Question

Consider an EOY geometric gradient, which lasts for eight years, whose initial value

at EOY one is $4000 and g = 6% per year thereafter. For an equivalent cash flow, find the

equivalent uniform gradient amount, G over the same period if the initial value of the cash flows

at the end of year one is $(3000). The interest rate is 18% per year. draw the cash flow diagram.

Answer 1

Rate of Interest = 18%

g = 6% which means cash flow increases by 6% every year

Present value if calculated as [Gradient cash flow / (1 + Rate of Interest)^^Year]

Year	Gradient Cash Flow	Present Value
1	4,000.00	3,389.83
2	4,240.00	3,045.10
3	4,494.40	2,735.43
4	4,764.06	2,457.25
5	5,049.91	2,207.36
6	5,352.90	1,982.88
7	5,674.08	1,781.23
8	6,014.52	1,600.09
		19,199.19

Uniform payment would be same for all 8 periods while there is uniform gradient amount which means cash flow rises by fixed amount every year

Year	Uniform payment	Present value of uniform payment	Increased amount	Present value of Increased Amount
1	3,000.00	2,542.37	-	-
2	3,000.00	2,154.55	1X	0.72X
3	3,000.00	1,825.89	2X	1.22X
4	3,000.00	1,547.37	3X	1.55X
5	3,000.00	1,311.33	4X	1.75X
6	3,000.00	1,111.29	5X	1.85X
7	3,000.00	941.78	6X	1.88X
8	3,000.00	798.11	7X	1.86X
		12,232.70		10.83X

To make both of these cash flow equivlent, present value of both of them should be equal to each other.

12,232.70 + 10.83X = 19,199.19

X = 643.25

Thus, fixed increased of amount should be 643.25