Question

An investment project costs $18,900 and has annual cash flows of $3,700 for six years.

  

Required :
(a)	What is the discounted payback period if the discount rate is zero percent?

  

(b)	What is the discounted payback period if the discount rate is 4 percent?

  

(c)	What is the discounted payback period if the discount rate is 22 percent?

Answer 1

Payback period is the time period required to cover the initial investment back. Discounted payback period takes in to account the time value of money of future cash inflows.

Note : It is assumed that all the cash flows are evenly distributed through out the year.

Initial investment cost = $18,900

Annual cash flows of $3,700 for six years.

We have to cover the initial investment back (18,900$).

a) Discount rate = 0 percent

In discounted payback period, each year cashflow is discounted at the the discount rate.

Present value of cash inflow can be calculated with help of below formula-

where,

FV = future value

PV = present value

r = rate of interest

n = no. of years

Cashflows are shown in table below-

year	cash inflow at end of year	Present value	present value of cashflow	Cumulative present value of cash flow
1	3,700	=3,700 / (1 + 0)1	3700	3,700
2	3,700	=3,700 / (1 + 0)2	3700	7,400
3	3,700	=3,700 / (1 + 0)3	3700	11,100
4	3,700	=3,700 / (1 + 0)4	3700	14,800
5	3,700	=3,700 / (1 + 0)5	3700	18,500
6	3,700	=3,700 / (1 + 0)6	3700	22,200

Discounted payback period = 5 + 400 / 3700

= 5 + 0.1081

= 5.11 years (approx)

( As the initial investment is $18900, we need to cover this cost, we will take year corresponding to the cumulative value which is less than or equal to 18900$, here it is 5 year. In 5 year 18,500 $ will be covered, we will be needing $400 (18,900 - 18,500) more to cover full initial cost. As the cash flows are evenly distributed, remaining 400 $ will be covered from 6th year inflows of amount 3700 $ . Hence 400 / 3700 is taken).

b) Discount rate = 4 percent

Present value of cash inflow can be calculated with help of below formula-

Cashflows are shown in table below-

year	cash inflow at end of year	Present value	present value of cashflow	Cumulative present value of cash flow
1	3,700	=3,700 / (1 + 0.4)1	3557.692308	3557.692308
2	3,700	=3,700 / (1 + 0.4)2	3420.857988	6978.550296
3	3,700	=3,700 / (1 + 0.4)3	3289.286527	10267.83682
4	3,700	=3,700 / (1 + 0.4)4	3162.775507	13430.61233
5	3,700	=3,700 / (1 + 0.4)5	3041.130295	16471.74263
6	3,700	=3,700 / (1 + 0.4)6	2924.163745	19395.90637

Discounted payback period = 5 + 2428.25737 / 2924.163745

= 5 + 0.83

= 5.83 years

( As the initial investment is $18900, we need to cover this cost, we will take year corresponding to the cumulative value which is less than or equal to 18900$, here it is 5 year. In 5 year 16471.74263 $ will be covered, we will be needing $2428.25737 (18,900 - 16471.74263) more to cover full initial cost. As the cash flows are evenly distributed, remaining 2428.25737 $ will be covered from 6th year inflows of amount 2924.163745 $ . Hence 2428.25737 / 2924.163745 is taken).

c) Discount rate = 22 percent

Present value of cash inflow can be calculated with help of below formula-

Cashflows are shown in table below-

year	cash inflow at end of year	Present value	present value of cashflow	Cumulative present value of cash flow
1	3,700	=3,700 / (1 + 0.22)1	3032.786885	3032.786885
2	3,700	=3,700 / (1 + 0.22)2	2485.89089	5518.677775
3	3,700	=3,700 / (1 + 0.22)3	2037.615483	7556.293258
4	3,700	=3,700 / (1 + 0.22)4	1670.176626	9226.469883
5	3,700	=3,700 / (1 + 0.22)5	1368.997234	10595.46712
6	3,700	=3,700 / (1 + 0.22)6	1122.12888	11717.596

If disount rate is 22 %, the will not be able to cover the initial investment back as at the end of 6 years, the cumulative cash inflow incurred is less than the initial investment. The investment is not a good capital budgeting decision.

Hope it helps!