Tutorial 5 Flashcards
1) An investor invests £500 on 1 January 2013 to receive a lumpsum of £650 on 30 September
2016. Calculate the annual effective interest rate earned over the period.
650 = 500(1+i)^3.75 i = (650/500)^1/3.75 -1 = 7.25%
2) Consider the following security (P = (Ia)n¬ + Rv^n) to solve the following:
i. Calculate P if I =60, n = 6, R =200 and i = 4%
ii. Calculate I if P = £1,000, n = 10, R =1,250 and i = 6%
iii. Calculate R if P = £2,000, I = 100, n =8, and i =5%
iv. Calculate n if P = £1,500, I = 200, R = 1,150 and i =6%
v. Calculate i if P = £1,250, I = 75, n = 9, R = 1,300
i) P = 60a6¬ + 200v^6 = 60(1-1.04^-6)/0.04 + 200(1.04^-6) = £472.59
ii) 1000 = Ia10¬ + 1250v^10 = [1000-1250(1.06^-10)]/[(1-1.06^-10)/0.06] = £41.03
iii) 2000 = 100a8¬ + Rv^8 = [2000-100(1-1.05^-8)/0.05]x1.05^8 = £2,000
iv) 1500 = 200(1-1.06^-n)/0.06 +1150(1.06^-n) = 3333.33(1-1.06^-n) + 1150(1.06^-n)
(1. 06^-n) = 1833.33/2183.33 = -nln(1.06) = ln(0.839694) = n = 3
v) 1250 = 75(1-(1+i)^-9)/i + 1300(1+i)^-9
Try 7%
75(1-1.07^-9)/0.07 + 1300(1.07^-9) = £1,195.76
Try 6%
75(1-1.06^-9)/0.06 + 1300(1.06^-9) = 1,279.59
So – 0.06 + (1250-1279.59)/(1195.76-1279.59)x(0.07-0.06) = 6.35%
3) Calculate the expected present value of the following as at 1 January 2013:
Cashflow Probability of payment Payment due
-200 1 1 January 2013
150 0.5 1 March 2013
350 0.4 1 August 2013
1000 0.1 1 December 2013
The effective rate of interest is 1% per month
-200 + 1500.5v^2 + 3500.4v^7 + 10000.1v^11 = £93.74
4) An individual has invested a lumpsum in a brand new company in return for a regular annual
payment of £2,000 per annum in arrears for the next 8 years. Given the current economic environment there is some uncertainty if the company will be able to meet these payments.
Calculate the PV of the payments @ i = 6% assuming:
i. All payments will be received in full
ii. Prob of receiving the 1st payment is 1, 2nd is 0.93, 3rd is 0.86 etc
iii. Increasing the force of interest by 0.0279
iv. The company requests to pay all 8 of the payments as a single lumpsum at the end
of the 8 years assuming the force of interest is increased by 0.03704 and allowing for inflation of 2.5% pa
i) 2000a8¬ @6% = 2000(1-1.06^-8)/0.06 = £12,419.59
ii) 2140a8¬ – 140(Ia)8¬
2140(1-1.06^-8)/0.06 – 140[(1-1.06^-8)/0.06x1.06 – 8(1.06^-8)]/0.06 = £9,641.77
iii) 1.06e^0.0279 = 1.09
2000a8¬ @9% = 2000(1-1.09^-8)/0.09 = £11,069.64
iv) 1.06e^0.03704 = 1.1, i’ = 1.1/1.025 – 1 = 7.317%
8x2000v^8@i = 7.317%
8x2000(1.07317^-8) = £9,094.35
Question to work through in tutorial
A company has just bought an office block for £5m, which will be rented out. The total rental income
for the first year will be £100k, increasing by 4% pa compound. It will be sold after 20 years for
£7.5m. Assuming the rent is paid half way through the year, calculate the yield obtained (ignore tax).
100v^0.5 + 100x1.04v^1.5 + 100x1.04^2v^2.5 + …. + 7500v^20
100v^0.5 (1 + 1.04v^1 + 1.04^2v^2 + ….) + 7500v^20
100v^0.5𝑎̈20¬@j + 7500v^20@i — 1+j = (1+i)/1.04
Try i =5% (j = 1.05/1.04 -1 = 0.009615)
100(1.05^-0.5)(1-1.009615^-20)/0.009615x1.009615 + 7500(1.05^-20) = £4,611.58
Try i = 4.5% (j =1.045/1.04 – 1 =0.004808%)
100(1.045^-0.5)(1-1.004808^-20)/0.004808x1.004808 + 7500(1.045^-20) = £4,979.85
0.045 + (5000-4979.85)/(4611.58-4979.85)x(0.05-0.045) = 0.0447 so 4.5%
