Rolling Down the Yield Curve: Horizon Return, Carry, and Risk [CFA Level 2 Notes #4]

Rolling down the yield curve means buying a bond whose maturity is longer than the investor’s holding period, then valuing or selling it after it has aged into a shorter maturity. If the relevant yield curve is upward sloping and remains unchanged, the shorter-maturity yield at the horizon can be lower than the bond’s initial yield. That lower horizon yield raises the projected sale price.

The phrase sounds like free return. It is not. The extra projected price gain is conditional on a curve scenario, and the position usually carries more interest-rate, curve-shape, credit-spread, and liquidity risk than a security matched to the holding period.

This article develops the mechanics for education. It is not investment advice, a recommendation to extend maturity, or a claim that any observed curve predicts realized returns.

Start with the horizon price

Suppose a bond is purchased at time 0 and sold after a holding period of $h$ years. Its holding-period return is

$$ HPR=\frac{P_h+I_h-P_0}{P_0}, $$

where:

$P_0$ is the dirty purchase price, including accrued interest;
$P_h$ is the dirty sale price at the horizon; and
$I_h$ is the value at the horizon of coupons received during the holding period, including any assumed reinvestment.

The central task is estimating $P_h$. At the horizon, discount the bond’s remaining cash flows using the rates that will then apply to those remaining maturities:

$$ P_h=\sum_{j:h\lt t_j\le T}\frac{CF_j}{\left(1+y_h(t_j-h)\right)^{t_j-h}}. $$

This compact notation uses one horizon yield for each remaining cash-flow maturity. In practice, the exact exponent and rate convention must match the market’s compounding and day-count rules. A yield-to-maturity shortcut discounts all remaining cash flows at one internal rate; a spot-curve valuation uses the appropriate zero rate for each cash flow.

Roll-down analysis is therefore a horizon-pricing exercise, not simply “initial yield minus shorter yield.” The maturity difference helps estimate the price change, but price is what enters return.

What the curve-static assumption actually says

A common teaching scenario assumes that the curve is unchanged over the holding period. Be precise about the assumption:

Today’s 4-year yield becomes the yield on the seasoned bond when it has four years left.
Today’s 3-year yield becomes the relevant 3-year horizon yield, and so on.
For a credit bond, a complete scenario must also say what happens to its spread, credit quality, liquidity, and embedded options.

“Unchanged curve” does not mean that the purchased bond keeps its original yield. It means the mapping from remaining maturity to yield is unchanged. As time passes, the bond moves to a different point on that mapping.

This is also different from assuming that today’s forward rates become future spot rates. Under no-arbitrage, today’s spot rates determine today’s implied forward rates. A static spot curve instead assigns the current shorter-maturity spot rate to the horizon. Those are different evolution scenarios and generally produce different horizon returns.

The original version of this note said that an upward-sloping spot curve necessarily makes both the YTM and forward curves upward sloping. That was too broad. Spot rates, par yields, coupon-bond YTMs, and forward rates are related but are not interchangeable:

a spot rate prices one cash flow at one maturity;
a par yield is the coupon rate that makes a coupon bond price at par;
a bond’s YTM is one internal discount rate fitted to all of that bond’s cash flows; and
a forward rate is implied by adjacent spot discount factors.

For example, with annual compounding, the one-year forward rate from year $n-1$ to year $n$ is

$$ 1+f_{n-1,n}=\frac{(1+s_n)^n}{(1+s_{n-1})^{n-1}}. $$

Even when spot rates rise with maturity, forward rates need not rise monotonically from one forward period to the next. Coupon effects and bond-specific cash flows also mean that a collection of observed YTMs is not the spot curve.

A recomputable zero-coupon example

Use annual compounding and a face value of $100$. Assume today’s relevant spot rates include:

Maturity	Spot rate
1 year	3.00%
4 years	3.50%
5 years	4.00%

Buy a default-free 5-year zero-coupon bond and hold it for one year.

1. Purchase price

$$ P_0=\frac{100}{(1.04)^5}=82.1927. $$

2. Horizon price under the static curve

After one year, the bond has four years remaining. If the spot curve is unchanged, its horizon discount rate is today’s 4-year spot rate, 3.50%:

$$ P_1^{\text{static}}=\frac{100}{(1.035)^4}=87.1442. $$

There is no coupon, so the one-year holding-period return is

$$ HPR^{\text{static}} =\frac{87.1442-82.1927}{82.1927} =6.0243\%. $$

The one-year zero-coupon alternative earns 3.00%, so this scenario’s excess return over that horizon-matched benchmark is about

$$ 6.0243\%-3.00\%=3.0243\%. $$

That is a scenario result, not an arbitrage profit or a forecast.

3. Separate yield carry from roll-down

“Carry” has several desk conventions, so always state the definition. For this zero-coupon example, define one-year yield carry as the accretion that would occur if the bond continued to earn its initial 4.00% yield for one year:

$$ P_0(1.04)=85.4804. $$

Measured against the original price:

$$ \text{yield carry}=4.0000\%, $$$$ \text{roll-down return} =\frac{87.1442-85.4804}{82.1927} =2.0243\%. $$

Thus,

$$ 6.0243\%=4.0000\%+2.0243\%. $$

This exact decomposition is especially clean because the bond has no coupons. For a coupon bond, the analyst must specify whether “carry” means coupon income, yield accrual, financing-adjusted income, reinvestment income, or some desk-specific combination. Otherwise two correct calculations can appear to disagree merely because they label the components differently.

As a cross-check, the return equals the one-year forward rate implied between years 4 and 5:

$$ f_{4,5}=\frac{(1.04)^5}{(1.035)^4}-1=6.0243\%. $$

That equality follows from the zero-coupon arithmetic. It does not mean the forward rate is a guaranteed forecast of next year’s 4-year spot rate.

Coupon-bond workflow

For a coupon bond, use this sequence:

Choose the exact settlement date and investment horizon.
List every coupon and principal cash flow.
Compute the dirty purchase price using today’s curve and the bond’s spread or other pricing inputs.
State the horizon scenario: static government spot curve, unchanged YTM curve, realized forward curve, a parallel shock, or a nonparallel curve move.
Age the bond and reprice only its remaining cash flows at the horizon.
Add coupons received and an explicit reinvestment assumption.
Subtract financing, transaction costs, taxes, or hedging costs if they are within the analysis scope.
Compare the result with a horizon-matched benchmark under the same conventions.

Using clean price in one part of the calculation and dirty price in another double-counts or omits accrued interest. Likewise, quoting an annualized return requires a stated annualization convention; the raw holding-period return is not automatically an annual rate.

Duration and convexity explain sensitivity, not certainty

The static-curve horizon price is only a base case. If the horizon yield differs from that scenario by $\Delta y$, a conventional option-free bond’s percentage price change can be approximated by

$$ \frac{\Delta P}{P}\approx-D_{\text{mod}}\Delta y+\frac{1}{2}C(\Delta y)^2, $$

where $D_{\text{mod}}$ is modified duration and $C$ is convexity under consistent units.

Duration gives the first-order sensitivity: a longer-duration position usually loses more when yields rise and gains more when yields fall. Convexity adds the curvature of the price-yield relationship. For an ordinary option-free bond, positive convexity makes the gain from a yield decline somewhat larger than the duration-only estimate and the loss from an equal yield increase somewhat smaller.

But this approximation has limits:

a large move makes higher-order effects more important;
a nonparallel shift cannot be represented well by one $\Delta y$;
key-rate durations are more useful when different maturity points move differently; and
callable, putable, and mortgage-related bonds can change cash flows and may exhibit negative convexity.

A projected 2% roll-down contribution can be overwhelmed by a modest adverse rate or spread move when duration is substantial.

The “shoulder effect” is not a universal law

Some fixed-income discussions call a locally steep or curved section near the shorter-to-intermediate part of a yield curve its “shoulder.” Rolling through such a segment can produce attractive projected roll-down when two things coincide:

the yield drop across the holding-period maturity interval is large; and
the bond still has enough price sensitivity for that yield change to matter.

It is not generally true that every short-maturity bond experiences an extreme yield collapse or an unusually large price jump merely because maturity approaches. Very short bonds have low duration, and their prices mechanically converge toward par only if credit and payment expectations remain intact. The most favorable roll-down point, if any, depends on the local curve slope multiplied by relevant price sensitivity, with convexity, financing, and transaction costs also considered.

The curve can be flat, inverted, humped, or kinked. Its apparent shoulder can migrate or disappear before the horizon. Treat “shoulder effect” as informal curve-shape language, not as a separate pricing law.

Risks that the static example suppresses

Level, slope, and curvature risk

Rates can rise or fall, and the move need not be parallel. A steep curve can flatten because long yields fall, short yields rise, or both; those paths have different effects on a seasoned bond. Key-rate or factor scenarios are more informative than a single duration number.

Credit-spread and migration risk

For a corporate or other credit-sensitive bond, the benchmark curve may remain unchanged while the issuer’s spread widens. Deteriorating credit quality can reduce the horizon price or create default loss. A government-curve roll-down calculation that holds credit spread constant is incomplete for such a bond.

Liquidity and transaction-cost risk

The model price is not necessarily executable. Bid-ask spreads, dealer capacity, issue size, security age, market stress, and the cost of financing or hedging can consume the projected advantage. A strategy requiring a sale at a particular horizon is exposed to the liquidity available on that date.

Reinvestment and financing risk

Coupon bonds require an assumption about reinvesting interim cash flows. Leveraged positions also depend on future funding rates and haircuts. Neither is captured by a simple purchase-yield-versus-sale-yield comparison.

Optionality and cash-flow risk

Calls, puts, prepayments, sinking funds, and floating coupons can change both expected cash flows and effective duration as rates move. A static cash-flow schedule is unsuitable when the issuer or borrower can alter the timing of payments.

Model and convention risk

Results vary with the selected curve, interpolation method, compounding, day count, spread measure, clean-versus-dirty pricing, and benchmark. Reproducible analysis records each convention instead of reporting “roll” as a standalone number.

A disciplined interpretation

Roll-down is best understood as one component of scenario-based total return:

$$ \text{horizon return} =\text{income and accrual} +\text{roll-down under the chosen curve scenario} +\text{effect of unexpected rate and spread changes} -\text{costs}. $$

The decomposition is useful because it forces the analyst to expose assumptions. It does not make the static curve likely, convert forward rates into forecasts, or remove risk. The practical question is not “Is the curve upward sloping?” but “What horizon price follows from an explicit curve-and-spread scenario, and how fragile is that price to plausible alternatives?”

Sources and archive note

CFA Institute, The Term Structure and Interest Rate Dynamics distinguishes spot, forward, and yield measures and describes roll-down under an unchanged spot curve.
U.S. TreasuryDirect, Understanding Pricing and Interest Rates explains the relationship among coupon, yield to maturity, par, premium, and discount pricing for Treasury securities.
U.S. SEC, What Are Corporate Bonds? summarizes interest-rate, credit, call, inflation, and liquidity risks for corporate bonds.

These authoritative materials were checked on 2026-09-23. The worked numbers above are deliberately hypothetical and fully recomputable; they are not current market quotes.

The 2024 version of this page was an informal study note. This edition preserves the original route, publication date, series position, author, and the old platform source identity while replacing overgeneralized curve claims and an unreproducible 30-year-bond anecdote with explicit definitions, formulas, assumptions, and risk boundaries.