Yield Curves and Bond Risk: What Rates Reveal and Conceal

Current curve reference · checked 8 October 2026

The US Treasury’s latest daily observation retrieved is 7 October 2026. Selected par yields are 4.77% at two years, 5.03% at five years, 5.28% at ten years and 5.67% at thirty years. These are par yields rather than a bootstrapped zero-coupon curve, so they should not be inserted directly as spot rates into the teaching examples below. The two-to-ten-year slope is 51 basis points, calculated from those published points. US Treasury.

The original five-point curve and bond examples remain explicitly illustrative. The new observed curve gives today’s reader a dated comparison while preserving the original distinction between yield quotations, cash-flow discount factors and maturity-specific risk. A higher long yield does not tell us by itself how much reflects expected short rates, inflation, term premium or market supply.

A bond is a schedule of cash flows before it is a quoted yield. Different cash flows occur at different dates and can carry different discounting, credit and reinvestment assumptions. Reading the yield curve well means keeping those distinctions intact.

This reference brings together curve fundamentals, fixed-income workbook examples and interest-rate futures conventions. Its purpose is to explain instruments and market risk; the retained numerical curves are teaching inputs, not current market quotes.

Three rate concepts serve different purposes

Three rate concepts and their distinct purposes

Equal numerical values do not make these rates interchangeable.

For continuously compounded zero rate $z(T)$, the discount factor for maturity $T$ is

$$D(T)=e^{-z(T)T}.$$

For a bond with cash flows $C_i$ at times $t_i$, present value is

$$P=\sum_i C_iD(t_i).$$

An annual par coupon $c_n$ for unit principal maturing at year $n$ satisfies

$$1=c_n\sum_{i=1}^{n}D(i)+D(n), \qquad c_n=\frac{1-D(n)}{\sum_{i=1}^{n}D(i)}.$$

In words: The coupon rate makes the discounted coupons and final principal equal the initial principal.

The coupon discounts at several maturities. It is therefore generally different from the zero rate at the final maturity.

Bootstrap cash flows rather than relabel quotes

Bootstrapping solves progressively longer discount factors from instruments whose earlier cash flows can already be valued. In the annual par-bond illustration,

$$D(n)=\frac{1-c_n\sum_{i=1}^{n-1}D(i)}{1+c_n}.$$

Real instruments add day counts, settlement lags, stub periods and different reset conventions. The principle remains the same: each quoted instrument should reprice using the curve constructed from it.

A retained workbook deliberately uses the numbers 6.0%, 6.5%, 6.8%, 7.0% and 7.1% once as continuously compounded zero rates and once as annual par-swap quotes. The two sets of labels define different curves even though the digits match. Treating them as the same inputs would obscure the very distinction the exercise is designed to teach.

Figure 01 · Conceptual framework

From a market quote to a valued cash flow

How the parts connect

1. AMarket quotationsinterpretContract conventions
2. BContract conventionsconstructDiscount factors
3. CDiscount factorsvalueBond value
4. DDiscount factorstest input instrumentsRepricing check
5. ERepricing checkresolve discrepanciesContract conventions

A curve can change shape within its maturity range

The retained five-point teaching curve is:

The retained teaching curve by maturity — Maturity: Spot rate

It declines through five years and then rises. Calling the entire curve simply “inverted” loses information. Short, middle and long maturities can move for different reasons.

Under an expectations interpretation, longer rates reflect expected future short rates plus premia and other influences. An inverted segment can be consistent with expected easing, but the curve is not a mechanically reliable recession forecast. Term premia, supply, hedging demand, liquidity and market segmentation can all affect its shape.

Government, overnight-indexed and credit-sensitive curves also represent different instruments and risks. A government curve is not universally risk-free in every currency or jurisdiction. A modern collateralised swap curve should not be described as carrying exactly the same unsecured bank-credit exposure as a legacy interbank benchmark.

Forward rates are implied prices, not guaranteed outcomes

For continuously compounded rates over $[T_1,T_2]$,

$$f_{1,2}=\frac{z(T_2)T_2-z(T_1)T_1}{T_2-T_1}.$$

Equivalently, it is the negative log discount-factor ratio divided by the interval length. With simple accrual fraction $\delta$, the corresponding forward rate is $F=(D(T_1)/D(T_2)-1)/\delta$.

The choice of compounding determines the numerical quote. A forward is consistent with today’s discounting relationships; interpreting it as an unbiased forecast of the future realised rate requires additional assumptions about risk premia and measure.

Interpolation affects forwards between quoted dates. Log-linear discount factors produce piecewise constant instantaneous forwards, which can jump at knots. Their simplicity should not be confused with universal smoothness or realism.

Bond yield compresses a cash-flow stream into one number

Yield to maturity is the single rate that reproduces the bond price under a specified compounding convention. It is convenient for quoting and comparison, but it does not mean every cash flow is economically discounted at one independently observed market rate.

A retained workbook example has a three-year bond, semiannual coupon of 2 per 100 principal, and bond-equivalent annual yield of 3.5%. Discounting six cash flows at 1.75% per half-year gives price approximately 101.41225. The final payment is 102, combining coupon and principal.

The bond trades above par because its 4% annual coupon exceeds the quoted yield under the stated convention. Its total holding-period return still depends on sale price, interim coupon reinvestment and any credit or liquidity changes.

The same workbook illustrates a 10-year 8% coupon bond quoted near 104.19 at a 7.4% yield, alongside alternative constant-spread discounting calculations. These are teaching comparisons. A quoted yield spread and a constant spread over a zero curve are different measures and should not be equated without matching the valuation setup.

Duration is a local sensitivity

Macaulay duration is the present-value-weighted average payment time. Modified duration converts it into sensitivity to a change in the quoted yield. For a small yield change $\Delta y$,

$$\frac{\Delta P}{P}\approx-D_{\mathrm{mod}}\Delta y +\frac12\mathcal C(\Delta y)^2,$$

In words: The percentage price change combines the first response to a yield move with an adjustment for its curvature.

where $D_{\mathrm{mod}}$ is modified duration and $\mathcal C$ is convexity under the same yield convention.

DV01 is the approximate currency price change for a one-basis-point yield move, often reported as a positive magnitude for a long fixed-rate bond. Systems can instead report signed sensitivity; the convention should be stated.

Duration captures a local move in a specified rate. A portfolio exposed to several maturities needs key-rate sensitivities or a fuller curve-risk representation. Two portfolios with the same aggregate duration can respond differently to steepening or a move concentrated in the curve’s middle.

Figure 02 · Qualitative comparison

One curve, three different rate concepts

One curve, three different rate concepts

Rate concept	What it describes	Interpretation limit
Spot / zero rate	What it describesDiscounting a single future payment	Interpretation limitNot the coupon of a multi-payment bond
Par yield	What it describesCoupon that values the specified bond at par	Interpretation limitNot necessarily its final-maturity zero rate
Forward rate	What it describesRate implied between future dates by current discount factors	Interpretation limitNot a guaranteed future realised rate

Level, slope and curvature are useful summaries

How the curve changes shape

Principal component analysis can estimate analogous statistical factors from a chosen history.

The factors depend on the data window, rate representation and scaling. A first principal component often resembles a level move, but the percentage of variance explained is an empirical result, not a universal constant. Historical factors also do not exhaust plausible future stress scenarios.

Portfolio hedging should therefore include both factor summaries and maturity-specific exposures. A compact factor hedge can be efficient while leaving basis or local-curve risks that require separate attention.

Futures add settlement and delivery conventions

A short-rate futures quote often uses a price convention of 100 minus a percentage rate. The retained workbook quote of 93.5 corresponds to 6.5% under that convention. It is a quote transformation, not a complete present-value formula.

Futures are settled through daily variation margin, while a forward generally settles according to its contract at a later date. When rates are stochastic, the timing of gains and financing creates a convexity difference. The adjustment depends on the model and contract; a futures-implied rate should not automatically be inserted as an identical forward rate.

Bond futures can include delivery choices and conversion factors. The cheapest-to-deliver bond, financing rate and delivery option affect the relationship between the futures price and a simple forward on one chosen bond. Hedging by nominal amounts alone can therefore leave conversion-factor and basis risk.

Reinvestment and pull to par complete the return picture

If a conventional bond survives to maturity and contractual payments are made, its price approaches redemption value as the remaining life shrinks, conditional on the relevant yield path. That pull to par is only one component of return.

Coupon reinvestment matters more for higher coupons and longer horizons. The retained workbook contrasts reinvestment-income contributions across 2-, 3-, 5-, 8- and 15-year bonds and several coupon rates. The contribution grows with the time available to reinvest; its realised amount depends on future reinvestment rates.

A fixed-income analysis should therefore specify cash flows, discounting curve, quoting convention, curve sensitivities, credit exposure and reinvestment assumptions. The market view becomes more useful when it identifies which of these components is expected to change instead of treating “the yield” as the entire investment.

Figure 03 · Conceptual framework

Why one duration number misses a curve twist

How the parts connect

1. ACurve shockshort-end changeShort maturities
2. BCurve shocklong-end changeLong maturities
3. CShort maturitiesweighted effectPortfolio impact
4. DLong maturitiesweighted effectPortfolio impact
5. EPortfolio impacttest residual riskHedge mismatch

Clean price and dirty price answer different questions

The full settlement amount for a conventional coupon bond is its dirty price. A clean quote removes accrued coupon interest under the market’s day-count and coupon conventions. Comparing a clean market quote with a discounted-cash-flow value that includes accrued interest produces an apparent pricing discrepancy.

Accrual is not simply the fraction of a calendar year since the last payment under every convention. Actual-day, 30/360 and other methods can differ. Settlement lag also changes which cash flows and accrued amount belong in the transaction.

The retained workbook includes fractional-period valuation and an accrued-interest adjustment, illustrating that pricing between coupon dates requires more than discounting an integer number of periods. A sound calculation records settlement date, previous and next coupon dates, day count and whether the quoted price is clean or dirty.

Duration and reinvestment risk can offset over a horizon

When yields rise, a fixed-rate bond’s immediate market price generally falls, while future coupons can be reinvested at higher rates. When yields fall, the price effect reverses and reinvestment income declines. The relative importance depends on the investor’s holding horizon and the cash-flow schedule.

Classical immunisation uses duration matching to offset these effects locally under restrictive assumptions about yield changes and reinvestment. It is not an exact hedge against arbitrary curve shifts, credit changes or changing liability cash flows.

Convexity and key-rate mismatches matter even when aggregate duration matches. Rebalancing may be necessary as time passes and yields change. A long-term liability should therefore be compared with the asset portfolio’s full cash-flow and curve-risk structure, not only a single duration number.

A curve move can leave the average yield unchanged

Consider an illustrative two-point exposure in which the short rate rises 20 basis points and the long rate falls 20 basis points. Their unweighted average change is zero. A portfolio concentrated at the long maturity can still gain materially, while one concentrated at the short maturity can lose.

A duration-weighted or key-rate analysis resolves the difference. If sensitivity at the two points has opposite signs, the same twist can create an even larger net effect than a parallel shift. This is why a single “rates unchanged on average” statement may be uninformative for a particular book.

Curve-risk reports should specify the bump construction. Raising one quoted instrument and rebuilding the curve is different from bumping one interpolated zero-rate node while holding all others fixed. Both can be useful, but they measure different sensitivities.

Spread measures should match the contract

Spread measures answer different valuation questions

These measures can differ substantially for callable or prepayable instruments.

An embedded option changes how cash flows respond to rates. A callable bond may appreciate less when rates fall because the issuer can refinance. A mortgage investor may receive principal back earlier when reinvestment opportunities are less attractive. A fixed-cash-flow duration can therefore misstate the effective sensitivity.

The appropriate market conclusion identifies whether the opportunity concerns the base curve, credit spread, liquidity, financing basis or embedded option. Clear decomposition makes both the expected return and the residual risk easier to assess.