Assignment 2 2020 PDF

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Assignment 2 Semester 2 2020Question 1a. Risk Premiums and risk-free rate of the two factors:A: 19 =λ 0 +( 2 )λ 1 +( 0 )λ 2 ... 1 B: 8 =λ 0 +(− 1 )λ 1 +( 2 )λ 2 ... 2 C: 5 =λ 0 ... 33 into 1,19 = 5 + 2 λ 1 7 = λ 1Into 2,8 = 5 − 7 + 2 λ 2 5 = λ 2Hence, 5%=λ 0 , 7% = λ 1 and 5% = λ 2b. APT-consistent ...

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Assignment 2 Semester 2 2020 Question 1 a. Risk Premiums and risk-free rate of the two factors: A: 19= λ0 + ( 2) λ1 +(0) λ2 … 1 B: 8=λ0 + (−1) λ1 +(2) λ2 … 2 C: 5= λ0 … 3 3 into 1, 19 = 5+2 λ1 7 = λ1 Into 2, 8 = 5−7 +2 λ2 5 = λ2 Hence, 5 %=λ0 , 7% =

λ1

and 5% = λ2

b. APT-consistent expected return on Portfolio D: D = λ0 + ( 0 ) λ1+(1)λ 2 = 5+( 0) 7+(1)5 = 10% c. Design an arbitrage strategy involving portfolio A, B, C and D Portfolio D is overpriced, so short sell to profit. Zero Risk: 0+W A (2 )+ W B (−1) +W C ( 0 )=0 …1 −1+W A ( 0 )+W B ( 2) +W C ( 0 ) =0 … 2 From 2, W B=

−1 2

Into 1, W A =−¿ 1 Zero initial investment: −1+ W A + W B + W C =0 W C=1 −W A −W B

W C=1+1+ W C=2

1 2

1 2

Question 2 a. From the question, it is noticeable that the coupon rate is lower than the yield to maturity. This shows that the bond is selling at a discount. Since investors obtain capital gains from the yield to maturity, it would be expected that the bond price to increase. b. FV = 1000, PMT = 60, N = 19, YTM = 9% PV =

(

)

60 1 1000 1− + 19 19 0.09 (1.09) (1.09 )

PV = $731.50 c. FV = 1000, PMT = 60, N = 20, YTM = 9% PV =

(

)

1 1000 60 1− + 20 0.09 (1.09 ) (1.09)20

PV = $726.14 FV = 1000, PMT = 60, N = 19, YTM = 7% PV =

(

)

60 1 1000 1− 19 + 0.07 (1.07 ) (1.07)19

PV = $896.64 Before-tax holding period return:

−726.1436 + 60 ( 896.6440726.1436 ) 100 %

d. After-tax holding period return: To be taxed at 40% : 731.50−726.14 + 60=65.36 To be taxed at 30% : 896.64 −731.50=165.14 After tax: 65.36 x 0.60 = 39.216 165.14 x 0.70 = 115.598 Total = 154.81 After-tax holding period return:

154.81 x 100 % =21.32 % 726.14

e. RCY = (Ending Wealth / Purchase Price)1/n – 1 FV = 1000, PMT = 60, N=18, YTM= 7%

= 31.74%

PV2=

(

)

60 1 1000 1− + 18 0.07 (1.07) 1.0718

RCY = (

= $899.41

( 60 X 1.03 )+ 60+ 899.41 12 ) −1=18.59 % 726.14

Question 3 a. Macaulay Duration Year 5

Cash Flow 10,000

6

10,000

7

10,000

8

10,000

9

10,000

PV 10,000/(1.1)^5 = $6,209.213 10,000/(1.1)^6 =$5,644.739 10,000/(1.1)^7 =$5,131.581 10,000/(1.1)^8 =$4,665.074 10,000/(1.1)^9 =$4,240.976 Total= $25,891.58

Duration 1.19908 1.30808 1.387366 1.44142 1.47418 Total= 6.810126

b. Invests in zero coupon bonds and their face values 1−W (¿ ¿B)( 5 ) 6.810126 = W B (10 ) +¿ 6.810126 = 10 W B + 5 - 5 W B 1.810126 = 5 W B 0.36203 = W B 0.63797 = 1- W B Hence, invest $16,518.16 in 5-year zero coupon bond and $9,373.52 in 10-year zero coupon bond. Face values: $16,518.16 x (1.1)^5 = $26,602.66 $9,373.52 x (1.1)^10 = $24,312.50

c. Refer to the table below Year

Cash

Payment

5

$26,602.66

$10,000

Balance to sell or reinvested $16,602.66

6

16,602.66 x (1.09) =$18,096.90

$10,000

$8,096.90

7

8,096.90 x (1.09) =$8,825.62

$10,000

-$1,174.38

8

0

$10,000

-$10,000

9

0

$10,000

-$10,000

PV of 10-year coupon bond 24,312.50/ (1.09)^5 =$15,801.46 24,312.50/ (1.09)^4 =$17,223.59 24,312.50/ (1.09)^3 =$18,7737.71 $1,174.38 =$17,599.33 17,599.33 x 1.09 =$19,183.27$10,000 =$9,183.27 9,183.27 x 1.09 =$10,009.76 $10,000 =$9.76

Question 4 a. Expected yields to maturity on 1-year and 2-year coupon bonds next year f = (1.112/ 1.1) – 1 = 12.01%

1 2

1

f =

1 3

(1.123 / 1.1) 2 - 1 = 13.01%

b. Rate of return over the next year Purchase price =

1000 (1.12)3

= $711. 78

Expected price over the year =

Rate of return =

783.01 711.78

1000 2 (1.1301)

= 783.01

- 1 = 10.01%

c. FV = 1000, PMT= 120 1120 120 120 + + 3 2 1 (1.1) (1.11) (1.12) = $1,003.68

PV =

d. Expected rate of return over the next year Expected price over the next year =

Rate of return over the next year =

1120 120 + 1 (1.1201) (1.1301)2 120 + 984.10 1003.68

= $984.10

- 1 = 10.01%

Question 5 a. Value of put option C =S N P =K e

d

1

d

2

=

ln

( d 1 ) - K e - rT N ( d 2 ) - rT N ( - d 2 ) - S N ( - d 1)

(S K )+(r+ σ 2) T 2

σ

=d

1

- σ

T

T

( )

0.52 0.5 2 ¿ ¿ 0.3182 50 1n +¿ 50 d 1=¿ d 2=0.3182−0.5 √ 0.5 −0.0354 ¿ 0.1+

( )

N( d 1 ¿ N( d 2 ¿

¿ ¿

0.6255 0.4840

C = 50 ( 0.6255) −50 e(−0.1 x 0.5 ) (0.4840 ) =8.2552 Using put-call parity: P = 8.2552 +50 e (−0.1 x 0.5)−50 =¿

5.8167

Hence, the value of put option is $5.8167 b. The call option is overpriced. Strategy: Short-sell call option, long put option, long stock and borrow bond. The arbitrage profit: 9 – 8.2552 = $0.7448

Short-sell call option Long put option Long stock Borrow bond Net

S > 50 50 – S 0 S -50 0

S< 50 0 50 –S S -50 0

c. The maximum payoff of this position could be explained based on the table below

Expiry

80 70 60 50 40 30 20 10 0 Stock sale 80 70 60 50 40 30 20 10

0 55 Short call -25 -15 -5 0 0 0 0 0 0 45 Long put 0 0 0 0 5 15

25 35 45 Net pay-off 55 55 55 50 45 45 45 45 45 Hence, based on the table above, we can see that the highest (maximum) net pay-off of this position is $55

Expiry 80 70 60 50 40 30 20 10 0 Stock sale 80 70 60 50 40 30 20

10 0 55 Short call -25 -15 -5 0 0 0 0 0 0 45 Long put 0 0 0 0 5

15 25 35 45 Net pay-off 55 55 55 50 45 45 45 45 45 Hence, based on the table above, we can see that the highest (maximum) net pay-off of this

position is $55 Expiry Stock sale 55 Short call 45 Long put Net pay-off

80 80 -25 0 55

70 70 -15 0 55

60 60 -5 0 55

50 50 0 0 50

40 40 0 5 45

30 30 0 15 45

20 20 0 25 45

10 10 0 35 45

0 0 0 45 45

Resultantly, it is proven that the maximum payoff of this position is $55.

Question 6 a. Arbitrage-free futures settlement price is: 50 x (1+ 0.1) – 3 = $52 Hence, the futures price is overpriced. The arbitrageur would sell the futures contract, borrow funds to purchase the underlying stocks and hold the stocks until the fair value is restored or until it reaches the settlement date of the futures contract. Now: Sell the overpriced futures contract at $55, borrow $52 and purchase a stock portfolio equivalent to $52. In 1 year: Receive the $3 dividend, settle the short futures position by delivering to a buyer for $52 and repay $57.2 ($52 x 1.1) to settle the loan. Arbitrage profit: $55 + $3 - $57.2 = $0.80. This strategy upholds until the futures price restores to fair value. b. Put-call parity: P = C + Ee-Rt – S Spot-futures parity: F = SeRt If X = F, E = F, Ee-Rt =S Hence, P = C + S – S = C...