Question

The future worth in year 10 of an arithmetic gradient cash flow series for years 1 through 10 is $725,000. If the gradient increase each year, G, is $1750, determine the cash flow in year 1 at an interest rate of 8% per year.

The cash flow in year 1 is $______

Answer 1

FV = Sum [ CF * FVF(r%, n) ]

FVF(r%, n) = ( 1 +r)^n

r = Int rate per anum

n - Bal Yrs.

Let X be the CF in Year 1.

FV of Base CF i.e X is X * FVAF (r%, n)

= X * 14.4866

= 14.4866X

FVAF = SUm [ FVF(r%, n) ]

FV of Increments:

Year	Bal Yrs	Inc CF	FVF @8%	FV of CFs
1	9	$               -	1.9990	$               -
2	8	$   1,750.00	1.8509	$   3,239.13
3	7	$   3,500.00	1.7138	$   5,998.38
4	6	$   5,250.00	1.5869	$   8,331.09
5	5	$   7,000.00	1.4693	$ 10,285.30
6	4	$   8,750.00	1.3605	$ 11,904.28
7	3	$ 10,500.00	1.2597	$ 13,226.98
8	2	$ 12,250.00	1.1664	$ 14,288.40
9	1	$ 14,000.00	1.0800	$ 15,120.00
10	0	$ 15,750.00	1.0000	$ 15,750.00
FV of Inc CFs				$98,143.55

Thus 14.4866X + 98143.55 = 725000

14.4866X = 725000 - 98143.55

= $ 626856.45 / 14.4866

= $ 43271.58

Year 1 CF is $ 43271.58

Proof:

Year	Bal Yrs	Inc CF	FVF @8%	FV of CFs
1	9	$ 43,271.58	1.9990	$    86,500.09
2	8	$ 45,021.58	1.8509	$    83,331.80
3	7	$ 46,771.58	1.7138	$    80,158.27
4	6	$ 48,521.58	1.5869	$    76,997.65
5	5	$ 50,271.58	1.4693	$    73,865.44
6	4	$ 52,021.58	1.3605	$    70,774.79
7	3	$ 53,771.58	1.2597	$    67,736.70
8	2	$ 55,521.58	1.1664	$    64,760.37
9	1	$ 57,271.58	1.0800	$    61,853.31
10	0	$ 59,021.58	1.0000	$    59,021.58
FV of Inc CFs				$7,25,000.00