Question

Suppose you are to receive the following series of payments,over time. Time (years) Payment made 150 375 220 145 900 627 812 11.5 13.7 20 In addition, you have the following information about interest rates: From time 0 to time 5, i(4)-5.5% (nominal rate of interest, per annum) . From time 5 to time 11, d-: 7.25% (per annum) . From time l 1 to time 20, i 3% (per annum) .From time 20 onwards, (t -20) 10t o(t- years, according to the table above) Calculate each of the following: (a) The present value at t 0, of all future payments (b) The accumulated value at time 10, of all payments made up to that time. (c) The accumulated value at time 30, of all payments made.

Answer 1

a) PV of all payments made is

PV = PMT1 / (1+r1)^t1 + PMT2/ (1+r2)^t2+...

where PMT1 is Payment 1, r1 is interest rate 1 and t1 is time 1 and so on

     $PV=\frac{150}{(1+0.055)^{0}}+\frac{375}{(1+0.055)^{1}}+\frac{220}{(1+0.055)^{4}}+\frac{145}{(1+0.725)^{8}}+\frac{900}{(1+0.0725)^{9}}+\frac{627}{(1+0.03)^{11.5}}+\frac{812}{(1+0.03)^{13.7}}+\frac{777}{(1+0.03)^{20}}$ PV= 2663.36

Present value of all future payments is 2663.36

b) Accumulated value at t=10,

$FV={150}*{(1+0.055)^{10}}+{375}*{(1+0.055)^{9}}+{220}*{(1+0.055)^{6}}+{145}*{(1+0.725)^{2}}+{900}*{(1+0.0725)^{1}}$

FV =2298.76

c) Since last payment is at t=20, first find the value at t=20

$FV={150}{(1+0.055)^{20}}+{375}{(1+0.055)^{19}}+{220}{(1+0.055)^{16}}+{145}{(1+0.725)^{12}}+{900}{(1+0.0725)^{11}}+{627}{(1+0.03)^{8.5}}+{812}{(1+0.03)^{6.3}}+{777}{(1+0.03)^{0}}$

FV = 6833.71

Now finding interest rate at each year from 21 to 30,

	(t-20)/10t
Year	Interest rate
21	0.476%
22	0.909%
23	1.304%
24	1.667%
25	2.000%
26	2.308%
27	2.593%
28	2.857%
29	3.103%
30	3.333%

Now Final Future Value will be

For year 21, FV= 6833.71*(1+0.0046) =6866.25

For year 22, FV= 6866.25*(1+0.00909)= 6928.67

For year 23, FV= 6928.67*(1+0.01304)= 7019.041

For year 24, FV=  7019.041*(1+0.01667)= 7136.025

For year 25, FV=  7136.025*(1+0.02)= 7278.75

For year 26, FV=  7278.75*(1+0.02308)= 7446.72

For year 27, FV=  7446.72*(1+0.02308)= 7639.78

For year 28, FV=  7446.72*(1+0.02857)= 7858.059

For year 29, FV=  7858.059*(1+0.03103)= 8101.93

For year 30, FV=  8101.93*(1+0.03103)= 8371.99 = 8372

Accumulated value at t=30 is 8372

Year	Interest rate
21	0.476%		6866.247
22	0.909%		6928.667
23	1.304%		7019.041
24	1.667%		7136.025
25	2.000%		7278.746
26	2.308%		7446.717
27	2.593%		7639.78
28	2.857%		7858.059
29	3.103%		8101.93
30	3.333%		8371.994