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219 ©2013 Pearson Education, Inc. Publishing as Prentice Hall Chapter 15 Financial Engineering and Security Design Question 15.1 Let R = e 0.06 . The present value of the dividends is R 1 + (1.50) R 2 + 2R 3 + (2.50) R 4 + 3R 5 = 8.1317. (1) The note originally sells for 100 8.1317 = 91.868. With the 25 cent permanent increase, the present value of dividends rises by 1 2 3 4 –5 4 R R R R R + + + + =1.04785 (2) to 9.1796 leading the note value to fall to 100 9.1796 = 90.8204. Question 15.2 For this problem let B = 1 1.03 . a) The prepaid forward price is 1,200e 0.015(3) = 1,147.20. b) We have to solve the coupon, c, that solves 6 1 i i c B = + 1,147.20 = 1,200 c = 52.8 7 1 B B B = 9.7467. (3) c) The prepaid forward price for one share at time t is P t F = 1,200e 0.015t ; for each semiannual share, we can write the relevant prepaid forward price as 1,200D i where D = e 0.015/2 . With this formulation we have a similar analysis for the fractional shares, c : 6 1 i i c D = 1,200 + 1,147.20 = 1,200 c = 7 52.8 1 1, 200 D D D = 0.007528 shares. (4) Note this is interpreted as we will receive 0.007528 units of the index every six months. This has a current value of 1,200(0.007528) = 9.0336. We could quote c in dollars ($9.0336) instead of units. Question 15.3 a) S 0 e δT = 1,200e 0.015(2) = 1,164.5.
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Page 1: McDonald ISM3e Chapter 15 - Texas Christian Universitysbufaculty.tcu.edu/mann/!_AdvInv-Fall2016/solution/Chapter 15.pdf · Chapter 15 Financial Engineering and Security Design Question

219 ©2013 Pearson Education, Inc. Publishing as Prentice Hall

Chapter 15

Financial Engineering and Security Design

Question 15.1

Let R = e0.06. The present value of the dividends is

R−1 + (1.50) R−2 + 2R−3 + (2.50) R−4 + 3R−5 = 8.1317. (1)

The note originally sells for 100 − 8.1317 = 91.868. With the 25 cent permanent increase, the present value of dividends rises by

1 2 3 4 –5

4R R R R R− − − −+ + + + =1.04785 (2)

to 9.1796 leading the note value to fall to 100 − 9.1796 = 90.8204.

Question 15.2

For this problem let B = 11.03

.

a) The prepaid forward price is 1,200e−0.015(3) = 1,147.20.

b) We have to solve the coupon, c, that solves

6

1

i

i

c B=

⎛ ⎞⎜ ⎟⎝ ⎠∑ + 1,147.20 = 1,200 ⇒ c = 52.8 7

1 BB B−⎛ ⎞

⎜ ⎟−⎝ ⎠ = 9.7467. (3)

c) The prepaid forward price for one share at time t is PtF = 1,200e−0.015t ; for each semiannual

share, we can write the relevant prepaid forward price as 1,200Di where D = e−0.015/2. With this formulation we have a similar analysis for the fractional shares, c∗:

6

1

i

i

c D∗

=

⎛ ⎞⎜ ⎟⎝ ⎠∑ 1,200 + 1,147.20 = 1,200 ⇒ c∗ = 7

52.8 11,200

DD D−⎛ ⎞

⎜ ⎟−⎝ ⎠= 0.007528 shares. (4)

Note this is interpreted as we will receive 0.007528 units of the index every six months. This has a current value of 1,200(0.007528) = 9.0336. We could quote c∗ in dollars ($9.0336) instead of units.

Question 15.3

a) S0e−δT = 1,200e−0.015(2) = 1,164.5.

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b) As in equation (15.6),

c =( )0.015(2)

08

1

1, 200 1 7.4475

PT

tii

eS FP

=

−−=

∑ = 4.762. (5)

c) As in the problem 15.2c, letting D = e−0.015/4,

8

1

i

i

c D∗

=

⎛ ⎞⎜ ⎟⎝ ⎠∑ 1,200 + 1,200D8 = 1,200 ⇒ c∗ =

8

8

1

1i

i

DD

=

∑= 0.003757 shares, (6)

which is currently worth 0.003757 ($1,200) = $4.5084.

Question 15.4

The relevant two year interest rate is ln (1/0.8763) /2 = 6.6%.

a) The embedded option is worth 247.88. The prepaid forward is worth 1,200e−0.015(2) = 1,164.53. The bond price is worth the sum 1,164.53 + 247.88 = 1,412.41.

b) λ must solve 1,164.53 + λ247.88 = 1,200 ⇒ λ = 35.47/247.88 = 0.1431.

Question 15.5

As in the previous question, we use r = 6.6%.

a) The embedded option is worth 247.88. The bond price is worth 1,200 (0.8763) + 247.88 = 1,299.44.

b) λ must solve 1,200 (0.8763) + λ247.88 = 1,200 =⇒ λ = 0.59884.

Question 15.6

We continue to use 6.6 percent as the relevant two-year interest rate.

a) The out of the money option (i.e., K = 1,500) is worth 141.54, making the bond have a value of 1,164.53 + 247.88 − 141.54 = 1,270.9.

b) We must solve 1,164.53 + λ (247.88 − 141.54) = 1,200 for a solution of λ = 0.3336.

c) If λ = 1, we have to adjust the strike (from part (a), we know we have to lower K ) to make the out of the option worth C (K ) = 1,164.53 + 247.88 – 1,200 = 212.41 ⇒ K ≈ 1,284.

Question 15.7

Let B = e−0.06×5.5 be the relevant discount factor. The equity linked CD is worth 1,300B + 0.7C. Let the two-year forward price be F0 = 1,300e(r − q )2. By put-call parity, C − P = (F0 – 1,300) B. This implies the CD is equivalent to a long forward position on 0.7 on the index (zero cost), a

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long position of 0.7 at the money puts, and an investment of 0.3 (1,300B ) + 0.7F0 (B ) dollars in the risk free bond (i.e., 0.3 + 0.7F0 bonds). The final payoff will be

0.7 (max (1,300 − S5.5, 0) + S5.5 − F0) + 0.7F0 + 0.3 (1300). (7)

If S5.5 < 1,300, this equals 0.7 (1,300 − F0) + 0.7F0 + 0.3 (1,300) = 1,300, and if S5.5 ≥ 1,300, it is equal to

0.7 (S5.5 − F0) + 0.7F0 + 0.3 (1,300) = 1,300 + 0.7 (S5.5 – 1,300), (8)

which is the same as the CD.

Question 15.8

Using a semiannual coupon of c, if the value of the CD is 1,300, c and γ must solve

1,300 = 1,300e−0.06×5.5 + c11

0.03

1

i

i

e−

=

⎛ ⎞⎜ ⎟⎝ ⎠∑ + γ441.44. (9)

This implies the participation rate, as a function of c is γ = 0.82774 − 0.0209c (i.e., a line with intercept 0.82774 and slope −0.0209). For example, a $10 semiannual coupon will require 0.618 74 call options to have the CD be worth 1,300. If the bank would like to earn 5 percent, c and γ must solve

1,300 (1 − 0.05) = 1,300e−0.06×5.5 + c11

0.03

1

i

i

e−

=

⎛ ⎞⎜ ⎟⎝ ⎠∑ + γ441.44. (10)

This implies γ = 0.6805 − 0.0209c; this is a parallel line (to the previous answer’s line) with a lower intercept.

Question 15.9

For notation clarity, let the six-month discount factor be B = e−0.06/2. The ATM put option will be worth 178.99. We need to find the cash payment, c, to solve

1,300 = 1,300e−0.06×5.5 − 178.99 + c12

1B B

B⎛ ⎞−⎜ ⎟−⎝ ⎠

(11)

for a solution of c = 58.984.

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Question 15.10

The 2,600-strike call has a value of 162.48. The 1,300-strike put is worth 178.99. This implies λ must solve

1,300 = 1,300e−0.06×5.5 − 178.99 + λ (162.48) (12)

implying λ = 3.3505.

Question 15.11

λ must solve 1,300 = 1,200e−0.06×5.5 + λ (441.44) for a solution of λ = 0.9906.

Question 15.12

See Table One for the numerical solution.

a) The value is 1,300e−r ×5.5 + 0.7 × BSCall (1300, 1300, σ, r, 5.5, δ).

b) We must solve for γ ,

1,300(1 − 0.043) = 1,300e−r ×5.5 + γ × BSCall (1300, 1300, σ, r, 5.5, δ) (13)

Question 15.13

Using the information from the Problems Table:

a) The prepaid forward price is Pt8 Ft8 = 0.8763 (19.8) = 17.351.

b) The cash payment solves

c(0.9388 + 0.8763) = 20.90 − 17.351 ⇒ c = 1.9553 (14)

Question 15.14

The two-year, prepaid forward price is 17.351.

a) c∗ must solve

c∗20.5 (0.9388) + c∗19.8 (0.8763) + 19.8 (0.8763) = 20.9 (15)

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Hence c∗ = 20.9 17.35119.245 17.351

−+

= 0.097 barrels. In dollars, this is currently worth 0.097 (20.9) =

$2.0273.

b) Similarly,

8

1i

Pt

i

c F∗

=

⎛ ⎞⎜ ⎟⎝ ⎠∑ = 20.9 − 17.351 (16)

hence c∗ =20.9 17.351

152.1556− = 0.023325 barrels, currently worth 0.0233 (20.9) = $0.4875.

Question 15.15

We can value the option as a futures option (or use a spot option with a dividend yield equal to the lease rate). Either way, the option is worth 1.073 (use 19.8e−0.066×2 = 17.352 in the nondividend BSCall). Hence λ must solve 20.9 = 17.352 + λ1.073 for a solution of λ = 3.3066.

Question 15.16

For the options, we can use the following answer from part (a) as the underlying in the nondividend BSCall, the interest rate being ln (1/0.9388) = 6.32%.

a) The prepaid forward price is Fi Pi = 20.5 (0.9388) = 19.2454.

b) With K1 = 19.577 and K2 = 21.577, a put with K1 strike is worth 0.7436 and a call with strike K2 is worth the same. Hence the cost is the same as the prepaid forward contract, 19.2454. With this contract, we receive the spot price of oil if S1 is between 19.577 and 21.577. We receive a lower bound of 19.577 (if S1 < 19.577) and an upper bound of 21.577 (if S1 > 21.577). This payoff is similar to a bull spread with a risk free bond. The prepaid forward has the holder owning a barrel of oil; whereas, this contract involves owning a barrel of oil only if S1 is between K1 and K2. It will do worse than the prepaid forward if S1 > K2 and better if S1 < K1 (trading off upside to protect downside).

c) The value of this claim is zero: S1 − 20.50 is a (zero cost) forward contract and the two options have equal premiums. If S1 is between K1 and K2, we pay 20.5 and receive a barrel of oil worth S1. If S1 < K1 = 19.577, we pay 20.5, receive a barrel of oil, and sell it for K1 leading to a cash flow of 19.577 − 20.5 = −0.923. If S1 > K2 = 21.577, we pay 20.5, receive a barrel of oil, and sell it for K2 leading to a cash flow of 21.577 − 20.5 = 1.077. This is the profit from a bull spread.

Question 15.17

The contract has the party receiving a K2 call and writing a K1 put. We will assume K2 = K1 + 2 (otherwise there are an infinite number of solutions).

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a) The strikes must solve C (K1 + 2) = P (K1). K1 = 19.577 satisfies this condition. If K2 − K1 ≠ 2, there are infinite solutions to this. One solution is K2 = 22 and K1 = 19.229, another being K1 = 18 and K2 = 23.607.

b) We must solve ( ) ( )8 81 11 1

2, ,i ii iC K t P K t

= =+ =∑ ∑ . We can solve numerically for K1 =

19.456 and K2 = 21.456.

Question 15.18

Since the contract has zero value, it doesn’t matter which side of the contract we examine. Consider the “seller” of the contract. Each quarter i, the seller receives a cash flow

19.90 − F + max (Si − 19.90, 0) − max (Si − 21.90, 0) (17)

To check, if Si < 19.90, the seller has a cash flow 19.90 − F . If Si is between 19.90 and 21.90, there is a cash flow of Si − F , and if Si > 21.90, the cash flow if 21.90 − F . Each quarter the bull spread will have a value

Vi = BSCall (Fi Pi , 19.90, 0.15, ri, ti , 0) − BSCall (FiPi , 21.90, 0.15, ri, ti , 0) (18)

where Pi is zero coupon bond price from Problems Table and i ir te = Pi . Table Two on the next page shows the values for Vi and ri . We now must solve for F in

Vcontract = 0 = ( )8

1

19.90 i

F=

−∑ Pi + 8

1i

iV

=∑ (19)

implying

8

18

1

6.361619.907.4475

ii

ii

VF

P=

=

= + =∑∑

+ 19.90 = 20.754. (20)

Question 15.19

An investor needs to hold the stock (perhaps tracking an industry) but is pessimistic about the prospects of the company is an investor who prefers a PEPS. If the stock goes down, the PEPS outperforms the stock, while if the stock rises sufficiently, the stock outperforms the PEPS.

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Question 15.20

The main textbook gives the following table of payment at maturity of the Times Mirror PEPS:

Underlying price of Netscape at expiration Payoff to DECS holder

S(T) <$39.25 S(T)

$39.25 < S(T) < $45.14 $39.25

$45.14 < S(T) $39.25 + 0.8696 × (S(T) − $45.14)

The above table can be replicated with the following instruments: buy one share of Netscape, sell one Netscape call with a strike of $39.25, and buy 0.8696 calls with a strike of $45.14.

Furthermore, we need to take care of the dividends of the PEPS. We are entitled to the annual cash dividend of $1.67 (which is the issue price of $39.25 times the coupon rate of 4.25 percent) of the PEPS (which is missing from the above strategy). We therefore need to add zero-coupon bonds characterizing the PEPS dividend payments.

PPEPS = ST − C (39.25,…) + 0.8696 × C (45.14) + PV (DivPEPS)

We can value the annual PEPS dividend

R 0.07 PEPS Div 1.668125

P(t) disc. DECS Div 1 0.932 1.555

2 0.869 1.450

3 0.811 1.352

4 0.756 1.261

5 0.705 1.176

6.794

Therefore, we can solve: PPEPS = 39.25 − 16.302 + 0.8696 × 18.153 + 6.794 = 42.067. The PEPS is worth $42.067 today.

Question 15.21

Suppose a DECS (Debt Exchangeable for Common Stock) contract pays two shares if S(T ) < 27.875, 1.6667 shares if S(T) > 33.45, and $55.75 otherwise. The DECS contract pays a quarterly dividend of $0.87. Value this DECS assuming that S(0) = 26.70, sigma = 35%, r = 9%, and T = 3.25 and the underlying stock pays a quarterly dividend of $0.10. Further assume that

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the DECS and the underlying just paid their dividend (i.e., that the next dividend arises precisely three months from today).

The payoff of the DECS contract can be classified as follows:

Underlying share price at expiration Payoff to DECS holder

S(T) < $27.875 2 × S(T)

$27.875 < S(T) < $33.45 $55.75

$33.45 < S(T) $55.75 + 1.6667 × (S(T) − $33.45)

The above table can be replicated with the following instruments: buy two shares of the underlying, sell two calls with a strike of $27.875, and buy 1.6667 calls with a strike of $33.45.

Furthermore, we need to take care of the dividends of both the DECS and the underlying stock. We are entitled to the $0.87 cash dividend of the DECS (which is missing from the above strategy), and we are NOT entitled to the cash dividend of the stock during the life of the DECS (which we erroneously receive via the above strategy). We, therefore, need to add zero-coupon bonds characterizing the DECS dividend payments and subtract zero-coupon bonds characterizing the dividend payments of the two shares we bought.

When we value the stock option, we remove the present value of the dividends from today’s stock price, and do as if the stock was ex dividend. This is more accurate than converting the discrete dividend to a dividend yield.

The value of the DECS contract is:

PDECS = 2 × ST − 2 × C (27.875,…) + 1.6667 × C (33.45) + PV (DivDECS) − 2 × PV (DivStock)

We have to value the dividends of the DECS and the stock:

r 0.09

rquarterly 0.0225

DECS Div 0.87

Stock Div 0.1

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P(t) disc. DECS Div 2 × disc. Stock Div

1 0.978 0.851 0.196

2 0.956 0.832 0.191

3 0.935 0.813 0.187

4 0.914 0.795 0.183

5 0.894 0.777 0.179

6 0.874 0.760 0.175

7 0.854 0.743 0.171

8 0.835 0.727 0.167

9 0.817 0.711 0.163

10 0.799 0.695 0.160

11 0.781 0.679 0.156

12 0.763 0.664 0.153

13 0.746 0.649 0.149

9.696 2.229

We need to discount the stock price by the present value of one dividend, or $1.1145 to $25.5855. The option prices are:

C(27.875) = 8.428

C(33.45) = 6.572

The value of the DECS contract is:

PDECS = 2 × 26.70 − 2 × 8.428 + 1.6667 × 6.572 + 9.696 − 2.229 = 54.965

Question 15.22

a) We have to value a forward contract with a dividend yield of zero. We know that the price for such a forward contract is:

F0,T = S0 × exp (rt ) = 100 × exp (0.03 × 3) = 109.417

b) We could see in part (a) that the forward price is taking into account the interest accruing to the $100. The annual payment, if the price agreed upon is $100, would reflect that interest. The coupon would have to be 3 percent, continuously compounded, or $3.045, at the end of Years 1, 2, and 3.

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c) Now, we can depict the payoff of the stock purchase contract as follows:

Underlying stock price at expiration Payoff to stock purchase contract holder

S(T) < $100.00 1 × S(T)

$100 ≤ S(T) < $116.20 S(T) + ($100 − S(T)) = $100

$100$116.20

≤ S(T) S(T) +($100 − S(T)) + 100/116.20 × (S(T) − $116.20)

Therefore, we can value this complicated stock purchase contract as a simple stock purchase contract for a price of $100 plus the simultaneous purchase of 100/116.20 call options with a strike of 116.20 and a sale of one call option with a strike of $100. We valued the simple stock purchase contract in part (b), and now have to value the two stock options:

Based on the information given in the exercise, we obtain:

C(100) = 24.2068

C(116.20) = 18.1729

Overall, we have:

Payoff = ST − 100 + max (0, ST − 100) + 100116.20

× max (0, ST − 116.20)

and a price today of:

Price =3

1 $3.045

t=−∑ × exp (−0.03 × t) + 24.2068 − 0.8606 × 18.1729

= −8.6056 + 24.2068 − 15.6396

= −$0.0384

We would need to be paid 3.84 cents to enter into the contract.

d) Now, we are not entitled to receive the dividend the stock pays. This is an inconvenience, and we have to take this into account when calculating the forward price:

F0,T = S0 × exp [(r − δ) × t] = 100 × exp [(0.03 − 0.03) × 3] = 100.00

$100 is now the fair price, and we would not have to pay a coupon in part (b). For part (c), we would pay up to 24.2068 − 15.6396 = 8.5672 to enter the stock purchase contract with the contractual features described.

Question 15.23

In Chapter 3, we have seen that the payoff to the Marshal & Ilsley stock purchase contract can be described by:

Payoff = 0.6699 × ( ) ( )1max 0, 37.32 max 0, 46.281.24MI MI MIS S S⎡ ⎤− − + × −⎢ ⎥⎣ ⎦

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In order to value the two call options, we need the six inputs necessary to value the option:

Initial stock price: 37.32

K = 37.32, 46.28

Sigma = 15%

r = 3%

Dividend yield = 2%

T = 3 year

We have for the two option prices:

Call(37.32) = 4.122

Call(46.28) = 1.394

We can refer to problem 15.22 to solve the issue at hand. We have engaged a stock purchase contract for 0.6699 shares of M&I for a price of $25, and we have simultaneously sold 0.6699 call options with a strike of 37.32 and bought 1/1.24× 0.6699 call options with a strike of $46.28.

First, let’s value the stock purchase contract. Based on our forward price formula, we should accept a price of:

F0,T = 0.6699 × S0 × exp [(r − δ) × t] = 0.6699 × 37.32 × exp (0.01 × 3) = 25.76205

Buying the share at $25 is, therefore, slightly advantageous. The difference, 0.76205, has a present value of 0.69646.

Furthermore, we receive an annual coupon payment of 2.6 percent. That annual coupon payment we receive is valued at:

0.6699 × 37.32 × 0.026 × ( )0.03 0.06 0.09e e e− − −+ + = 1.83704

Now, we can sum all payments up to get an initial price for the stock purchase agreement of:

Price = −0.69646 + 1.83704 + 4.122 − 0.54024 × 1.394

= 4.5095