Calculate carry roll-down under the interest rate assumptions most commonly made for it, and outline what those assumptions are.
The idea
Carry roll-down depends on a path assumed for rates over the horizon, and two assumptions are standard. First, the term structure is unchanged. Each spot rate stays where it is, so the bond ages and is repriced on today's curve at its shorter maturity. Carry roll-down return = (price at horizon + coupons received − today's price) ÷ today's price. Subtracting the financing cost gives the net figure. On an upward-sloping curve roll-down is positive. On an inverted curve it is negative. Second, forward rates are realized: the future curve equals today's implied forwards, so every bond earns exactly the short-term financing rate and net carry roll-down is zero. The gap between the two is the view that forwards will not be realized. Frequent mistakes are forgetting the coupons or the financing cost, and confusing the YTM with the horizon return.
Formulas
Carry roll-down return over a horizon
CRD = (Ph + C − P0) ÷ P0
- Ph
- the bond's price at the horizon under the assumed rate path
- C
- coupons received over the horizon
- P0
- today's price
Under an unchanged curve Ph uses today's spot rates at the shorter maturity. Subtract the financing rate for the net figure, which is zero if forwards are realised.
A practice question
Which two assumptions about future interest rates are most commonly used when computing carry roll-down?
- The term structure stays unchanged, or the forward rates are realized. - correct
- Rates shift in parallel by one standard deviation up or down.Why not: These are risk-scenario shocks for measuring rate risk, not carry assumptions.
- Rates mean-revert to a long-run level, or follow a random walk with drift.Why not: These are stochastic rate models; carry roll-down uses deterministic paths for the curve.
- The curve is flat, or it is perfectly inverted.Why not: Curve shape is an input observed in the market, not an assumption about how rates evolve.
Carry roll-down needs a path for the curve over the horizon. The two usual choices are an unchanged curve (the bond just rolls down it) and forwards realized (the future curve equals today's implied forwards).
A harder one
A 3-year bond with face value 100 pays a 5% annual coupon. Spot rates (annual compounding) are 2%, 3% and 4% for one, two and three years. Assume the curve stays unchanged. What is the carry roll-down return over the next year (coupon plus price change, as a percentage of today's price), closest to?
- 0.89%Why not: Counts only the roll-down price change and leaves out the coupon.
- 3.93%Why not: Reports the bond's YTM as the return, which ignores that the unchanged curve reprices the bond.
- 5.74% - correct
- 2.00%Why not: Gives the 1-year spot rate, which is the return under forwards realized, not under an unchanged curve.
Price now = 102.96. In a year the bond has two years left; on the unchanged curve its price is 5/1.02 + 105/1.03² = 103.87. Return = (103.87 + 5 - 102.96) / 102.96 = 5.74%.
The hardest tier
A 4-year bond with face value 100 pays a 3.5% annual coupon. Spot rates (annual compounding) are 1.0%, 1.8%, 2.5% and 3.0% for one to four years, and they stay unchanged. The position is financed at the 1-year rate of 1.0%. What is the one-year net carry roll-down return (excess over financing), as a percentage of today's price, closest to?
- 3.31% - correct
- 4.31%Why not: Gives the gross carry roll-down return and does not deduct the 1.0% financing cost.
- 2.43%Why not: Uses coupon income less financing and omits the roll-down price gain.
- 1.95%Why not: Uses the YTM less financing, which ignores that the bond is repriced as it ages on the curve.
Price now = 102.05. With three years left on the unchanged curve the price is 102.95. Gross return = (102.95 + 3.50 - 102.05) / 102.05 = 4.31%. Net of 1.0% financing = 3.31%.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
Start free →More in Bond Yields and Return Calculations
- Distinguish gross realized returns from net realized returns and calculate a bond's realized return over a holding period, reinvestments included.
- Define a bond's spread, interpret it, and explain its derivation from the bond price together with a term structure of rates.
- Explain what a bond's yield to maturity (YTM) is, interpret it, and use it in pricing the bond.
- Explain the way a bond's YTM is calculated when its structure and price are known.
- Determine the price of an annuity as well as that of a perpetuity.
- Explain how spot rates and YTM are related.
- Explain what the coupon effect is and how coupon rate, YTM and bond prices relate to one another.
- Break a bond position or portfolio's profit and loss (P&L) down into separate drivers, namely carry roll-down, rate change and spread change effects, and explain each.