Lesson-4

Chapter 7: Stocks, Stock Valuation, and Stock Market Equilibrium

Chapter 8: Financial Options and Applications in Corporate Finance

Discussion Questions (DQs)

1. Do Problems 7-1 thru 7-7 on pp. 296-297.

2. Do Thomson ONE Problem on pp. 300-301.

3. Do Problems 8-1 thru 8-7 on pp. 329-330.

Problem 7-1. Solution:

Here given,

Current Dividend (D₀) = $ 1.50 per share

Dividend growth rate for 3 years (g₀) = 5 % =0.05

Dividend growth rate after 3 years (g₁) = 10% =0.10

We know that,

Expected Dividend for the t^th year (D_t) = D₀ (1+g) ^t    

For the first year:

Expected Dividend for first year (D₁) = D₀ (1+g) ¹

= 1.50(1+0.05)¹

= $ 1.5750

For the second year:

Expected Dividend for the second year (D₂) = D₀ (1+g) ²

= 1.50(1+0.05)²

= $ 1.6537

Similarly, for the third year:

 Expected Dividend for third year (D₃) = D₀ (1+g) ³

= 1.50(1+0.05)³

= $ 1.7364

Now, for the fourth year:

Expected Dividend for year four (D₄) = D₃ (1+g) ⁴

= 1.7364(1+0.10)¹

= $ 1.9101

For the fifth year:

Expected Dividend for year five (D₅) = D₄ (1+g) ⁵

= 1.9101(1+0.10)¹

= $ 2.1011

Therefore, expected dividend per share for each of the next 5 years will be $ 1.5750, $ 1.6537, $ 1.7364, $ 1.9101 and $ 2.1011 respectively.

Problem 7-2. Solution:

Here given,

Expected dividend for year 1(D₁) = $ 1.50

Growth rate (g_c) = 7% = 0.07

Return on stock (r_s)= 15% = 0.15

We know that,

r_s = (D₁/P₀)+ g_c

0.15 = (1.50/P₀) + 0.07

0.08 = (1.50/ P₀)

P₀ = $ 18.75

Therefore, Value per share of Boehm’s stock is $ 18.75 per share.

Problem 7-3. Solution:

 Here given,

Stock value P₀ = $ 20 per share

Current Dividend (D₀) = $ 1.00 per share

Growth rate (g_c) = 10 % = 0.10

Expected stock price one year from now,

We know that,

Expected dividend for year 1,(D₁) = D₀(1+g)^t

=1.00(1+0.1)¹

D₁= $ 1.1 per share

P₁ = P₀ (1+g_c)

= 20(1+0.1)

=$ 22

r_s = (D₁/P₀)+ g_c

= (1.1/20) + 0.10

=0.055+0.10

=0.1550 = 15.50%

Thus, expected stock price 1 year from now is $22 and required rate of return of Woidtke $ is 15.50%.

Problem 7-4. Solution:

Here given,

Stock value of Nick’s Enchiladas Inc, P₀ = $ 50 per share

Preferred dividend at the end of each year or Current Dividend (D₀) = $ 5 per share

Required rate of return if there is no growth rate is given,

We know that,

P₀ = D₀/r_ps

50 = 5/ r_ps

r_ps = 0.10 or 10 %

Thus, required rate of return of stocks of Nick’s Enchiladas Inc. is 10%.

Problem 7-5. Solution:

Here given,

Current dividend, D₀ = $2

Growth rate for next two years, g = 20%

Similarly, Growth rate thereafter, g_c= 7 %

Stock beta,  β = 1.2

Risk free rate, r_f =7.5%, market risk premium, (r_f  - r_m) = 4%

Stock current price, P₀

Required rate of return, r_s =  r_f + (r_f  - r_m)β

=7.5% + (4%) 1.2

=12.3 %

Expected Dividend for year one (D₁) = D₀ (1+g)^t

= 2(1+0.20)¹

= $ 2.4

Similarly, for year 2

Expected Dividend for year two (D₂) = D₀ (1+g)^t

= 2(1+0.20)²

= $ 2.88

Similarly, for year 3

Expected Dividend for year three (D₃) = D₀ (1+g)^t

= 2.88(1+0.07)¹

= $ 3.0816

We know that

Current stock price(P₀) =  D₁/(1+r_s)¹ +  D₂/(1+r_s)² + D₃/(1+r_s)³ + D₃ (1+g_n)/ r_s –g/(1+r_s)³

=2.4/1+0.123 + 2.88/(1+0.123)² + 3.0816/(1+0.123)³ + 3.0816(1+0.07)/ 0.123 –0.07/(1+0.123)³

=2.1371 + 2.28367 +2.1759 + 3.2973/0.053/1.4162

=6.59667+ 43.9296

= $ 50.52

Therefore, the estimated current stock price will be $ 50.52.

Problem 7-6. Solution:

Here given,

Expected dividend, D₁ = $4

Similarly, constant growth rate, g_c= ?

Stock current price, P₀ =$ 80

Required rate of return, r_s = 14 % =0.14

r_s = (D₁/P₀)+ g_c

0.14 =( 4/80) + g_c

g_c = 0.14-0.05

g_c = 0.09 or 9%

Therefore, the constant growth rate of the stock will be 9 %.

Problem 7-7. Solution:

Here given,

We have given

Expected dividend, D₁ = $4

Stock beta,  β = 0.9

Risk free rate, r_f =5.6 %, market risk premium, (r_f  - r_m)=6%

Stock current price, P₀ = $ 25

Required rate of return, r_s =  r_f + (r_f  - r_m)β

=5.6% + (6%) 0.9

Required rate of return, r_s = 11% or 0.11

Similarly, constant growth rate, g_c =?

r_s = (D₁/P₀)+ g_c

0.11 = ( 2/25) + g_c

g_c = 0.11-0.08

g_c = 0.03 or 3%

Therefore, the constant growth rate of the stock will be 3 %.

D₁ = D₀ (1+g)^t

2 = D₀ (1+0.03)¹

D₀ = 2/1.03 = 1.9417

Similarly, for year 2

Expected Dividend for year two (D₂) = D₁ (1+g)^t

= 2(1+0.03)

= $ 2.06

Similarly, for year 3

Expected Dividend for year three (D₃) = D₂ (1+g)^t

= 2.06(1+0.03)¹

= $ 2.1218

Now,

Current stock price (P₃) = D₃ (1+g_n)/ r_s -g_c

=2.1218(1+0.03)/ 0.11 –0.03

=2.185454/0.08

=$ 27.32

Therefore, Crisp Cookware’s stock price at the end of 3 years will be $ 27.32.

References

C. Paramasivan, T. Subramanian. (2015). Financial Management. New Delhi: New Age International Publishers.

Eugene F. Brigham, Michael C. Ehrhardt. (2011). Financial Management: Theory and Practice . Natorp Boulevard Mason, USA: South-Western Cengage Learning.

Problems 8-1. Solution:

Here given,

Current price of stock (V_s)  = $ 30

Strike price (E) = $ 25

Exercise value of call option (V_c) = $7

Using formula:

Exercise value of the call option (V_o) = Max [V_s – E, 0]

= [30-25, 0] = $5

Now,

Options time value = Market price of call option (V_c) - Exercise value of the call option(V_o)

= $7 - $5

= $ 2

Therefore, Bedrock Boulders exercise value of call option and options time value are $ 5 and $ 2 respectively.

Problems 8-2 Solution: 

Here given,

Stock exercise value, E = $ 15

Exercise value of option (V_o) = $22

Options time value= $ 5

Market price of call option (V_c) =?

Current stock price, V_s =?

We know,

Options time value = Market price of call option (V_c) - Exercise value of the call option(V_o)

$ 5 = V_c -22

V_c = $ 27

Now,

Exercise value of the call option, V_o= Max [V_s – E, 0]

22 = V_s – 15

V_s = $ 37

Therefore, Flanagan Company’s Market price of call option and price of stock are $ 27 and $ 37 respectively.

Problems 8-3 Solution:

Here given,

Current Stock price (V_s) = $ 15

Strike price of option, E = $ 15

Time to maturity of option, t = ½ year

Risk free rate, r_f = 6% = 0.06

Variance of stock return = 0.12

d₁= 0.24495, N (d₁) = 0.59675

d₂ =0.00000, N (d₂) = 0.5000

According to Black-Scholes option pricing model:

Option value, V₀ = V_s * N(d₁) – E *e ^-rf*t * N(d₂)

=`15*0.59675 -15*e^-0.06*0.5 * 0.5000

= 8.95125-7.278

= $ 1.67

Therefore, value of stock option under Black-Scholes model is $1.67.

Problem 8-4. Solution:

Here given

Current Stock price (V_s) = $ 33

Strike price of option, E = $ 32

Time to maturity of option, t = 1 year

Risk free rate, r_f = 6% = 0.06

Value of call option V_c = $ 6.56

According to Put – Call Parity Equation:

Put Option + Stock = Call option + Present value of exercise price

or Put option = V_c - V_s+ e ^-rf*t

= 6.56 - 33 + 32 e ^-0.06*1

= - 26.44 + 32*0.94176

= 30.1364 – 26.44

= $ 3.6964

Therefore, the value put option is $ 3.6964 under Put-Call parity.

Problems 8-5. Solution:

Here given

Current Stock price (V_s) = $ 30

Strike price of call option, E = $ 35

Time to maturity of option, t = 4 months or 1/3 year

Risk free rate, r_f = 5% = 0.05

Variance of stock return = 0.25 standard deviation, SD = √0.25 = 0.50

d₁= ln (V_s / E) + [r_f + 0.5(SD²)]t

= [ln (30 /35) + {0.05+ 0.5(0.25)} 0.3333]/SD√t

= [-0.1542 +0.0583333]/0.5√0.3333

=0.3317

Referring the normal distribution table:

At 0.30, the tail value is 0.3821

At 0.35, the tail value is 0.3632

By interpolation

The tail value at 0.3317 =0.3821 – (0.3317-0.3000)*(0.3821-0.3632)/(0.35-0.30)

=0.3821-0.634*0.0189

=0.3701174

d₂ = d₁ - SD√t

= -0.3317 – 0.50√0.3333

=0.6204

Referring the normal distribution table

At 0.60, the tail value is 0.2743

At 0.65, the tail value is 0.2578

By interpolation

The tail value at 0.6204 =0.2743 – (0.6204-0.6000)*(0.2743-0.2578)/(0.65-0.60)

=0.2743-(0.0204*0.0165)/0.05

=0.2676

Using Black-Scholes option pricing model:

Option value, V₀ = V_s * N(d₁) – E *e ^-rf*t * N(d₂)

=`30*0.3701 -35*e^-0.05*0.3333 * 0.2676

= 11 .103 - 34.4214*0.2676

=$ 1.89

Therefore, value of stock option under Black-Scholes model is $1.89.

Problem 8-6 Solution:

Here given,

Valuation of call option under binomial option pricing model can be done as follows:

Current Stock price (Vs) = $ 20

Price of stock Up, UVs = $ 26

Price of stock down, LVs= $ 16

Risk free rate, rf = 5% p.a = 0.05

 Strike price of call option, E = $ 21

Step 1

Set up binomial tree and calculate the option values at expiration for each ending stock price

Price of option Up, UV0 = $ 5

Price of option down, LV0 = $ 0

                $26 (Value of option) = [Vs-E, 0] = [26-21, 0] = 5

$21

                $16 (Value of option) = [Vs-E, 0] = [16-21, 0] = 0

Step 2

Solve for the amount to invest in the stock and the amount to borrow in order to replicate the option gives in the up state and its value in the down state,

Now we know that,

Hedge ratio = UV0 - LV0 / UVs - LVs

= (5-0)/ (26-16)

=0.5

Amount of borrowing (B) = Present value (hLVs-LV0)/1+rf

= (0.5*16-0)/(1+(0.05/365))365  (note- interest compounding daily)

=8/1.05126

= $ 7.60986

Step 3

Now, use the values derived in step 2 to solve for the value of the option at the beginning of the period.

Value of call option = (h*Vs-B)

=0.5*20 -7.60986

= $2.3901

Therefore, the price of call option on the stock is $2.3901.

Problem 8-7. Solution:

Here given,

Valuation of call option under binomial option pricing model can be calculated as follows:

Current Stock price (Vs) = $ 15

Price of stock Up, UVs = $ 18

Price of stock down, LVs= $ 13

Risk free rate, rf = 6% p.a = 0.06/2 = 0.03 (6 months to maturity)

 Strike price of call option, E = $ 14

Time to maturity of option, t = 4 months or 1/3 year

Variance of stock return = 0.25 standard deviation, SD = √0.25 = 0.50

Step 1

Set up binomial tree and calculate the option values at expiration for each ending stock price

Price of option Up, UV0 = $ 4

Price of option down, LV0 = $ 0

                      $18 (Value of option) = [Vs-E, 0] = [18-14, 0] = 4

$14

                   $13 (Value of option) = [Vs-E, 0] = [13-14, 0] = 0

Step 2

Solve for the amount to invest in the stock and the amount to borrow in order to replicate the option gives in the up state and its value in the down state,

Now we know that,

Hedge ratio = UV0 - LV0 / UVs - LVs

=(4-0)/ (18-13)

=0.8

Amount of borrowing (B) = Present value (hLVs-LV0)/1+rf

= (0.8*13-0) / (1+ (0.03/365)) 365(note- interest compounding daily)

=10.4/1.0304532

= $ 10.0926

Step 3

Now, use the values derived in step 2 to solve for the value of the option at the beginning of the period.

Value of call option = (h*Vs-B)

=0.8*15 -10.0926

= 12- 10.0926

= $ 1.9074

Hence, the price of call option on the stock is $ 1.9074.

References

C. Paramasivan, T. Subramanian. (2015). Financial Management. New Delhi: New Age International Publishers.

Eugene F. Brigham, Michael C. Ehrhardt. (2011). Financial Management: Theory and Practice . Natorp Boulevard Mason, USA: South-Western Cengage Learning.