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Convertible bonds

Convertible bonds

: face value

: coupon rate

: conversion ratio; number of shares received upon conversion

: conversion price

: Stock market price

: conversion value (parity)

: time to maturity of the convertible bond

Assume coupon is 0; conversion only at maturity; the bond is non callable; no default risk

The payoff to the convertible bond at maturity is This can be written as

Therefore, the payoff to the convertible bond at maturity is the sum of the face value of the bond and the payoff to European call options on the firms stock with exercise price equal to the conversion price and time to maturity ,

The value of the convertible today is then

Alternately, the payoff to the convertible at maturity can be expressed as

Therefore, the payoff to the convertible bond at maturity is the sum of the conversion value of the bond at maturity and the payoff to European put options on the firms stock with exercise price equal to the conversion price and time to maturity ,

The value of the convertible today is then

If we assume that the dividend during the life of the bond is known with certainty then the value today of receiving the stock price, , at is If the bond pays coupons at then the two valuations can be written as

With default risk the value of the straight bond will also depend upon the likelihood of default.

This implies that the straight bond value is a lower bound or a floor to the value of the convertible. However, this floor is not fixed and can fall to the recovery value in the event of default.

The coupon rate on a par value convertible bond will be less than the coupon rate on a par value straight bond. The coupon rate on the convertible bond will increase with the extent to which the conversion price exceeds the current stock price.

This implies that the convertible cannot sell for less than its conversion value.

With default risk the put is not default risk free.

The put and call are actually warrants and not options since the company will issue new shares. While valuing warrants the dilution effect of the shares issued have to be taken into account. This is not necessary in the case of options since no new shares are issued.

The conversion option is American since it can be exercised prior to maturity. If the stock is non dividend paying then it will not be worthwhile to exercise the conversion option prior to maturity. This incentive will be stronger if the coupon on the bond is taken into account. Early exercise may be worthwhile if the dividend exceeds the coupon. Coupons on convertible bonds are usually set higher than the dividend yield of the company.

Convertible bonds are callable by the issuer, usually after an initial call protection period, at a price as a percentage of face value. The price is specified at the time of issue and may vary with time to exercise. The issuer can use the call provision to force conversion of the convertible bond. Calling the bond and forcing conversion will be optimal if the conversion value exceeds the call price.