warrant

							USING THE OPTION-PRICING TECHNIQUE

TO VALUE CORPORATE SECURITIES

WE SAW MANY SITUATIONS WHERE

WARRANTS ARE USED:

- UNDERWRITER COMPENSATION IN

 IPOS (WARRANTS)

- PART OF VENTURE CAPITALIST’S

 EQUITY STAKE IN A FIRM

- COMPENSATION PAID BY ACQUIRING

 FIRM IN TAKEOVERS TO TARGET FIRM

 SHARE HOLDERS MAY INVOLVE

 WARRANTS (DEBT WITH WARRANTS,

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 CONVERTIBLE DEBT, OR

 CONVERTIBLE PREFERRED EQUITY)

- OUTSIDER’S STAKE IN LEVERAGED

 BUYOUTS MAY INVOLVE WARRENTS

MANY RECENT FINANCIAL

INNOVATIONS INVOLVE TRADITIONAL

SECURITIES PACKAGED WITH ONE OR

MORE OPTION-LIKE SECURITY:

(1) EQUITY NOTES: STRAIGHT, COUPON-

  BEARING DEBT WITH THE

  OBLIGATION TO CONVERT TO

  EQUITY AT A CERTAIN STRIKE PRICE,

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  AFTER A CERTAIN NUMBER OF

  YEARS.

(2) PERCS: PREFERRED EQUITY THAT

  MUST BE CONVERTED TO EQUITY

  AFTER A CERTAIN NUMBER OF

  YEARS.

  E.g: ISSUED BY RJR NABISCO IN 1991.

I. WARRANT VALUATION

IMPORTANT DIFFERENCE BETWEEN

WARRANTS AND CALL OPTIONS:

- CALLS ARE “SIDE-BETS” BETWEEN

 OUTSIDERS.

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- WARRANTS ARE ISSUED BY THE FIRM

 ITSELF.

TWO CONSEQUENCES OF THIS:

(1) THE ISSUE OF WARRANTS AFFECTS

  EQUITY VALUE  SINCE WARRANT

  PROCEEDS GO TO THE FIRM, SHARE

  PRICE HAS TO GO UP (CASH GOES

  INTO THE FIRM-UPON ISSUE).

(2) WHEN WARRANTS ARE EXERCISED

  (IF THEY ARE EXCERCISED), NEW

  SHARES ARE ISSUED, DILUTING FIRM

  EQUITY.

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WE HAVE TO ADJUST FOR BOTH

EFFECTS IF WE ARE TO VALUE

WARRANTS ACCURATELY (IF

DILUTION EFFECTS ARE

SIGNIFICANT).

RELATIONSHIP BETWEEN WARRANT

VALUE AND THAT OF AN ORDINARY

CALL ON AN EQUIVALENT FIRM.

CONSIDER A FIRM WITH:

n  SHARES OUTSTANDING (NO.)

m  WARRANTS OUTSTANDING

  m/n, “DILUTION FACTOR”.
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TO VALUE WARRANTS, FIRST START

WITH PAYOFF AT EXPIRATION:

WT = MAX [ST –X, 0]             (1)

ST  STOCK PRICE AT T (TIME)

X  EXERCISE PRICE

ST = (VT + mX)/(n + m)              (2)

SUBSTITUTING FOR ST FROM (2) IN (1):

WT = MAX [(VT + mX)/(n + m) – X, 0]

= MAX [(VT/n – X)/(1 + ), 0]

= [1/(1 + )] * MAX [VT/n – X, 0]




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BUT VT/n CAN BE THOUGHT OF AS

THE SHARE PRICE OF AN

EQUIVALENT, ALL EQUITY FIRM,

WITH SHARE PRICE S = VT/n,

n SHARES OUTSTANDING.

 WT = [1/(1 + )] * MAX [S – X, 0]

BUT CT = MAX [S – X, 0] ORDINARY

CALL VALUE

 AT EXPIRATION,

WT(S,X) = [1/(1 + )] * CT(S,X)




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BUT, IF THERE IS NO RISKLESS

ARBITRAGE, THIS RELATIONSHIP

HAS TO HOLD TODAY AS WELL:

W(S,X) = [1/(1 + )] * C(S,X)   (6)

WHAT IS THE RELATIONSHIP

BETWEEN S AND S?

S = (V – mW)/n, S = V/n

 S = S + (m/n)W

S = S + W                (7)




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VALUING A WARRANT ON A

DIVIDEND PAYING FIRM WITH THE

BLACK-SCHOLES FORMULA

PROBLEM-1:

S = $50/SHARE

X = $60

T = 5 YEARS

 = 0.2/YEAR

R = 10% (RISK-FREE)

y = DIVIDEND YIELD = 2%

ASSUMPTION: CONTINUOUS

DIVIDEND YIELD y (GOOD

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APPROXIMATION FOR LONG-TERM

OPTIONS)

BLACK-SCHOLES FORMULA FOR AN

OPTION ON A STOCK PAYING A

CONTINUOUS DIVIDEND YIELD:

C = Se-yTN(d1) – Xe-RTN(d2)

d1= [Ln(S/X)

          + (R + (1/2)2 – y)T]/(T)

d2 = d1 - T

*HOWEVER NO NEED TO USE THIS

HERE; EASIER TO USE OPTION

TABLES, INCLUDING IN FINAL EXAM.
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THIS GIVES A VERY GOOD

APPROXIMATION.

STEP-1: ADJUST FOR DIVIDENDS

S* = Se-yT = 50e-0.02(5) = $45.24

STEP-2: COMPUTE DILUTION FACTOR,



 = m/n = 500,000/1,000,000 = 1/ 2 (VERY

SIGNIFICANT).

STEP-3: COMPUTE ORDINARY CALL

VALUE CORRESPONDING TO (S*, r).

S*/PV(X) = 45.24/60e-0.1(5) = 1.243

T = 0.25 = 0.4472
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FROM OPTION TABLES, CALL VALUE

= 27.9% OF $45.24 = 0.279(45.24) = $12.62

WARRANT VALUES, W = C/(1 + )

= 12.62/1.5 = $8.41

STEP-4: ITERATE, USING S = S + W

(NOTE: USE W = W – FEES INSTEAD

OF W IF ISSUE COSTS/FEES > 0)

S = 50 + (1/2)8.41 = 54.21

S* = 54.21[e-0.02(5)] = 49.05

S*/PV(x) = 49.05/60e-0.1(5) = 1.348

T = 0.4472 (UNCHANGED)

FROM TABLES:
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CALL VALUE, C(S,X) = 31.7% OF 49.05

= $15.55

WARRANT VALUE = 15.55/(1 + ) =

15.55/1.5 = $10.35

STEP-5: ITERATE SEVERAL TIMES

UNTIL TWO SUCCESSIVE VALUES

ARE VERY CLOSE

S = 50 + (1/2)(10.35) = $55.18

S* = 55.18e-0.02(5) = $49.92

S*/PV(x) = 49.92/60e-0.1(5) = 1.372

T = 0.4472 (UNCHANGED, SAME

ALWAYS)
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FROM TABLES, CALL VALUE = 32.6%

OF 49.92 = 0.326(49.92) = $16.27

WARRANT VALUE = 16.27/(1 + ) =

16.27/1.5 = $10.85

IF YOU ITERATE AGAIN,

S = 55.42                 S* = 50.15

S*/PV(X) = 1.38,          T = 0.4472

CALL VALUE = 32.6% OF 50.15 = $16.34

 WARRANT VALUE = 16.34/1.5 =

$10.89




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SO NO SIGNIFICANT CHANGE FROM

PREVIOUS ITERATION (i.e.,

CONVRGENCE IS ACHIEVED), SO:

W = $10.87 (APPROX)




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