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Outline what warrants are, calculate a warrant's value, and work out the dilution cost it imposes on existing shareholders.

FRM® Part I · Valuation and Risk Models · The Black-Scholes-Merton Model · FRM1.VR.15.h

The idea

A warrant is a call option issued by a company on its own stock. When it is exercised the company creates new shares and receives the strike price, which dilutes existing shareholders. If N shares are outstanding and M warrants each give the right to one share, the value of a warrant is N ÷ (N + M) times the BSM value of an otherwise identical call. The share price used already reflects the market's expectation of dilution. The same factor applies to the payoff: N ÷ (N + M) × (ST − K), where ST is the share price just before exercise. The dilution cost to existing shareholders is the total value of the warrants, M times the per-warrant value, which is less than M times the plain call value. Common errors are using M ÷ (N + M) or applying no dilution factor.

Formulas

Value of a warrant

W = N ÷ (N + M) × c

N
the number of shares outstanding
M
the number of warrants, each for one new share
c
the BSM value of an otherwise identical ordinary call

The total, M × W, is the dilution cost to existing shareholders. Using M ÷ (N + M), or no factor, are the traps.

Payoff of a warrant at exercise

Payoff = N ÷ (N + M) × (ST − K)

N
the number of shares outstanding
M
the number of warrants exercised
ST
the share price just before exercise
K
the strike price

The new shares are issued at K, so the post-exercise price falls; the factor captures that. Taking ST − K without it overstates the payoff.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which of the following best describes a warrant?

  1. A call option written by an investor and traded on an exchange, settled with existing shares.Why not: Exchange-traded options are contracts between investors and do not create new shares.
  2. A call option issued by a company on its own stock, which on exercise leads the company to issue new shares and receive the strike price. - correct
  3. A right to sell the company's shares back to the company at a fixed price.Why not: Describes a put or buyback right; a warrant is a call.
  4. A forward contract on the company's shares that must be settled in cash.Why not: A warrant is an option with no obligation on the holder, and exercise is usually into new shares.

Because exercise creates new shares, the company receives the strike but the claim on the firm is spread over more shares, which is the source of dilution.

A harder one

A company has 10 million shares at USD 30 each and issues 2 million warrants, each for one share. An ordinary BSM call with the same terms is worth USD 6.00. The dilution cost to existing shareholders, per existing share, is closest to:

  1. USD 1.20Why not: Values the warrants as ordinary calls (2 million x 6.00) without the factor N/(N + M).
  2. USD 0.83Why not: Spreads the total warrant value over N + M = 12 million shares instead of the 10 million existing shares.
  3. USD 5.00Why not: Reports the per-warrant value instead of the cost per existing share.
  4. USD 1.00 - correct

Warrant value = 10/12 x 6.00 = 5.00. Total cost = 2 million x 5.00 = USD 10 million. Per existing share = 10 / 10 million shares = USD 1.00.

The hardest tier

A company has 8 million shares outstanding and 2 million warrants, each giving the right to buy one share at USD 20. Just before exercise, the share price is USD 35 and all the warrants are exercised. The value of each warrant at exercise is:

  1. USD 15.00Why not: Ignores dilution and takes the payoff as S - K = 35 - 20.
  2. USD 32.00Why not: Reports the post-exercise share price (8 x 35 + 2 x 20)/10 rather than the warrant payoff.
  3. USD 12.00 - correct
  4. USD 3.00Why not: Applies M/(N + M) = 0.2 rather than N/(N + M) = 0.8 to the payoff of 15.

Payoff = N/(N + M) x (S - K) = 0.8 x 15 = 12. Check: after exercise the share price is (8 x 35 + 2 x 20)/10 = 32, and 32 - 20 = 12.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

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