Callable Bonds
A callable bond lets the issuer redeem early, so its price depends on which call date occurs — to guarantee a yield, an investor prices it at the single worst-case (price-to-worst) date.
By the end you'll be able to tell, from the coupon rate versus the yield alone, which call date sets a callable bond's price-to-worst, and price the bond at that date.
Predict: if the bond would sell at a premium, which call date gives the investor the worst yield — the earliest or the latest? Drag the yield and check which date sets the price.
This is a 1000 par bond with an 8% annual coupon, callable at par on any coupon date from year 5 through its year-10 maturity. Drag the yield i and watch the price at each call date. The ringed point is the lowest price across all six dates — the price-to-worst — and pricing there guarantees the investor at least yield i no matter when the issuer actually calls.
Price at earliest call (n = 5): 1084.25
Price at latest date (n = 10): 1147.20
Price-to-worst (guaranteed-minimum-yield price): 1084.25 — assume the earliest call (n = 5)
The issuer picks the call date, not you — so to guarantee a yield you have to price against whichever date is worst for you, not whichever one you'd prefer.
At a fixed date n, the price is the ordinary bond formula \(P(n) = Fr\cdot a_n + C\cdot v^n\), rewritten as \(P(n) = C + (Fr-Ci)\cdot a_n\) — the same premium/discount form from premium/discount pricing. Since \(a_n\) increases with n, the sign of \((Fr-Ci)\) tells you which direction price moves as n grows: if the bond is a premium (Fr > Ci), price rises with n, so the lowest price — the price-to-worst — sits at the smallest n, the earliest call. If it's a discount (Fr < Ci), price falls with n, so the price-to-worst sits at the largest n, the maturity date. This shortcut assumes every call date redeems at the same value; if redemption amounts differ across dates, price every candidate and take the minimum directly.
Think of it from the issuer's side: they'll only call the bond when it benefits them, which is exactly when it hurts you. If your coupons are rich (premium), the issuer wants to call as soon as possible to stop overpaying you — so plan for that. If your coupons are cheap (discount), the issuer is happy to let the bond run, so the pain of collecting below-market coupons drags out to maturity — plan for that instead. Either way, price to the case where the issuer's incentive and your outcome are most misaligned.
F = C = 1000, 8% annual coupon, callable at par from year 5 through year-10 maturity, priced to yield i = 6%. Since 8% > 6%, this is a premium bond, so assume the earliest call, n = 5: \(a_5\) at 6% ≈ 4.212364, \(v^5\) ≈ 0.747258, so \(P = 80(4.212364) + 1000(0.747258) = 336.99 + 747.26 =\) 1084.25. Paying no more than 1084.25 guarantees at least a 6% yield, however the issuer ultimately calls.
Same bond, a higher yield: F = C = 1000, 8% annual coupon, callable from year 5 through year-10, priced to yield i = 9%. Since 8% < 9%, this is now a discount bond, so assume the latest date, n = 10: \(a_{10}\) at 9% ≈ 6.417658, \(v^{10}\) ≈ 0.422411, so \(P = 80(6.417658) + 1000(0.422411) = 513.41 + \) ____
Reveal the answer
\(P = 513.41 + 422.41 = \) 935.82. Set the slider above to i = 9.0% and check the price-to-worst readout against this figure — it should land on the n = 10 point, not n = 5.
More info — when the shortcut breaks down
The earliest/latest shortcut relies on every call date sharing the same redemption value C, so that only \(a_n\) changes across dates. Real callable bonds sometimes have a declining call schedule — an early call redeems at a premium above par (say 105% of face), stepping down toward par as maturity nears. When that happens, \(a_n\) and the redemption term both change from date to date, and the direction of price movement is no longer guaranteed by the coupon-vs-yield comparison alone. The safe approach is always the definition: price the bond at every candidate call date and take the minimum. See the Wikipedia link in Dive deeper below for how call schedules are typically structured.
Check your understanding
For a premium bond (coupon rate above the yield) with several possible call dates, which call date sets the price-to-worst — the price an investor should pay to guarantee at least that yield?
A 1000 par, 6% annual coupon bond is callable at par on any date from year 6 through its year-15 maturity, priced to yield 4%. At 4%, \(a_6 \approx 5.2421\) and \(v^6 \approx 0.79031\). Since 6% > 4%, this is a premium bond — what price guarantees the investor at least a 4% yield?
A callable bond's early call dates redeem at a premium above par, while later dates redeem closer to par. What should you do to find its price-to-worst?
A callable bond's coupon rate is below its yield. Combining what you know about premium/discount pricing with the price-to-worst rule, which call date should the investor assume?
Recap
- A callable bond's price depends on which call date occurs, so its price at yield i is \(P(n) = Fr\cdot a_n + C\cdot v^n\) evaluated at each candidate n.
- Price-to-worst is the minimum price across all call dates — pay no more than that to guarantee at least yield i.
- Premium (coupon rate > yield): price rises with n, so assume the earliest call.
- Discount (coupon rate < yield): price falls with n, so assume the latest (maturity) date.
- If redemption values differ across call dates, the shortcut can fail — price every date and take the minimum directly.
Dive deeper
- Callable bond — Wikipedia Read how the issuer's right to redeem early on call dates works.
- Marcel Finan — Exam FM study guide (PDF) Study the price-to-worst rule for pricing callable bonds.
Sources
- Callable Bonds and Price-to-Worst