Sensitivities for Make-whole Callable Bonds

A callable bond is a bond that provides the issuer with the right to exchange the bond for its call value in cash. There are a few ways one can value a callable bond by extending the expression of a vanilla bond:

  • Yield-to-x: Calculate the yield-to-maturity, yield(s)-to-call, and/or yield-to-worst and take the lowest of this set. Revalue the bond with this yield to obtain the market value.
  • Option-Adjusted Bond: Add an option-adjusted spread to the interest rates of each coupon and revalue the bond to obtain the market value.
  • Binomial Pricing: Populate a binomial tree with interest rates from now until maturity. Obtain the price of a bond at each node by starting at maturity. “Step back” for each node in the tree to obtain the value of the bond at each coupon date. For each call date, evaluate whether the price of the bond is greater than the call value. If it is, set the node to the call value and “step back” each node in the tree to obtain the updated value of the bond.
  • Replication via Swaption: A long callable bond can be replicated as a long asset swap and short receiver’s swaption.
  • Replication via Option: A long callable bond can be replicates as a vanilla bond and a short call option on said bond.
  • Taylor’s Series Expansion: The value of a callable bond can be decomposed into a linear combination of its partial derivatives with respect to each underlying variable.

A bond provision sets the value of the callable bond at redemption. Some common provisions associated with callable bonds are

  • fixed call price
  • floating call price
  • make-whole call price

Simply put, the optionality of the bond is contingent on a constant value in the first case and a non-constant value in the second and third case. Make-whole provisions are unique in that the call price is generally the maximum of a fixed and floating rate, ensuring that the bond holder is "made-whole" upon the issuer exercising the bond.

Make-whole provision bonds, while not as popular in other jurisdictions, are quite popular in Canada. Unfortunately, measuring the risk of these instruments is typically done by making crude approximations to fixed-price callable bonds, or using discretizations of valuation methods listed above to obtain sensitivities (which can be computationally expensive).

Sensitivities can be used for hedging, position analysis, and in VaR methodologies (such as sensitivity-based historical simulation). Under the Taylor's series expansion method indicated above, the work of [1] solve for the sensitivities of a fixed price callable bond. However, with a floating or make-whole call price, the sensitivities are not available. By making a simplification of the problem (ignoring time effects and higher order terms) we can determine what the sensitivity of the yield is under a floating or make-whole call price. Note that the approach outlined below can be extended for the remaining terms and time sensitivities.

We first define the replication relationship between the prices of a callable bond (CB), non-callable bond (NCB), and option on a non-callable bond (O): P_{CB}=P_{NCB}-P_O. The change in price (i.e. return) of a callable bond can then be expressed as

\frac{\Delta P_{CB}}{P_{CB}}=\frac{\Delta P_{NCB}}{P_{NCB}}}-\frac{\Delta P_{O}}{P_{O}}

Here,

P_{CB}^t=\sum_{i=1}^n \frac{CF_i}{Y_t^{i-t}}

where Y_t =G_t+S_t is the yield of the bond decomposed as a sum of a reference curve and a spread. Recall that by Taylor’s Theorem we can view the bond return as a sum of partial derivatives. First we make a simplifying assumption by removing non-first-order terms and time. Then, we can express the return on a callable bond as a function of the reference yield and spread: P_{CB}=P_{CB}(G_t,S_t).

\frac{\Delta P_{CB}}{P_{CB}} \simeq \frac{\delta P_{CB}}{\delta G_t} \frac{\Delta G_t}{P_{CB}} + \frac{\delta P_{CB}}{\delta S_t} \frac{\Delta S_t}{P_{CB}}

 = ( \frac{\delta P_{NCB}}{\delta G_t} - \frac{\delta P_{O}}{\delta G_t} ) \frac{\Delta G_t}{P_{CB}} + ( \frac{\delta P_{NCB}}{\delta S_t} - \frac{\delta P_{O}}{\delta S_t} ) \frac{\Delta S_t}{P_{CB}}

 := A+B

The bond option P_O (P_{NCB},K) is assumed by Black’s formula for European options, which is treated a function of the non-callable bond price and the call price. Indeed, while the optionality of callable bond is of American style, this simplifying assumption provides an approximation without adding non-closed-form terms. Then,

P_O=P_{NCB} N (d_1) - K N(d_2)

Where N is the cumulative normal distribution function, K is the call price, P_{NCB} is the price of the non-callable bond as before,

d_1 = \frac{1}{\sigma \sqrt{t}} ( \ln \frac{P_{NCB}}{K} + \frac{1}{2} \sigma^2 t )

d_2=d_1-\sigma \sqrt{t}

The call price will depend on the provision of the bond. For a make-whole bond with a fixed price of C, K=\max {\sum_{i=1}^n \frac{CF_i}{(G_t+spread)^{i-t}}, C} . To solve for the bond return, we first focus on the term A, representing the sensitivity with respect to the reference rate G.

A= ( \frac{\delta P_{NCB}}{\delta G_t} - \frac{\delta P_{O}}{\delta G_t} ) \frac{\Delta G_t}{P_{CB}}

 = (\frac{\delta P_{NCB}}{\delta G_t} - ( \frac{\delta P_{O}}{\delta P_{NCB}} \frac{\delta P_{NCB}}{\delta G_t} + \frac{\delta P_O}{\delta P_K} \frac{\delta P_K}{\delta G_t} ))\frac{\Delta G_t}{P_{CB}}

By defining

 D_{NCB}^G= - \frac{1}{P_{NCB}} \frac{\delta P_{NCB}}{\delta G_t}

 =\frac{1}{P_{NCB} G_t} \sum_{i=1}^n \frac{(i-t)CF_i}{G_t^{i-t}}

as the duration of the non-callable bond with respect to the reference yield, we may rearrange to obtain

 \frac{\delta P_{NCB}}{\delta G_t} = - P_{NCB} D_{NCB}^G

Similarly, in a make-whole bond the duration for the reference bond in the call price is given by D_{RB}^G=-\frac{1}{P_{RB}} \frac{\delta P_{RB}}{\delta G_t} yielding

 

 \frac{\delta K}{ \delta G_t}= \begin{cases} -P_{RB} D_{RB}^G, & P_{RB} > C \\ 0, & P_{RB} \leq C \end{cases}

The remaining terms of A can be attained from Black’s formula:

\frac{\delta P_{O}}{\delta P_{NCB}} = N(d_1)

\frac{\delta P_{O}}{\delta P_{K}} = -N(d_2)

Thus,

 A = \begin{cases} (-P_{NCB}D_{NCB}^G(1-N(d_1))-N(d_2)P_{RB}D_{RB}^G) \frac{\Delta G_t}{P_{CB}}, & P_{RB} > C \\ (-P_{NCB} D_{NCB}^G (1-N(d_1))) \frac{\Delta G_t}{P_{CB}} , & P_{RB} \leq C \end{cases}

Following similar steps as above, the derivation of B can be obtained as

 B = \begin{cases} (-P_{NCB}D_{NCB}^S(1-N(d_1))-N(d_2)P_{RB}D_{RB}^S) \frac{\Delta S_t}{P_{CB}}, & P_{RB} > C \\ (-P_{NCB} D_{NCB}^S (1-N(d_1))) \frac{\Delta S_t}{P_{CB}} , & P_{RB} \leq C \end{cases}

Combining A and B yields an expression for the return on a make-whole (or floating rate) callable bond:

 \frac{\Delta P_{CB}}{P_{CB}} = \begin{cases} V , & P_{RB} > C \\ -\frac{P_{NCB}}{P_{CB}}(D_{NCB}^S \Delta S_t + D_{NCB}^G \Delta G_t)(1-N(d_1)) , & P_{RB} \leq C \end{cases}

where  V=(-P_{NCB}(D_{NCB}^S \Delta S_t + D_{NCB}^G \Delta G_t)(1-N(d_1))-N(d_2)P_{RB}(D_{RB}^S \Delta S_t + D_{RB}^G \Delta G_t)) \frac{1}{P_{CB}}

For a callable bond with fixed call price, P_{RB}=0 and hence we obtain the expression in [1]:

\frac{\Delta P_{CB}}{P_{CB}} = -\frac{P_{NCB}}{P_{CB}}(D_{NCB}^S \Delta S_t + D_{NCB}^G \Delta G_t)(1-N(d_1))

 

[1] An accurate measure of callable bond price sensitivity to interest rates and passage of time. J. H. CHOU, H. F. YU AND C. Y. CHANG.  517–543.