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Given information 1) Investment fund A:
πΏπ΄ π‘ = οΏ½
11000
π‘ + 10 ; 0 β€ π‘ β€ 10
2π‘
100 + π‘2; 10 < π‘ β€ 20
2) Investment fund B: continuously compounded rate π π
πΏπ΅ π‘ =π
100
MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4
Worapol Ratanapan A0074997E
Given information 1) Investment fund A:
πΏπ΄ π‘ = οΏ½
11000
π‘ + 10 ; 0 β€ π‘ β€ 10
2π‘
100 + π‘2; 10 < π‘ β€ 20
2) Investment fund B: continuously compounded rate π π
πΏπ΅ π‘ =π
100
MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4
Worapol Ratanapan A0074997E
(i) If at π‘ = 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 π‘ = 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 π‘ = 20, the accumulated value of $1 invested in fund A equals 2.5 times the accumulated value of $1 invested in fund B.
$1 Γ ππ΄ 20 = 2.5 Γ $1 Γ ππ΅ 20
MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4
Worapol Ratanapan A0074997E
ππ΄ 20 = πππ οΏ½ πΏπ΄ π‘20
0
ππ‘
MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4
Worapol Ratanapan A0074997E
ππ΄ 20 = πππ οΏ½ πΏπ΄ π‘20
0
ππ‘
πΏπ΄ π‘ = οΏ½
11000
π‘ + 10 ; 0 β€ π‘ β€ 10
2π‘
100 + π‘2; 10 < π‘ β€ 20
Recall
MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4
Worapol Ratanapan A0074997E
ππ΄ 20 = πππ οΏ½ πΏπ΄ π‘20
0
ππ‘
πΏπ΄ π‘ = οΏ½
11000
π‘ + 10 ; 0 β€ π‘ β€ 10
2π‘
100 + π‘2; 10 < π‘ β€ 20
ππ΄ 20 = πππ οΏ½1
1000π‘ + 10
10
0
ππ‘ + οΏ½2π‘
100 + π‘2
20
10
ππ‘
Recall
MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4
Worapol Ratanapan A0074997E
ππ΄ 20 = πππ οΏ½ πΏπ΄ π‘20
0
ππ‘
πΏπ΄ π‘ = οΏ½
11000
π‘ + 10 ; 0 β€ π‘ β€ 10
2π‘
100 + π‘2; 10 < π‘ β€ 20
ππ΄ 20 = πππ οΏ½1
1000π‘ + 10
10
0
ππ‘ + οΏ½2π‘
100 + π‘2
20
10
ππ‘
ππ΄ 20 = πππ1
1000π‘2
2+ 10π‘
0
10
+ ln π‘2 + 100 1020
Recall
MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4
Worapol Ratanapan A0074997E
ππ΄ 20 = πππ οΏ½ πΏπ΄ π‘20
0
ππ‘
πΏπ΄ π‘ = οΏ½
11000
π‘ + 10 ; 0 β€ π‘ β€ 10
2π‘
100 + π‘2; 10 < π‘ β€ 20
ππ΄ 20 = πππ οΏ½1
1000π‘ + 10
10
0
ππ‘ + οΏ½2π‘
100 + π‘2
20
10
ππ‘
ππ΄ 20 = πππ1
1000π‘2
2+ 10π‘
0
10
+ ln π‘2 + 100 1020
ππ΄ 20 =52π3/20
Recall
MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4
Worapol Ratanapan A0074997E
(i) If at π‘ = 20, the accumulated value of $1 invested in fund A equals 2.5 times the accumulated value of $1 invested in fund B.
$1 Γ ππ΄ 20 = 2.5 Γ $1 Γ ππ΅ 20
52π3/20 = 2.5 Γ ππ΅ 20
MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4
Worapol Ratanapan A0074997E
ππ΅ 20 = πππ οΏ½ πΏπ΅ π‘20
0
ππ‘
MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4
Worapol Ratanapan A0074997E
ππ΅ 20 = πππ οΏ½ πΏπ΅ π‘20
0
ππ‘
πΏπ΅ π‘ =π
100 Recall
MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4
Worapol Ratanapan A0074997E
ππ΅ 20 = πππ οΏ½ πΏπ΅ π‘20
0
ππ‘
ππ΅ 20 = πππ οΏ½π
100
20
0
ππ‘
πΏπ΅ π‘ =π
100 Recall
MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4
Worapol Ratanapan A0074997E
ππ΅ 20 = πππ οΏ½ πΏπ΅ π‘20
0
ππ‘
ππ΅ 20 = πππ οΏ½π
100
20
0
ππ‘
ππ΅ 20 = ππ /5
πΏπ΅ π‘ =π
100 Recall
MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4
Worapol Ratanapan A0074997E
(i) If at π‘ = 20, the accumulated value of $1 invested in fund A equals 2.5 times the accumulated value of $1 invested in fund B.
$1 Γ ππ΄ 20 = 2.5 Γ $1 Γ ππ΅ 20
52π3/20 = 2.5 Γ ππ /5
MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4
Worapol Ratanapan A0074997E
(i) If at π‘ = 20, the accumulated value of $1 invested in fund A equals 2.5 times the accumulated value of $1 invested in fund B.
$1 Γ ππ΄ 20 = 2.5 Γ $1 Γ ππ΅ 20
52π3/20 = 2.5 Γ ππ /5
π =34
= 0.75
MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4
Worapol Ratanapan A0074997E
(ii) If $1 invested in fund A at π‘ = 5 is worth $2 at π‘ = π, find the exact value of π, giving your answer in the form of π΄ππΌ + π΅ where π΄,π΅ππ and πΌππΌ.
MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4
Worapol Ratanapan A0074997E
(ii) If $1 invested in fund A at π‘ = 5 is worth $2 at π‘ = π, find the exact value of π, giving your answer in the form of π΄ππΌ + π΅ where π΄,π΅ππ and πΌππΌ.
MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4
Worapol Ratanapan A0074997E
(ii) If $1 invested in fund A at π‘ = 5 is worth $2 at π‘ = π, find the exact value of π, giving your answer in the form of π΄ππΌ + π΅ where π΄,π΅ππ and πΌππΌ.
$1 Γ οΏ½πΏπ΄ π‘π
5
ππ‘ = $2
MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4
Worapol Ratanapan A0074997E
(ii) If $1 invested in fund A at π‘ = 5 is worth $2 at π‘ = π, find the exact value of π, giving your answer in the form of π΄ππΌ + π΅ where π΄,π΅ππ and πΌππΌ.
$1 Γ οΏ½πΏπ΄ π‘π
5
ππ‘ = $2
MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4
Worapol Ratanapan A0074997E
(ii) If $1 invested in fund A at π‘ = 5 is worth $2 at π‘ = π, find the exact value of π, giving your answer in the form of π΄ππΌ + π΅ where π΄,π΅ππ and πΌππΌ.
$1 Γ οΏ½πΏπ΄ π‘π
5
ππ‘ = $2
Itβs obvious that πΏπ΄ π‘ >0 .
MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4
Worapol Ratanapan A0074997E
(ii) If $1 invested in fund A at π‘ = 5 is worth $2 at π‘ = π, find the exact value of π, giving your answer in the form of π΄ππΌ + π΅ where π΄,π΅ππ and πΌππΌ.
$1 Γ οΏ½πΏπ΄ π‘π
5
ππ‘ = $2
Itβs obvious that πΏπ΄ π‘ >0 . Observe thatβ« πΏπ΄ π‘10
5 ππ‘ = π7/80 =< 2, therefore π > 10.
MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4
Worapol Ratanapan A0074997E
Let π > 10
MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4
Worapol Ratanapan A0074997E
Let π > 10,
2 = πππ οΏ½πΏπ΄ π‘π
5
ππ‘
MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4
Worapol Ratanapan A0074997E
Let π > 10,
2 = πππ οΏ½πΏπ΄ π‘π
5
ππ‘
2 = πππ οΏ½1
1000π‘ + 10
10
5
ππ‘ + οΏ½2π‘
100 + π‘2
π
10
ππ‘
MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4
Worapol Ratanapan A0074997E
Let π > 10,
2 = πππ οΏ½πΏπ΄ π‘π
5
ππ‘
2 = πππ οΏ½1
1000π‘ + 10
10
5
ππ‘ + οΏ½2π‘
100 + π‘2
π
10
ππ‘
2 = π7/80 π2 + 100200
β π2 = 400πβ7/80 β 100
MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4
Worapol Ratanapan A0074997E
Let π > 10,
2 = πππ οΏ½πΏπ΄ π‘π
5
ππ‘
2 = πππ οΏ½1
1000π‘ + 10
10
5
ππ‘ + οΏ½2π‘
100 + π‘2
π
10
ππ‘
2 = π7/80 π2 + 100200
β π2 = 400πβ7/80 β 100
π = 400πβ7/80 β 100
MA3269: Mathematical Finance I Tutorial 2/Discussion Problem 4
Worapol Ratanapan A0074997E