Tutorial 3 Flashcards
1) State which of the following equations are correct, if not state the correct version:
i. x| 𝑎y¬ =(1+i)^(-x) 𝑎y¬
ii. (Ia)n¬ = (𝑎n¬ +nv^n)/i
iii. (I𝑎̈)n¬ = (i/d)x(Ia)n¬
iv. (I𝑎̅)n¬ = (𝑎̅n¬ -nv^n)/𝛿
i. Yes
ii. No, (Ia)n¬ = (𝑎̈n¬ -nv^n)/i
iii. Yes
iv. No, (I𝑎̅)n¬ = (𝑎̈n¬ -nv^n)/𝛿 = (i/𝛿)(Ia)n¬
2) An annuity certain is payable monthly in arrears for 20 years. The annuity is to be paid at a rate
of £1,200 pa for the first 10 years and £1,500 pa for the next 6 years and £2,000 pa for the last 4
years. Calculate the present value of the payments assuming i = 6% pa
1200𝑎10¬(12) + 1500v^10𝑎6¬(12) + 2000v^16𝑎4¬(12)
i(12) = 0.05841
1200(1-1.06^-10)/0.05841 + 1500(1.06^-10)(1-1.06^-6)/0.05841+ 2000(1.06^-16)(1-1.06^-4)/0.05841 = £16,105.67
3) Evaluate the following at i = 9% pa and at i = 11% pa
i. (Ia)40¬
ii. (I𝑎̈)40¬
iii. (I𝑎̅)60¬
iv. (𝐼𝑎̅)60¬ [bar over I and a]
i. (Ia)40¬ = [(1-1.09^-40)/0.09x1.09^-40(1.09^-40)]/0.09 = 116.1335
= [(1-1.11^-40)/0.11x1.11^-40(1.11^-40)]/0.11 = 84.73
ii. (I𝑎̈)40¬ =[(1-1.09^-40)/0.09x1.09^-40(1.09^-40)]/0.09x1.09 = 126.5856
=[(1-1.11^-40)/0.11x1.11^-40(1.11^-40)]/0.11x1.11 = 94.05
iii. (I𝑎̅)60¬ =[(1-1.09^-60)/0.09x1.09^-60(1.09^-60)]/ln(1.09) =135.7829
=[(1-1.11^-60)/0.11x1.11^-60(1.11^-60)]/ln(1.11) = 95.4117
iv. (𝐼𝑎̅)60¬ [bar over I and a] =
[(1-1.09^-60)/ln(1.09)^-60(1.09^-60)]/ln(1.09) =129.931
=[(1-1.11^-60)/ln(1.11)^-60(1.11^-60)]/ln(1.11) = 90.5465
4) a) Calculate the present value of an annuity payable annually in advance for a term of 20 years,
such that the payment is £1,000 in year 1, £1,100 in year 2, £1,200 in year 3 etc. Assume i =6%
pa.
b) Calculate the present value of the above annuity assuming i = 6% pa for the first five years and
5% pa thereafter
a) 900𝑎̈20¬ + 100(I𝑎̈)20¬
900(1-1.06^-20)/0.06x1.06 + 100[(1-1.06^-20)/0.06x1.06^-20(1.06^-20)]/0.06x1.06 = £21,404.54
b) 900𝑎̈5¬ @6% + 100(I𝑎̈)5¬ @6% + v^5@6%(1400𝑎̈15¬@5% + 100(I𝑎̈)15¬ @5%)
900(1-1.06^-5)/0.06x1.06 + 100[(1-1.06^-5)/0.06x1.06^-5(1.06^-5)]/0.06x1.06 + 1.06^-5(1400(1-1.05^-15)/0.05x1.05 + 100[(1-1.05^-15)/0.05x1.05^-15(1.05^-15)]/0.05x1.05) = £22,488.03
5) a) Calculate the present value of an annuity payable annually in arrears for a term of 10 years
such that the payment is £2,000 in year 1, £1,950 in year 2, £1,900 in year 3 etc. Assume i =4%
pa.
b) Calculate the present value of the above annuity assuming that from the end of the fourth
year the annuity remains level for the remainder of the term.
a) 2050𝑎10¬ - 50(Ia)10¬
2050(1-1.04^-10)/0.04 - 50[(1-1.04^-10)/0.04x1.04^-10(1.04^-10)]/0.04 = £14,527.72
b) 2050𝑎4¬ - 50(Ia)4¬ + 1850v^4𝑎6¬
2050(1-1.04^-4)/0.04 - 50[(1-1.04^-4)/0.04x1.04
-4(1.04^-4)]/0.04 + 1850(1.04^-4)(1-1.04^-6)/0.04 = £15,286.29
6) Calculate the present value of an annuity payable annually in advance for a term of 9 years, such
that the first payment is £1,000 and each subsequent payment increases at 3% pa compound.
Assume i = 6% pa.
Cashflows
1000 + 1000x1.03v +1000x1.03^2v^2 +.. .+1000x1.03^8v^8
1000(1+1.03v +1.03^2v^2.+…+ 1.03^8v^8)
1000(1+v’ + v’^2+…+ v’^8)
1000𝑎̈9¬’ using i’ = 1.06/1.03-1 = 0.029126
1000(1-1.029126^-9)/0.029126x1.029126 = £8045.64
Question to work through in tutorial
An investor wishes to calculate the present value of the following cashflows:
- Income initially of £10,000 pa, paid quarterly in advance
- Payments remain fixed for 5 year periods at a time. At the end of the 5 year period, the payments rise in line with total inflation growth over the previous 5 years
Assume the following
- Inflation is 2.5% pa
- i = 10% pa
- Term is 25 years
PV = 10000𝑎̈5¬(4) +10000x1.0255v^5𝑎̈5¬(4) +10000x1.025^10v^10𝑎̈5¬(4) + 10000x1.025^15v^15𝑎̈5¬(4) + 10000x1.025^20v^20𝑎̈5¬(4)
=10000𝑎̈5¬(4) @i=10%(1 + 1.025^5v^5 + 1.025^10v^10 + 1.025^15v^15 + 1.025^20v^20)
Note = 1.025^5v^5 = 1.025^5/1.1^5 = 1/(1+j) j = 1.1^5/1.025^5 -1 = 42.3456% d(4) = (1-(1.1)^-1/4)4 = 0.094184
=10000𝑎̈5¬(4)@i=10% 𝑎̈5¬ @j
=10000(1-1.1^-5)/0.094184x(1-1.423456^-5)/0.423456x1.423456 = £112,146
