20 Callable Bonds

20.1 Definitions

Callable bonds are bonds that are redeemable early at the option of the issuer on specified call dates at specified call prices. We refer to a call price at time $t$ as the redemption value at time $t$.

The most common reason for a bond issuer to call a bond early is that the market interest rate at which the issuer can issue new bonds has fallen below the coupon rate on the callable bond, which allows the bond issuer to refinance at a lower interest rate. As a result, investors of callable bonds often require a higher yield to compensate for the call risk as compared to non-callable bonds.

In order to attract investors, some callable bonds offer a call protection period. The issuer is not allowed to call the bond before the ending date of the protection period. The first call date is the date after which the bond is fully callable.

The uncertainty of the possible call date is the main difficulty in pricing a callable bond. Here we consider a defensive pricing approach for the investor. Under this approach, an investor assumes that the issuer will call the bond at a date which will maximize the call benefit.

20.2 Pricing Callable Bonds

In general, when the call price varies in a pre-fixed relation with the possible call dates, we can apply the defensive pricing approach. Under this strategy, the investor will pay the lowest price among the prices calculated by assuming all possible call dates.

Example: Consider a $\$1,000$ face value 15-year bond with coupon rate of $4.0\%$ convertible semiannually. The bond is callable, and the first call date is the date immediately after the 15th coupon payment. Assume that the issuer will only call the bond at a date immediately after the $n$-th coupon $(15 \le n \le 30)$ and the call price is: \[C = \begin{cases} 1,000 \qquad & \text{if }15 \le n \le 20 \\ 1,000+10(n−20) \qquad & \text{if } 20 < n \le 30 \end{cases}\] Find the price of the bond if the investor wants to achieve a yield of at least $5\%$ compounded semiannually.

Solution: First, we compute the prices of the bond assuming all possible call dates, with $r = 0.02, i = 0.025, F = 1,000$ and the value of $C$ following the given call price formula.

If the bond is called at time $15 \le n \le 20$, then the price of the bond is $\begin{align} P(n) &= 1000 (1.025)^{-n} + 20 \times a_{\overline{n|}0.025} = 1000 (1.025)^{-n} + 20 \times \frac{1-(1.025)^{-n}}{0.025} \\&= 1000(1.025)^{-n} + 800(1-1.025^{-n})\end{align}$ We can see that, as $n$ grows, $P$ gets smaller due to the fact that the coupon rate is smaller than the yield rate. Thus, $\arg\min_{n\in[15,20]} P(n) = 20$

If the bond is called at time $20 < n \le 30$, then the price of the bond is: \[\begin{align} P(n) &= (1000 + 10(n-20)) (1.025)^{-n} + 800 (1 - 1.025^{-n}) \\ &= 800 + (1.025^{-n}) (200 + 10(n-20)) = 800 + 1.025^{-n} \times 10n \end{align}\] We can show that $\arg\min_{n\in [21, 30]} P(n) = 21$, since $n \times 1.025^{-n}$ is increasing for $n < 30$.

Thus, the minimum price is $P(20) = \$922.05$, which assumes that the bond will be called after the 20th payment. This is the price of the callable bond a defensive investor is willing to pay if she wants to obtain a yield of at least $5\%$ compounded semiannually.

Remark: For a par value bond,

if the bond sells at a discount, i.e. $r < i$, the minimum yield rate for the callable bond is calculated based on a call at the latest possible date.
if the bond sells at a premium, i.e. $r > i$, the minimum yield rate for the callable bond is calculated based on a call at the earliest possible date. These dates are the most disadvantageous to the bond holder in each case.