualised standard deviation of the proportional change in the yield over a chosen interval. If the interval is a month, the interval length is 1/12 of a year. The resulting quantity is written σ(t, T), the volatility at time t of the rate on a security maturing at T.

The sample deliberately ends before the 2008 financial crisis, which produced unusual volatility magnitudes.

Yield curve risk is the risk to portfolio value from unanticipated changes in the yield curve. Managing it means measuring the current exposures, deciding what they should be, and trading securities or derivatives to close the gap. Three measures are available, and they differ in how much of the curve they can see.

• Effective duration measures the sensitivity of a bond price to a small parallel shift in a benchmark yield curve. It handles level risk and nothing else.
• Key rate duration measures sensitivity to a small change in the benchmark curve at one specific maturity segment, holding the others fixed.
• Factor sensitivities measure sensitivity to the level, steepness and curvature movements described in the factor model.

Only the last two can capture shaping risk. Effective duration adequately addresses parallel changes and is silent about everything else.

A worked portfolio

Consider a portfolio holding $100 each of one-year, five-year and ten-year zero-coupon bonds, so the total value is $300. Consider also the following set of hypothetical factor movements, expressed in units of one.

A parallel movement shifts all rates by the same amount. A steepness movement raises the long rate and lowers the short rate by one unit each. A curvature movement raises both the short and the long rate by one unit while leaving the medium-term rate unchanged. The three movements must be defined so that none of them is a linear combination of the other two.

The two representations carry the same information, and either can be derived from the other. Decomposing the term structure changes into level, slope and curvature gives the following identities for this example:

• Level sensitivity = KeyDur₁ + KeyDur₅ + KeyDur₁₀
• Steepness sensitivity = −KeyDur₁ + KeyDur₁₀
• Curvature sensitivity = KeyDur₁ + KeyDur₁₀

Everything so far has been machinery. The last question is what to do with it. Implied forward rates are the market-neutral reference point: if today’s forward rates are realised, bond values simply roll down the curve and no maturity outperforms any other. Active participants form their own view of how rates will develop and position portfolios to profit from the difference between that view and the market consensus. An accurate forecast generates return the portfolio would not otherwise have earned.

The bond risk premium is the expected excess return of a default-free long-term bond over an equivalent short-term bond, or over the one-period risk-free rate. It is also called the term premium or duration premium, and it is usually measured with government bonds so that the measurement captures uncertainty about default-free rates alone, while credit, liquidity and other risks add further premiums for a specific bond. Unlike a historical return observed after the fact, the bond risk premium is a forward-looking expectation and has to be estimated.

What drives yields

Inflation, GDP growth and monetary policy explain most of the variance of bond yields, but not in the same proportions at every maturity. Short-term and intermediate-term yields are driven mostly by monetary policy, which accounts for roughly two-thirds of their variation, with the remaining third split roughly equally between economic growth and factors including inflation. Long-term yields work the other way: inflation explains nearly two-thirds of the variation, and most of the rest is attributable to monetary policy.

The policy channel produces recognisable curve shapes. During expansions, monetary authorities raise benchmark rates to contain inflation. Short-term yields rise more than long-term yields, so the curve flattens while yields rise, which is bearish flattening. During recessions, or when one is anticipated, the authority cuts benchmark rates to stimulate activity. Short rates fall by more than long rates, so the curve steepens while yields fall, which is bullish steepening. Short-term rates therefore move procyclically.

Central banks have also turned increasingly to their balance sheets. Large-scale purchases of government bonds and mortgage-backed securities are intended to stimulate activity by expanding the money supply and by driving down the bond risk premium, which encourages capital to move into higher-risk assets. Such purchases affect the term structure by raising demand within particular maturity segments.

Supply, demand and flight to quality

Government bonds are how nations fund their cumulative budget deficits, so fiscal policy feeds directly into supply. Larger deficits require more borrowing, which raises yields; smaller deficits reduce supply and lower them. In the late 1990s, when market participants expected the US government to run surpluses, the Treasury stopped issuing new 30-year bonds for four years, and the anticipated reduction in supply pushed long-maturity yields down.

The maturity structure of outstanding debt matters as well. Longer government debt maturity structures predict greater excess bond returns, which is a segmented market effect: a greater supply of long-dated bonds raises the yield in that specific segment.

On the demand side, pension funds and insurance companies use long-dated government bonds to match expected future liabilities, so stronger domestic demand raises prices and compresses the bond risk premium. Non-domestic demand, arising from reserve holdings or from currency management operations, works the same way: inflows bid prices up and lower the risk premium, outflows do the reverse. Because these flows can be large, they move prices meaningfully.

In periods of high uncertainty investors sell higher-risk assets such as equities and commodities and buy default-risk-free government bonds. This flight to quality is typically associated with bullish flattening, in which the curve flattens because long-term rates fall by more than short-term rates.

Turning a view into a trade

Trades built on rate forecasts often use bond futures, which allow the exposure to be changed without heavy turnover in the underlying portfolio. Whatever the instrument, any view must be judged against the current short rate and the forward curve, because those already embed the returns available to an investor who simply rolls down the curve under the prevailing set of implied forward rates. A forecast that merely reproduces the forward curve is worth nothing.

Making a steepening or flattening trade duration neutral is what separates a curve view from a level view. Without that adjustment the position also carries a bet on the level of rates, and the level factor is the one that usually dominates, so an accurate curve call can easily be swamped by an inaccurate level call.