Callable Bond Pricing

Price a callable bond for the full term and it gets called early (your yield drops. Price it for the call date and it runs to maturity) you overpaid. Solution: price at the worst-case call date.

HIGH-FREQUENCY: Callable bond pricing appears regularly on FM. The worst-case principle is essential.

The issuer can redeem before maturity at a call price on designated call dates. The call premium = call price minus par.

Common structures:

• Callable at par from some date onward.
• Declining call schedule (e.g., 105, 102, 100 over different windows).

The investor wants a minimum yield of $i$. The issuer calls when it hurts the investor most. Price at the worst case:

$$P=\underset{k\in \text{call dates}}{\min }\left[Fr\cdot a_{\overline{k|i}}+C_k\cdot v^k\right]$$

Use the premium/discount formula at each call date:

$$P_k=C_k+(Fr-C_{k}i)\cdot a_{\overline{k|i}}$$

When $Fr>C_{k}i$ (premium relative to call price): extending the term increases $a_{\overline{k|}}$, which increases $P_k$. Minimum is at the shortest term, earliest call.

Common mistakes

• Pricing at maturity for a premium bond. Premium = earliest call. Using maturity gives a higher price, and the investor overpays. If called early, yield falls below target. Trap: $50-$100 too high.
• Pricing at earliest call for a discount bond. Discount = maturity. Earliest call gives the highest price, wrong direction. Trap: $20-$50 too high.
• Ignoring the call price. If call price is $1,050, use $C_k=1050$, not $1,000. This affects both the PV of redemption and the premium/discount classification.

Bottom line

• Worst-case pricing: $P={\min }_k[Fr\cdot a_{\overline{k|i}}+C_k\cdot v^k]$ over all call dates.
• Price each call date as one TVM (N to that date, FV = call price), then take the minimum price, which guarantees the minimum yield.
• Premium bond ($Fr>Ci$): worst case = earliest call.
• Discount bond ($Fr<Ci$): worst case = maturity.

Exam shortcut

Compute $C^{\ast }=Fr/i$ immediately. All call prices below (price at earliest. All above) price at maturity. Mixed, compute at each tier boundary. Classification takes 10 seconds. "Premium Early, Discount Late." Threshold: $C^{\ast }=Fr/i$. Below = premium (earliest). Above = discount (latest). Compute, take minimum.

The full lesson (about 1,705 words, 11 min read) adds 2 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 4c

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