MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4

Worapol Ratanapan A0074997E

Given information 1) Investment fund A:

𝛿𝐴 𝑡 = �

11000

𝑡 + 10 ; 0 ≤ 𝑡 ≤ 10

2𝑡

100 + 𝑡2; 10 < 𝑡 ≤ 20

2) Investment fund B: continuously compounded rate 𝑅𝑅

𝛿𝐵 𝑡 =𝑅

100

MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4

Worapol Ratanapan A0074997E

Given information 1) Investment fund A:

𝛿𝐴 𝑡 = �

11000

𝑡 + 10 ; 0 ≤ 𝑡 ≤ 10

2𝑡

100 + 𝑡2; 10 < 𝑡 ≤ 20

2) Investment fund B: continuously compounded rate 𝑅𝑅

𝛿𝐵 𝑡 =𝑅

100

MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4

Worapol Ratanapan A0074997E

(i) If at 𝑡 = 20, the accumulated value of $1 invested in fund A equals 2.5 times the accumulated value of $1 invested in fund B.

MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4

Worapol Ratanapan A0074997E

(i) If at 𝑡 = 20, the accumulated value of $1 invested in fund A equals 2.5 times the accumulated value of $1 invested in fund B.

MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4

Worapol Ratanapan A0074997E

(i) If at 𝑡 = 20, the accumulated value of $1 invested in fund A equals 2.5 times the accumulated value of $1 invested in fund B.

$1 × 𝑎𝐴 20 = 2.5 × $1 × 𝑎𝐵 20

MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4

Worapol Ratanapan A0074997E

𝑎𝐴 20 = 𝑒𝑒𝑒 � 𝛿𝐴 𝑡20

0

𝑑𝑡

MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4

Worapol Ratanapan A0074997E

𝑎𝐴 20 = 𝑒𝑒𝑒 � 𝛿𝐴 𝑡20

0

𝑑𝑡

𝛿𝐴 𝑡 = �

11000

𝑡 + 10 ; 0 ≤ 𝑡 ≤ 10

2𝑡

100 + 𝑡2; 10 < 𝑡 ≤ 20

Recall

MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4

Worapol Ratanapan A0074997E

𝑎𝐴 20 = 𝑒𝑒𝑒 � 𝛿𝐴 𝑡20

0

𝑑𝑡

𝛿𝐴 𝑡 = �

11000

𝑡 + 10 ; 0 ≤ 𝑡 ≤ 10

2𝑡

100 + 𝑡2; 10 < 𝑡 ≤ 20

𝑎𝐴 20 = 𝑒𝑒𝑒 �1

1000𝑡 + 10

10

0

𝑑𝑡 + �2𝑡

100 + 𝑡2

20

10

𝑑𝑡

Recall

MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4

Worapol Ratanapan A0074997E

𝑎𝐴 20 = 𝑒𝑒𝑒 � 𝛿𝐴 𝑡20

0

𝑑𝑡

𝛿𝐴 𝑡 = �

11000

𝑡 + 10 ; 0 ≤ 𝑡 ≤ 10

2𝑡

100 + 𝑡2; 10 < 𝑡 ≤ 20

𝑎𝐴 20 = 𝑒𝑒𝑒 �1

1000𝑡 + 10

10

0

𝑑𝑡 + �2𝑡

100 + 𝑡2

20

10

𝑑𝑡

𝑎𝐴 20 = 𝑒𝑒𝑒1

1000𝑡2

2+ 10𝑡

0

10

+ ln 𝑡2 + 100 1020

Recall

MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4

Worapol Ratanapan A0074997E

𝑎𝐴 20 = 𝑒𝑒𝑒 � 𝛿𝐴 𝑡20

0

𝑑𝑡

𝛿𝐴 𝑡 = �

11000

𝑡 + 10 ; 0 ≤ 𝑡 ≤ 10

2𝑡

100 + 𝑡2; 10 < 𝑡 ≤ 20

𝑎𝐴 20 = 𝑒𝑒𝑒 �1

1000𝑡 + 10

10

0

𝑑𝑡 + �2𝑡

100 + 𝑡2

20

10

𝑑𝑡

𝑎𝐴 20 = 𝑒𝑒𝑒1

1000𝑡2

2+ 10𝑡

0

10

+ ln 𝑡2 + 100 1020

𝑎𝐴 20 =52𝑒3/20

Recall

MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4

Worapol Ratanapan A0074997E

(i) If at 𝑡 = 20, the accumulated value of $1 invested in fund A equals 2.5 times the accumulated value of $1 invested in fund B.

$1 × 𝑎𝐴 20 = 2.5 × $1 × 𝑎𝐵 20

52𝑒3/20 = 2.5 × 𝑎𝐵 20

MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4

Worapol Ratanapan A0074997E

𝑎𝐵 20 = 𝑒𝑒𝑒 � 𝛿𝐵 𝑡20

0

𝑑𝑡

MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4

Worapol Ratanapan A0074997E

𝑎𝐵 20 = 𝑒𝑒𝑒 � 𝛿𝐵 𝑡20

0

𝑑𝑡

𝛿𝐵 𝑡 =𝑅

100 Recall

MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4

Worapol Ratanapan A0074997E

𝑎𝐵 20 = 𝑒𝑒𝑒 � 𝛿𝐵 𝑡20

0

𝑑𝑡

𝑎𝐵 20 = 𝑒𝑒𝑒 �𝑅

100

20

0

𝑑𝑡

𝛿𝐵 𝑡 =𝑅

100 Recall

MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4

Worapol Ratanapan A0074997E

𝑎𝐵 20 = 𝑒𝑒𝑒 � 𝛿𝐵 𝑡20

0

𝑑𝑡

𝑎𝐵 20 = 𝑒𝑒𝑒 �𝑅

100

20

0

𝑑𝑡

𝑎𝐵 20 = 𝑒𝑅/5

𝛿𝐵 𝑡 =𝑅

100 Recall

MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4

Worapol Ratanapan A0074997E

(i) If at 𝑡 = 20, the accumulated value of $1 invested in fund A equals 2.5 times the accumulated value of $1 invested in fund B.

$1 × 𝑎𝐴 20 = 2.5 × $1 × 𝑎𝐵 20

52𝑒3/20 = 2.5 × 𝑒𝑅/5

MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4

Worapol Ratanapan A0074997E

(i) If at 𝑡 = 20, the accumulated value of $1 invested in fund A equals 2.5 times the accumulated value of $1 invested in fund B.

$1 × 𝑎𝐴 20 = 2.5 × $1 × 𝑎𝐵 20

52𝑒3/20 = 2.5 × 𝑒𝑅/5

𝑅 =34

= 0.75

MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4

Worapol Ratanapan A0074997E

(ii) If $1 invested in fund A at 𝑡 = 5 is worth $2 at 𝑡 = 𝑇, find the exact value of 𝑇, giving your answer in the form of 𝐴𝑒𝛼 + 𝐵 where 𝐴,𝐵𝜖𝜖 and 𝛼𝜖𝛼.

MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4

Worapol Ratanapan A0074997E

(ii) If $1 invested in fund A at 𝑡 = 5 is worth $2 at 𝑡 = 𝑇, find the exact value of 𝑇, giving your answer in the form of 𝐴𝑒𝛼 + 𝐵 where 𝐴,𝐵𝜖𝜖 and 𝛼𝜖𝛼.

MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4

Worapol Ratanapan A0074997E

(ii) If $1 invested in fund A at 𝑡 = 5 is worth $2 at 𝑡 = 𝑇, find the exact value of 𝑇, giving your answer in the form of 𝐴𝑒𝛼 + 𝐵 where 𝐴,𝐵𝜖𝜖 and 𝛼𝜖𝛼.

$1 × �𝛿𝐴 𝑡𝑇

5

𝑑𝑡 = $2

MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4

Worapol Ratanapan A0074997E

(ii) If $1 invested in fund A at 𝑡 = 5 is worth $2 at 𝑡 = 𝑇, find the exact value of 𝑇, giving your answer in the form of 𝐴𝑒𝛼 + 𝐵 where 𝐴,𝐵𝜖𝜖 and 𝛼𝜖𝛼.

$1 × �𝛿𝐴 𝑡𝑇

5

𝑑𝑡 = $2

MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4

Worapol Ratanapan A0074997E

(ii) If $1 invested in fund A at 𝑡 = 5 is worth $2 at 𝑡 = 𝑇, find the exact value of 𝑇, giving your answer in the form of 𝐴𝑒𝛼 + 𝐵 where 𝐴,𝐵𝜖𝜖 and 𝛼𝜖𝛼.

$1 × �𝛿𝐴 𝑡𝑇

5

𝑑𝑡 = $2

It’s obvious that 𝛿𝐴 𝑡 >0 .

MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4

Worapol Ratanapan A0074997E

(ii) If $1 invested in fund A at 𝑡 = 5 is worth $2 at 𝑡 = 𝑇, find the exact value of 𝑇, giving your answer in the form of 𝐴𝑒𝛼 + 𝐵 where 𝐴,𝐵𝜖𝜖 and 𝛼𝜖𝛼.

$1 × �𝛿𝐴 𝑡𝑇

5

𝑑𝑡 = $2

It’s obvious that 𝛿𝐴 𝑡 >0 . Observe that∫ 𝛿𝐴 𝑡10

5 𝑑𝑡 = 𝑒7/80 =< 2, therefore 𝑇 > 10.

MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4

Worapol Ratanapan A0074997E

Let 𝑇 > 10

MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4

Worapol Ratanapan A0074997E

Let 𝑇 > 10,

2 = 𝑒𝑒𝑒 �𝛿𝐴 𝑡𝑇

5

𝑑𝑡

MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4

Worapol Ratanapan A0074997E

Let 𝑇 > 10,

2 = 𝑒𝑒𝑒 �𝛿𝐴 𝑡𝑇

5

𝑑𝑡

2 = 𝑒𝑒𝑒 �1

1000𝑡 + 10

10

5

𝑑𝑡 + �2𝑡

100 + 𝑡2

𝑇

10

𝑑𝑡

MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4

Worapol Ratanapan A0074997E

Let 𝑇 > 10,

2 = 𝑒𝑒𝑒 �𝛿𝐴 𝑡𝑇

5

𝑑𝑡

2 = 𝑒𝑒𝑒 �1

1000𝑡 + 10

10

5

𝑑𝑡 + �2𝑡

100 + 𝑡2

𝑇

10

𝑑𝑡

2 = 𝑒7/80 𝑇2 + 100200

↔ 𝑇2 = 400𝑒−7/80 − 100

MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4

Worapol Ratanapan A0074997E

Let 𝑇 > 10,

2 = 𝑒𝑒𝑒 �𝛿𝐴 𝑡𝑇

5

𝑑𝑡

2 = 𝑒𝑒𝑒 �1

1000𝑡 + 10

10

5

𝑑𝑡 + �2𝑡

100 + 𝑡2

𝑇

10

𝑑𝑡

2 = 𝑒7/80 𝑇2 + 100200

↔ 𝑇2 = 400𝑒−7/80 − 100

𝑇 = 400𝑒−7/80 − 100

MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4

Worapol Ratanapan A0074997E

Conclusion (i) 𝑅 = 0.75 (ii) 𝑇 = 400𝑒−7/80 − 100

MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4

Worapol Ratanapan A0074997E