Convertible Bond Valuation
The Valuation of Convertible and Reverse Convertible Bond
Overview
Convertible bonds can be thought of as normal corporate bonds with embedded options, which enable the holder to exchange the bond asset for the issuer's stock. Having properties of both stocks and bonds, convertibles can be an attractive choice for investors and have tended to have lower risk.
A convertible bond has an embedded call option that gives bondholders the right to convert their bonds into equity at a given time for a predetermined number of shares in the issuing company. Whereas a reverse convertible bond has an embedded put option that gives the issuer the right to convert the bond's principal into shares of equity at a set date.
Convertible bonds typically have lower yields than the yields on similar bonds without the convertible option. Reverse convertible bonds usually have shorter terms to maturity and higher yields than most other bonds.
Most convertible bonds are subordinated debt of the issuer. In the event of bankruptcy, the claims of other bondholders take priority over convertible bondholders, who themselves have priority over owners of the preferred and common stock.
Issuers have several reasons to use convertible financing. By issuing convertibles they can lower their cost of debt funding compared to straight debt alone. Lower-credit companies who may not be able to access the straight debt market can often still issue convertible debt. Companies who anticipate equity appreciation can use convertibles to defer equity financing to a time when growth has been achieved.
Investors find several features of convertibles appealing. They offer greater satiability of income than common stock. They provide a yield that is often higher than the dividend yield of common stock. Finally, because they are often theoretically underpriced, they may provide a cheap source of common stock volatility. This presentation gives an overview of convertible bond and valuation model.
Keyword
Convertible bond, reverse convertible bond, conversion ratio, hybrid security, valuation, PDE
    1.
    Introduction
    A convertible bond has an embedded call option that gives bondholders the right to convert their bonds into equity at a given time for a predetermined number of shares in the issuing company.
    An reverse convertible bond (RCB) has an embedded put option that gives the issuer the right to convert the bond's principal into shares of equity at a set date.
    Convertible bonds typically have lower yields than the yields on similar bonds without the convertible option.
    Reverse convertible bonds usually have shorter terms to maturity and higher yields than most other bonds
    Convertible bonds are hybrid securities that have both debt and equity features.
    Most convertible bonds are subordinated debt of the issuer. In the event of bankruptcy, the claims of other bondholders take priority over convertible bondholders except the preferred and common stock owners.
    1.
    The Use of convertible bonds
    By issuing convertibles they can lower their cost of debt funding compared to straight debt alone.
    Lower-credit companies who may not be able to access the straight debt market can often still issue convertible debt.
    Companies who anticipate equity appreciation can use convertibles to defer equity financing to a time when growth has been achieved.
    Convertible bonds offer greater satiability of income than common stock.
    They provide a yield that is often higher than the dividend yield of common stock.
    Given the optionality, convertibles have tended to have lower risk.
    1.
    Valuation
    Convertible bonds are hybrid securities that have both debt and equity features.
    The valuation of convertible or reverse convertible bonds can be quite complex because of its dual nature as a normal bond and as an equity call/put option.
    There is no closed-form solution for convertibles.
    Convertible prices can only be solved by numerical methods, such as, Monte Carlo simulation, tree/lattice approaches, or partial differentiao equation (PDE) solutions.
    Three sources of randomness exist in a convertible bond: the stock price, the interest rate, and the credit spread.
    Interest rate is assumed to be constant as the effect of a stochastic interest rate on convertible bond prices is so small that it can be neglected.
    Accurately modeling the equity process appears crucial.
    Since convertible bonds are issued mainly by start-up or small companies (while more established firms rely on other means of financing), credit risk plays an important role in the valuation.
    FinPricing uses PDE to price convertible and reverse convertible bonds, and use Monte Carlo simulation to value convertibles with exotic path-dependent trigger provisions.
    The value of the convertible at each node is divided into two components: a component of bond and a component of stock
    The PDE of the equity component G is given by
where
S the stock price
r the interest rate
q the dividend
h the hazard rate
s the equity recovery rate
    The PDE of the bond component B is
where b is the bond recovery rate
    The final conditions at maturity T can be generalized as
where
N denotes the bond principal
C denotes the coupon
denotes the call price,
denotes the put price
denotes the conversion ratio.
    The final conditions tell us that the convertible bond at the maturity is either a debt or an equity.
The upside constraints at time
are
where
is the continuation value of the convertible bond
is the continuation value of the bond component
is the continuation value of the equity component.
    1.
    Implementation
    The valuation can be done via backward induction. The procedure is as follows.
For i = penultimateTime to currentTime
// determine accrual interest and call/put prices
// determine boundary nodes
// use the PSOR (Projected Successive over Relaxation) method to obtain the
continuation value of the bond component
and the continuation value of the equity component
, applying the constraints (31).
EndFor
    The value at node[0][y] is the convertible bond price where the equity price at node[0][y] is equal to the current market stock price.
    1.
    A Real World Example
Underlying Equity
EFN.TO
Conversion Ratio
52.35
Convertible Bond ISIN
CA286181AB88
Conversion Start Date
1/1/2016
Currency
CAD
Conversion End Date
6/30/2020
Face Value
100
Coupon
0.0425
Percent Redemption
1
First Coupon Date
12/31/2015
DayCount
dc30360U
First Settle Date
5/29/2015
Call Start Date
6/30/2018
Coupon Frequency
SEMIANNUAL
Call End Date
6/29/2020
Roll Type
Following
Call Price
100
Issue Price
100
Call Notice Period
30
Market Quote Type
CLEAN
Call Trigger
125
Maturity
6/30/2020
Trigger days
20
Par Amount
1000
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