# Finance

Created 7/1/1996
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### Warrants and Convertibles

Basics:

• Present value of a perpetuity: C/r
• Present value of a growing (or shrinking) perpetuity: C/(r-g)
• Present value of C dollars t years from now: C/[(1+r)t]
• Present value of a C-dollar t-year annuity: C[(1/r)-(1/[r(1+r)t])
• "Rule of 72": (1+r)t = 2 (approximately) whenever rt=.72
• beta = [E((r1-r*1)(rm-r*m)]/[(rm-r*m)2]
• r*i = r*f + betai(r*m-rf)
• r*a=(D/V)r*d+(E/V)r*e
• Expected return of a portfolio with N securities, a share 1/N invested in each security:
• Standard deviation of a portfolio with N securities, a share 1/N invested in each security:

Types of real options:

• The option to make follow-on investments
• The option to abandon
• The option to delay and learn
• The option to change the scale of operations

Real options allow managers to add value to their firms by acting to amplify good fortune or to mitigate losses. Why "real"? "Real" as opposed to "financial". Vlue of management.

The Value of Follow-on Opportunities:

Mark I microcomputer: negative expected NPV of \$46 million at a 20% per year discount rate.

• But investment in the Mark I gives you the option to invest in a Mark II--with twice the scale--in three years.

Mark II investment is \$900 million; forecasted cash flows of the Mark II have a present value of \$800 million three years hence (thus \$463 million today)--a negative \$100 million NPV investment. Standard deviation of value of the project is \$463 million.

Asset Value/PV(EP) = 0.68

sigma-root t = .61

Call value/asset value = .119; call value = .119 x \$463 = \$55 million.

DCF analysis implicitly assumes that firms hold real assets passively. You could say that DCF does not reflect the value of management. Adding options pricing to DCF allows you to incorporate the value of alternative investment opportunities.

The Option to Abandon:

1. Technology A uses computer-controlled machinery custom-designed to produce the complex shapes required for Wankel (rotary) engines in high volumes and at low costs. But if the Wankel engine doesn't sell, this equipment will be worthless.

2. Technology B uses standard machine tools. Labor costs are much higher, but the tools can be sold or shifted to another use if the engine doesn't sell.

You won't be surprised to hear that technology B is like holding a put option...

The timing option:

Suppose a project is not "now or never". Suppose you can invest right away or wait. If the project is truly a winner, waiting means loss or deferral of positive early cash flows. If the project is a loser, waiting could prevent a bad mistake.

The opportunity to invest in a positive NPV project is equivalent to an in-the-money call option. Optimal investment means exercising that call at the best time.

A call option on a stock that pays dividends...

Current value = \$200 million; If demand is high in the future, project value rises to \$250 million; if demand is low, project value falls to \$160 million.

If you delay the investment, you miss out on the first year's cash flow (\$16 or \$25 million). If you delay, you avoid a mistake in which you invest \$180 million in a plant that turns out to be worth only \$160 million.

Call option. Worth \$70 million in the good state next year; worth zero in the bad state; call option today worth \$22.9 million which is more than the \$20 million payoff from immediate exercise.

Is the binomial method for valuing options merely another application of decision tree analysis? Yes--with the proviso that option pricing theory is a very compact and powerful way of summarizing certain kinds of decision trees.

Flexible Production: the opportunity to change your scale of operations.

Option Value at a glance:

• American calls--no dividends. Don't exercise it before maturity; treat it as a European call.
• European puts--no dividends. Value of put = Value of call - value of underlying + PV(EP)
• American puts--no dividends. It can sometimes pay to exercise an American put before maturity in order to reinvest the exercise price. For example, suppose that immediately after you buy an American put the price of the underlying falls to zero. In this case it is certainly better to exercise the put immediately; valuing it is not easy: Black-Scholes does not apply.
• European calls on dividend-paying stocks; reduce the price of the stock in Black-Scholes by the PV of the dividends paid before the option matures, because some of today's stock value is made up of those dividends.
• American calls on dividend paying stocks. If the dividend is sufficiently large, you might want to capture it and exercise just before the ex-dividend date.

Real options encountered in practice areusually pretty complicated...

Warrants and Convertibles

A warrant is an American call option issued by a company on its own stock.

Complications of warrants:

• Exercise before maturity to capture the dividends
• Dilution. If the warrants are exercised, the stock price goes down because the number of shares goes up.
• If there are warrants outstanding, then ownership of a stock entails the obligation to satisfy a call from the holders of the warrants.

A convertible bond is a close relative of the bond + warrant package. In 1991 Wendy's International issued \$100 million of 7% convertible bonds due in 2006 convertible at any time to 81.3 shares of common stock. Face value of \$1000; 1000/81.3 = \$12.30 as the conversion price of the bond.

A LYON: a callable and putable, convertible zero-coupon bond.

Chemical Waste Management's 1990 LYON; price of 30.7%; 20-year zero coupon bond convertible at any time into shares; ocmpany had the option to call he bond for cash; bondholders had an option to put the bond for cash; exercise prices for both increased each year.

Position diagram of a convertible bond (p. 625).

If you want to estimate a convertible's value, first value the bond part, then add the value of the option to convert.

"Calling" a bond ;by a company means that the company "calls" the bond back; it is not a call option written on the company. It is a "call" option held by the company.

Why Do Companies Issue Warrants and Convertibles?

Convertibles tend to be issued by saller and more speculative firms; they are almost invariably unsecured and generally subordinated. What can the management of such a firm do to ensure yu that its intentions are honorable? Give you a share of the upside potential for your bond.

Convertibles also provide tax shields that common stocks do not. Convertible as a contingent issue of equity.

Summary of Options Pricing