Perhaps the best way to understand the impact of including a call option on a municipal bond is to look at the value of a callable bond as if it were the value of a noncallable bond minus the value of the embedded call.

where CB is the value of a callable bond, NCB is the value of a noncallable bond, and OVB is the option value of the embedded call.

The greater the value of the call option, the greater the difference in value between callable and noncallable bonds and the greater the spread between callable and noncallable bonds.

We consider two types of bonds: 1) a bond with a make-whole call provision and 2) an option-free or noncallable bond. The call price at which the firm can call its debt is calculated as the present value of the remaining coupon payments c and balloon payment F discounted at the risk-free rate r plus some prespecified premium m > 0.

With a noncallable bond, if a situation arises where the firm must retire the bond early, the firm will normally attempt a bond tender offer.

In contrast, the median noncallable bond has a rating of A-, 10 years to maturity, and an offering amount of $250 million.

The value of the implied put is the price of the callable bond minus the price of a noncallable bond that matures on the first call date.

To control for term effects and market movements not associated with the call,(5) we selected a matched-control sample of independent noncallable bonds with a maturity date on or immediately following the call date.

The coupon range for noncallable bonds is slightly larger than that of callable bonds, although the standard deviation is substantially smaller.

The parameter values chosen are the same as those for noncallable bonds. The call price K was assumed to be equal to the par value of the bond.

This is in contrast to the results for noncallable bonds (see, for example, Exhibit 5) where the yield spread increases with maturity over a wider range.

Initially, suppose that the firm issues a two-period default-free noncallable bond. Choosing a coupon payment C, such that the bond sells for par, B, gives B = [p.sub.01][C (1 - [[Tau].sub.C] + [B.sub.1] - [[Tau].sub.C]([B.sub.1] - B)] + [p.sub.02] [C (1 - [[Tau].sub.C]) + [B.sub.2] - [[Tau].sub.C] ([B.sub.2] - B)], (3) where [B.sub.1] and [B.sub.2] are the time 1 market values of debt in states 1 and 2, and are equal to [B.sub.1] = [p.sub.11] [C (1 - [[Tau].sub.C]) + B - [[Tau].sub.C](B - [B.sub.1])], (4) and [B.sub.2] = [p.sub.12] [C(1 - [[Tau].sub.C] + B - [[Tau].sub.C](B - [B.sub.2])], (5) respectively.

Suppose that instead of issuing a noncallable bond, the firm issues a callable bond.

Our tests are based on a sample of 958 bonds issued during 1987-1991, when issuance of noncallable bonds was quite common.

Of the three, only the Kish and Livingston study has a sufficient number of noncallable bonds to estimate a zero/one equation on the inclusion of a call option, although they are forced to overweight the sample of noncallable bonds.