In: Economics
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.