Tutorial 2

Question 1

1) A new parent invests £Y on the birth of his twins to provide a payment of £25 each on their
birthdays forever. If i =4% pa calculate Y?

Answer 1

Y = 2x25x𝑎∞¬ = 2x25x1/i = 2x25x1/0.04 = £1,250

Question 2

When i = 11%, calculate: 
 𝑎5¬  
 𝑎̈6¬ 
 ᾱ10¬ 
 𝑠5¬    
 𝑠̈6¬ 
 ӯ10¬
 𝑎5¬(2)
 𝑠̈6¬(4)

[ӯ = s bar]

Answer 2

𝑎5¬  = (1-v^5)/i = (1-1.11^-5)/0.11 = 3.696 
𝑎̈6¬  = (1-v^6)/d = (1+i)(1-v^6)/i = 1.11x(1-1.11^-6)/0.11 = 4.696 
ᾱ10¬ = (1-v^10)/δ = (1-v^10)/ln(1+i) = (1-1.11^-10)/ln(1.11) = 6.208 
𝑠5¬   = (1.11)^5(1-v^5)/i = (1.11)^5 (1-1.11^-5)/0.11 = 6.228 
𝑠̈6¬  = (1.11)^6(1-v^6)/d = (1.11)^7 (1-1.11^-6)/0.11 = 8.783 
ӯ10¬  = (1.11)^10(1-v^10)/δ = (1.11)^10(1-v^10)/ln(1+i) = (1.11)^10(1-1.11^-10)/ln(1.11) = 17.627 
𝑎5¬(2)= [i(2) = 2(1.11^0.5  -1) = 0.1071] = (1-v^5)/i(2) = (1-1.11^-5)/0.1071 = 3.796  
𝑠̈6¬(4) = [d(4) = 4(1-1.11^-0.25) = 0.103] = 1.11^6(1-v^6)/d(4) = 1.11^6(1-1.11^-6)/0.103 = 8.451

Question 3

3) An investor, who has a sum of £20,000 to invest, wishes to purchase an annuity certain
with a term of 10 years. Calculate the amounts of each payment that can be provided if
the annuity takes each of the following forms, assuming i = 6% pa
i. A level annuity payable annually in advance
ii. A level annuity payable monthly in arrears
iii. A level annuity payable quarterly in advance

Answer 3

20000 = X𝑎̈10¬ , X = 20000/[1.06(1-1.06^-10)/0.06] = £2,563.55 pa

20000 = 12X 𝑎10¬(12) [i(12)=12(1.06^1/12 -1)=0.05841], X = 20000/[12(1-1.06^-10)/0.05841 = £220.45 pm

20000 = 4X𝑎̈10¬(4) [d(4) = 4(1-1.06^-0.25) = 0.05785], X = 20000/[4(1-1.06^-10)/0.05785 = £654.96 pq

Question 4

4) Calculate the accumulated value of a series of payments of £500 payable:
i. Annually in arrears
ii. Monthly in advance
iii. Continuously each year
For a period of 6 years with i = 5% pa.

Answer 4

500 𝑠6¬ = 500(1.05^6)(1-1.05^-6)/0.05 = £3,400.96

500x12x𝑠̈6¬(12) [d(12) = 12(1-1.05^-1/12) = 0.04869] = 500x12(1.05^6)(1-1.05^-6)/0.04869 = £41,909.51

500𝑠6¬ = 500(1.05^6)(1-1.05^-6)/ln(1.05) = £3,485.29

Question 5

5) Calculate the present value of the following cashflows, with i = 4% pa

Payment date Amount
1 January 2013 £500
1 May 2013 £500
1 September 2013 £500
1 January 2014 £500
1 May 2014 £500
1 September 2014 £500
1 January 2015 £500
1 May 2015 £500
1 September 2015 £500

Answer 5

= 500x3x𝑎̈3¬(3) [d(3) = 3(1-1.04^-1/3) = 0.03897] = 500x3x(1-1.04^-3)/0.03897 = £4,272.66

Question 6

Question to work through in tutorial, with i12 = 12%

1) PV of annuity payable monthly in arrears of £2,000pa for first 6 years, £800pa for next 4
years and lumpsum of £4,000 at end of 10 years.

2) Calculate the level annuity payable continuously for 10 years having same PV?

Answer 6

2000a6¬(12) + 800v^6 a4¬(12) + 4000v^10 at i =0.126825 OR
2000/12a72¬ + 800/12v^72 a48¬ + 4000v^120 at i =0.01

2000(1-1.126825^-6)/0.12 + 800(1.126825^-6)(1-1.126825^-4)/0.12 + 4000(1.126825^-10) = £10,973.72

2000/12(1-1.01^-72)/0.01 + 800/12(1.01^-72)(1-1.01^-48)/0.01 + 4000(1.01^-120) = £10,973.72

10973.72 = Xᾱ10¬, 10973.72 = X(1-1.126825^-10)/ln(1.126825)  
X = 10973.72/5.83737 = £1,879.91 pa