Tuesday, June 5, 2012

Pricing Convertible Securitie


This article appeared in The Edge in Jan 2012

By Jasvin Josen

Convertible securities are traded widely in global debt markets. The most common convertible securities are convertible bonds and convertible preferred stocks. The Convertible Bond (CB) is an equity-linked instrument that gives the holder of the bond the right to convert the bond into a predetermined number of common stock in the future. It is attractive to investors as it provides downside protection of a straight bond with coupons and principal payback at maturity plus the upside return of equities.

Being a hybrid of a bond and equity, the price behaviour of a CB is rather unique. The pricing of CBs is also not very clear-cut. In this article, I describe the features of the CB and its pricing behaviour and illustrate two common pricing methods in the market.

Features of a Convertible Bond
Suppose Company Q issues a convertible bond with a conversion ratio of 25.32 shares. The par value of the bond is $1000. This means that for each $1000 of the par value of this issue that the bondholder exchanges for Company Q’s stock, he will receive 25.32 shares.

The stated conversion price is therefore:
= Par Value of the CB / Conversion Ratio
= $1000 / 25.32
= $39.49

If this CB pays an annual coupon of 6% with a maturity of 5 years, the CB has an investment value. Assuming the risk free discounting rate is 2.5% and the credit spread is zero, the Investment Value of the bond can be derived by discounting its cash flows at the risk free discount rate, as shown in Chart 1.


Chart 1: Investment Value of Company Q’s Convertible Bond


Now, the conversion price, realistically, is revised to:
= Investment Value of the CB / Conversion Ratio
= $1162.60 / 25.32
= $45.92

The CB example I provide above is one of its most basic kinds. Frequently, CBs are accompanied with a redeemable feature that allows the issuer to call back the bond after a certain period, at a predetermined price. In some cases the issuer can only call back the bond if the price of the underlying stock is above the conversion price by a certain percentage. The callable or redeemable feature makes the CB to some extent cheaper than a non-redeemable CB.

Price Behaviour of Convertible Bonds
The price of the CB is affected by movement in interest rates (and credit spreads) as well as movements in the stock prices. Taking the example of Company Q’s CB, there are two extreme cases:
(i)             When the stock price of Q is relatively small to the conversion price of $45.92, the CB is very unlikely to be converted and therefore it is effectively a straight bond that can be priced using the standard bond pricing method.

(ii)           When the stock price is very high relative to the conversion price, the CB will certainly be converted into stock. The CB price will then be the conversion value, which is the stock price multiplied by the conversion ratio.

For example, if the stock price is $60, this CB is very likely to be converted. Its price will behave very much like the stock. The price of the CB in the market is likely to be near $1,519.20 (stock price of $60 multiplied by the conversion ratio of 25.32)

Chart 2: Convertible Bond Price Chart


Source: Convertible Bonds, Zhi Da (2004)
In Chart 2, the solid line corresponds to the conversion value, which linearly increases with the stock price. The horizontal dashed line corresponds to the price of a straight bond and it is not affected by the stock price. Generally, since you have the option to make the convertible bond either a bond or a stock, its value should be at least the value of the bond or the stock. This is represented by the dotted line.

Pricing a CB as a straight bond plus a call option on the stock
Intuitively, a CB must be worth at least the value of a non-convertible bond of similar characteristics. If the holder does not convert, he will receive the coupons and the principal back. However, the CB is worth more than this as the holder has a chance in participating in the stock price movement. 
This conversion feature resembles a call option on the stock. The conversion premium increases as the spot price or the volatility of the stock increases.
Chart 3: CB priced as a straight bond plus a call option on the stock


In Chart 3, say the current stock price is $40. This price is not too far from the conversion price (which effectively now is the strike price of the call option) of $45.92. The CB holder effectively is holding the option to buy the stock at $45.92. As the stock price increases beyond $40, we expect the call option to get more expensive.
The call option here is priced using the Black Scholes model. The parameters are:
Current stock price:             $40
Strike Price:            $45.92
Risk Free rate:            2.5%
Stock volatility:            20%
Maturity:             5 years

The CB value of $1169.47 is then derived by adding the call option value of $6.87 to the Investment Value of the Bond of $1162.60.

Pricing a CB as a stock plus put option on the stock
Owning a CB can also be seen from another angle. The holder of the CB will only choose not to convert the CB into stock if the stock price is below the strike price of $45.92. This is as good as saying that the owner already has the stock, but has the option to sell the stock if its price goes down beyond a certain level. In return, he gets the returns from the bond.
Chart 4: CB priced as a stock plus put option on the stock


In Chart 4, the stock value is simply the conversion value (stock price of$40 multiplied by the conversion ratio of 25.32). The put option is arrived using the Black Scholes model, with the same parameters used to arrive at the call option on the stock in the first pricing method.
Multiplying the put option value ($7.39) with the conversion ratio (25.32) would give us the put value embedded in the CB. Indirectly, this put value represents the fixed income value of the convertible.

Conclusion
There are also other more advanced pricing techniques (but less commonly used) in the market like multi-factor models that take into account of several stochastic factors that simultaneously affect the price of the CB, namely interest rates, credit spreads, stock price and in some cases, foreign exchange rates.
We see that convertibles can be theoretically priced using a few techniques to resemble their real market prices. However in certain situations (e.g. in periods of high volatility) theoretical prices can drift away away from real market prices. This is because the models used to obtain theoretical prices are subject to many assumptions and are therefore, at best, only an estimate of the real prices. The real market price is determined by random but collective behaviour of market players reacting simultaneously to interest rates, stock price and the issuer’s credit risk.

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