5. Numerical Analysis
In this section, we first discuss the characteristics of FCCBs, synthesis straight bonds,
and call options on FCCB under the following situation: (1) FCCBs without the call and put provisions. (2) Without call or put provisions prior to the maturity date of FCCB asset swap.
(3) With call or put provisions prior to the maturity date of FCCB asset swap. We also show that the relationship of FCCBs, synthesis straight bonds, and call options on FCCBs coincides with Theorem 2. Finally, we also investigate the characteristics of suitable swap rate in FCCB asset swap.
In the following section, we assume that a German corporation issues a ten-year maturity FCCB with face value equal to US$ 100. Each FCCB can convert to 3 shares of the underlying stock. The initial stock price is EUR 25.4. Initial exchange rate is: USD 1 = EUR 0.8155. Hence, the stock price is US$ 31.1465 (equal to 25.4/0.8155). The volatility of the stock return and the instantaneous correlation coefficient of stock return and exchange rate are 20% and 15%, respectively. We also assume that the time-t price function of default-free
zero coupon bond in America is , where . The volatility
of the default-free interest rate in America is equal to 5% and the maturity date of FCCB asset swap (call options on FCCBs) is at the third year.
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1 (
* 025 . 0 015 . 0 exp[
) , 0
( t = − − t−
pα t≥1
5.1 FCCBs without Call and Put Provisions
Using the pricing method for coupon-bearing FCCBs without call or put provisions, we report the numerical values of FCCBs, synthesis straight bonds, and call options on FCCBs by varying the different levels of coupon rate, hazard rate, and volatility of exchange rate (or equivalently, the volatility of stock return) in Exhibit 7.
Since that the higher volatility of exchange (or stock return) makes the conversion right more valuable and the higher level of coupon rate makes the debt component of FCCBs more costly, it is reasonable that the values of FCCBs are increasing function of volatility of exchange (or stock return) and coupon rate in Exhibit 7A. Meanwhile, due to the fact that the
higher level of hazard rate results in higher default probability, and hence the values of FCCBs are decreasing function of hazard rate. Similarly, from Exhibit 7B, the values of synthesis straight bonds are increasing function of coupon rate, and are decreasing function of the hazard rate. Nevertheless, the synthesis straight bonds are debt part of FCCBs, their values are irrelevant with the volatility of exchange rate (or stock return).
The results from Exhibit 7C indicate that the values of three-year maturity call options on FCCBs have a positive relationship with the volatility of exchange rate (stock return). The influence of coupon rate is also clear. If the coupon rate increases, and hence the values of synthesis straight bonds (the strike price) increase, the values of call options on FCCBs decrease. Meanwhile, the higher level of hazard rate causes the higher credit risk and results in the higher option value. However, the higher credit risk makes the stock price jump to zero with a higher probability. The tradeoff causes the relationship between values of call options on FCCBs and the level of hazard rate to be a humped-shaped curve. By the way, from Exhibit 7A, 7B, and 7C, we can see that (19), the FCCB Parity, holds.
5.2 Without Call and Put Provisions Prior to Maturity Date of FCCB Asset Swap We assume that FCCBs are embedded with call and put provisions. The call prices at the 4th, 7th, and 9th years are at par, and the put price at the 3rd, 6th, and 10th year are US$ 107, US$ 110, and US$ 123, respectively. In Exhibit 8, the numerical results are given for different levels of hazard rate, coupon rate, and the volatility of exchange rate. General speaking, the properties are similar to the case of FCCBs without call and put provisions. For example, the values of FCCBs and synthesis straight bonds are increasing function of the coupon rate, and are decreasing function of hazard rate. The values of synthesis straight bonds are also indifferent with the volatility of exchange rate. However, the call and put provisions make the relationship between coupon rate and the values of call options on
FCCBs uncertain. For example, from Exhibit 8C, we can see that their relationship is V-shaped curve at the case that hazard rate is equal to 0.4. Otherwise, they are positive relationship for different levels of hazard rate. Since that the call or put provisions will influence the strike price of call options on FCCBs, it is reasonable for those results.
Similarly, if there is no call and put provisions prior to maturity date of FCCB asset swap, the FCCB parity also holds even though the FCCBs are embedded with call and put provisions.
5.3 Call and Put Provisions Prior to Maturity Date of FCCB Asset Swap
Exhibit 9A, 9B, and 9C report the numerical results of FCCBs, synthesis straight bonds, and call options on FCCBs for the case that call and put provisions are prior to maturity date of FCCB asset swap. The properties of them are the same as the above case. However, the only difference is the FCCB parity is invalid and (18) holds. The numerical results verify Theorem 2.
5.4 Swap Rate for FCCB Asset Swap
For simplicity, we assume that the maturity dates of FCCBs and their asset swap are five and three years, respectively. We discuss three cases as described as follows: (1) Without call and put provisions (2) Put prices at third and fifth years are equal to 110% of the principal, and call price at the fourth and fifth years are at par. 3) Put prices at third and fifth years are equal to 110% of the principal, and call price at the second and fifth years are at par. Hence, there is no call and put provisions prior to maturity date of FCCB asset swap for the first and second cases. In Exhibit 10, we summarize the suitable swap rates of FCCB asset swaps by varying the different levels of coupon rate, hazard rate, and the volatility of stock return (exchange rate) for the above three cases.
In the first and second cases, as the level of coupon rate increases, the present value of
synthesis straight bond also increases, and hence the initial cash outflow paid by credit investor decreases. As a result, it is clear that the swap rate is decreasing function of coupon rate. For the impact of hazard rate, since that the higher level of hazard rate means the high default probability, the swap rate is increasing function of hazard rate. It is deserve to be mentioned that the volatility of stock return is irrelevance for the suitable swap rate, and hence the credit investors only face the credit risk and is free of equity exposure. For the third case, the higher level of hazard rate also makes the suitable swap rate increase. However, the call provision prior to maturity of FCCB asset swap makes the influence of coupon rate and volatility of stock return uncertain.
In practice, the maturity date of FCCB asset swap is the nearness put date of FCCBs, if the issuer do not call or there is no call provision prior to maturity, the suitable swap rate is positive relationship with hazard rate, negative relationship with coupon rate, and unrelated with volatility of stock return. Or equivalently, the credit investor is free of equity exposure, nevertheless, they require high swap rate for the high credit risk exposure.
It is worth to note that the above examples can be explained as a German CB with principals linked to CPI in Germany. The initial CPI is set as 0.8155, the real prices of default-free and risky zero-coupon bonds correspond to the prices of default-free and risky zero coupon bonds in America. Hence, using the same pricing algorithm, we can obtain the theoretical values of inflation-indexed CBs.