[Theoretical Analysis] When will convertible bonds be converted?

This article will discuss when rational market participants choose to convert convertible bonds into shares.
The roughest view is to treat a convertible bond as a combination of an ordinary bond and an American call option. Accordingly, the conversion of a convertible bond is approximately equivalent to exercising an option. Under American call option theory with a non-dividend-paying stock as the underlying, it is theoretically impossible to exercise early—first, if you plan to hold the underlying for a long time, you’d be better off putting the money in the bank to earn interest and then exercise at maturity; second, even if you plan to take profit, you can still simply sell and close the position to recover the time value. Similarly, a rational convertible bond holder will not choose to convert before maturity, and will only make a decision at maturity.
Of course, the reality of convertible bonds is more complex. First, exercising an American option turns a fixed amount of cash into a fixed number of shares, whereas converting a convertible bond turns a fixed number of bonds into a fixed number of shares. The pure bond value of a convertible bond is affected by a series of factors such as remaining term and market interest rates, and therefore it is not strictly equal to par value. Thus, the strike price of the option embedded in a convertible bond is not a fixed conversion price; rather it is a floating value—specifically, conversion price * par value / pure bond value. Second, during the life of a convertible bond, certain special events will occur, such as downward adjustment, compulsory redemption, repurchase, interest payment on the convertible bond, and dividends on the underlying shares. All of these will affect the value of the convertible bond.
Below, we may start with a simple scenario: assume the market is full of rational investors, such that the price of the convertible bond accurately and fairly reflects its value, and the holder only needs to consider whether to convert or to keep holding. Cases where the convertible bond market misprices the instrument can be discussed later.
0. Trading days with no special events
On day T, buy the convertible bond; on day T, you can submit a conversion. On day T+1, you can sell the underlying shares. If on day T you buy the underlying shares, then on day T+1 you can sell the underlying shares. Therefore, the right to buy a convertible bond on day T is no worse than buying the corresponding number of underlying shares. The value of the convertible bond is therefore greater than or equal to its conversion value. On trading days with no special events, a convertible bond holder will certainly choose to continue holding rather than convert.

1. Downward adjustment (downward adjustment)
   A downward adjustment lowers the strike price, increasing the option’s intrinsic value. Therefore, if the holder expects a downward adjustment to occur, he should be even less willing to convert before it happens. After the downward adjustment ends, the convertible bond holder obtains a new option with a lower strike price. Whether this option is newly obtained due to the downward adjustment or it was originally already held does not affect the holder’s investment decision. He will continue holding by following the same logic as on ordinary trading days.
2. Compulsory redemption (compulsory redemption)
   Compulsory redemption shortens the remaining term, reducing the option’s time value. However, compulsory redemption only causes loss of part of the time value, while conversion would cause loss of all the time value. Judging from this, the sensible approach still is to hold until the very last moment to make the choice.
   However, in practice there is a subtle detail: the compulsory redemption price and the pure bond value are not strictly equal. Therefore, compulsory redemption not only shortens the term, but also brings additional gains or losses to the party that still holds the option and does not exercise. Quantitative analysis of the value of this kind of exotic option is difficult; but qualitatively, if the compulsory redemption price is less than the pure bond value, then the option’s value will drop more than it would from merely shortening the term, and the decline could even exceed the original time value. Therefore, if the holder expects that compulsory redemption will occur in the future, and the compulsory redemption price is clearly below the pure bond value, then he will choose to convert before the compulsory redemption announcement date.
3. Repurchase (repurchase)
   Repurchase gives convertible bond holders one more chance to choose, improving the option value; therefore, holders will not convert if they expect repurchase to occur. After repurchase ends, it of course will not affect the decision anymore.
4. Interest payment on the convertible bond (interest payment on the convertible bond)
   Assume that day T is the bondholder record date. If a holder submits a conversion on day T, he will receive a certain amount of underlying shares on day T+1. If he submits a conversion on day T+1, then he will receive the same amount of underlying shares on day T+2, along with the interest for that year. Suppose holding the underlying shares one additional day allows an extra 0.1%-3% in interest (depending on the specific situation); this is equivalent to an annualized 28.39%-161822.12%. Surely, a normal person would not convert on day T. Similarly, converting a few days before an interest payment is obviously extremely unwise; after the interest payment, it returns to the logic of ordinary trading days.
5. Dividends on the underlying shares (dividends on the underlying shares)
   Suppose we are discussing options: dividends on the underlying shares reduce the option’s intrinsic value, and thus the option holder might exercise before the dividend. However, the rules of convertible bonds regarding dividends are different. After the dividend, the convertible bond’s conversion price is also adjusted downward by the per-share dividend amount.
   This protects the interests of convertible bond holders, but it will significantly reduce their willingness to convert. Even so, early conversion is not entirely impossible.
   Due to the author’s limited ability, the discussion below does not consider how the option value of a convertible bond changes before and after dividends on the underlying shares. Instead, it presupposes the time value of the convertible bond’s option after the dividend, and uses that to calculate the necessary conditions under which a holder would convert before the dividend.
   Assume that on day T a convertible bond holder holds the convertible bond. Day T+1 is the start date for the conversion suspension; day T+2 is the equity record date; and day T+3 is the ex-rights/ex-dividend date. Let the conversion price on day T be P, the underlying share price be Q, the cash dividend per share be X, the time value on day T+3 for each convertible bond be U, and the pure bond value for each convertible bond be V. Since we only involve a few trading days, the short-selling fee rate can basically be ignored. According to the no-arbitrage principle, the mathematical expectation of the share price on day T+3 and after is Q - X.
   The necessary condition for a holder to convert on day T is that the final assets obtained by converting on day T are higher than the value of the convertible bond on day T+3. The value of a convertible bond equals intrinsic value plus time value: max{conversion value, pure bond value} + time value. Therefore, we can write the following two inequalities:
   (100 / P) * (Q - X) + (100 / P) * X > (100 / (P - X)) * (Q - X) + U
   (100 / P) * (Q - X) + (100 / P) * X > V + U
   Solving yields:
   ((V + U) / 100) * P < Q < (1 - (U * (P - X)) / (100 * X))* P
   For such a Q to exist, we need (V + U) / 100 < 1 - (U * (P - X)) / (100 * X). Solving gives X > (U * P) / (100 - V).
   Thus, in order for holders to convert before the dividend, the following must be satisfied: the pure bond value is less than par value, the time value is very small, the dividend amount is relatively large, and the underlying share price is appropriate (slightly lower than the conversion price). There are many conditions; in practice, they may be difficult to meet.
   That concludes the discussion of the efficient market. In reality, the market sometimes makes mistakes, causing convertible bonds to trade at a discount—its price is less than its conversion value. In this case, rational underlying-share investors should of course sell the underlying shares, buy the convertible bond, and submit a conversion to realize risk-free arbitrage.
   To summarize: rational market participants convert before maturity mainly in the following three situations:
   (a) when the pure bond value is higher than the compulsory redemption price, exercising the compulsory redemption;
   (b) when the pure bond value is far below 100 yuan and the underlying share market price is slightly lower than the conversion price, conducting large-scale underlying share dividends;
   © when the convertible bond price is below its conversion value.
   Finally, let’s talk about what this article does not cover—the credit risk of convertible bond default is insufficiently considered; the scenario where the convertible bond issuance size is so large that it clearly dilutes existing shareholders’ equity is insufficiently considered; the shock/impact costs of large capital operations are insufficiently considered; the tax treatment of related actions is insufficiently considered; and the voting rights and control associated with stock representation are insufficiently considered as well.
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   Note: The author has participated in securities trading for less than 24 months, does not meet the conditions specified in documents such as the Shanghai Stock Exchange’s “Notice on Matters Concerning the Suitability Management of Investors in Convertible Corporate Bonds” and the Shenzhen Stock Exchange’s “Notice on Improving Matters Concerning the Suitability Management of Investors in Convertible Corporate Bonds,” and has not opened the permissions for convertible bond trading, nor has actually participated in convertible bond trading. The above content is purely for theoretical derivation. If there are any errors, please point them out and let’s discuss.