Question

You are buying a perpetuity with annual payments as follows Payment of X at the end of the first year and every three years thereafter. Payment of X+1 at the end of the second year and every three years thereafter. Payment of X+2 at the end of the third year and every three years thereafter The interest rate is 5% convertible semi-annually. If the present value is 40, Calculate

Calculate X .

Answer 1

The cash flows are:

Year	1	2	3	4	5	6
Cash flow	X	X+1	X+2	X	X+1	X+2

Future value of Year 1 cash flow in Year 2 = X * (1 + 5%/2)^2 = 1.05 X

Similarly the value of Year 3 cash flow in Year 2 = (X+2)/(1 + 5%/2)^2 = 0.95 X

Essentially these cash flows can be reduced to one single perpetuity, as follows:

Year	1	2	3	4	5	6
Cash flow		Y			Y

A cash flow of Y is received every 3 years starting from year 2, where Y = 1.05 X + (X+1) + 0.95* (X+2)

Therefore, Y = 3X + 2.9

The future value of these cash flows at the end of year 2 will be:

Y + Value of perpetuity = Y + Y/ (1.025^6) = Y + 0.86 Y = 1.86 Y

Present value of this will be = 1.86 Y/ (1.025^4) =  1.86 * 0.90 * Y = 1.674 Y

Its given that present value of these cash flows = 40

Therefore, 1.674 Y = 40

So, Y = 23.89

=> 3X + 2.9 = 23.89 or ~ 23.9

=> 3X = 21

=> X = 7

Final answer, X = 7