Chapter 24, Problem 16QP

Summary Introduction

To determine: Number of warrant, company should issue.

Warrant:

Warrant is given to the security holder those who purchase the shares. It is not the obligation of the share holder it’s a right of the share holder to purchase the share at a fixed price.

Explanation of Solution

Formula to calculate the number of the warrants,

$\text{Present}\text{value}\text{of}\text{liability}=\text{AW}\left(\frac{\text{Common}\text{stock}}{\text{Common}\text{stock}+\text{AW}}\right)\times \text{Call}\left(\text{Value}\text{of}\text{stock}\text{market}\right),\$95$

Where,

• AW is the number of the warrant.
• Call(S, E) is the call option.

Substitute $17,468,019.6 for the PV liability, $2,700,000 for the common stock, $88.89 for the value of the stock market.

$\begin{array}{c}\text{17,468,019}\text{.6}=\left(\text{AW}\right)\left[\frac{2,700,000}{2,700,000+\text{AW}}\right]\times \text{Call}\left(\$88.89,\$95\right)\\ \text{AW}=3,756,683\end{array}$

Working notes:

Calculate the present value of the liability,

$\begin{array}{c}\text{Present}\text{value}\text{of}\text{liability}=\text{Present}\text{value}\text{of}\text{the}\text{share}-0.6\left(\text{Period}\text{time}\right)\\ =\$18,000,000\text{e}-0.06\left(0.5\right)\\ =\$17,468,019.60\end{array}$

Calculate the value of the stock price,

$\begin{array}{c}\text{Stock}\text{price}=\frac{\text{Market}\text{value}\text{of}\text{the}\text{assets}}{\text{Commom}\text{stock}}\\ =\frac{\$240,000,000}{2,700,000}\\ =\$88.89\end{array}$

Calculate the value of the single warrant,

$\text{W}=\left[\frac{\text{A}}{\text{A}+\text{AW}}\right]\times \text{Call}\left\{\text{S}=\left(\frac{\text{V}}{\text{A}}\right),\text{E}=\text{EW}\right\}$

Where,

• A is the number of shares.
• AW is the number of warrants.
• Call(S, E) is the call option.
• V is the value of the debt.
• EW is the strike price of each debt.

$\begin{array}{c}\text{W}=\left[\frac{2,700,000}{2,700,000+\text{AW}}\right]\times \text{Call}\left\{\left(\frac{\$240,000,000}{2,700,000}\right),\text{E}=\$95\right\}\\ =\left[\frac{2,700,000}{2,700,000+\text{AW}}\right]\times \text{Call}\left\{\left(\$88.89\right),\text{E}=\$95\right\}\end{array}$

Since the firm must raise $17,468,019.60 as a result of the warrant issued, and $\text{AW}\times \text{W}$ must equal $17,468,019.60.

So,

$\begin{array}{l}\$17,468,019.60=\left(\text{AW}\right)\left(\text{W}\right)\\ \$17,468,019.60=\left(\text{AW}\right)\left(\frac{2,700,000}{\left(2,700,000+\text{AW}\right)}\right)\times \text{Call}\left(\$88.89,\$95\right)\end{array}$

Calculate the value of the warrant by using the black schools, for the call option,

Calculate the ${\text{D}}_{\text{1}}$,

${\text{D}}_{\text{1}}=\left[\frac{\text{In}\left(\frac{\text{S}}{\text{K}}\right)+\left(\text{R}+\frac{{\text{σ}}^{\text{2}}}{2}\right)\times \text{t}}{\sqrt{{\text{σ}}^{\text{2}}}\times \text{t}}\right]$

Where,

• S is the value of the stock.
• K is the exercise price.
• R is the risk rate.
• T is the tenure of the option.
• $\sigma$ is the standard deviation.

Calculate D₁,

$\begin{array}{c}{\text{D}}_{\text{1}}=\left[\frac{\text{In}\left(\frac{\$88.89}{\$95}\right)+\left(.06+\frac{0.50}{2}\right)\times .5}{\left(\frac{\sqrt{0.50}\times 5}{2}\right)}\right]\\ =0.736\end{array}$

Calculate the D2,

$\begin{array}{c}{\text{D}}_{\text{2}}={\text{D}}_{\text{1}}-\left(\frac{\text{σ}\times \sqrt{\text{t}}}{2}\right)\\ =0.736-\left(\frac{0.5\times 0.5}{2}\right)\\ =0.611\end{array}$

Calculate the $\text{N}\left(\text{d1}\right)$,

$\begin{array}{c}\text{N}\left(\text{d1}\right)=\text{N}\left(0.736\right)\\ =0.27004\end{array}$

Calculate the $\text{N}\left(\text{d2}\right)$,

$\begin{array}{c}\text{N}\left(\text{d2}\right)=\text{N}\left(0.611\right)\\ =0.378162\end{array}$

Black scholes formula to calculate the value of one option,

$\begin{array}{c}\text{C}={\text{S}}_{\text{x}}\times \text{N}\left(\text{D}\right)-{\text{K×e}}^{\text{-Rt}}\times \text{N}\left({\text{D}}_{\text{2}}\right)\\ =\left(\$88.89\right)\left(0.27004\right)-\left(\$95\right)\times 0.06\left(0.5\right)\left(0.378162\right)\\ =\$24.004-\$1.078\\ =\$22.92\end{array}$

Conclusion

Hence, the number of the warrants is 3,756,683.