Definition
A bullet bond is a fixed‑income security in which the issuer repays the entire principal (face or par value) in one single payment at maturity. Interest (the coupon) may be paid periodically during the life of the bond, but the principal is not amortized over time. Bullet bonds are typically non‑callable, meaning the issuer does not have the contractual right to redeem the bond early.
Key terms (defined)
– Par (face) value: the amount repaid at maturity (commonly $1,000 for corporate issues).
– Coupon: the periodic interest payment, usually stated as an annual percentage of par.
– Yield: the investor’s required return, expressed as an annual rate; used to discount future cash flows.
– Callable: a feature that allows the issuer to repay debt before maturity; bullet bonds are generally non
-callable.
Key advantages and disadvantages
– For issuers
– Advantage: Simpler cash‑flow planning because principal is repaid in one lump sum at maturity; no amortization schedule to manage.
– Disadvantage: Large cash outflow at maturity requires refinancing or cash reserves; refinancing risk if market conditions are unfavorable at maturity.
– For investors
– Advantage: Predictable principal repayment date and fixed coupon schedule; easier modeling of cash flows and yield.
– Disadvantage: Higher interest‑rate sensitivity (duration concentrated toward maturity) compared with amortizing structures; full principal exposed to credit risk until maturity.
How bullet bonds are priced (formulas and steps)
– Price formula (annual coupon, par F, coupon C, yield y, n periods):
– Price = C * [1 − (1 + y)^−n] / y + F * (1 + y)^−n
– First term = present value (PV) of the coupon annuity. Second term = PV of the single principal repayment.
– Yield to maturity (YTM)
– YTM is the internal rate of return that makes the price equal to the PV of the bond’s cash flows. Solve for y in the price equation—typically numerically.
Worked numeric example (step‑by‑step)
– Assumptions:
– Par (F) = $1,000
– Annual coupon rate = 5% → C = $50
– Maturity n = 5 years
– Market yield (y) = 4% (annual)
– Step 1 — PV of coupons:
– Annuity factor = [1 − (1 + y)^−n] / y = [1 − (1.04)^−5] / 0.04 = 4.45182
– PV(coupons) = 50 * 4.45182 = $222.59
– Step 2 — PV of principal:
– PV(par) = 1,000 * (1.04)^−5 = $821.93
– Step 3 — Price:
– Price = 222.59 + 821.93 = $1,044.52 (bond priced above par because coupon > yield)
– Duration (interest‑rate sensitivity)
– Macaulay duration = (Σ t * PV(CF_t)) / Price ≈ 4.56 years (calculated from the PVs at each year)
– Modified duration = Macaulay / (1 + y) = 4.56 / 1.04 ≈ 4.38
– Interpretation: A small parallel
shift in yields of Δy will change the bond’s price approximately by −(modified duration) × Δy (expressed in decimal form). Using the numbers above:
– Modified duration ≈ 4.38.
– For a 100 basis‑point (1.00% = 0.01) increase in yield: approximate percent price change ≈ −4.38% → dollar change ≈ −0.0438 × $1,044.52 ≈ −$45.78. New approximate price ≈ $1,044.52 − $45.78 ≈ $998.74.
– For a 100 basis‑point decrease in yield: approximate percent price change ≈ +4.38% → dollar change ≈ +$45.78. New approximate price ≈ $1,090.30.
Limitations of the linear (duration) approximation
– Duration gives a first‑order (linear) estimate. For larger yield moves the error grows because the price–yield relationship is convex (curved).
– Convexity is the second‑order term that adjusts the duration estimate: ΔP/P ≈ −MD·Δy + 0.5·Convexity·(Δy)^2, where Convexity is measured in 1/(yield)^2 units. Convexity makes price gains larger than losses for symmetric yield moves (positive convexity).
– Bullet (noncallable) bonds typically have positive convexity. Callable bonds can show negative or reduced convexity when yields fall (because calls limit price appreciation).
Practical checklist for using duration on a bullet bond
1. Confirm cash‑flow timing and frequency (annual vs. semiannual). Duration formula depends on the period.
2. Convert yield to the same compounding period as cash flows (e.g., annual yield for annual coupons).
3. Compute Macaulay duration = Σ[t × PV(CF_t)] / Price.
4. Compute modified duration = Macaulay / (1 + y_per_period).
5. Estimate price change for small Δy with −MD × Δy. Use convexity adjustment for larger Δy.
6. Remember reinvestment risk: Macaulay duration also reflects the weighted average time to receive cash flows, which matters for reinvesting coupons.
Worked numeric recap (with our bond)
– Inputs: Face = $1,000; Coupon = 5% annual ($50); n = 5; market yield y = 4% (0.04).
– Price = $1,044.52 (computed earlier). Macaulay ≈ 4.56 years; Modified ≈ 4.56 / 1.04 ≈ 4.38.
– Approximate price if y → 5% (+1.00%): $998.7 (using duration). Expect small additional error; include convexity if precision is needed.
When to use duration vs. full re‑pricing
– Use duration for quick, small‑move approximations and risk comparisons across bonds.
– Recompute full PV of cash flows (full repricing) when yield moves are large, when the bond is callable/embedded options exist, or when high precision is required.
Key takeaways
– A bullet bond pays coupons and returns principal at maturity; its price sensitivity is captured well by Macaulay and modified duration for small, parallel yield changes.
– Duration is linear and should be supplemented by convexity for larger yield moves.
– Always match compounding and payment periodicity when calculating duration