Callable Bond

Study guide: Exam FM-2

A callable bond gives the issuer the right (but not the obligation) to redeem the bond before maturity at a specified Call Price on or after a specified call date. Issuers call bonds when interest rates fall, allowing them to refinance at a lower rate.

P = \min_{m}\left(Fr\,a_{\overline{m}|j} + C_m\,v_j^{\,m}\right)

• The investor faces reinvestment risk: if the bond is called early, the proceeds must be reinvested at the lower prevailing rates. The investor therefore assumes the issuer calls on the date worst for the investor. To guarantee a yield of at least j per period, the most the investor will pay is the smallest price over all the possible call dates.
• In the formula, m runs over every possible call date (in coupon periods), including maturity; C_m is the call price at date m (the redemption value at maturity), Fr the coupon and v_j = 1/(1+j). Given a price instead, the investor's yield is the smallest of the yields at the possible call dates.
• Shortcut, when the call price is the same on every call date, maturity included:
  • If the bond is at a premium (price > redemption value, i.e. the yield is below the modified coupon rate g = Fr/C): assume the earliest call date.
  • If the bond is at a discount (price < redemption value): assume the latest possible redemption date, which is maturity.
• When the call price differs between dates, the shortcut does not hold: the worst date need not be the earliest or the latest. Price the bond at each possible call date and take the minimum. Within a stretch of dates sharing one call price K, the price moves steadily one way (up with m when Fr > Kj, down when Fr < Kj), so only the two ends of each stretch need checking.

Callable Bond Price A 1{,}000 bond with 8% annual coupons matures in 10 years but is callable at par on any coupon date from year 5 on. Find the price to guarantee a minimum yield of 6%.

Answer The call price is par on every call date and at maturity, so the shortcut applies. Price will exceed par (coupon > yield), so assume a call at year 5 (earliest call, worst case): P = 80 \cdot a_{\overline{5}|6\%} + 1000(1.06)^{-5} = 80(4.2124) + 747.26 = 336.99 + 747.26 = 1084.25

Call Prices That Step Down A 100 par bond with a 4% coupon rate payable semiannually is callable at 109 from 5 to 9 years after issue, at 104.50 from 10 to 14 years, and matures at par in 15 years. What should an investor pay at issue to be sure of a yield of at least 3% convertible semiannually?

Answer Per half-year the coupon is 2 and j = 1.5\%. Since 2 > 109(0.015), 2 > 104.50(0.015) and 2 > 100(0.015), the bond is at a premium against every call price, so within each stretch the earliest date is the worst. Price at the start of each stretch and at maturity:

\begin{align*}
\text{Call at 5 years: } P &= 2\,a_{\overline{10}|1.5\%} + 109\,v^{10} \\
&= 112.37 \\
\text{Call at 10 years: } P &= 2\,a_{\overline{20}|1.5\%} + 104.50\,v^{20} \\
&= 111.93 \\
\text{Maturity: } P &= 2\,a_{\overline{30}|1.5\%} + 100\,v^{30} \\
&= 112.01
\end{align*}

The investor pays at most 111.93, the price for a call at 10 years — neither the earliest call date nor maturity.