FI 1 – The Term Structure and Interest Rate Dynamics
- Published
- Updated
Interest rates do two jobs at once. They report on the state of the economy and they are the lever by which policymakers try to steer it. For anyone valuing a bond, a swap or an option on either, the whole schedule of market rates across maturities, which is the term structure of interest rates, is the single most important input. This reading builds the machinery: how rates at different maturities relate to one another, how they evolve, how that evolution is measured, and how a portfolio can be positioned for a view about it.
Everything starts with the simplest possible security. Consider a promise to pay one unit of currency after N periods, with no default risk and no option attached. The price of that promise today is the discount factor for maturity N, written DFN. The annualised yield implied by that price is the spot rate, written zN.
Plot DFN against maturity and you have the discount function. Plot zN against maturity and you have the spot curve, also called the zero-coupon yield curve. Because each is recoverable from the other, they are two views of one object: the market price of time. When the term structure of interest rates is referred to without further qualification, the spot curve is what is meant.
Why the spot curve is the reference curve
A zero-coupon bond has one cash flow, at maturity. That single feature removes the awkwardness that afflicts every coupon-paying instrument. There is nothing to reinvest, so no reinvestment rate has to be assumed, and no reinvestment risk is borne. An investor who buys a zero and holds it to maturity earns exactly the quoted yield, no more and no less. For that reason the yield on a zero maturing in year T is treated as the cleanest available statement of the T-year interest rate, and the spot curve is treated as the most basic term structure of all.
Forward rates and the forward pricing model
A forward rate is a rate agreed today for a loan that begins on a future date. Collect the forward rates for loans of differing tenors that all start on the same future date and you have a forward curve. Nothing about a forward curve requires a separate market: forward rates are already implicit in the spot curve and can be extracted from it.
Write fA,B−A for the rate on a loan beginning A periods from today and running for a further B − A periods, and write FA,B−A for the corresponding forward price. In the forward contract the buyer commits to pay that price at time A and to receive one currency unit at time B. No cash changes hands at inception, because the whole agreement concerns the future.
The link between forward prices and discount factors follows from a single idea: two tradable packages that deliver the same cash flow on the same date must cost the same today.
The reasoning is easiest to see with numbers. Suppose a two-year zero costs DF2 = 0.93 and a one-year zero costs DF1 = 0.95. One route to owning a unit of currency in two years is to buy the two-year zero outright for 0.93. A second route is to buy a one-year zero today and enter a forward contract to buy, one year from now, a one-year zero at a price fixed today. Both routes deliver the same single unit at the same date, so the forward price must satisfy 0.93 = 0.95 × F1,1, which gives F1,1 = 0.93 ÷ 0.95 = 0.9789. At any other price a trader could short the expensive package, buy the cheap one with the proceeds and lock in a profit with no capital at risk.
Consider a two-year loan beginning in one year, so A = 1 and B = 3. The one-year spot rate is z1 = 7% and the three-year spot rate is z3 = 9%.
From forward prices to forward rates
The forward price is a price. The forward rate is the discount rate that produces that price over the life of the forward loan, so that the present value at time A of a unit received at time B equals the forward contract price. Substituting the discount factor definition into the forward pricing model turns a statement about prices into a statement about rates.
This equation is the workhorse of the whole reading. It says that any two spot rates fix the forward rate that spans the gap between them. Forward rates are therefore not extra information from a separate market. They are extrapolated from today’s spot curve and are implicit in it at every moment.
A forward rate can be read in two equivalent ways. Suppose f7,1, the rate contracted today for a one-year loan starting seven years out, is 3%. First, 3% is the reinvestment rate that would leave an investor exactly indifferent between buying an eight-year zero and buying a seven-year zero and rolling the proceeds for one more year. On that reading a forward rate is a breakeven rate. Second, 3% is the incremental one-year rate that can be locked in today simply by choosing the eight-year zero over the seven-year zero. On that reading a forward rate is the price of extending maturity by one year.
Three hypothetical zero-coupon bonds carry the following spot rates.
| Maturity (T) | 1 | 2 | 3 |
|---|---|---|---|
| Spot rate | z1 = 9% | z2 = 10% | z3 = 11% |
f1,1 = (1.10)2 ÷ 1.09 − 1 = 11.01%.
f2,1 = (1.11)3 ÷ (1.10)2 − 1 = 13.03%.
f1,2 = [(1.11)3 ÷ 1.09]1/2 − 1 = 12.01%.
Spot rates as geometric averages
Apply the forward rate model repeatedly, substituting each result into the next, and a longer-dated spot rate unwinds into the first-period spot rate followed by a chain of one-period forward rates.
That single line explains a great deal of what follows. A spot rate is an average; a forward rate is a marginal quantity. Whenever an average is rising, the next marginal observation must sit above the average, and whenever an average is falling the next marginal observation must sit below it. This is why an upward-sloping spot curve always carries a forward curve above it, and a downward-sloping spot curve always carries a forward curve below it.
The same relationship matters directly to an active manager. Implied forward rates are not provably unbiased estimates of what the market expects future spot rates to be, and whether they are unbiased remains contested. They are, however, the most accessible and generally the best available proxy for market expectations. If a manager can find a sequence of short-dated bonds whose realised returns beat the forward rates quoted today, then rolling that sequence will beat a matched-maturity buy-and-hold position provided the curve stays reasonably stable.
Using the results already obtained, z1 = 9%, f1,1 = 11.01% and f2,1 = 13.03%.
z2 = [(1 + 0.09)(1 + 0.1101)]1/2 − 1 ≈ 10%.
For three years, add the second forward rate and take the cube root:
z3 = [(1 + 0.09)(1 + 0.1101)(1 + 0.1303)]1/3 − 1 ≈ 11%.
Both reproduce the original spot rates, which is the arithmetic check that the forward rates were extracted correctly.
Slope, restated
A rearrangement of the forward rate model isolates the comparison between a forward rate and the long spot rate that bounds it.
Take A = 1, B = 5, z1 = 2% and z5 = 3%. The left side becomes (1.03 ÷ 1.02)1/4 × 1.03 = 1.0024 × 1.03 = 1.0325, so f1,4 = 3.25%, which sits above z5 = 3.00%. The general results fall straight out of the ratio in brackets:
- Upward sloping, so zB > zA: the forward rate exceeds the long spot rate, and forward rates rise as the start date moves further out.
- Downward sloping, so zB < zA: the forward rate is below the long spot rate, and forward rates fall as the start date moves further out.
- Flat: the bracket equals one and every one-period forward rate equals the spot rate.
One practical limit deserves emphasis. Forward rates cannot reach beyond the longest maturity on today’s curve, because there is no spot rate available to anchor the far end. If the curve stops at 30 years, then a three-year forward curve can extend only 27 years, and four years from now the longest forward rate that today’s data supports is f4,26.
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.
The following par rates are observed on annual-coupon sovereign debt: one-year 5%, two-year 5.97%, three-year 6.91% and four-year 7.81%. Derive the zero-coupon rates.
1 = 0.0597 ÷ (1.05) + (1 + 0.0597) ÷ (1 + z2)2.
Here 0.0597 is the interest payment per unit of principal and 1.0597 is principal plus interest. Solving gives z2 = 6%.
1 = 0.0691 ÷ (1.05) + 0.0691 ÷ (1.06)2 + (1 + 0.0691) ÷ (1 + z3)3, giving z3 = 7%.
For four years:
1 = 0.0781 ÷ (1.05) + 0.0781 ÷ (1.06)2 + 0.0781 ÷ (1.07)3 + (1 + 0.0781) ÷ (1 + z4)4, giving z4 = 8%.
The bootstrapped zero curve is therefore z1 = 5%, z2 = 6%, z3 = 7% and z4 = 8%.
| Maturity | Par rate | Zero-coupon rate |
|---|---|---|
| One year | 5.00% | 5% |
| Two years | 5.97% | 6% |
| Three years | 6.91% | 7% |
| Four years | 7.81% | 8% |
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.
| Shape | Frequency | Common interpretation |
|---|---|---|
| Upward sloping, flattening at the long end | Most common in developed markets | Rising or stable expected inflation with reasonable growth, plus a premium for interest rate risk |
| Downward sloping (inverted) | Less common | Expected decline in inflation from a high level; often precedes recessions |
| Flat | Brief | Transition between an upward and a downward slope |
| Humped | Rare | Intermediate rates 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.
The spot rates are again z1 = 9%, z2 = 10% and z3 = 11%.
Price = $60 ÷ (1.09) + $1,060 ÷ (1.10)2 = $931.08.
The same price expressed with a single rate is $60 ÷ (1 + y2) + $1,060 ÷ (1 + y2)2 = $931.08, and solving gives y2 = 9.97%.
Because the bond has only one price, y2 must lie between the two spot rates it averages, so z1 = 9% < y2 = 9.97% < z2 = 10%.
Price = £5 ÷ (1.09) + £5 ÷ (1.10)2 + £105 ÷ (1.11)3 = £85.49.
Setting £5 ÷ (1 + y3) + £5 ÷ (1 + y3)2 + £105 ÷ (1 + y3)3 = £85.49 gives y3 = 10.93%.
Again the yield to maturity is a weighted average and must lie between the highest and lowest spot rates: 9% < 10.93% < 11%. With an upward-sloping curve, y3 < z3.
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:
- Interest rates are volatile, so reinvestment will not occur at the assumed rate.
- The yield curve slopes, upward or downward, which again means reinvestment happens at rates other than the yield to maturity.
- Default risk is significant, so actual cash flows may differ from the promised ones used in the calculation.
- 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.
Assume z1 = 5%, z2 = 6%, z3 = 7%, z4 = 8% and z5 = 9%. A five-year annual-coupon bond pays 10%. The one-period forward rates extrapolated from those spot rates are f1,1 = 7.0%, f2,1 = 9.0%, f3,1 = 11.1% and f4,1 = 13.1%. Priced off the spot curve, the bond is worth 105.43 per 100 of par and its yield to maturity is 8.62%.
10(1.07)(1.09)(1.111)(1.131) + 10(1.09)(1.111)(1.131) + 10(1.111)(1.131) + 10(1.131) + 110 ≈ 162.22.
The total return over five years is therefore (162.22 − 105.43) ÷ 105.43 = 53.87%, and the annualised rate of return solves (1 + x)5 = 1.5387, giving x = 9.00%.
The yield to maturity was 8.62%, so the two differ even under the generous assumption that forward rates are realised. The yield to maturity of 8.62% would be the expected return only if every coupon were reinvested at 8.62% itself, which is what discounting every cash flow at a single rate implicitly assumes. Reinvesting instead at 7.0%, 9.0%, 11.1% and 13.1% produces 9.00%. The yield to maturity is a realistic estimate of expected return only when the yield curve is flat.
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 DFt+T divided by DFt. Rewrite the forward price at the new date using the shortened remaining terms, substitute the assumed new discount factors, and the common factor DFt cancels from numerator and denominator. What survives is DFB divided by DFA, 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 F2,1 = 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.
Return to z1 = 9%, z2 = 10% and z3 = 11%, with f1,1 = 11.01% and f1,2 = 12.01%. All prices are per 100 of par, and prices and forward rates are rounded to the nearest hundredth, so results are approximate; without rounding every answer would be exact.
One-year zero. Bought at 100 ÷ 1.09 = 91.74 and redeemed at 100. The return is 100 ÷ 91.74 − 1 = 9%.
Two-year zero. Bought at 100 ÷ (1.10)2 = 82.64. One year later it has a year left and is priced off f1,1: 100 ÷ 1.1101 = 90.08. The return is 90.08 ÷ 82.64 − 1 = 9%.
Three-year zero. Bought at 100 ÷ (1.11)3 = 73.12. One year later it has two years left and is priced off f1,2: 100 ÷ (1.1201)2 = 79.71. The return is 79.71 ÷ 73.12 − 1 ≈ 9%.
The two-year zero returns 10%: (100 ÷ 1.10) ÷ [100 ÷ (1.10)2] − 1 = 10%.
The three-year zero returns 13.03%: [100 ÷ (1.10)2] ÷ [100 ÷ (1.11)3] − 1 = 13.03%.
The realised one-year spot curve was below the forward curve, so the longer bonds beat the risk-free 9%. Only the maturity-matched bond is immune.
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.
Assume a flat yield curve at 8%. A trader holds a three-year bond with an 8% annual coupon, priced at 100 on a par value of 100 because coupon and yield coincide. Should the spot curve one year from now land exactly on the forward curve implied today, the position returns 8%.
[8 + 8 ÷ 1.09 + 108 ÷ (1.09)2] ÷ 100 − 1 = 6.24%, which is less than 8%.
[8 + 8 ÷ 1.07 + 108 ÷ (1.07)2] ÷ 100 − 1 = 9.81%, which is more than 8%.
| Future spot rate relative to today’s forward rate | Effect on the forward contract price | Trade indicated |
|---|---|---|
| Equal to the forward rate | Unchanged | None |
| Below the forward rate | Rises, because the payments are discounted at a lower rate than the market assumed | Buy the forward contract |
| Above the forward rate | Falls | Sell the forward contract |
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.
Annual-coupon bonds trading at par offer the following yields.
| Maturity | One year | Three years | Four years | Five years | Six years |
|---|---|---|---|---|---|
| Yield | 2% | 4% | 5% | 6% | 7% |
An investor with a five-year maturity target expects the curve to be unchanged over the next two years. She compares holding the matched-maturity 6% five-year bond against buying the 7% six-year bond, in each case measuring the outcome after two years.
Sale price = 6 ÷ 1.04 + 6 ÷ (1.04)2 + 106 ÷ (1.04)3 = 105.55, using the unchanged three-year yield of 4%.
Total = (6 × 1.02) + 6 + 105.55 = 117.67.
The annualised return solves 117.67 ÷ 100 = (1 + r)2, giving r = 8.476%.
Sale price = 7 ÷ 1.05 + 7 ÷ (1.05)2 + 7 ÷ (1.05)3 + 107 ÷ (1.05)4 = 107.09.
Total = (7 × 1.02) + 7 + 107.09 = 121.23.
The annualised return is 10.10%.
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.
A bank raises funds with two certificates of deposit. The first raises $10 million at 1.5% for two years, the second raises $10 million at 1.70% for three years. The bank then enters two swaps, each on a notional of $10 million: on the first it receives 1.50% fixed and pays MRR minus 10 bps for two years; on the second it receives 1.70% fixed and pays MRR minus 15 bps for three years.
[10 × (−10 bps) + 10 × (−15 bps)] ÷ 20 = MRR minus 12.5 bps on $20 million for the first two years.
The point of the exercise is the last step: once liabilities are expressed as a margin over a floating benchmark, that margin becomes the yardstick against which the whole funding cost is measured and against which asset yields can be judged.
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 sT.
A government spot curve implies the following discount factors: DF1 = 0.9524, DF2 = 0.8900, DF3 = 0.8163 and DF4 = 0.7350.
z1 = (1 ÷ 0.9524)1 − 1 = 5.00%
z2 = (1 ÷ 0.8900)1/2 − 1 = 6.00%
z3 = (1 ÷ 0.8163)1/3 − 1 = 7.00%
z4 = (1 ÷ 0.7350)1/4 − 1 = 8.00%
For T = 1: (s1 + 1) ÷ (1 + 0.05) = 1, so s1 = 5%.
For T = 2: s2 ÷ (1 + 0.05) + (s2 + 1) ÷ (1 + 0.06)2 = 1, so s2 = 5.97%.
For T = 3: s3 ÷ (1 + 0.05) + s3 ÷ (1 + 0.06)2 + (s3 + 1) ÷ (1 + 0.07)3 = 1, so s3 = 6.91%.
For T = 4: adding the fourth term discounted at 8% gives s4 = 7.81%.
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.
An investor is pricing a US$1 million position in GE Capital notes with a coupon of 1 5/8%, that is 1.625%, maturing on 2 July 2024 with semiannual coupons. The evaluation date is 12 July 2021, so the remaining maturity is 2.97 years, computed as 2 + (350/360). Treasury rates are 0.525% for two years and 0.588% for three years. The swap spread for the same maturity is 0.918% and the I-spread is assumed to be zero.
0.525% + (350/360)(0.588% − 0.525%) = 0.586%.
Add the swap spread: 0.918% + 0.586% = 1.504%.
Accrued interest = 1,000,000 × (0.01625/2) × (10/180) = US$451.39.
Clean price = 1,003,954.12 − 451.39 = US$1,003,502.73.
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.
| Measure | Definition | What it gauges |
|---|---|---|
| Swap spread | Swap rate less the on-the-run government yield of the same maturity | Credit risk relative to default-risk-free debt; widens counter-cyclically |
| I-spread | Bond yield to maturity less the interpolated swap rate of the same maturity | Credit and liquidity risk of an individual bond |
| Z-spread | Constant spread over a benchmark spot curve that reprices the bond | Same, but measured against the whole curve rather than one point |
| TED spread | MRR less the matched-maturity Treasury bill yield | Economy-wide credit and liquidity risk |
| MRR–OIS spread | MRR less the overnight indexed swap rate | Bank funding stress in the money market |
Four traditional theories offer economic explanations for the shape of the yield curve. None is complete on its own, and each carries different implications for whether a forward rate can be trusted as a forecast.
Unbiased expectations theory
The oldest of the four, also called pure expectations theory, holds that the forward rate is an unbiased predictor of the future spot rate. Read broadly, it says bonds of every maturity are perfect substitutes: buying a five-year bond and holding it three years has the same expected return as buying a three-year bond, or as rolling three consecutive one-year bonds.
That conclusion requires risk neutrality. In a risk-neutral world uncertainty does not trouble investors and risk premiums do not exist, so every security is effectively risk free and yields the risk-free rate for its maturity. The assumption produces elegant results and conflicts with a large body of evidence that investors are risk averse.
Local expectations theory
A more rigorous relative of the same idea, the local expectations theory does not claim that every maturity strategy earns the same return over a given horizon. It claims something narrower: over short periods, the expected return on every bond is the risk-free rate. That follows from a no-arbitrage condition on bond pricing rather than from an assumption about preferences.
The narrower claim is also more useful, because it survives in a world with risk. Risk premiums are ruled out only for very short holding periods; nothing restricts longer-term investments, so the theory applies to risky bonds as well as risk-free ones. Empirically, though, short-holding-period returns on long-dated bonds routinely exceed those on short-dated bonds. Demand for liquidity and for hedging capacity keeps demand for short-term securities above demand for long-term securities, so both yields and realised returns on short-dated paper tend to be lower.
Liquidity preference theory
Where the expectations theories leave no room for risk aversion, liquidity preference theory is built on it. It asserts that liquidity premiums exist to compensate investors for the additional interest rate risk of lending long, and that those premiums grow with maturity, with each successive period carrying a premium no smaller than the one before. Given an expectation that short-term spot rates will not change, the theory therefore predicts an upward-sloping curve, and it implies that a forward rate is an estimate of the expected spot rate biased upward by the size of the premium. That bias is exactly what invalidates the unbiased expectations theory.
Consider the 30-year Treasury bond. Very few investors have a 30-year horizon. To hold one, they demand extra return for the risk that the curve moves and that they must sell at an uncertain price before maturity. That increment is the liquidity premium, and it should not be confused with a yield concession for a thinly traded bond. It applies to all long-term bonds, including those with deep and active markets.
The theory explains the existence of premiums but does not explain the whole term structure. A downward-sloping curve remains consistent with liquidity premiums if, for example, deflation is expected, or if spot rates are expected to fall sharply enough that the decline outweighs the premiums.
Segmented markets theory
This theory sets expectations and premiums aside entirely and lets the preferences of lenders and borrowers determine the curve. Yields at each maturity are set purely by the supply of and demand for funds at that maturity, and each maturity sector behaves as a separate market whose yield is determined independently of the others.
It describes a world of binding asset and liability management constraints, whether regulatory or self-imposed, in which investors confine themselves to the maturity sector that best matches their liabilities and thereby avoid a mismatch. Life insurers, whose liabilities are long-dated insurance contracts, are most active at the long end. Pension plans, with long-dated liabilities, likewise invest long: investing short would expose them to falling returns while the cost of their liabilities stayed fixed. Money market funds, at the other extreme, are generally limited to debt maturing within one year.
Preferred habitat theory
Preferred habitat theory accepts that borrowers and lenders have strong maturity preferences but rejects the claim that yields at different maturities are set independently. Its distinctive contention is that a large enough expected return advantage will induce institutions to leave their habitat. An intermediate-term bond fund will lengthen its assets if long-term securities offer enough extra expected return; a life insurer will move part of its portfolio into shorter maturities if short-term securities offer enough.
By accepting parts of both the segmented markets theory and the unbiased expectations theory while rejecting the extreme position of each, preferred habitat theory comes closest to describing observed behaviour. Both market expectations and institutional factors shape the term structure.
Quantitative easing seen through preferred habitat
Quantitative easing is an unconventional policy used to expand the money supply when policy and interbank rates are already near zero. The first of several US Federal Reserve programmes began in late 2008, after a near-zero target range for the federal funds rate was established, and successive asset purchase programmes greatly expanded holdings of long-term securities with the aim of pushing long-term rates down and easing financial conditions further.
| Holding | 20 Sep 2007 | 29 Oct 2014 |
|---|---|---|
| Total held outright | 780 | 4,219 |
| Of which US Treasury | 780 | 2,462 |
| T-bills | 267 | 0 |
| Nominal notes and bonds | 472 | 2,347 |
| Inflation-indexed notes and bonds | 36 | 115 |
| Compensation for inflation | 5 | 16 |
| Agency debt | 0 | 40 |
| Mortgage-backed paper | 0 | 1,718 |
The 2014 column is dated at the end of the third round of quantitative easing. Figures are as reported.
In September 2007 the portfolio was entirely US Treasury issuance, and about 34% of it was short term in the form of T-bills. By October 2014 only about 58% of the holdings were Treasury securities, none of them bills, and mortgage-backed securities of well over US$1.7 trillion accounted for 41% of everything held.
Yields on mortgage-backed securities had typically run in the 5% to 6% range before quantitative easing and fell below 2% by the end of 2012. Preferred habitat offers an explanation. Buying mortgage-backed securities reduced the supply available for private purchase. If many investors in that market are unwilling or unable to leave it, because their comparative advantage lies in managing the interest rate and prepayment risks specific to those securities rather than option-free bonds, then they occupy a preferred habitat there. Unable to meet demand without bidding more aggressively, they drove yields down. Prepayment behaviour reinforced the move, since a homeowner who prepays sends the payment through to investors pro rata, and prepayment becomes more likely as rates decline.
| Theory | Core claim | Implication for forward rates | Implied curve shape |
|---|---|---|---|
| Unbiased expectations | Investors are risk neutral; all maturities are perfect substitutes | Unbiased predictors of future spot rates | Any shape, driven purely by rate expectations |
| Local expectations | Expected return over very short periods is the risk-free rate for every bond | Unbiased only over very short horizons | Any shape; extends to risky bonds |
| Liquidity preference | Premiums compensate for the interest rate risk of lending long and rise with maturity | Upwardly biased estimates of future spot rates | Typically upward sloping |
| Segmented markets | Each maturity sector clears independently on its own supply and demand | Carry no information about expectations | Any shape, set by institutional flows |
| Preferred habitat | Maturity preferences exist but yield incentives can overcome them | Biased by habitat effects, not independent of expectations | Any shape; expectations and institutions both matter |
Match each observation to the theory that best explains it.
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.
| Maturity (years) | 0.25 | 0.50 | 1 | 2 | 3 | 5 | 7 | 10 | 20 | 30 |
|---|---|---|---|---|---|---|---|---|---|---|
| σ(t,T) | 0.3515 | 0.3173 | 0.2964 | 0.2713 | 0.2577 | 0.2154 | 0.1885 | 0.1621 | 0.1332 | 0.1169 |
The sample deliberately ends before the 2008 financial crisis, which produced unusual volatility magnitudes.
0.1015 ÷ (1/12)1/2 = 0.3515, that is 35.15%.
Dividing by the square root of one twelfth is the same as multiplying by the square root of twelve, which is the standard square-root-of-time scaling.
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.
| Year | 1 | 5 | 10 |
|---|---|---|---|
| Parallel | 1 | 1 | 1 |
| Steepness | −1 | 0 | 1 |
| Curvature | 1 | 0 | 1 |
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.
0.333(1 + 5 + 10) = 5.333.
1 ÷ [(300)(0.01)] = 0.3333.
The same calculation at the other two maturities gives 5 ÷ [(300)(0.01)] = 1.6667 at five years and 10 ÷ [(300)(0.01)] = 3.3333 at ten years.
The three key rate durations sum to 5.333, which is the effective duration. That is not a coincidence: if every key rate moves by the same amount the curve has shifted in parallel, so the proportional change in value must agree with what effective duration predicts.
Level. By definition this is the parallel sensitivity:
(1 + 5 + 10) ÷ [(300)(0.01)] = 5.3333.
Steepness. An upward steepness shift of 100 bps lowers the one-year rate, producing a gain of $1, and raises the ten-year rate, producing a loss of $10. The change in value is (1 − 10), and the sensitivity is the negative of the proportional change per unit of movement:
−(1 − 10) ÷ [(300)(0.01)] = 3.0.
Curvature. Both ends move up by one unit and the middle is unchanged:
(1 + 10) ÷ [(300)(0.01)] = 3.6667.
−5.3333(−0.0050) − 3.0(0.002) − 3.6667(0.001) = 0.026667 − 0.006 − 0.003667 = +1.7%.
The fall in the level factor dominates, as it usually does.
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
Key rate changes for a portfolio are defined at one, five and ten years. The estimated key rate durations are 0.50, 0.70 and 0.90 respectively.
−0.50(−0.005) − 0.70(−0.005) − 0.90(−0.005) = 0.0105 = +1.05%.
Because the shift is parallel, the same answer follows from summing the key rate durations to 2.10 and multiplying by 0.005.
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.
| View | Position | Reasoning |
|---|---|---|
| Rates will fall | Extend portfolio duration relative to the benchmark | Prices rise as rates fall, and longer duration magnifies the gain |
| Rates will rise | Shorten portfolio duration | Reduces exposure to falling bond prices |
| Curve will steepen, with long rates rising relative to short rates | Short long-term bonds and buy short-term bonds | The long end loses more; the position can be made duration neutral |
| Curve will flatten, with short rates rising relative to long rates | Buy long-term bonds and sell short-term bonds short | The short end loses more; again the position can be made duration neutral |
| Bullish flattening expected, long-only mandate | Shift from a bullet portfolio to a barbell portfolio | A bullet concentrates in a single maturity; a barbell of similar duration combines short and long maturities and gains more when the long end rallies |
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.
A fixed-income analyst advises clients on bond trading opportunities. The economy is currently in recession, the level of government bond yields is low and the term structure is nearly flat. The research team forecasts that after a brief recession, economic growth will return quickly over the coming 12 months.