Question

You invest $50,000 now and receive $10,000 per year for 15 years starting at the end of the first year. What is the discounted payback period? Use i = 9% annual rate compounded annually. Give the discounted payback period between two consecutive integers.

Answer 1

investment of 50000 and return of 10000 every year for 15 years

i = 9%

We need to find net present value of each cash flow and then find cumulative cash flow

We need to find year when cumulative cash flow turns positive

using excel

Year	Investment	Return	Discount factor	Present Value	Cumulative cash flow
0	-50000		1.000000	-50,000.00	-50,000.00
1		10,000.00	0.917431	9,174.31	-40,825.69
2		10,000.00	0.841680	8,416.80	-32,408.89
3		10,000.00	0.772183	7,721.83	-24,687.05
4		10,000.00	0.708425	7,084.25	-17,602.80
5		10,000.00	0.649931	6,499.31	-11,103.49
6		10,000.00	0.596267	5,962.67	-5,140.81
7		10,000.00	0.547034	5,470.34	329.53
8		10,000.00	0.501866	5,018.66	5,348.19
9		10,000.00	0.460428	4,604.28	9,952.47
10		10,000.00	0.422411	4,224.11	14,176.58
11		10,000.00	0.387533	3,875.33	18,051.91
12		10,000.00	0.355535	3,555.35	21,607.25
13		10,000.00	0.326179	3,261.79	24,869.04
14		10,000.00	0.299246	2,992.46	27,861.50
15		10,000.00	0.274538	2,745.38	30,606.88

Cumulative cash flow turns positive in 7th year

Discounted pay back period = year bef cash flow turns positive + (cumulative cash flow in year bef the positive value/net present worth of cash flow in the year where cumulative cash flow turn positive)  

= 6+(5,140.81/5470.34)

= 6+ 0.93976

= 6.939 years = 6.94 years

Showing formula in excel

Year	Investment	Return	Discount factor	Pressent Value	Cumulative cash flow
0	-50000		=1/((1+0.09)^A19)	=B19	=E19
1		10000	=1/((1+0.09)^A20)	=C20*D20	=E20+F19
2		10000	=1/((1+0.09)^A21)	=C21*D21	=E21+F20
3		10000	=1/((1+0.09)^A22)	=C22*D22	=E22+F21
4		10000	=1/((1+0.09)^A23)	=C23*D23	=E23+F22
5		10000	=1/((1+0.09)^A24)	=C24*D24	=E24+F23
6		10000	=1/((1+0.09)^A25)	=C25*D25	=E25+F24
7		10000	=1/((1+0.09)^A26)	=C26*D26	=E26+F25
8		10000	=1/((1+0.09)^A27)	=C27*D27	=E27+F26
9		10000	=1/((1+0.09)^A28)	=C28*D28	=E28+F27
10		10000	=1/((1+0.09)^A29)	=C29*D29	=E29+F28
11		10000	=1/((1+0.09)^A30)	=C30*D30	=E30+F29
12		10000	=1/((1+0.09)^A31)	=C31*D31	=E31+F30
13		10000	=1/((1+0.09)^A32)	=C32*D32	=E32+F31
14		10000	=1/((1+0.09)^A33)	=C33*D33	=E33+F32
15		10000	=1/((1+0.09)^A34)	=C34*D34	=E34+F33