Question

In: Economics

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

Solutions

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.


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