Ch22 23 29

Dr. Ekaterina Chernobai p. 1

FRL 367

Instructor: Dr. Ekaterina Chernobai

Ch.22: “Options”


Key concepts

Stock options - call options - put options - etc

Calculating the value of options


Lecture outlineSlides

4 – 12

13 – 21

22 – 32

33 – 38

39 – 53

54 – 95

69 – 76

77 – 85

86 – 95

Financial “derivatives” & stock options

Call options

Put options

Option quotes

Combinations of options, and “put-call parity”

Option value before expiration date - Binomial option pricing model - Black-Scholes option pricing model Example, similarity with binomial, Put option value calculations


What are stock options? (1 of 9)

In finance, like in real world, we always deal with risk.

Risk is concerned with uncertainty. E.g., will stock price go up or down in 3 months?

Upside risk = possibility of gain in the future

Downside risk = possibility of loss in the future



Question: Can we somehow eliminate future uncertainty?

Answer: Yes. (Although we will have to pay for this.) We can lock in an agreed-upon price at which the transaction will take place in the future.

Example:

I think stock price on Google will go up beyond $700 per share by the end of the next 3 months. I find a person with whom I agree today that in 3 months I will buy from him Google’s stock at $700 per share.

This way, I eliminate the future Google stock price uncertainty by locking in today the future purchase price of $700.



In general,

We can buy the right to buy or sell some asset at a certain point of time in the future at a price that we agree upon today.

= options



In general, there are many types of “options” in finance:

- real options (recall Ch.8)

- commodity options

- currency exchange options

- bond options

- stock options we will focus on these



Stock options are also called derivatives because they are “derived” from stock. This makes stock an underlying asset.

Think of the following:

• Stock prices are “derived” from firm value. Firm is the underlying asset. Stock is the derivative.

Similarly,• Stock options are “derived” from stock. Stock is the underlying asset. Stock option is the derivative.



A stock option gives the holder of it the right, but not the obligation, to buy or sell a given quantity of stock on (or before) a given date in the future, at prices agreed upon today.

European option: if can be bought or sold ON a given date in the future American option: if can be bought or sold BEFORE a give date in the future

“Exercising the option” = buying or selling the underlying asset

Strike price or Exercise price = fixed price in the option contract at which the holder can buy or sell the underlying asset.

Expiration date = the maturity date of the option = the date on (or before) which the holder can buy or sell stock.



today point of time in the future

Expiration date

No transaction takes place.

Agree on price (exercise price) at which transaction will take place in the future.

Transaction takes place.

American option:

Can buy (or sell) stock at exercise price on or before the expiration date.

European option:

Can buy (or sell) stock at exercise price on the expiration date.



In-the-Money = when exercising the option would result in a positive payoff.

At-the-Money = when exercising the option would result in a zero payoff (i.e., exercise price equal to spot price).

Out-of-the-Money = when exercising the option would result in a negative payoff. Option holder can walk away from the option.

Recall that stock options are only “options”. They can be exercised only if it’s profitable to do so.



Stock options

Call options Put options

Its holder:

has the right, but not the obligation, to buy a given amount of a company’s stock on or before some exercise date at the pre-agreed exercise price.

Its holder:

has the right, but not the obligation, to sell a given amount of a company’s stock on or before some exercise date at the pre-agreed exercise price.

Its seller:

has the obligation to sell a given amount of a company’s stock if the holder exercises the option.

Its seller:

has the obligation to buy a given amount of a company’s stock if the holder exercises the option.


Call options (1 of 9)

Call option gives the holder of it the right, but not the obligation, to buy a given quantity of some company’s stock on or before some time in the future (exercise date), at prices agreed upon today (exercise price, or strike price).

When exercising a call option, you “call in” the asset.



Exercise price, agreed upon today, is $50.

Exercise date is 3 months from today.

In (or during) 3 months:

• If stock price is $80, option is in-the-money. So, exercise the option: should buy stock from option writer for $50 per share. Can sell immediately for the market price of $80. Payoff = $80 - $50 = $30 per share.

• If stock price is $40, option is out-of-the-money. So, do not exercise the option. Payoff = $0. (NOT -$10.)

EXAMPLE


Call options (3 of 9)At expiration date, an American call option is worth the same as a European call option with the same characteristics.

If the call is in-the-money, it is worth ST – E.

If the call is out-of-the-money, it is worth 0.

C = max [ ST – E, 0 ]

whereC is the value of the call option at expiration

ST is the value of the stock at expiration (time T)

E is the exercise price



C = max [ ST – E, 0 ] C is the value of the call option at expirationST is the value of the stock at expiration (time T)E is the exercise price

Call option value at expiration equals ST – E or zero, whichever is higher

Example:

Exercise price is $50.

If stock price at expiration is $80, then ST–E = $80–$50 = $30. C = max[30, 0]= $30

If stock price at expiration is $40, then ST–E = $40–$50 = -$10. C = max[-$10, 0]= 0



–2012020 40 60 80 100

20

40

60

Actual stock price ($) at expiration date, ST

Option payoff ($) to option’s holder at expiration date

50

Option holder’s payoff at expiration date

Exercise price

If actual stock price at expiration date < $50, do not exercise the option. Payoff =$0

If actual stock price at expiration date > $50, exercise the option. I.e., buy stock for $50. If immediately sell stock for current price, then payoff = actual stock price - $50



–2012020 40 60 80 100

20

40

60

Stock price ($) at expiration date, ST


50

Option holder’s payoff at expiration date

Exercise price If stock price at expiration date = $80, then

exercise the call option. I.e., buy stock from option writer for $50. If immediately sell stock in the market for $80, then payoff = $80 - $50 = $30.

30



We’ve just looked at option payoffs.

However, buying an option costs money.

Option profits will be reduced by the amount of option premium.



option premium =$10

Exercise price


Option payoff ($) to option’s holder at expiration date Option holder’s profit

at expiration date

10 20 30 40 50 60 70 80

40

30

20

10

If call option premium is $10 per share, then:

Now, if stock price at expiration date = $80, then exercise the call option. I.e., buy stock from option writer for $50. If immediately sell stock in the market for $80, then profit = $80 - $50 - $10 = $20.


10 20 30 40 50 60 70


What about payoffs and profits to SELLERS of call options?

Call option seller’s payoffs: Call option seller’s profits:

Exercise price

Option payoff ($) to option’s seller at expiration date

40

30

20

10 10 20 30 40 50 60 70

40

30

20

10

Option profit ($) to option’s seller at expiration date

STST

Exercise price

premium


Put options (1 of 8)

Put option gives the holder the right, but not the obligation, to sell a given quantity of some company’s stock on or before some time in the future (exercise date), at prices agreed upon today (exercise price, or strike price).

When exercising a put, you “put” the asset to someone.



Exercise price, agreed upon today, is $50.

Exercise date is 3 months from today.

In (or during) 3 months:

If stock price is $20, option is in-the-money. So, exercise the option: should sell stock to option writer for $50 per share. Can buy stock on the market immediately for $20. Payoff = $50 - $20 = $30 per share.

If stock price is $60, option is out-of-the-money. So, do not exercise the option. Payoff = $0. (NOT -$10.)

EXAMPLE


Put options (3 of 8)At expiration date, an American put option is worth the same as a European option with the same characteristics.

If the put is in-the-money, it is worth E – ST.

If the put is out-of-the-money, it is worth 0.

P = max [E – ST, 0 ]

where P is the value of the put option at expiration

ST is the value of the stock at expiration (time T)

E is the exercise price

Put option value at expiration

equals E – ST or 0, whichever is higher



–2012020 40 60 80 100

20

40

60



50

Put option holder’s payoff at expiration date

Exercise price

If actual stock price at expiration date > $50, do not exercise the option. Payoff =$0

If actual stock price at expiration date < $50, exercise the option. I.e., sell stock for $50. If immediately buy stock for actual current price, then payoff = $50 - actual stock price

50



–2012020 40 60 80 100

20

40

60



50

Put option holder’s payoff at expiration date

Exercise price

50

If stock price at expiration date = $20, then exercise the put option. I.e., sell stock to option writer for $50. If immediately buy stock in the market for $50, then payoff = $50 - $20 = $30.

30



We’ve just looked at option payoffs.

Just like with call options, buying a put option also costs money.

Like with call options, put option profits to its holder will be reduced by the amount of put option premium.



option premium =$10

Exercise price


Option payoff ($) to option’s holder at expiration date Put option holder’s

profit at expiration date

10 20 30 40 50 60 70 80

40

30

20

10

If put option premium is $10 per share, then:

Now, if stock price at expiration date = $20, then exercise the put option. I.e., sell stock to option writer for $50. If immediately buy stock in the market for $20, then profit = $50 - $20 - $10 = $20.


10 20 30 40 50 60 70


What about payoffs and profits to SELLERS of put options?

Put option seller’s payoffs: Put option seller’s profits:

Exercise price

Option payoff ($) to option’s seller at expiration date

40

30

20

10 10 20 30 40 50 60 70

40

30

20

10

Option profit ($) to option’s seller at expiration date

STST


Call vs. put options (1 of 3)

Comparison of call options & put options

Call option Put option

Holder (buyer) Seller Holder (buyer) Seller

Payoff at expiration date

Profit at expiration date

payoff

ST

payoff

ST

payoff

ST

payoff

ST

profit

ST

profit

ST

profit

ST

profit

ST

E

E

E E E

EEE



EXAMPLEQuestion: What type of option(s) should you buy if you are betting that stock price will move away from $100 in either direction?

Answer: Buy both call option and put option with exercise price of $100.

Allows the holder to profit based on the magnitude of stock price movement, regardless of the direction of price movement.

Holder’s profit

ST

100

Buy callHolder’s profit

ST

100

Buy putHolder’s profit

ST

100



EXAMPLEQuestion: What type of option(s) should you buy if you are betting that stock price will NOT move far away from $100 in either direction?

Answer: Sell both call option and put option with exercise price of $100.

Allows the holder to profit based on the magnitude of stock price movement, regardless of the direction of price movement.

Seller’s profit

ST

100

Sell callSeller’s profit

ST

100

Sell putSeller’s profit

ST

100


Option quotes (1 of 6)

Option/Strike Exp. Vol. Last Vol. LastIBM 130 Oct 364 15¼ 107 5¼138¼ 130 Jan 112 19½ 420 9¼138¼ 135 Jul 2365 4¾ 2431 13/16138¼ 135 Aug 1231 9¼ 94 5½138¼ 140 Jul 1826 1¾ 427 2¾138¼ 140 Aug 2193 6½ 58 7½

--Put----Call--Let’s say you are looking at the following option quote table…




--Put----Call--

This option has a strike price of $135

a recent price for the stock is $138.25

July is the expiration month




--Put----Call--

This makes a CALL option with this exercise price in-the-money by $3.25 = $138¼ – $135 (i.e., payoff)

PUTS with this exercise price are out-of-the-money since E > ST




--Put----Call--

On the day of this quote, 2,365 CALL options with this exercise price were traded.




--Put----Call--

The CALL option with a strike price of $135 is trading for $4.75 (premium)

Option contracts usually consist of 100 shares of stock. Buying this contract would cost 100 x $4.75 = $475 plus commissions.




--Put----Call--

On this day, 2,431 PUT options with this exercise price were traded.

The PUT option with a strike price $135 is trading for 13/16 = $0.8125

(premium)Since the option contract is on 100 shares of stock, buying this PUT option would cost 100 x $0.8125 = $81.25 plus commissions.


Combinations of options (1 of 15)

Puts and calls can serve as the building blocks for more complex option contracts.

For example, we can use put options as “insurance”.

(see next slides)



Protective put strategy:

Stock payoff

(stock value at expiration)

Actual stock price at expiration, ST

Put value at expiration


Exercise price = $50

value of combination of stock & put at expiration


$50 $50

Buy stock Buy put option Combination

As if we are buying insurance for the stock.

$50 $50



Can explain this result using numbers:

Stock value (payoff)


Put value (payoff)



value (payoff) of combination

Actual stock price at expiration, ST$50 $50


$50 $50

ST 0 10 20 30 40 50 60 70 80 90 100payoff if purchased stock (1)

0 10 20 30 40 50 60 70 80 90 100

payoff if purchased put option (2)

50 40 30 20 10 0 0 0 0 0 0

payoff if purchased combination (1) + (2)

50 50 50 50 50 50 60 70 80 90 100



Consider another strategy:

Bond value at expiration


Call option value at expiration

Actual stock price at expiration, STExercise

price = $50

value of combination of 0-coupon bond & call at expiration


Buy a risk-free 0-coupon bond

Buy call option Combination

Zero-coupon bond pays no coupon payments. Only face value at maturity.E.g., T-bills.

Here, face value = $50.

$50$50



Again, let’s use numbers:

Bond value (payoff)


Call option value (payoff)



value (payoff) of combination


Buy a risk-free 0-coupon bond

Buy call option Combination

$50$50

ST 0 10 20 30 40 50 60 70 80 90 100payoff if purchased bond (1)

50 50 50 50 50 50 50 50 50 50 50

payoff if purchased call option (2)

0 0 0 0 0 0 10 20 30 40 50

payoff if purchased combination (1) + (2)

50 50 50 50 50 50 60 70 80 90 100



Stock payoff (stock value at expiration)


Option value at expiration


E = $50

value of combination of stock & put at expiration


$50 $50





Actual stock price at expiration, STE = $50

value of 0-coupon bond & call at expiration


$50 $50

Buy call option CombinationSAME !Buy a risk-free

0-coupon bond

$50

$50

$50



So, we got the following result:

Investors get the same payoff from:

• Strategy 1: buy a put option & buy a share of the underlying stock.

• Strategy 2: buy a call option & buy a risk-free zero-coupon bond.



Since payoffs from both strategies are the same, the two strategies must have the same cost today.

Strategybuy buy buy risk-free buy

stock put option 0-coupon bond call option

stock put option PV of call optionprice price exercise price price S0 P0 E/(1+R)T C0

Put-call parity

& &

Cost today



“Put-call parity” holds only when:

- put and call have the same exercise price

- put and call have the same expiration date

- the maturity date of the 0-coupon bond is the same as the expiration date of both call and put options.

- the face value of the 0-coupon bond is the same as the exercise price of both call and put options.



EXAMPLE“Synthetic” stock

A call option with 1 year to maturity and a $110 exercise price sells for $15. A put with the same terms sells for $5.

Risk-free rate on a zero-coupon bond is 10%.

Based on the put-call parity, what is the current price of the underlying stock?

Put-call parity: S0 + P0 = E/(1+R)T + C0 E/(1+R)T + C0 - P0

= $110/(1+0.1) + $15 - $5

= $110

Rearrange: S0 =Thus, by buying the bond, buying a call option, and selling a put option, investor has “purchased” a synthetic stock.





$110

Thus, by buying the bond, buying a call option, and selling a put option, investor has purchased a “synthetic stock”.

Graphically:

$110


$110

Put option value to seller at expiration

$110

Stock value at expiration

$110

$110

buy bond buy call sell put get “synthetic” stock







ST$110

Using numbers:

$110


$110

Put option value to seller at expiration

$110

Stock value at expiration

$110

$110

buy bond buy call sell put get “synthetic” stock

ST ST ST


110 110 110 110 110 110 110 110 110 110 110


0 0 0 0 0 0 20 40 60 80 100

payoff if sold put (3)

-100 -80 -60 -40 -20 0 0 0 0 0 0

payoff from combination, or synthetic stock (1) + (2)+(3)

10 30 50 70 90 110 130 150 170 190 210



EXAMPLEGiven:

Shares of stock of ABC company are selling today for $70.

A 3-month call option goes today for $10.

The risk-free rate is 0.5% per month.

Based on the put-call parity, what is the value of a 3-month put option with an $80 strike price?

Put-call parity: S0 + P0 = E/(1+R)T + C0 E/(1+R)T + C0 - S0

= $80/(1+0.005)3 + $10 - $70

= $18.81

Rearrange: P0 =Thus, by buying the bond, buying a call option, and selling a share of stock, investor has “purchased”a put option.




Actual stock price, ST

$80

In other words, by buying the 0-coupon bond, buying a call option, and selling a share of stock, we can replicate the payoff from a put option.

Graphically:

$80


$80

Stock value to seller at expiration

$80Actual stock price, ST


Put value at expiration


$80

$80

buy bond buy call sell stock get put option



Bond value (payoff)

$80

Using numbers:

$80

Call option value (payoff)

$80

Stock value (payoff)

$80

Put value (payoff)

$80

$80

buy bond buy call sell stock get put option

STST ST ST


80 80 80 80 80 80 80 80 80 80 80


0 0 0 0 0 0 0 0 10 20 30

payoff if sold stock (3)

-10 -20 -30 -40 -50 -60 -70 -80 -90 -100 -110

payoff from combination, or put option (1) + (2)+(3)

70 60 50 40 30 20 10 0 0 0 0


Valuing options prior to expiration (1 of 14)

2 cases in which we can calculate the value of an option

Value of an option ON the expiration date

Value of an option PRIOR TO the expiration date

What we’ve been doing so far.

Next slides.

- more difficult

- We want to get an exact formula for calculating current price (or value) of an option, without using the “put-call parity” formula.

today at expiration

“Call” holder

i.e., how much it should cost to buy



We will focus on call option values PRIOR TO their expiration (i.e., how much they should cost).

First, let’s see what the minimum call option price (lower bound) and the maximum call option price (upper bound) should be.



American Call LOWER BOUND:

Stock price – exercise price, S – E when in-the-money,

0 when out-of-the-money.

E

Actual stock price prior to expiration, SIntrinsic value

If lower than this lower bound, then we have an arbitrage opportunity (riskless profit):

Example: call option costs $9. Current stock price is $60. Exercise price is $50.Can make immediate profit if (i) buy call for $9, (ii) immediately exercise call & buy stock at $50 per share, (iii) immediately sell stock for $60 per share. Immediate profit = -$9 - $50 + $60 = $1.So, to eliminate arbitrage opportunities, call option should cost AT LEAST $10.

Value of call option prior to expiration ($), C0



American Call UPPER BOUND:

Stock price, S

E

Actual stock price prior to expiration, S

If higher than this upper bound, then we have an arbitrage opportunity (riskless profit):Example: call option costs $41. Current stock price is $40. Can make immediate profit if (i) sell call for $41, (ii) immediately buy stock at $40 per share. Immediate profit = $41 - $40 = $1.This way on expiration date you will have a share of stock to satisfy a call, if exercised, PLUS the $1 with accrued interest.So, to eliminate arbitrage opportunities, call option should cost AT MOST current stock price ($40).




American CallLOWER BOUND

Stock price – exercise price, S–E, or zero

UPPER BOUND

Stock price, S

E


Call option price

Actual call option price will be somewhere between the lower and upper bounds. The exact formula of the blue curve – Black-Scholes formula.

It’ll be the most complicated formula you will probably ever see in finance!




American Call UPPER BOUND

Stock price, S

E


Time value

Intrinsic value

Market Value

Call option price

Actual call option market value = intrinsic value + time value premium

LOWER BOUND

Stock price – exercise price, S–E, or zero




So, we have:

Call option market (total) value = intrinsic value + time value premium

What does the “time value premium” of the call option price depend on?

Lower bound.

Current stock price – Exercise price, S – E

or zero



5 main determinants of call option value PRIOR TO the expiration date:

1.) Current stock price, S

2.) Exercise price, E

3.) Expiration date, T (i.e., time left until option expires)

4.) Variability of stock price, σ

5.) Interest rate, R



1.) Current stock price, S

We’ve already looked at this.

Recall the lower bound.

The higher the current stock price, the more valuable the call option gets.



2.) Exercise price, E

The higher the exercise price, the lower the value of the call.

Example: You can choose between two calls, the first one with E=$60, the second one with E=$40.

Question: Which one would you rather have?

Answer: The second one, because it is $20 more in-the-money.

And so the second call is more valuable.



3.) Expiration date, T

The longer the time left until the expiration date, the higher the value of the call.

Example: You can choose between two calls: the first one expires in 6 months, the second one expires in 9 months.

Question: Which one would you rather have?

Answer: The second one, because it has 3 extra months during which you have freedom to exercise the call option.

And so the second call is more valuable.



4.) Variability of stock price, σ

Example: Exercise price is $100. You can choose between call options on two stocks: the first stock price can be either $120 or $60 with equal probabilities, the second one can be either $85 or $95 with equal probabilities. (Both give expected price of $90.)

Question: Which stock will have a more expensive call option?

Answer: The 1st stock. With 50% probability the stock price can be $20 above the exercise price. Can exercise the call option and make $20.

In case of the 2nd stock, the price will not rise above the exercise price in either outcome, thus no payoff can be expected.

Call option on the 1st stock is valued higher.

The greater the variability of stock price, the higher the value of the call.



5.) Interest rate, R

The higher the interest rate, the higher the value of the call.

The higher the interest rate, the lower the PV today of the exercise price that will need to be paid at the expiration date.



For PUT options:

1.) Current stock price, S The higher the current stock price, the lower the value of put.

2.) Exercise price, E The higher the exercise price, the higher the value of put.

3.) Expiration date, T The longer the time left until the expiration date, the higher the value of put.

4.) Variability of stock price, σ The more volatile the stock price, the higher the value of put.

5.) Interest rate, R The higher the interest rate, the lower the value of put.

Same effects as on call

option value


Option pricing formula

Option value before expiration

Binomial model Black-Scholes model

-Assumes that stock price can take only two possible values in the future

-Unrealistic

-Easy calculations

-Stock prices can take many different values

-More realistic

-Worst formula you’ll ever see in your life!

We will now look at the formulae that calculate the current option price.

There are 2 of them.


Option pricing formula: Binomial (1 of 8)

Binomial Option Pricing ModelSuppose a stock is worth $25 today. At expiration, it will cost either 15% more or 15% less. So, S0= $25 today and at expiration S1 will be either $28.75 or $21.25.

The risk-free rate is 5%. Exercise price is $25 (i.e., option is now at-the-money)

$25

$21.25

$28.75

S1S0

EXAMPLE

Question: What is the current value of this call option?



Strategy 1: Buy a call option on this stock with exercise price of $25 (at-the-money)

$25

$21.25

$28.75

S1S0 C1 (payoff at expiration)

$3.75 = $28.75 - $25 Exercise the call option

$0Don’t exercise the call option



Strategy 2: Borrow PV of $21.25 today and buy 1 share of stock for $25

$25

$21.25 – $21.25 = $0

$28.75 – $21.25 = $7.50S1 – debtS0

We can replicate the payoffs of the call option with a levered position in the stock (i.e., buy stock & take loan)

Value of portfolio at expiration



This is twice the call option’s payoff ($3.75 and $0), so the levered portfolio is worth twice the call option value

$25$21.25 – $21.25 = $0

$28.75 – $21.25 = $7.50S1 – debtS0 C1

$3.75

$0

from previous slides:

So, to replicate exactly the call option payoffs $3.75 and $0, buy HALF a share and borrow HALF of PV of $21.25 today.



Since such portfolio perfectly replicates the call option payoffs $3.75 and $0, the present value of the call option C0 must also equal $2.38:

$2.380.05)(1

$21.25$2521

0C

So, the Present Value of the portfolio is:

1/2 shares of stock at $25 per share ……..……… ½ $25 = $12.5

Borrowed …………………………………………… ½

Portfolio Present Value = $12.5–$10.12 = $2.38

$10.120.051

$21.25

rf = 5% (given)

!



Another way to figure out that we need to buy only ½ shares of stock:

Calculate the “delta” of the call.

21

$7.5$3.75

$21.25$28.750$3.75

Swing of call Call payoff in state 1 – Call payoff in state 2

(The Delta of a put option is negative, because the numerator of the Delta will be 0 – 3.75 = -3.75)

Means: the risk of buying ½ shares of stock ( ½ $28.75 – ½ $21.25 = $3.75) is the same as the risk of buying a call ($3.75 – $0 = $3.75).

Swing of stock Stock price in state 1 – Stock price in state 2

In our case,



From previous slides, today’s value of call option C0 = $2.38, which is equivalent to buying “delta” shares of stock and borrowing a certain amount of money.

Today’s value of a call = Stock price × Delta – Amount borrowed $2.38 = $25 × ½ – $10.12



Something interesting to notice:

We didn’t even need to know the probabilities of the two states!

With the correct # of shares and amount borrowed, we can replicate call option payoffs in the future regardless of how likely this or that state is.


Option pricing formula: Black-Scholes (1 of 9)

Binomial model vs. Black-Scholes model

Binomial model Black-Scholes model

-Assumes that stock price can take only two possible values in the future.

-Unrealistic.

(what we have just looked at)

-More realistic model: stock prices can take many different values.

-Shows that we can indeed duplicate a call option, although only over a very small time interval.

-Was developed in 1973.



)N(dEe)N(dSC 2RT

10

C0 = the value of an option at time 0 (today)

S = current stock priceR = annual risk-free interest rate. In decimals. Usually T-bill rate.T = time (in years) until the expiration date. Can be a fraction.

Tσ

)T2

σ(Rln(S/E)d

2

1

Tσdd 12

N(d) = Probability that a standardized, normally distributed, random variable will be less than or equal to d.

(Use Table 22.3 in textbook, or “normsdist” function in Excel)

N(d)

d

σ = annual stock return volatility (standard deviation). In decimals.



Tσ

)T2

σ(Rln(S/E)d

2

1

Tσdd 12

N(d) = Probability that a standardized, normally distributed, random variable will be less than or equal to d.

(Use table 22.3 in textbook, or “normsdist” function in Excel)

N(d)

d

How to calculate N(d1) or N(d2):

Method #1: Use “normsdist” function in Excel. Easy

Method #2: Use Table 22.3 in the textbook. Harder on the Final exam



Method #2: Using Table 22.3 in the textbook to calculate N(d)

0 d

this area:

d = ▒. ▒ ▒

d = ▓. ▓ ▒

d = 0. ▒ ▓



0 d

So, Table 22.3 numbers show this area:

0 d

But what we need in the Black-Scholes formula is N(d), which is this area:



Total area under the curve equals 1

0



0

Either half of the area under the curve equals 0.5

0


0 d

If d>0 then

N(d) = 0.5 + number from Table

If d<0 then

N(d) = 0.5 – number from Table


d 0

Example: d = 0.53

N(0.53) = 0.5 + 0.2 = 0.7

Example: d = -0.53

N(-0.53) = 0.5 – 0.2 = 0.3



Also, see in Ch.22 how to calculate N(d) more precisely when you have more than 2 decimal points in d.


Black-Scholes example (1 of 7)

EXAMPLE Given:The current value of a share of Microsoft is $160The risk-free interest rate available in the US is 5%The option maturity is 6 monthsThe volatility of Microsoft stock returns is 30% per annum

Find the value of a six-month call option on Microsoft with an exercise price of $150

S=160R=0.05T=0.5σ=0.3

E=150



The intrinsic value (i.e., the lower bound) of the option is $160 - $150 = $10 — our answer must be at least that amount.

Tσ)T0.5σ(Rln(S/E)d

2

1

First, calculate N(d1) and N(d2):

Tσdd 12

0.528150.50.30

)0.5.5(0.30)(0.05)ln(160/150 2

0

N(d1) = N(0.52815) = 0.7013

0.316020.50.300.52815

N(d2) = N(0.31602) = 0.62401



)N(dEe)N(dSC 2RT

10

Finally, use the Black-Scholes formula to calculate the current call option value:

$20.920.62401e $1500.7013$160 0.50.05



E=$150 Current stock price prior to expiration, S

Time value=$10.92Intrinsic value = $160 - $150 = $10

Market Value = $20.92

Call option price

Market (total) value of call option = intrinsic value + time value premium

$20.92 = $10 + time value premium

S0=$160




Similarity between the Binomial & the Black-Scholes models:

Black-Scholes:

Binomial:

)N(dEe)N(dSC 2RT

10

C0 = S x Delta – Amount borrowed value of current # shares a call stock price

N(d1) is “Delta”, or the number of stocks that needs to be purchased,

E e-RT N(d2) is the amount that needs to be borrowed in order to replicate the payoffs from the call option.



Based on this,

We can duplicate the call of $20.92 by following this strategy:

Strategy: Buy N(d1) = 0.7013 shares of stock for $160 each,

and

Borrow E e-RT N(d2) = $150*e-0.05*0.5 * 0.62401 = $91.29



What about the value of a PUT option, that has the same terms as the call option?

Earlier we had “put-call parity” formula:

S0 + P0 = E/(1+R)T + C0

Rearrange: P0 = E/(1+R)T + C0 – S0

P0 = $150 e -0.05(0.5) + $20.92 – $160

= $7.216

Change into continuous compounding: P0 = Ee-RT + C0 – S0

Calculate put option price P0:


Review: Black-Scholes call option formula (1 of 3)

Value of call option prior to expiration,

C0

LOWER BOUND

=max [S – E, 0]

UPPER BOUND

= Current stock price, S

Exercise price at expiration,

E

Current stock price prior to expiration,

S

Call option price

)N(dEe)N(dSC 2RT

10 Equation of the blue line:

Black-Scholes call option pricing model



What does this Black-Scholes formula show?

It shows how much you a call option on a particular company’s stock should cost today, before the call option will expire.

In this formula we use:

- the known exercise price at which stock can be purchased at the expiration date in the future (E)

- the known time left until the option expires (T)

- the known current stock price (S)

- the annual risk-free interest rate (R). Also known.

- the annual volatility of stock returns (σ). For this often look at past stock returns.


from Black-Scholes formula


What if we were interested in the price of PUT option?

For this we can plug in the Black-Scholes call option price into the Put-Call Parity formula.

Put-Call Parity formula connects current price on call option with current price on put option:

current stock + current put = current call + PV of price option price option price exercise price

S + P0 = C0 + E*e-RT

P0 = C0 + E*e-RT - S



FRL 367


Ch.22, 23: “Applications of options”


Key concepts

• How can we express stocks & bonds as options ?

• How can we apply options to merger valuation?

• How can we apply options to start-up valuation?



5 – 15

16 – 27

28 – 37

38 - 42

Option applications:

- Equity value & debt value as options (Ch.22)

- Mergers as call options (Ch.22)

- Start-up valuation as call options (Ch.23)

Executives’ stock options (Ch.23)


Option applications

• So far we’ve been looking at options written on stocks.

• The same concept of “options” can also be applied to:

- equity value & debt value expressed as options,

- mergers & acquisitions expressed as options,

- capital budgeting decisions expressed as options,

- other.

Ch.22

Ch.22

Ch.23


Equity value & debt value as options (1 of 11)

It turns out that we can express firm’s equity value & debt value as options:

- Equity value & debt value expressed as CALL options on the firm.

- Equity value & debt value expressed as PUT options on the firm.



EXAMPLE We have a levered firm, with debt and equity, which is formed today but will shut down in one year.

In one year it will pay off the debt (principal & interest) in the amount of $1,000, and whatever is left will be distributed among its stockholders.

5 possible equally-likely scenarios for next year’s earnings, before interest & principal are paid:

$1,500, $1,200, $1,000, $800, $200

(Assume no corporate taxes.)



Next year’s earnings under different possible scenarios

complete failure

somewhat unsuccessful

so-so big success

very big success

Earnings before debt is paid off

$200 $800 $1,000 $1,200 $1,500

CF to debt-holders

CF to stock-holders

Next year need to pay off the debt (principal & interest) in the amount of $1,000.

given given given given given

CF to bondholders

Firm earns 200 1000 1500 800 1200

1000 800 200

CF to stockholders

Firm earns 200 1000 1500 800 1200

500 200

$200 $800 $1,000 $1,000 $1,000

$0 $0 $0 $200 $500=200-200 =800-800 =1,000-0 =1,200-1,000 =1,500-1,000


500

200


CF to bondholders next year

Firm earns next year 200 800 1000 1200 1500

1000

800

200

Question: What do these resulting graphs look like to you??

Answer: That’s right! They look like options!

CF to stockholders next year




500

200



1000

800

200



Moreover, each line can be expressed as a call option OR as a put option, with exercise price = $1,000.



There is one more way to look at these two graphs:

1500

1200

1000

800

200

CF to the firm next year

Firm earns next year

200 800 1000 1200 1500

500

200



1000

800

200





1500

1200

1000

800

200

CF to the firm next year


200 800 1000 1200 1500

500

200



200 800 1000 1200 1500

1000

800

200



200 800 1000 1200 1500

Can be expressed as call option or put option with E=$1,000

Can be expressed as call option or put option with E=$1,000



Let’s see, for example, how we can express CF to stockholders as a call option.

SKIP: CF to stockholders as a put option

CF to bondholders as a call option

CF to bondholders as a put option

In our case,

- the underlying asset is not stock, but firm’s earnings

- the time to expiration is 1 year when debt needs to be repaid

- the exercise price is the debt that needs to be paid off next year to bondholders ($1,000)



Recall that CF to stockholders are:

500

200




holdcall option


First, Put-call parity says that

hold hold hold risk-free holdstock put option 0-coupon bond call option& &

Rearrange:

hold hold owe risk-freestock put option 0-coupon bond& &

stockholders’ cash flows expressed as CALL option

stockholders’ cash flows expressed as PUT option SKIP



holdcall option

hold hold owe risk-freestock put option 0-coupon bond& &

stockholders’ cash flows expressed as CALL option stockholders’ cash flows

expressed as PUT option

500

200



View bondholders as “owning the firm”,& stockholders as “holding call option on the firm”If firm earns <$1000, stockholders won’t exercise the call option. As a result, get nothing.If firm earns >$1000, stockholders exercise the call option. “Buy” firm from bondholders for $1000. As a result, stockholders get the difference between firm’s earnings & $1000.


Mergers as call options (1 of 12)

Ch. 29 looks at mergers.

One of the examples looked at:

Diversification is a frequently mentioned reason for mergers.Diversification reduces risk and, therefore, volatility of equity.However, when the goal of merger is only diversification, and when no synergy (=additional value) is generated, stockholders lose.

Such mergers, which create no synergy and only reduce risk through diversification, are known as “financial” mergers.



In Ch.22 we saw that:

Equity can be expressed as a call option on firm’s assets.

So, we can use the Black-Scholes formula to calculate more precisely how equity value will change with a merger!One of the examples in Ch. 29 illustrated that value of equity decreases with merger. Let’s prove this using the Black-Scholes call option value formula.



EXAMPLEConsider the following two merger candidates, A and B.The merger is for diversification purposes only with no synergies involved (i.e., “financial” merger).Risk-free rate is 4%.

Firm A Firm B Firm AB (after merger)Market value of assets

$40 million $15 million 40 + 15 = 55 million(no synergy)

Face value of zero-coupon debt

$18 million $7 million 18 + 7 = 25 million (combined debt)

Debt maturity in 4 years in 4 years in 4 yearsAsset return standard deviation

40% 50% 30%(diversification effect)

Question: Use Black-Scholes formula to calculate equity value (expressed as call option) before and after the merger.



Recall that the value of call option depends on 5 things:

S, E, T, R, σ

If volatility σ decreases, call option becomes less valuable.

In our example, volatility falls after merger. (Given: new volatility of 30% is less than both 40% for firm A and 50% for firm B.)

And so call option value should fall.

Since value of equity can be expressed as call option value, value of equity should fall with merger. So stockholders should lose. Let’s show this.



Firm A

Market value of assets

$40 million


$18 million

Debt maturity

in 4 years

Asset return standard deviation

40%

Market value of equity of firm A before the merger.

(given)

S = $40 million, E = $18 million, T = 4, σ = 0.4, R = 0.04

Risk-free rate 4%

d1 = [ln(S/E) + (R+σ2/2)*T ] / [σ*√T ]

=[ln(40/18) + (0.04+0.42/2)*4] / [0.4* √4] = 1.5981

)N(dEe)N(dSC 2RT

10

N(d1) =

d2 = d1 - σ√T

= 1,5981 – 0.4*√4 = 0.7981N(d2) = 0.7876

0.945

Value of Firm A’s equity before merger =

= C0 =

Black-Scholes:

40*0. 945– 18(e-0.04*4)0.7876 = $25.72 million



Firm B


$15 million


$7 million

Debt maturity

in 4 years


50%

Market value of equity of firm B before the merger.

(given)


Risk-free rate 4%

d1 = [ln(S/E) + (R+σ2/2)*T ] / [σ*√T ]

=[ln(15/7) + (0.04+0.52/2)*4] / [0.5* √4] =1.4221

)N(dEe)N(dSC 2RT

10

N(d1) =

d2 = d1 - σ√T

= 1.4221 – 0.5*√4 = 0.4221N(d2) = 0.6635

0.9225

Value of Firm B’s equity before merger =

= C0 =

Black-Scholes:

15*0.9225 – 7(e-0.04*4)0.6635 = $9.88 million



Market value of the debt = market value of assets – market value of equity

Firm A Firm B

Market value of assets $40 mil $15 mil

Market Value of Equity $25.72 mil $9.88 mil

Market Value of Debt

Face Value of Debt $18 mil $7 mil

=$14.28 mil = $40 - $25.72

=$5.12 mil = $15 - $9.88

What we’ve just calculated from Black-Scholes formula (last 2 slides)



Firm AB


$55 million=40 + 15


$25 million=18 + 7

Debt maturity

in 4 years


30%

Market value of equity of firm AB after merger.

(given)


Risk-free rate 4%

d1 = [ln(S/E) + (R+σ2/2)*T ] / [σ*√T ]

=[ln(55/25) + (0.04+0.32/2)*4] / [0.3* √4] = 1.8808

)N ()N ( 210 dEedSC RT

N(d1) =

d2 = d1 - σ√T

= 1.8808 – 0.3*√4 =1.2808 N(d2) = 0.9

0.97

Value of Firm AB’s equity after merger =

= C0 =

Black-Scholes:

55*0.97 – 25(e-0.04*4)0.9 = $34.18 million



Firm A Firm B Firm AB (merger)


$40 mil $15 mil $55 mil = $40+ $15

Market Value of Equity

$25.72 mil $9.88 mil $34.18


Face Value of Debt

$18 mil $7 mil $25 = $18 + $7

=$14.28 mil = $40 - $25.72

=$5.12 mil = $15 - $9.88

After the merger:

We’ve just calculated all three using Black-Scholes formula

=$20.82 mil =$55-$34.18





$40 mil $15 mil $55 mil = $40+ $15


$25.72 mil $9.88 mil $34.18


Face Value of Debt

$18 mil $7 mil $25 = $18 + $7

=$14.28 mil =$5.12 mil =$20.82 mil

Do stockholders lose or gain from the “financial” merger?

Market value of equity of separate firms A and B = = $25.72 + $9.88 = = $35.60 million

Market value of AB’s equity = = $34.18 million

So, stockholders lose $35.60 - $34.18= =$1.42 million





$40 mil $15 mil $55 mil = $40+ $15


$25.72 mil $9.88 mil $34.18


Face Value of Debt

$18 mil $7 mil $25 = $18 + $7

=$14.28 mil =$5.12 mil =$20.82 mil

Do bondholders lose or gain from the “financial” merger?

Market value of debt of separate firms A and B = = $14.28 + $5.12 = = $19.40 million

Market value of AB’s debt = = $20.82 million

So, bondholders gain =$20.82 - $19.40= =$1.42 million



Results:

Stockholders lose $1.42 million after merger,

Bondholders gain $1.42 million after merger.(This is the same result as what was shown in the “no synergy” merger example on the lecture on Ch.29. The only difference is: the Ch.29 example used a “binomial” case with only two possible firm values in the future. Now we applied the Black-Scholes formula for the general case with many possible firm values in the future.)

We have a wealth transfer from stockholders to bondholders in the amount of $1.42 million.


A start-up & a call option (1 of 10)

One more example in which we can apply call options!

start-up valuation



Every new business is risky. High chance that it will fail.And so NPV may be negative.

However, an important option is the option to expand.

This option may increase NPV and make the business worthwhile.

This option can be calculated using Black-Scholes formula.

Let’s look at an example.



Imagine a start-up firm, Campusteria, Inc., which plans to open private dining clubs on college campuses.

The test market will be this campus, and if the concept proves successful, expansion will follow nationwide.

Nationwide expansion will occur in year 4.

The start-up cost of the test dining club is $30,000.

EXAMPLE



Year 0 Years 1~4

Revenues $60,000

Variable Costs -$42,000Fixed Costs -$18,000Depreciation $7,500

Operating Cash Flow =(Revenue–All Costs)(1-0.34) + Depreciation*Tc

$2,550

Total Cash Flow -$30,000 $2,550

assume no change in

NWC & no salvage

value

The start-up company plans to sell 25 meal plans at $200 per month with a 12-month contract. So, annual revenue = $200*25*12 = $60,000

Variable costs are projected to be $3,500 per month. Per year: $3,500*12 = $42,000Fixed costs are projected to be $1,500 per month. Per year: $1,500*12 = $18,000

The initial $30,000 investment will be depreciated on a straight line over 4 years. Annual depreciation = $30,000/ 4 years = $7,500

Also given:



$21,916.840.10)(1

$2,550$30,0004

1tt

At 10% discount rate,

Is the start-up project worth it?

Not worth it.

NPV =

This calculations are most likely based on expected sales.

If the start-up is not popular, it will probably shut down in a couple of years.

If however the start-up turns out really popular, the company may decide to expand the business. In other words, it has an OPTION TO EXPAND.



Let’s say, the option to expand involves opening 60 more dining clubs across the country. Each will have same initial investment, same annual cash flows, and will also last 4 years.

This option to expand will change the NPV of the start-up!

NPV of the start-up NPV of the start-up Value of the option with the option = without the option + to expand to expand to expand

From previous slides this equals

$ -21,916.84

Use Black-Scholes formula

(next slide)



$331,2540.10)(1

$484,9890.10)(1

0.10)(1$2,55060

44

4

1tt

)N(dEe)N(dSC 2RT

10 Black-Scholes:

•What is “S”?

It is the Present Value of all future cash flows from the 60 locations that will be opened in 4 years:

• What is “E”?

It is the price that will need to be paid in 4 years, i.e. the total initial investment.

E =

S=

60 x $30,000 = $1,800,000

• What is “R”?

It’s 10% (given)

• What is “T”?

It’s 4 years (given)

• What is “σ”?

Let’s say it equals 48% per year (given here)



d1 = [ln(S/E) + (R+σ2/2)*T ] / [σ*√T ]

= [ln(331,254/1,800,000) + (0.1+0.482/2)*4] / [0.48* √4] = -0.8665

N(d1) =

d2 = d1 - σ√T

= -0.8665 – 0.48*√4 = -1.8265

N(d2) = 0.0339

0.1931

Current value of the option to expand to 60 more restaurants = = C0

= S x N(d1) – E x e-RT x N(d2) = 331,254 x 0.1931 – 1,800,000 x (e-0.1 x 4) x 0.0339 = $23,079.97

Calculate the value to expand using Black-Scholes formula:

S = $331,254

E = $1,800,000

R = 0.1

T = 4

σ = 0.48



The NPV of the start-up:

NPV of the start-up NPV of the start-up Value of the option with the option = without the option + to expand to expand to expand

= -21,916.84

+ $23,079.97

= $1,163.126

THE START-UP IS WORTH IT.



Notice:

You may have thought that the expansion wouldn’t be worth it, since initial investment = $1,800,000, which is several times higher than the PV of future cash flows from 60 new locations of $331,254. Right?

What’s the trick here??

The reason why it IS worth it is due to the high expected volatility (σ=48%) of the future cash flows. I.e., there’s a high chance that cash flows will be extremely high, which (from the Black-Scholes formula) will in turn increase the Present Value of the option to expand by a lot.


Executive stock options (1 of 5)

Firms’ executives’ compensation usually consists of base salary plus one or more of the following:

- annual bonuses,

- retirement contributions,

- stock options.



Question: What’s the benefit of giving stock options to an executive?

Answer:

1.) Makes executive work in the interests of stockholders. The better you work is for the firm, the higher the firm’s stock price. (And so the higher the potential profit from exercising call options.)

2.) Allows company to lower the executive’s base salary.

3.) A more tax-efficient way to pay the executives. Under the current tax law, when the received call options are at-the-money, they are not considered as taxable income to the employee. Taxes are due only when the option is exercised.



Stock options given to an executive:- Exercise price is generally set equal to the market price of

the stock on the date the executive receives the options.

- The company requires that the executives hold the options for at least a “freeze out” period, rather than letting them sell the options and make an immediate profit.

Typical “freeze out” period: 3 ~ 5 years.

During this period the executive cannot sell or transfer the received options.



2005 top option grants:Company CEO # of options

grantedStock price at

that time

Wells Fargo & Co. Richard Kovacevich

1,853,000 $58.71

Gillette Company James Kilts 2,000,000 $39.71… … … …

How much did J. Kilts’s option grant was worth at that time?

We can use the Black-Scholes formula to calculate it.


Total value of call option grant =$12.01*2,000,000 shares = $24,020,000


S = $39.71, E = $39.71, T = 5, σ = 0.2168, R = 0.05

d1 = [ln(S/E) + (R+σ2/2)*T ] / [σ*√T ]

= [ln(39.71/39.71) + (0.05+0.21682/2)*5] / [0.2168* √5] = 0.7581

)N(dEe)N(dSC 2RT

10

N(d1) =

d2 = d1 - σ√T

= 0.7581 – 0.2168*√5 = 0.2733

N(d2) = 0.6077

0.7758

Value of J. Kilts’ call option grant (per share) = = C0

=

Black-Scholes:Stock price at that time was $39.71 per share

Exercise price was $39.71 (because stock options are always at-the-money when received as part of compensation)

Let’s say we also know:

“Freeze out” period was 5 yrs

Risk-free rate was 5%

Volatility was 21.68% 39.71*0.7758 – 39.71(e-0.05*5)0.6077 = $12.01


Summary

• Option applications: - Equity value & debt value can be expressed as options - Mergers can be expressed as call options - Start-up valuation can be expressed as a call option

Executives’ stock options

• For all of the above we used Black-Scholes formula.


FRL 367


Chapter 29:“Mergers & Acquisitions”


Key concepts

• Reasons for M&A

• How M&A can increase value of combined firm

• How M&A can be a bad idea

• Different methods for paying for an acquisition

• Friendly vs. hostile takeovers

• Do mergers add value in real life?



4

5 – 11

12 – 20

21

22 – 32

33 – 49

50 – 51

52

Takeovers, mergers, acquisitions, …

Varieties of takeovers

Synergy

Bad reasons for mergers

Why stockholders may lose from mergers

The NPV from a merger - merger through cash offer - merger through stock acquisition

Friendly vs. hostile takeoversIn real life, do mergers add value (synergy)?


2 companies get combined into 1 larger company

1 company splits into 2 smaller companies

-Takeovers-Mergers-Acquisitions-consolidation-“going private”-etc

-demerger-“Sell-offs”-“spin-offs”-“carve-outs”-etc

We’ll focus

on these


Varieties of takeovers (1 of 7)

Takeovers

Acquisition

Going Private(leveraged buyouts, LBO)

Merger, consolidation

Acquisition of Stock

Acquisition of Assets= a general term referring to the transfer of control of a firm from one group of shareholders to another.



Merger– One firm is acquired by another– Acquiring firm retains name and acquired firm ceases to exist– Advantage: legally simple– Disadvantage: must be approved by stockholders of both firms

Consolidation– Same as mergers, except an entirely new firm is created from

combination of existing firms– Stockholders of the two firms exchange their shares for shares

of a new firm

Mergers & consolidations are very similar. Often both are referred to as “mergers”.



Acquisitions:

1.) Stock acquisition =A firm can be acquired by another firm or individual(s) purchasing

voting shares of the firm’s stock

“Tender offer” – public offer to buy shares of a target firm• Unlike mergers, no stockholder vote required• Can deal directly with stockholders, even if management is unfriendly• May be delayed if some target shareholders hold out for more money• Complete absorption requires a merger

2.) Asset acquisition = One firm acquires another by buying all of its assets. ▪ Formal vote of target firm’s stockholders is required



Classifications of acquisitions:

Horizontal acquisition = both firms are in the same industry Examples: Exxon acquired Mobil in 1998 Google acquired YouTube Macy’s acquired Robinson’s May stores AOL acquired Time Warner Cable

Vertical acquisition = firms are in different stages of the production process Example: travel agency acquires an airline company

Conglomerate type acquisition = firms are unrelated Example: Mitsubishi (in Japan) produces automobiles, TVs, etc



Going-private transactions = A.k.a. leveraged buyouts (LBO)

A private group purchases all equity of a firm. The firm’s stock gets taken off the market.

In LBO case, purchase is financed with large amounts of debt.

Usually highly risky due to large amount of debt involved.



In this Chapter, we will be referring to all forms of takeovers as “mergers” or “acquisitions”, or “M&A”.



BIDDER = acquiring firm

= a firm that has decided to take over another firm

TARGET = acquired firm

= a firm that gives up control over its stock or assets to the bidder


Synergy (1 of 9)

Question: Why do some firms decide to acquire other firms?

Answer: Because they believe they can create additional value.

This additional value is called synergy.


Synergy (2 of 9)

Most acquisitions, however, fail to create value for the acquirer.

Why? The main reason lies in failures to integrate two companies after a merger.

– Intellectual capital often walks out the door when acquisitions are not handled carefully.

– Traditionally, acquisitions deliver value when they allow for scale economies or market power, better products and services in the market, or learning from the new firms.


Synergy (3 of 9)

Synergy:

We have firm A with value VA and firm B with value VB.

Suppose firm A is contemplating acquiring firm B.

The synergy from the acquisition is :

Synergy = VAB – (VA + VB)

I.e., acquisition is a success if the value of combined firm (VAB) is greater than the sum of the individual firms’ values (VA + VB).!


Synergy (4 of 9)

)V(VVSynergy BAAB

T

1tt

BAAB

R)(1)CF(CFCF

Question: Where does this synergy come from?

Answer: Increases in Cash Flows create this additional value

“ΔCFt” is the difference in each year t between the cash flows of the combined firm AB and the sum of cash flows of the two separate firms A and B.

T

1tt

t

R)(1ΔCF

Example: if A makes $200 each year and B makes $100, the CFA+CFB = $200+$100 = $300. If the combined firm AB makes $350, then annual gain from synergy equals $350-$300 = $50.


Synergy (5 of 9)

Question: What are these “ΔCF”s ?

Answer: From capital budgeting we know that this equals

ΔCF = ΔRevenue - ΔCosts - ΔTaxes - ΔCapital RequirementsIncludes Net Working Capital & Initial Investment

So, sources of synergy fall into 4 categories: 1.) higher revenue ………………..ΔRevenue…….e.g., monopoly power allows firm to increase prices

2.) lower costs ………………..…..-ΔCosts………..e.g., economies of scale (average production costs fall), replacement of ineffective managers 3.) lower taxes…………………..... -ΔTaxes……….e.g., see next slide 4.) lower capital requirements….-ΔCap.Requir...e.g., get rid of redundant production facilities!


Synergy (6 of 9)

EXAMPLEExample of synergy from tax gains due to merger

We have two firms, A and B.

Two states of the economy each year.

Firm A: State 1: profits = $200

State 2: losses = -$100

Firm B: State 1: losses = -$100

State 2: profits = $200

Corporate tax rate is 34%.

Show tax gains from merger.


Synergy (7 of 9)

Before merger After merger

Firm A Firm B Firm AB

state 1 state 2 state 1 state 2 state 1 state 2

Taxable income

$200 -$100 -$100 $200

Taxes

Net Income

A and B together (no merger) State 1 State 2

(given)

$68=200*0.34

$0 $0 $68=200*0.34

$132=200-68

-$100 -$100 $132=200-68

$100 =200+(-100)

$100 =200+(-100)

$34=100*0.34

$34=100*0.34

$66=100-34

$66=100-34

$68 $68taxes drop with

merger, and so profits rise

TaxesNet Income $32 $32

= 132+(-100) = 132+(-100)


Synergy (8 of 9)

Results:

Taxes paid by firm AB < taxes paid by firms A and B together

Because of this,

profit of Firm AB > profits of firms A and B together


Synergy (9 of 9)

Additional tax gains that may take place:

• Unused debt capacity Target firm may have less debt than is optimal (recall Ch.17) due to poor

management. An acquiring firm may be smarter, and by raising the amount of debt after the merger create bigger tax shield, i.e., larger tax savings.


“Bad” reasons for mergers• Merger can create the appearance of earnings growth.

Investors may get fooled. How this can happen: If merger creates no synergy (i.e., VAB=VA+VB) then earnings of the new firm may double, but the number of new company’s shares of stock will less than double. This creates higher earnings per share. Fooled investors may take it as a signal of growth.

• Diversification = increase in value? Not necessarily.

It is true that unsystematic risk can be diversified away through mergers. But stockholders might as well create diversifications on their own at much lower cost (with one phone call to their broker) by buying stock from different companies.


Why stockholders may lose from merger (1 of 11)

Mergers often bring the following two results:

Good: Diversification. Volatility of newly created firm’s earnings is less than volatility of separate firms’ earnings.

Bad: The reduction of risk from this diversification is beneficial to bondholders, now that their debt is “insured” by two firms, not just one. But stockholders will be hurt.

Show these two effects in the next example…



EXAMPLEGiven:

Firm A wants to acquire Firm B

Firm A before merger: Firm value in state 1 equals $80 Firm value in state 2 equals $20

Firm B before merger: Firm value in state 1 equals $5 Firm value in state 2 equals $45

The two states are equally likely.

Calculate gains to stockholders of the newly created firm AB:Case 1: Firms hold no debt.Case 2: Face value of debt held by Firm A is $30 Face value of debt held by Firm B is $15



Case 1: Neither firm has debtFirm Value

State 1 State 2 Expected value Volatility (i.e., variance)

Probability 0.5 0.5

before merger

Firm A $80 $20

Firm B $5 $45

after merger

Firm AB $85 = 80+5

$65 = 20+45

$50 =0.5*80+0.5*20

$25 =0.5*5+0.5*45

$75 =0.5*85+0.5*65

σ2=900=0.5(80-50)2+0.5(20-50)2

σ2=400=0.5(5-25)2+0.5(45-25)2

σ2=100=0.5(85-75)2+0.5(65-75)2

(no synergy) a.k.a. financial merger, since the only good result is risk reduction (diversification)

Notice that risk falls. (Why?) 100 is less than both 400 and 900. “Diversification”.



Summarize results:

Result 1: If stockholders of former firm B still receive stock in the amount of $25, then stockholders of former firm A receive $75 - $25 = $50. No change. So stockholders of both firms are indifferent to the merger.

Result 2: Volatility of new firm AB’s value is lower than the volatilities of the individual firms: σ2

AB = 100 versus σ2A = 900 and σ2

B = 400.

This is the diversification effect.



Firm ValueState 1 State 2 expected value volatility

Probability 0.5 0.5

before merger

Firm A debt equity

$80 $20 $50 σ2=900

Firm B debt equity

$5 $45 $25 σ2=400

after merger

Firm AB debt equity

$85 $65 $75 σ2=100

Case 2: Debt of firm A is $30 , debt of firm B is $15

30 50

20 0

25=0.5*30+0.5*20

25=0.5*50+0.5*0

5 0

15 30

10=0.5*5+0.5*15

15=0.5*0+0.5*30

45= 40=85-45

4530=0.5*40+0.5*20

45= 20=65-45

σ2=…=25 σ2=…=625

σ2=…=25 σ2=…=225

σ2=…=0 σ2=…=100

(no synergy)

30+15 30+15




Probability 0.5 0.5

before merger

Firm A debt equity

$80 $20 $50 σ2=900

Firm B debt equity

$5 $45 $25 σ2=400

after merger

Firm AB debt equity

$85 $65 $75 σ2=100

Result 1: Bondholders benefit from the merger:

30 50

20 0

25=0.5*30+0.5*20

25=0.5*50+0.5*0

5 0

15 30

10=0.5*5+0.5*15

15=0.5*0+0.5*30

45=30+15 40=85-45

4530=0.5*40+0.5*20

45=30+15 20=65-45

σ2=25 σ2=625

σ2=25 σ2=225

σ2=0 σ2=100

Bondholders get paid in full in each state

And so risk becomes zero




Probability 0.5 0.5

before merger

Firm A debt equity

$80 $20 50 σ2=900

Firm B debt equity

$5 $45 25 σ2=400

after merger

Firm AB debt equity

85 65 75 σ2=100

Result 1: Bondholders benefit from the merger:

30 50

20 0

25=0.5*30+0.5*20

25=0.5*50+0.5*0

5 0

15 30

10=0.5*5+0.5*15

15=0.5*0+0.5*30

45=30+15 40=85-45

4530=0.5*40+0.5*20

45=30+15 20=65-45

σ2=25 σ2=625

σ2=25 σ2=225

σ2=0 σ2=100

Moreover, bondholders get 10 more under merger. 45 > 25 + 10



Intuition:

Without the merger, when A is not earning much in state 2, B does not help A pay for its debt. Similarly, when B is not earning much in state 1, A does not help B pay for its debt.

With the merger, when one of the divisions of the combined firm fails to pay for the debt, creditors can be paid from the profits of the other division.

This makes debt less risky (in our case σ2 =0).

Coinsurance effect

Why do bondholders receive more under the merger?




Probability 0.5 0.5

before merger

Firm A debt equity

$80 $20 50 σ2=900

Firm B debt equity

$5 $45 25 σ2=400

after merger

Firm AB debt equity

85 65 75 σ2=100

Result 2: Stockholders lose from the merger:

30 50

20 0

25=0.5*30+0.5*20

25=0.5*50+0.5*0

5 0

15 30

10=0.5*5+0.5*15

15=0.5*0+0.5*30

45=30+15 40=85-45

4530=0.5*40+0.5*20

45=30+15 20=65-45

σ2=25 σ2=625

σ2=25 σ2=225

σ2=0 σ2=100

On the one hand, equity becomes safer after merger.

100 < 625 & 225 (diversification)

On the other hand, AB’s stockholders get less than A’s and B’s combined. 30<25+15




Probability 0.5 0.5

before merger

Firm A debt equity

$80 $20 50 σ2=900

Firm B debt equity

$5 $45 25 σ2=400

after merger

Firm AB debt equity

85 65 75 σ2=100

Result 2: Stockholders lose from the merger:

30 50

20 0

25=0.5*30+0.5*20

25=0.5*50+0.5*0

5 0

15 30

10=0.5*5+0.5*15

15=0.5*0+0.5*30

45=30+15 40=85-45

4530=0.5*40+0.5*20

45=30+15 20=65-45

σ2=25 σ2=625

σ2=25 σ2=225

σ2=0 σ2=100

It can be shown that both A’s and B’s stockholders lose from the merger. 30 < 25+15



Summarize all results:

- Bondholders benefit from the merger due to the “coinsurance effect”. Debt becomes less risky.

- Stockholders may lose from the merger. Even though we have “diversification effect”, stockholders under merger receive less than stockholders in two separate firms combined. Both firms’ stockholders lose.


The NPV of a merger (1 of 13)

• Typically, a firm would use NPV analysis when making acquisitions.

• The two most common methods of acquisition are 1.) cash offer, and 2.) stock acquisition (stock-for-stock transaction).

• The analysis is straightforward with a cash offer, but gets complicated when the consideration is stock-for-stock exchange.



Cash offer:

Let’s say, A is the acquirer (bidder) and B is the acquired firm (target).

Recall that synergy = VAB – (VA + VB)

Also, premium paid for target (firm B) = Price paid for B – VB

NPV of merger to acquirer (firm A) = synergy – premium

= VAB – (VA+VB) – (price paid for B – VB)

= VAB – VA – VB – price paid for B + VB

= VAB – VA – price paid for B



EXAMPLE Given:

Firm A value before merger = $500. Number of shares = 25

Firm B value before merger = $300. Number of shares = 10

Synergy from merger = $100

Firm B will sell its assets only for $330

Find NPV of the merger to Firm A (acquiring firm):

1.) Merger through cash offer

2.) Merger through stock acquisition


The NPV of a merger (4 of 17)1.) Cash offer

VAB =

Also, premium =

VA + VB + synergy = 500 + 300 + 100 = 900

NPV of the merger to firm A =

Price paid for B – VB = 330 – 300 = 30

VAB – VA – price paid for B= 900 – 500 – 330 = 70

1.) Merger through CASH OFFER

Find NPV of the merger to firm A (acquiring firm):

NPV of the merger to firm A = VAB – VA – price paid for B

Or simply:synergy–premium=

= 100 – 30 == 70



Price per share:

Firm A:• Before merger: Price per share =

• After merger: New value of firm A (i.e., firm AB) with the merger = =

New price per share =

Firm B: • Before merger: Price per share = • After merger:

VA / #shares = $500 / 25 shares = $20

initial firm A value + NPV of merger to firm A = $500 + $70 = $570

new VA / #shares = $570 / 25 shares = $22.8 better off

VB / #shares = $300 / 10 shares = $30 N/A



Comment:

The market value of the acquiring firm A prior to acquisition should include the possibility of successful merger.And so it will be higher than the original value of $500.

Firm A’s value after merger = original value

of firm A + NPV of merger to firm A

Firm A’s value without merger

(given)

If there is, say, 30% of successful merger, thenMarket value of Firm A prior to acquisition = 0.3*_________ + 0.7*____ = ____0.3*($500+$70) + 0.7*$500 = $521


The NPV of a merger (7 of 17)2.) stock acquisition

2.) Merger through STOCK ACQUISITION (stock-for-stock transaction)

The analysis gets a little more complicated.Because we need to consider the post-merger value of those shares that we’re giving away.



Recall that

number of shares in firm A = 25

number of shares in firm B = 10

We have earlier calculated stock prices:

Firm A’s stock price before merger = $500 / 25 shares = $20 per share

Firm B’s stock price before merger = $300 / 10 shares = $30 per share



If firm A needs to pay $330 to acquire firm B, it may want to issue $330/ $20 per share = 16.5 shares and sell them to firm B in exchange for the entire 10 shares of firm B.

(i.e., exchange ratio is 1.65 : 1)

Under the merger: New Firm A (i.e, firm AB) will have 25 + 16.5 = 41.5 shares, Firm B will own 16.5/41.5 = 40% of the newly created firm AB, and

Firm A will own 25/41.5 = 60% of the newly created firm AB.

Firm B’s stockholders will own 40% x $900 = $358 (Recall: we got VAB = 900)

This is greater than $330 that Firm B asks for (This is due to synergy)! This means that we have made an incorrect stock-for-stock transaction!

Firm A needs to issue fewer shares. Also, with these calculations, the value of A’s stock after merger is $900/41.5 shares

= $21.69 only. Recall that under the cash offer it goes up from $20 to $22.8.



Question: What should the correct number of firm A’s newly issued shares be, in order to make stock acquisition equivalent to cash offer ($330)?

Answer:

Let α be the proportion of the shares in the combined firm AB that firm B’s stockholders own:

α = new shares issued by acquirer

old shares of acquirer + new shares issued by acquirer



α = new shares issued by firm A = 25 + new shares issued by firm A

In our case, we want α to be such that the value of the new firm AB that B’s stockholders will own is exactly $330:

In other words,

Solve for “α”: α =

Next, using the equation from the last slide, we can calculate how many new shares firm A needs to issue:

Rearrange and solve for “new shares issued by firm A”:

New shares issued by firm A = α*25 / (1 – α) = 0.367*25/(1 – 0.367) = 14.5 shares

Less than 16.5 shares calculated earlier

α * $900 = $330

$330 / $900= 36.7%

0.367



So, the correct exchange ratio is 1.45 : 1 instead of 1.65 : 1

What is the new price per share?

There are now _______________shares, and the new firm A (i.e., firm AB) value is $900.

So, Price per share = ___________________________________

Exactly the same stock price as with cash offer!

25 + 14.5 = 39.5

$900 / 39.5 shares = $22.8 per share



Results:

• NPV of a merger through cash offer is positive to the acquiring firm if synergy exceeds premium paid for the target firm.

In this case, acquiring firm’s stockholders benefit because stock price goes up.

• Due to synergy, the number of shares needed for merger through stock acquisition is less than the price for firm B divided by the pre-merger stock price of firm A.


Summary:

CASH OFFER type merger STOCK ACQUISITION type merger

• Bidder buys all Target’s assets

• If, say, Target is worth $300 but is willing to accept the merger offer only if offered $330, then Bidder pays $30 premium.

As long as synergy > premium, merger is worth it, i.e., will create positive NPV of merger to the Bidder.

• Bidder issues new shares of stock and sells them to Target in exchange for all Target’s shares

• If Bidder’s current stock price is $20, then exchanging $330/$20 = 16.5 shares for all Target’s shares is too much!

Why? – Since synergy increases combined firms’ value, fewer Bidder’s shares will be equivalent to $330 cash offer.

To find the exact # of shares needed, use the “alpha” calculations, as was shown in the earlier slides.



Question: When does Bidder want to pay with cash, and when with stock?

Answer: No easy answer.

Depends a lot on Bidder’s stock price.



Bidder’s managers – who have a lot more information about their company than investors do – may believe that their stock is overpriced.

In our example: Bidder’s stock is priced in the market at $20 per share, while “true” value may only be, say, $15.

Cash offer case: -No change.-Target firm still gets $330

Stock acquisition case: -Big change! -Target receives $330 worth of Bidder’s stock, based on market prices. If market soon realizes that the stock is overpriced, price per share will drop to $15, and Target firm’s stockholders end up with less money (below $330).


EXAMPLE


Bidder has an incentive to offer stock, rather than cash, if it believes that its stock may be overpriced.

Empirical studies show:

Since stock-for-stock offer is a signal of a possible price overvaluation, Bidder’s stock price tends to fall upon the announcement of a stock-for-stock deal.



Friendly vs. hostile takeovers (1 of 2)

Friendly vs. hostile takeovers

Friendly takeovers Hostile takeoversBoth companies’ managers are receptive.

CEO of one company calls CEO of another & proposes a merger.

Agree on price, terms of payment, etc.

Target’s management resists to merger.

Toehold = Bidder buys Target’s stock in secret.

Tender offer = Bidder makes an offer directly to Target’s stockholders to buy their shares of stock at a premium.

Street sweep = Bidder keeps buying Target’s shares through “tender offer”, until control over Target is achieved.

Proxy fight = Bidder gains majority of seats on Target’s board of directors.


Friendly vs. hostile takeovers (2 of 2)

Some of Target’s defensive tactics:

Golden parachutes = raise the cost of takeover by having to offer Target’s managers generous compensation packages (e.g., $5 million each).

Poison pills = e.g., once Bidder has purchased a certain % of Target’s stock, all Target’s stockholders, except for the Bidder, can buy new shares from Target at half price. Since price per each share drops, Bidder’s % of the firm drops also.

Greenmail = Target’s managers repurchase own stock from potential Bidder at a premium, & agree that Bidder won’t buy Target’s stock again for a specified period.

White knight = Target finds a more friendly acquirer that offers a higher price, or who promises not to lay off employees, not to fire managers, etc.

White squire = Target invites a third party (another company) to buy large number of its shares under condition that it won’t buy any additional shares & will support Target’s management. E.g., Warren Buffet (the most successful billionaire-investor) has acted as a white squire to Gillette & other firms.


Do mergers add value in real life?

In real life, do mergers add value? Does synergy exist?

Empirical studies for the US mergers in 1980-2001 show:

• Small vs. large mergers: value increases for small mergers, value decreases for large mergers.

• Small vs. large bidders: large bidders did worse than small ones.

• Targets’ stockholders: benefit. Merger premium is about 40%! So, Target’s management should never resist takeovers if it acts in the interest of its

stockholders

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