List the components of a bond's return, then calculate what a risk-averse investor can expect to earn from holding it.
The idea
A bond's expected return over a short horizon has four pieces. The first is the short-rate carry for holding it. The second is the price effect of expected rate changes, either rolldown if the curve stays unchanged or the move relative to forwards. The third is a risk premium. The fourth is a convexity effect that is positive by Jensen's inequality. A risk-averse investor is paid for interest rate risk, so longer bonds earn more. Let D be the duration, λ the market price of risk and σ the rate volatility. For a bond of duration D, the expected excess return over the short rate is Premium = D × λ × σ. Let r be the short rate, Conv the convexity effect and Δr the expected rate change. Adding the four pieces gives Return = r + D × λ × σ + Conv − D × Δr. Risk-neutral pricing sets the premium to zero and is a valuation device, not a forecast of what a holder earns. Common errors: dropping duration from the premium, forgetting the short rate, and treating the yield as the expected return.
Formulas
Expected excess return of a bond over the short rate
Premium = D × λ × σ
- D
- the bond's duration
- λ
- the market price of interest rate risk
- σ
- the annual rate volatility
The reward a risk-averse investor demands for rate risk; it is zero under risk-neutral pricing. Forgetting the duration is the usual slip.
Expected short-horizon return of a default-free bond
Return = r + D × λ × σ + Conv − D × Δr
- r
- the short rate
- D
- the bond's duration
- λ
- the market price of risk
- σ
- the rate volatility
- Conv
- the convexity effect, about ½ × convexity × σ²
- Δr
- the expected change in the rate
Adds carry, risk premium, convexity and the expected rate move. An expected rise in rates subtracts; the yield alone is not the expected return.
A practice question
Which list gives the components of the expected return on a default-free bond over a short horizon?
- Short-rate carry, the price effect of expected rate changes, a risk premium for interest rate risk, and a convexity effect. - correct
- Coupon income, a credit spread, expected default loss and recovery.Why not: Lists credit components; a default-free bond has no credit spread or default loss.
- Short-rate carry and a risk premium only, since convexity affects risk but not expected return.Why not: Omits convexity, which raises expected return through Jensen's inequality.
- Carry, duration, convexity and value at risk.Why not: Mixes risk measures with return components; duration and VaR are not sources of return.
Expected return = carry + price effect of expected rate changes + risk premium + convexity effect. Credit terms apply to defaultable bonds, and duration, convexity and VaR are measures of risk.
A harder one
A default-free zero-coupon bond is held for a short period. If the yield curve is unchanged, the two-year zero (4.50%) rolls down to a one-year zero (4.00%) after one year. Annual compounding. The one-year holding-period return is closest to:
- 4.00%Why not: Gives the one-year rate; that is the return only if the curve rolls forward along the forwards, not if it is unchanged.
- 5.00% - correct
- 4.50%Why not: Counts only the carry at the two-year yield and omits the rolldown gain.
- 5.50%Why not: Adds twice the 0.50% yield slope to the 4.50% yield, a rule of thumb that treats rolldown as slope times maturity.
Buy at 100/1.045² = 91.57. After a year the bond is a one-year zero at 4.00%, worth 100/1.04 = 96.15. Return = 96.15/91.57 - 1 = 5.00%, which is the 4.50% yield plus about 0.50% rolldown.
The hardest tier
A risk-averse investor holds a five-year zero-coupon bond yielding 4.00% (annual compounding, modified duration 4.81, convexity 27.74) for one year. The short rate is 3.00%, rate volatility is 1.20% a year and the market price of risk is 0.25. The investor expects the yield to rise by 0.15% on average. Using expected return = short rate + duration x lambda x sigma + 0.5 x convexity x sigma² - duration x expected yield change, the expected return is closest to:
- 3.72%Why not: Leaves out the convexity effect of about 0.20%.
- 3.52%Why not: Subtracts the convexity effect, treating it as a cost rather than a gain.
- 3.92% - correct
- 5.36%Why not: Adds the price effect of the expected yield rise instead of subtracting it; rising yields lower prices.
Premium = 4.81 x 0.25 x 1.20% = 1.44%. Convexity = 0.5 x 27.74 x 0.012² = 0.20%. Rate effect = -4.81 x 0.15% = -0.72%. Total = 3.00 + 1.44 + 0.20 - 0.72 = 3.92%.
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