The term structure: one date, many maturities
Why does lending for ten years command a different rate from lending for one year? The term structure of interest rates describes the relationship, observed at a single point in time, between interest rates and maturity for comparable instruments. Its graphical representation is a yield curve.
For precise valuation, the fundamental curve is the zero-coupon or spot curve: each maturity has its own discount rate. A curve of coupon-bond yields to maturity is useful for quotation but is not identical to the spot curve.
A curve is a snapshot across maturities. A time series follows one maturity across different observation dates.
Four rates—and the discount factor
| Concept | Meaning | Principal use |
|---|---|---|
| Spot rate yt | Rate today for a single payment at time t | Discount an individual cash flow |
| Forward rate ft,T | Rate fixed today for the interval from t to T | Price future borrowing or lending |
| Yield to maturity z | Single IRR equating a bond’s cash flows to its price | Quote and compare bonds |
| Par yield cn | Coupon rate making an n-year bond worth par | Describe hypothetical par instruments |
All worked examples assume annual coupons, annual effective compounding, valuation on a coupon date and face value 100. Figures are illustrative, not live quotations.
Zero-coupon price = Face value × D(t)
Coupon-bond price = Σ CFtD(t)
Price = Σ CFt / (1 + z)t
cn = [1 − D(n)] / Σ D(t)
The par-yield formula above assumes annual payments at years 1 through n. YTM is a summary IRR; it does not supply a separate market discount rate for each coupon. Earning that YTM as a compound realised return requires holding to maturity, contractual payment and reinvesting coupons at that yield.
Reading the shapes of the yield curve
| Shape | Pattern | Possible explanation |
|---|---|---|
| Upward sloping | Long yields exceed short yields | Expected rate increases, positive term premium, or both |
| Inverted | Short yields exceed long yields | Expected easing, strong demand for long bonds, or lower term premia |
| Flat | Similar yields across maturities | Stable expectations, or offsetting expectations and premia |
| Humped | Middle maturities yield most | Expected tightening followed by easing, or maturity-specific supply and demand |
Why curves take different shapes: four theories
| Theory | Core idea | Interpretation and limitation |
|---|---|---|
| Pure expectations | In its simple teaching form, forwards are unbiased expectations of future short rates; there is no term premium. | Rising forwards suggest expected tightening. The assumption overlooks changing compensation for risk. |
| Liquidity preference | Investors prefer shorter commitments and require a positive premium for longer exposure. | An upward slope can exist with stable expected short rates. A sufficiently large expected fall in short rates can still produce inversion. |
| Market segmentation | Supply and demand within maturity buckets largely set their yields. | Explains local humps and institutional demand. Strict separation understates substitution and relative-value trading. |
| Preferred habitat | Investors have preferred maturities but move elsewhere for adequate compensation. | Combines maturity preferences with substitution. Habitat demand and required premia are difficult to estimate separately. |
A pension fund matching distant payments may prefer long bonds; a treasury desk managing near-term cash may prefer short instruments. Such preferences can influence yields alongside expectations.
This is an explanatory approximation, not the exact compounded no-arbitrage identity. The term premium is unobservable and model-estimated; in broader modern models it need not always be positive. Avoid adding an unexplained liquidity premium directly to a compounded accumulation factor.
Spot-to-forward rates: the no-arbitrage bridge
Compare investing to T directly with investing to t and simultaneously locking the remaining interval at a forward rate. With matched cash flows and consistent conventions, both strategies must have the same accumulation.
ft,T = [(1 + yT)T / (1 + yt)t]1/(T−t) − 1
ft,T = [D(t) / D(T)]1/(T−t) − 1
No-arbitrage links today’s tradable prices. Equating a forward rate to an expected future realised rate requires an additional expectations assumption. Rolling into an unknown future spot rate is not a locked investment strategy.
Worked example: the one-year rate starting next year
Given y1 = 5.00% and y2 = 5.80%:
= 1.119364 / 1.05 − 1
= 6.6061% ≈ 6.61%
Investing 100 for two years produces 111.9364. Investing for one year produces 105; lending that amount for the next year at the locked forward rate also produces 111.9364.
If an executable equivalent forward rate were 7%, an idealised arbitrage would borrow 100 for two years, invest for one year, and lock forward lending of 105 at 7%. At year two, 112.35 is received against 111.9364 owed: a surplus of 0.4136 before costs. This assumes frictionless access, matching credit exposure and no funding constraints.
Bootstrapping: recover spot rates from coupon bonds
Coupon-bond prices contain information about several payment dates. Bootstrapping strips away cash flows already valued using shorter discount factors, leaving the next unknown factor.
| Maturity | Annual coupon | Price | Final payment |
|---|---|---|---|
| 1 year | 4.00 | 100 | 104.00 |
| 2 years | 5.00 | 100 | 105.00 |
| 3 years | 5.75 | 100 | 105.75 |
Step 1 · Solve the one-year factor
D(1) = 100/104 = 0.961538462
y1 = D(1)−1 − 1 = 4.0000%
Step 2 · Strip the first coupon from the two-year bond
D(2) = [100 − 5(0.961538462)]/105
= 0.906593407
y2 = D(2)−1/2 − 1 = 5.025249%
Step 3 · Strip both earlier coupons from the three-year bond
100 = 5.528846154 + 5.212912088 + 105.75D(3)
D(3) = 89.258241758 / 105.75 = 0.844049567
y3 = D(3)−1/3 − 1 = 5.814215%
Retain full precision internally and round only displayed answers. These three securities reprice to 100 using the derived factors.
Practise it in Excel with Goal Seek
A practical way to bootstrap a real-looking curve: six government securities (two T-bills and four semi-annual G-secs). For each bond, price = PV of its cash flows; the earlier spot rates are already known, so Excel’s Goal Seek finds the one unknown rate. Each bond block tells you exactly which cell to set and which to change, and an answer key checks your six spot rates.
Par, spot and forward curves compared
| Maturity | Par yield | Spot rate | One-year interval | Forward rate |
|---|---|---|---|---|
| 1 year | 4.0000% | 4.0000% | Today → Year 1 | 4.0000%* |
| 2 years | 5.0000% | 5.0252% | Year 1 → Year 2 | 6.0606% |
| 3 years | 5.7500% | 5.8142% | Year 2 → Year 3 | 7.4100% |
*The first interval starts today, so its rate is the one-year spot rate.
f2,3 = D(2)/D(3) − 1 = 7.409972%
In this example, the rising forward rates pull successive spot rates upward, while earlier low-rate coupons keep par yields below corresponding spot yields. This ordering is not a universal rule for every possible curve.
Apply the curve to price a different bond
Consider a three-year bond paying a 6% annual coupon with face value 100. Discount each payment at its own maturity factor:
| Year | Cash flow | Discount factor | Present value |
|---|---|---|---|
| 1 | 6 | 0.961538462 | 5.769231 |
| 2 | 6 | 0.906593407 | 5.439560 |
| 3 | 106 | 0.844049567 | 89.469254 |
The bond trades above par because its 6% coupon exceeds the three-year par coupon of 5.75%. Applying the three-year spot rate to every coupon would ignore the term structure.
Interactive forward-rate laboratory
Change annual effective spot rates and dates. The calculator derives the implied annual effective forward rate.
Implied forward rates from simple money-market quotations
Money-market deposit rates are often quoted as annualised simple-interest rates. For a deposit lasting d days on a basis B, the maturity value of 1 is 1 + R × d/B. This differs from the annual-effective accumulation (1 + y)t used earlier. Identify the instrument’s quotation and day-count convention before calculating.
1. Define the three accrual fractions
| Symbol | Meaning |
|---|---|
| Rt, RT | Annualised simple spot deposit rates from today to t and T |
| αt, αT | Accrual fractions from today to each maturity |
| αt,T | Accrual fraction for the future interval t to T |
| Ft,T | Annualised simple forward rate for that interval |
With one consistent Actual/B convention: αt = dt/B, αT = dT/B and αt,T = (dT − dt)/B. Use actual calendar-day counts between the applicable dates, rather than assuming every month has 30 days.
2. Derive the no-arbitrage forward rate
Invest directly from today to T, or invest to t and lock the reinvestment rate from t to T today. Equal final cash flows require:
Ft,T = {[(1 + RTαT)/(1 + Rtαt)] − 1}/αt,T
D(t) = 1/(1 + Rtαt)
Ft,T = [D(t)/D(T) − 1]/αt,T
The ratio determines the growth factor over the forward interval. Subtract 1 to obtain the interval return; divide by its accrual fraction to annualise it on a simple basis.
3. Worked example: 90 days to 180 days on Actual/360
| Input | Value |
|---|---|
| 90-day simple spot rate | 5.00% p.a. |
| 180-day simple spot rate | 5.80% p.a. |
| Day-count basis | Actual/360 |
| Forward interval | Day 90 to Day 180: 90 days |
| Initial investment | $10,000,000 |
This resembles a “3 × 6” forward illustration, but 90 and 180 days are explicit teaching assumptions. Actual 3-month and 6-month maturities need date and business-day adjustments.
αt = 90/360 = 0.25
αT = 180/360 = 0.50
αt,T = 90/360 = 0.25
Step B — Spot accumulation factors
A(t) = 1 + 0.05 × 0.25 = 1.0125
A(T) = 1 + 0.058 × 0.50 = 1.0290
Step C — Forward accumulation and interval return
A(T)/A(t) = 1.0290/1.0125 = 1.0162962963
Forward-period return = 1.0162962963 − 1 = 1.62962963%
Step D — Annualise on Actual/360
F = 0.0162962963/0.25
= 0.0651851852 = 6.51851852% p.a.
4. Reconcile both investment strategies
| Step | Calculation | Amount |
|---|---|---|
| Direct investment to Day 180 | $10,000,000 × 1.029 | $10,290,000 |
| Investment matures on Day 90 | $10,000,000 × 1.0125 | $10,125,000 |
| Interest on that amount for the forward interval | $10,125,000 × 6.51851852% × 90/360 | $165,000 |
| Rolled investment at Day 180 | $10,125,000 + $165,000 | $10,290,000 |
| Difference between strategies | $10,290,000 − $10,290,000 | $0 |
The forward reinvestment principal here is $10,125,000, because it includes the first deposit’s interest. It is not the original $10,000,000. A forward loan with a separately specified notional has the same implied rate, but different dollar interest.
5. Actual/365: two different comparisons
A. Recompute while keeping the numerical spot quotations at 5.00% and 5.80%. Now both spot growth factors change:
αT = 180/365 = 0.4931506849
αt,T = 90/365 = 0.2465753425
A(t) = 1 + 0.05 × 90/365 = 1.0123287671
A(T) = 1 + 0.058 × 180/365 = 1.0286027397
F = [(1.0286027397/1.0123287671) − 1] × 365/90
= 6.51962111% p.a. on Actual/365
B. Re-express the original Actual/360 forward on Actual/365 while preserving its cash flows. The forward-period return remains unchanged, so:
F365 = F360 × 365/360
= 6.51851852% × 365/360
= 6.60905350% p.a.
These answers differ because comparison A changes the underlying spot accumulation factors; comparison B changes only the quotation basis. To preserve all the original spot cash flows as well, convert both spot quotes by the same 365/360 factor before recomputing.
6. Additional illustration in rupees: 91 to 182 days
Assume simple spot rates of 6.50% for 91 days and 7.00% for 182 days, both on Actual/365. These are hypothetical rates, not current Indian market quotations.
A(182) = 1 + 0.070 × 182/365 = 1.0349041096
F91,182 = [(1.0349041096/1.0162054795) − 1] × 365/91
= 7.38039713% p.a.
₹1 crore invested to Day 91 → ₹10,162,054.79
Forward-period interest → ₹186,986.30
Rolled amount on Day 182 → ₹10,349,041.10
Direct investment: ₹10,000,000 × 1.0349041096 → ₹10,349,041.10
7. Try it in Excel
The whole calculation is one line:
Forward = [ (1 + Rlong × tlong) ÷ (1 + Rshort × tshort) − 1 ] × basis ÷ forward days
Here t = days ÷ basis (360 or 365). If you prefer months, use t = months ÷ 12 and finish with × 12 ÷ forward months.
| Example | Inputs | Plug in | Forward |
|---|---|---|---|
| Act/360 | 5.00% for 90 days, 5.80% for 180 days | [(1 + 0.058 × 180/360) ÷ (1 + 0.05 × 90/360) − 1] × 360/90 | 6.5185% |
| Months | 5.00% for 3 months, 5.80% for 6 months | [(1 + 0.058 × 6/12) ÷ (1 + 0.05 × 3/12) − 1] × 12/3 | 6.5185% |
| Act/365 (₹) | 6.50% for 91 days, 7.00% for 182 days | [(1 + 0.07 × 182/365) ÷ (1 + 0.065 × 91/365) − 1] × 365/91 | 7.3804% |
A full money-market curve. Apply the same formula to each pair of neighbouring tenors (Act/365, illustrative rates):
| Tenor | Days | Simple rate | Growth 1 + R × d/365 | Forward period | Forward days | Implied forward |
|---|---|---|---|---|---|---|
| 1M | 30 | 5.00% | 1.004110 | Today → 1M | 30 | 5.0000% |
| 3M | 91 | 5.20% | 1.012964 | 1M → 3M | 61 | 5.2767% |
| 6M | 182 | 5.50% | 1.027425 | 3M → 6M | 91 | 5.7258% |
| 9M | 273 | 5.70% | 1.042633 | 6M → 9M | 91 | 5.9372% |
| 12M | 365 | 5.90% | 1.059000 | 9M → 12M | 92 | 6.2280% |
Example, 3M → 6M: (1.027425 ÷ 1.012964 − 1) × 365/91 = 5.7258%. Because the curve slopes upward, each forward sits above the deposit rate for the longer tenor.
In Excel: the Companion’s Money Market sheet has exactly these inputs (short rate, short days, long rate, long days, basis) and one formula: =((1+C13*C14/C15)/(1+C11*C12/C15)-1)*C15/C17, where C17 = long days − short days. Below it, the same sheet builds this five-tenor curve with its implied forwards and a chart; change any rate or day count to see the forwards move.
8. Common errors to avoid
- Using the annual-compounding formula directly on simple deposit quotations.
- Confusing the annualised forward quote with the return for the forward interval.
- Subtracting spot rates or omitting the shorter deposit’s accumulation factor.
- Mixing Actual/360 and Actual/365 without preserving or deliberately changing the underlying cash flows.
- Treating a bank-discount yield as an investment yield. For discount quote d, price is Face × (1 − dα); the corresponding simple investment yield is d/(1 − dα).
- Assuming implied forwards forecast future spot rates without a risk-premium assumption.
Method reference: CME Group, Eurodollar Futures: Foundational Concepts (2018), implied-forward arithmetic. This historical source supports the calculation method, not the use of a discontinued benchmark for new business.
Curve movements and practical applications
| Movement | What changes? | Risk implication |
|---|---|---|
| Parallel shift | All maturities change by the same basis points | Duration and convexity provide useful approximations |
| Steepening | Long-minus-short spread increases | A duration-matched portfolio can still experience a curve loss |
| Flattening | Long-minus-short spread decreases | Exposure depends on which maturities move |
| Twist or curvature change | Different sections move differently | Key-rate sensitivities help locate exposure |
Example: short and long yields of 5% and 6% imply a 100 bp spread. If they move to 4.5% and 5.8%, the spread widens to 130 bp: a bull steepening. Both yields fell, but the short yield fell more.
Banks: the curve influences asset and liability repricing, net interest income and economic value of equity. “Borrow short, lend long” is only a starting description; deposit behaviour, hedges and repricing dates matter.
Investors: curves support cash-flow valuation, liability matching, relative-value analysis and bullet/barbell/ladder decisions. Higher long yields do not by themselves make long bonds more attractive: duration, horizon and reinvestment risk also matter.
Derivatives: discount factors and forwards underpin FRA and swap pricing. In collateralised markets, discounting and reference-rate projection may use different curves; the single-curve examples here establish the foundations.
Excel implementation and quotation conventions
Excel Companion: build and explore the term structure
A formatted, formula-driven workbook with ten sheets: Overview, Bootstrap, Bootstrap Steps, Non-Par Case, Forward Lab, Bond Value, Money Market, Worked Examples, Practice Q&A and Concept Q&A. It adds four extra worked examples (a 5-year par bootstrap, the coupon effect, pure expectations and rate conventions), ten practice questions with instant ✓/✗ checking, and ten concept questions with model answers. The Money Market sheet adds simple-interest forwards with editable day counts and a cash-flow reconciliation.
Yellow cells with blue numbers are inputs. Change coupon rates, market prices or spot rates and follow the results through to bond valuation.
Bootstrapping in plain words
The whole idea in one line: each bond's price = coupons you can already value + one final payment you cannot. Subtract the part you know, then divide by the final payment. That gives the next discount factor.
| Bond | ① Price | ② Subtract what you know | ③ Divide by final payment | Discount factor |
|---|---|---|---|---|
| 1-year | 100 | nothing to subtract | 100 ÷ 104 | D(1) = 0.961538 |
| 2-year | 100 | 5 × 0.961538 = 4.807692 | (100 − 4.807692) ÷ 105 | D(2) = 0.906593 |
| 3-year | 100 | 5.75 × (0.961538 + 0.906593) = 10.741758 | (100 − 10.741758) ÷ 105.75 | D(3) = 0.844050 |
Then two short conversions turn discount factors into rates:
| To get | Rule in words | Year 1 | Year 2 | Year 3 |
|---|---|---|---|---|
| Spot rate | Undo the discounting: D−1/t − 1 | 4.0000% | 5.0252% | 5.8142% |
| 1-year forward | Earlier D ÷ later D, minus 1 | 4.0000% | 6.0606% | 7.4100% |
Reading the workbook formula. In the Bootstrap sheet, cell F9 holds =(D9-E9*SUM(F$8:F8))/($C$13+E9). In words:
( Price − Coupon × sum of the discount factors already found ) ÷ ( Face value + Coupon )
The anchored range SUM(F$8:F8) grows as you copy the formula down. In row 10 it becomes SUM(F$8:F9), so the 3-year bond automatically subtracts both earlier coupons. That is the only trick in the sheet. The same steps are worked by hand in section 7.
In the workbook: the Bootstrap Steps sheet lays out exactly the columns in the tables above (price, known part, what is left, final payment, discount factor, then spot and forward rates), one plain step per cell, with check columns showing it matches the Bootstrap sheet.
Prices need not be 100. The same three steps work for bonds priced above or below par: use the market price in step ①. For example, at prices of 100.50, 99.80 and 101.20 the discount factors become 0.966346, 0.904460 and 0.855252, and the spot rates 3.4826%, 5.1491% and 5.3502%. A bond above par has a spot rate below its coupon, and a bond below par the reverse. The Companion’s Non-Par Case sheet works this example step by step.
Show the exact workbook cells and formulas (for tracing the Bootstrap and Bond Value sheets)
The worksheet excerpts below use the same sheet names, cell addresses and formulas as the downloadable workbook. Values shown are the base-case results.
| B | C | D | E | F | G | H | |
|---|---|---|---|---|---|---|---|
| 7 | Years | Coupon rate | Market price | Coupon amount | Discount factor | Spot rate | 1Y forward |
| 8 | 1 | 4.0000% | 100.000000 | 4.000000=$C$13*C8 | 0.961538462=D8/($C$13+E8) | 4.0000%=F8^(-1/B8)-1 | 4.0000%=G8 |
| 9 | 2 | 5.0000% | 100.000000 | 5.000000=$C$13*C9 | 0.906593407=(D9-E9*SUM(F$8:F8))/($C$13+E9) | 5.0252%=F9^(-1/B9)-1 | 6.0606%=F8/F9-1 |
| 10 | 3 | 5.7500% | 100.000000 | 5.750000=$C$13*C10 | 0.844049567=(D10-E10*SUM(F$8:F9))/($C$13+E10) | 5.8142%=F10^(-1/B10)-1 | 7.4100%=F9/F10-1 |
Face-value input: Bootstrap!C13 = 100. Columns B–E supply maturity, coupon rate, market price and coupon amount. Discount factors are in F8:F10; spot rates in G8:G10; one-year forwards in H9:H10. H8 equals the one-year spot rate because that interval starts today.
| B | C | D | E | F | G | |
|---|---|---|---|---|---|---|
| 12 | Year | Coupon | Principal | Cash flow | Discount factor | Present value |
| 13 | 1 | 6.000000=$E$8*$E$9 | 0.000000=0 | 6.000000=C13+D13 | 0.961538462=Bootstrap!F8 | 5.769231=E13*F13 |
| 14 | 2 | 6.000000=$E$8*$E$9 | 0.000000=0 | 6.000000=C14+D14 | 0.906593407=Bootstrap!F9 | 5.439560=E14*F14 |
| 15 | 3 | 6.000000=$E$8*$E$9 | 100.000000=$E$8 | 106.000000=C15+D15 | 0.844049567=Bootstrap!F10 | 89.469254=E15*F15 |
Bond inputs: 'Bond Value'!E8 = 100 (face value) and 'Bond Value'!E9 = 6% (annual coupon). The price in 'Bond Value'!G18 is =SUM(G13:G15), giving 100.678045.
Reprice the original bonds
| Workbook cell | Exact formula | Base-case value |
|---|---|---|
| Bootstrap!D20 | =($C$13+E8)*F8 | 100.000000 |
| Bootstrap!D21 | =E9*SUM(F$8:F9)+$C$13*F9 | 100.000000 |
| Bootstrap!D22 | =E10*SUM(F$8:F10)+$C$13*F10 | 100.000000 |
Repricing differences are in Bootstrap!E20:E22 and equal zero in the base case. The workbook’s percentage formats apply to G8:H10.
Match the quotation convention before calculating
Continuous compounding: D(t) = e−r(t)t
Continuous forward: fct,T = [r(T)T − r(t)t]/(T−t)
Simple money-market forward: F = [D(t)/D(T) − 1]/δ
Here j is a nominal annual rate, r a continuously compounded zero rate, and δ the applicable accrual fraction. Real-market bootstrapping also requires settlement dates, accrued interest, coupon schedules, day counts and interpolation between available instruments. Treasury securities in actual markets need not pay annual coupons.
Check your understanding and explore further
Does default-free mean a bond cannot lose market value?
No. A default-free bond can decline in price when market discount rates rise. Its nominal promised payment and its current market price are different concepts.
Is an implied forward rate a guaranteed future spot rate?
No. It is a rate derived from today’s prices for a future interval. Future realised spot rates remain uncertain.
Can liquidity preference coexist with an inverted curve?
Yes. Expected short-rate declines can outweigh a positive maturity premium.
Why are the par and spot rates equal at one year in this case?
The one-year bond has only one payment date, so its coupon and principal are discounted using the same one-year factor.
How can you validate a bootstrapped curve?
Reprice every input instrument from its cash flows using the derived discount factors. Residuals should be zero apart from rounding or a deliberately fitted estimation error.
Continue: Mastering Spot Rates vs Forward Rates →
Explore the Barbell Strategy of Bonds →
Return to Fixed Income Securities →
References and teaching basis
Adapted and expanded from the supplied notes, with numerical calculations independently recomputed. Additional reference: Moorad Choudhry, An Introduction to Banking Liquidity Risk and Asset-Liability Management, Chapter 3, “The Yield Curve”, especially printed pp. 71–79.
- Bank of England: Yield curve terminology and concepts
- Federal Reserve: Expectations and term-premium decomposition
Search concepts: term structure · yield curve · spot rate · zero curve · forward rate · par yield · expectations theory · liquidity preference · market segmentation · preferred habitat · bootstrapping · discount factor · ALM.