Pricing Convertible Securities

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THEEDGE MALAYSIA | JANUARY 30, 2012capital  32

derivative world

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onvertible securities are traded wide-ly in global debt markets. The mostcommon convertible securities areconvertible bonds (CBs) and con-

 vertible preferred stocks. The CB isan equity-linked instrument thatgives the holder of the bond the right to con- vert the bond into a predetermined number of common stocks in the future. It is attractiveto investors as it provides the downside pro-tection of a straight bond with coupons andprincipal payback at maturity plus the upsidereturn of equities.

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

Features of a CBCompany Q issues a convertible bond with aconversion 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.

T h e s t a t e d c o n v e r s i o n p r i c e i stherefore: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 five years,the CB has an investment value. Assuming the risk-free discounting rateis 2.5% and the credit spread is zero, the invest-ment value of the bond can be derived by dis-counting its cash flow at the risk-free discountrate, as shown in Chart 1.

Now, the conversion price, realistically, is revised to:Investment value of the CB / conversion ratio

= $1162.60 / 25.32= $45.92The CB example above is one of the most

 basic. CBs are often accompanied by a redeem-able feature that allows the issuer to call backthe bond after a certain period, at a predeter-mined price. In some cases, the issuer can onlycall back the bond if the price of the underlyingstock is above the conversion price by a certainpercentage. The callable or redeemable featuremakes the CB to some extent cheaper than anon-redeemable CB.

Price behaviour of CBsThe price of the CB is a ected by movementsin interest rates (and credit spreads) as wellas movements in the stock prices. Taking theexample of Company Q’s CB, there are two ex-treme cases:

(i) When the stock price of Q is relatively smallto the conversion price of $45.92, the CB is very unlikely to be converted and,therefore,it is e ectively a straight bond that can bepriced using the standard bond-pricingmethod.

(ii) When the stock price is very high relative tothe conversion price, the CB will certainly be converted into stock. The CB price willthen be the conversion value, which is thestock price multiplied by the conversionratio.

For example,if the stock price is $60,this CBis very likely to be converted. Its price will be-have very much like the stock. The price of theCB in the market is likely to be near $1,519.20(stock price of $60 multiplied by the conversionratio of 25.32)

In Chart 2, the solid line corresponds to theconversion value,which linearly increases withthe stock price.The horizontal dashed line corre-sponds to the price of a straight bond and is nota ected by the stock price. Generally, since youhave the option to make the convertible bondeither a bond or a stock, its value should be atleast the value of the bond or the stock. This isrepresented by the dotted line.

Pricing a CB as a straight bond plus a calloption on the stockIntuitively,a CB must be worth at least the valueof a non-convertible bond of similar charac-teristics. If the holder does not convert, he willreceive the coupons and the principal back.However, the CB is worth more than this as theholder has a chance to participate in the stockprice movement.

This conversion feature resembles a calloption on the stock. The conversion premiumincreases as the spot price or the volatility of the stock increases.

In Chart 3, say the current stock price is $40.This price is not too far from the conversion price(which e ectively now is the strike price of thecall option) of $45.92.The CB holder e ectively is

holding the option to buy the stock at $45.92.Asthe stock price increases beyond $40,we expectthe call option to get more expensive.

The call option here is priced using the Black Sc-holes model. The parameters are:Current stock price: $40 Strike price: $45.92Risk-free rate: 2.5%Stock volatility: 20%Maturity: five yearsThe CB value of $1,169.47 is then derived by

adding the call option value of $6.87 to the in- vestment value of the bond of $1,162.60.

Pricing a CB as a stock plus put option onthe stockOwning a CB can also be seen from another an-gle. The holder of the CB will only choose notto convert the CB into stock if the stock price is

 below the strike price of $45.92. This is as goodas saying that the owner already has the stock, but has the option to sell the stock if its pricegoes down beyond a certain level. In return, hegets the returns from the bond.

In Chart 4,the stock value is simply the con- version value (stock price of$40 multiplied bythe conversion ratio of 25.32). The put option isarrived using the Black Scholes model,with thesame parameters used to arrive at the call optionon the stock in the first pricing method.

Multiplying the put option value ($7.39) withthe conversion ratio (25.32) would give us theput value embedded in the CB. Indirectly, thisput value represents the fixed income value of the convertible.

ConclusionThere are also other more advanced pricing tech-

niques (but less commonly used) in the marketlike multi-factor models that take into accountseveral stochastic factors that simultaneouslya ect 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 theoreti-cally priced using a few techniques to resem- ble their real market prices.However in certainsituations (such as in periods of high volatility)theoretical prices can drift away away from realmarket prices. This is because the models usedto obtain theoretical prices are subject to manyassumptions and are, therefore, at best, onlyan estimate of the real prices. The real marketprice is determined by the random but collec-tive behaviour of market players reacting si-multaneously to interest rates, stock price andthe issuer’s credit risk.

Jasvin Josen is an ex-investment banker fromEurope, specialising in the areas of valuationand risk in financial derivatives. Currently,she is back in Malaysia providing consultancyand training. Readers can follow her at http://derivativetimes.blogspot.com/ and send theircomments to Jasvin@souqmatters.com

Pricing convertible securities

 Jasvin Josen

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

Chart 2: Convertible Bond Price Chart

CONVERTIBLE BONDS,ZHI DA 2004

Chart 3: CB priced as a straight bond plus a call option on the stock

Chart 4: CB priced as a stock plus put option on the stock

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