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

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 it the initial value of the cash flows at the end of year one is 3000$ .The interest rate is 5% per year

Expert Solution

Interest rate = 5% per year

Initial value = $4,000

g = 6% per year which means payment each year rises by 6%

Present value is calculated as: [Payment / (1 + Rate of Interest^^Year]

Year	Payment	Present value
1	4,000.00	3,809.52
2	4,240.00	3,845.80
3	4,494.40	3,882.43
4	4,764.06	3,919.41
5	5,049.91	3,956.73
6	5,352.90	3,994.42
7	5,674.08	4,032.46
8	6,014.52	4,070.86
		31,511.65

Equivalent uniform gradient amount rises by $X every year.

Year	Payment	Present value	Additional Payment	Present value of additional payment
1	3,000.00	2,857.14	-	-
2	3,000.00	2,721.09	X	0.907X
3	3,000.00	2,591.51	X	0.864X
4	3,000.00	2,468.11	X	0.823X
5	3,000.00	2,350.58	X	0.784X
6	3,000.00	2,238.65	X	0.746X
7	3,000.00	2,132.04	X	0.711X
8	3,000.00	2,030.52	X	0.677X
		19,389.64		5.511X

Present value of geometric gradient series and equivalent uniform gradient amount must be zero: 5.511Z + 19,389.64 = 31,511.65

X = 2,200

Annual payment must rise by 2,200.

Rahul Sunny answered 2 years ago

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