A.H.Y. Chen et al. International Review of Economics and Finance 8 1999 327–338 333
As shown in Eq. 4, three modifications to the conventional Black and Scholes option pricing model are necessary. First, S 1 MNW must be substituted for the
stock price, which represents the equity per share of stock. Second, the standard deviation used in Eq. 4 is the standard deviation of equity, i.e., S 1 MNW, rather
than the stock volatility. Finally, the entire conventional Black and Scholes model is multiplied by NNQ 1 M. Furthermore, a direct solution in Eq. 4 for the warrant
price is not possible as W appears on both sides of the equation. Hence, the solution must be solved through an iterative process.
Substituting the dilution-adjusted Black and Scholes option pricing model in Eq. 4 into Eq. 3 for C yields:
V 5 S 2 {W[S,Q
1
5 1,X
1
5 20,T] 2 FW[S,Q
2
5 0.806,X
2
5 24.81,T]} 1 I, 5
Where W[.] 5 current warrant price as a function of current stock price, the number of shares that can be purchased by each warrant, exercise price, and time to maturity.
Furthermore, since DECS are also redeemable at the discretion of Masco Tech on or after July 1, 1996, we follow Finnerty 1993 in handling early redemption and
conversion issues. Specifically, each DECS share is valued using Eq. 5 based on six alternative redemption dates, i.e., the initial redemption date, the four dates when
the redemption price steps down, and the maturity date. Since Masco Tech controls the optional redemption provision and will rationally choose the exercise date that
minimizes the value of the outstanding DECS, the lower of the six estimated values outlined above is used as the predicted DECS value.
3. Application of DECS
The sample period covers approximately twelve months after issuance, spanning from July 2, 1993 to June 30, 1994. As indicated in Eq. 5, in addition to the number
of outstanding common stock shares and warrants, exercise price, and dilution factor, the theoretical DECS value is determined by five other factors: 1 Masco Tech’s
common stock price, S; 2 the time to maturity of the DECS, T; 3 the risk-free rate, r; 4 the volatility of the equity of Masco Tech, s; and 5 the discount rate for
incremental dividend payments.
Among the aforementioned five factors, the last three factors must be estimated. First, the risk-free rate, r, is proxied by the annualized yield to maturity for the
Treasury note with a maturity date closest to that of the DECS. Second, following Finnerty 1993, the present value of the incremental dividend stream is calculated
by discounting incremental dividend payments at the quarterly yield to maturity for preferred stock instruments as reported in the Standard Poor’s Security Price Index
Record.
The Standard Poor’s preferred stock yield index is calculated from the yields at which ten high-grade preferred stock issues are trading. Finally, because the
equity volatility cannot be directly observed, we have to estimate the equity volatility implied by the warrant i.e., DECS price observed in the market. Specifically, for a
given sample date one day before pricing, the implied standard deviation ISD of
334 A.H.Y. Chen et al. International Review of Economics and Finance 8 1999 327–338
Fig. 2. Implied ISD and historical HSD volatility plots for DECS.
equity is computed using Eq. 5, the closing DECS price, and values of other inputs. This value is then used for the next day to price the DECS.
Fig. 2 depicts a time-series plot for the ISD estimates for the Masco Tech DECS. For comparison purposes, the historical standard deviation HSD of common stock
estimates, based on the standard deviation of the logs of the previous prior to valuation 100 trading days’ percentage changes in Masco Tech’s stock prices, are also depicted
in Fig. 2. As shown, the ISDs the equity volatilities appear large relative to the HSDs the stock volatilities over the entire sample period, and this finding reinforces that
equity volatilities, not stock volatilities, should be used to price warrants.
Table 2 presents a comparison between the market prices and the theoretical values for the Masco Tech DECS using the ISD measure in Eq. 5. To allow comparisons
of prices across various dollar levels, the statistics are computed on a percentage, rather than absolute, basis. The sample period is further subdivided into two six-
month intervals to examine the stationarity of the sample statistics. As shown in Table 2, the mean pricing errors are positive and statistically significant at the 1 level for
all three periods, ranging from 0.77 to 1.30. The mean pricing error in the first subperiod is slightly smaller than its counterpart in the second subperiod.
Overall, the above results indicate that, on average, the Masco Tech DECS issue had been slightly overpriced during the sample period. The magnitude of the overpricing
reported here is very similar to the first few PERCS issues as documented by Finnerty
A.H.Y. Chen et al. International Review of Economics and Finance 8 1999 327–338 335
Table 2 Statistical results of comparisons between market and theoretical prices for Masco Tech’s DECS
Panel A Panel B
Panel C 7293–63094
7293–123193 1193–63094
N 5 249 N 5 124
N 5 125 Mean
SD
a
Mean SD
a
Mean SD
a
Masco Tech stock price 21.55
3.35 22.02
2.03 21.07
4.23 DECS market price
20.76 2.51
21.22 1.29
20.31 3.24
Theoretical DECS price using ISD
20.55 2.47
21.05 1.29
20.04 3.17
a. Bull call spread value 3.51
1.01 3.70
0.68 3.31
1.23 Difference
b
2 and 3 1.03
1.55 0.77
1.08 1.08
1.87 10.52
c
7.88 7.75
ISD 44.64
7.96 45.41
7.37 43.88
8.47 HSD
35.43 7.20
30.75 3.08
40.08 7.10
Present value of incremental dividends
d
2.51 0.26
2.73 0.13
2.28 0.13
a
SD 5 standard deviation.
b
Difference 5 Market Price 2 Theoretical PriceMarket Price.
c
Numbers in parentheses are t-values.
d
Up to either redemption date or maturity date whichever is applied in the empirical test. Significant at the 0.01 level.
1993. This overpricing is in accordance with Tufano’s 1989 argument that mispricing tends to be more evident in the first few issues than it is in later issues, as a seasoning
process will typically take place when an innovative security enters the market. Further- more, although the magnitudes in overpricing resulted from using the equity volatility
measure are statistically significant, they are, however, quantitatively small. Recogniz- ing transaction costs and bid-ask spreads, the mispricing reported in Table 2 is unlikely
to result in any profitable arbitrage.
In addition, the overpricing can be attributed to the following four factors. First, DECS are listed on the New York Stock Exchange and Salomon Brothers makes an
active secondary market in DECS, thus providing liquidity for investors. Second, as in the primary market, most of the buyers were equity income funds. The high dividend
rate that DECS provide attracts investors who would not be attracted by the lower yield on the underlying yet still want exposure to the common stock. Third, following
Chen, Kensinger, and Pu’s 1994 argument, DECS provide “one-stop shopping” for a covered bull-call-spread strategy. This could theoretically save investors transaction
costs and help reduce their hedging costs. Especially for the institutional investors, it will be difficult, if not impossible, for them to replicate the payoff for the DECS.
Finally, dividends paid on the DECS will qualify for a 70 intercorporate dividends- received deduction subject to the minimum holding period generally at least 46 days.
DECS also offer advantages to the issuer. First, the issuer benefits from effectively selling its common stock at the point that it issues the security. If its stock performs
well, the conversion conditions insure that the company only issues a partial 0.806
336 A.H.Y. Chen et al. International Review of Economics and Finance 8 1999 327–338
Table 3 Sensitivity of the DECS values to changes in the stock price and changes in the equity volatility, the
risk-free rate, and the discount rate for the incremental dividend stream
a
Panel A. Sensitivity of the DECS values to changes in the equity volatility Equity volatility per year
b
Stock price 20
30 40
50 60
10.0 13.19
12.73 12.36
12.04 11.75
15.0 16.80
16.45 16.91
16.56 16.24
20.0 20.99
20.72 20.41
20.09 19.76
25.0 23.93
23.92 23.75
23.48 23.19
30.0 26.91
27.07 26.99
26.86 27.67
Panel B. Sensitivity of the DECS values to changes in the risk-free rate Risk-free rate per year
c
2.78 3.78
4.78 5.78
6.78 10.0
12.49 12.42
12.36 12.29
12.22 15.0
17.52 16.99
16.91 16.82
16.73 20.0
20.65 20.53
20.41 20.29
20.17 25.0
24.03 23.88
23.75 23.61
23.47 30.0
27.31 27.15
26.99 26.87
26.76 Panel C. Sensitivity of the DECS values to changes in the discount rate for the incremental
dividend stream Discount rate per year
d
6.92 7.92
8.92 9.92
10.92 10.0
12.50 12.43
12.36 12.29
12.22 15.0
17.05 16.98
16.91 16.84
16.77 20.0
20.55 20.48
20.41 20.34
20.28 25.0
23.88 23.81
23.75 23.68
23.61 30.0
27.13 27.06
26.99 26.92
26.86
a
The simulated DECS values in Panels A, B and C assume: 1 an annual incremental dividend payment of 1.12; 2 a maturity of four years; and 3 Q
1
5 1, X
1
5 20, Q
2
5 0.806, X
2
5 24.81.
b
Base stock price volatility is 0.40 or 40 per year.
c
Base risk-free rate is 4.78 per year.
d
Base discount rate for the incremental dividend stream is 8.92 per year.
share at maturity, thus benefiting from a portion of the upside. If its stock falls in value, the issuer is guaranteed a minimum price 20 for the share. Furthermore,
DECS get the best equity credit next to issuing common stock from rating agencies due to their mandatory conversion feature.
Because the pricing results presented above may be affected by the magnitude of the variables required, it is instructive to examine the sensitivity of these results to
changes in the input values. Table 3 presents the results of sensitivity tests for the impacts of changes in the stock price and volatility Panel A, the risk-free rate Panel
B, and the discount rate for the incremental dividend stream Panel C, respectively,
A.H.Y. Chen et al. International Review of Economics and Finance 8 1999 327–338 337
on DECS values. The simulated values in Panels A, B, and C of Table 3 assume a DECS with the following characteristics: 1 an annual incremental dividend payment
of 1.12; 2 a maturity of four years; and 3 Q
1
5 1, X
1
5 20, Q
2
5 0.806, X
2
5 24.81. The base values for the three inputs are: 1 an equity volatility of 40 per
year; 2 a risk-free rate of 4.78 per year; and 3 a discount rate for the incremental dividend stream of 8.92. These three base input values approximate their initial
levels at issuance.
As shown in Panel A, the DECS values are not sensitive to changes in equity volatility. When the stock price is 20, an increase of volatility from 40 to 50 an
increase of 25 only decreases the value of the DECS from 20.41 to 20.09. This result is not surprising because there are long and short positions in call options with
different exercise prices embedded in the DECS value. Thus, a counter movement in the short call option’s value largely offsets an increase in the long call option’s
value due to an increase in the equity volatility. Panels B and C of Table 3 illustrate the sensitivity of the theoretical DECS values to changes in the risk-free rate and the
discount rate for the incremental dividend stream. As anticipated, changes in both interest rates have a negligible impact on the DECS value.
4. Conclusions
