Question

You can purchase an optical scanner today for $490. The scanner provides benefits worth $75 a year. The expected life of the scanner is 10 years. Scanners are expected to decrease in price by 25% per year. Suppose the discount rate is 11%. When is the best time to purchase the scanner? (Round your answer to the nearest cent.)

                    

NPV is maximized when you wait  (Click to select)  0  1  2  3  4  6  5  7  8  9  10  years to purchase the scanner. At this date NPV will be equivalent to $  today.

Answer 1

If scanner is purchased today that is waiting time=0

Present value of cash inflow=$75*PVAF(11%,10)=$75*5.8892=$442

Present value of cash Outflow=$490

NPV=Present value of cash inflow-Present value of cash Outflow

=$442-$490

=(-)$48

If scanner is purchased after 1 year that is waiting time=1

Present value of cash inflow from year 2 to year 10 at 1st year end=$75* [PVAF(11%,10)-PVIF(11%,1)] =$75*[5.8892-0.9009]=$75*4.9883=$374

So it's present value today=$374/0.9009=$415

Present value of cash Outflow at 1st year=$490*75%=$368

present value of cash outflow today=$368/0.9009=$408

NPV today=Present value of cash inflow today-Present value of cash Outflow today

=$415-$408

=$7

Similarly,If scanner is purchased after 2 years that is waiting time=2

Present value of cash inflow from year 3 to year 10 at 2nd year end=$75* [PVAF(11%,10)-PVAF(11%,2)] =$75*[5.8892-1.7125]=$75*4.1767=$313

So it's present value today=$313/PVIF(11%,2)=$374/0.8116=$386

Present value of cash Outflow at 2nd year=$490*75%*75%=$276

present value of cash outflow today=$276/0.8116=$340

NPV today=Present value of cash inflow today-Present value of cash Outflow today

=$386-$340

=$46

Similarly,If scanner is purchased after 3 years that is waiting time=3

Present value of cash inflow from year 4 to year 10 at 3rd year end=$75* [PVAF(11%,10)-PVAF(11%,3)] =$75*[5.8892-2.4437]=$75*3.4455=$258

So it's present value today=$258/PVIF(11%,3)=$258/0.7312=$353

Present value of cash Outflow at 3rd year=$490*75%*75%*75%=$207

present value of cash outflow today=$207/0.7312=$283

NPV today=Present value of cash inflow today-Present value of cash Outflow today

=$353-$283

=$70

Similarly,If scanner is purchased after 4 years that is waiting time=4

Present value of cash inflow from year 5 to year 10 at 4th year end=$75* [PVAF(11%,10)-PVAF(11%,4)] =$75*[5.8892-3.1024]=$75*2.7868=$209

So it's present value today=$209/PVIF(11%,4)=$209/0.6587=$317

Present value of cash Outflow at 4th year=$490*75%*75%*75%*75%=$155

present value of cash outflow today=$155/0.6587=$236

NPV today=Present value of cash inflow today-Present value of cash Outflow today

=$317-$236

=$81

Again,If scanner is purchased after 5 years that is waiting time=5

Present value of cash inflow from year 6 to year 10 at 5th year end=$75* [PVAF(11%,10)-PVAF(11%,5)] =$75*[5.8892-3.6959]=$75*2.1933=$165

So it's present value today=$165/PVIF(11%,5)=$165/0.5935=$278

Present value of cash Outflow at 5th year=$490*75%*75%*75%*75%*75%=$116

present value of cash outflow today=$116/0.5935=$195

NPV today=Present value of cash inflow today-Present value of cash Outflow today

=$278-$195

=$83

Again,If scanner is purchased after 6 years that is waiting time=6

Present value of cash inflow from year 7 to year 10 at 6th year end=$75* [PVAF(11%,10)-PVAF(11%,6)] =$75*[5.8892-4.2305]=$75*1.6587=$124

So it's present value today=$124/PVIF(11%,6)=$124/0.5346=$232

Present value of cash Outflow at 6th year=$490*75%*75%*75%*75%*75%*75%=$87

present value of cash outflow today=$87/0.5346=$163

NPV today=Present value of cash inflow today-Present value of cash Outflow today

=$232-$163

=$69

If scanner is purchased after 7 years that is waiting time=7

Present value of cash inflow from year 8 to year 10 at 7th year end=$75* [PVAF(11%,10)-PVAF(11%,7)] =$75*[5.8892-4.7122]=$75*1.1770=$88

So it's present value today=$88/PVIF(11%,7)=$88/0.4817=$183

Present value of cash Outflow at 7th year=$490*75%*75%*75%*75%*75%*75%*75%=$65

present value of cash outflow today=$65/0.4817=$135

NPV today=Present value of cash inflow today-Present value of cash Outflow today

=$183-$135

=$48

If scanner is purchased after 8 years that is waiting time=8

Present value of cash inflow from year 9 to year 10 at 8th year end=$75* [PVAF(11%,10)-PVAF(11%,8)] =$75*[5.8892-5.1461]=$75*0.7431=$56

So it's present value today=$56/PVIF(11%,8)=$56/0.4339=$129

Present value of cash Outflow at 8th year=$490*75%*75%*75%*75%*75%*75%*75%*75%=$49

present value of cash outflow today=$49/0.4339=$113

NPV today=Present value of cash inflow today-Present value of cash Outflow today

=$129-$113

=$16

If scanner is purchased after 9 years that is waiting time=9

Inflow can be generated for next year that is for year 10 only.So,Present value of cash inflow for year 10 at 9th year end=$75* PVIF(11%,10) =$75*0.3522=$26

So it's present value today=$26/PVIF(11%,9)=$26/0.3909=$67

Present value of cash Outflow at 9th year=$490*75%*75%*75%*75%*75%*75%*75%*75%*75%=$37

present value of cash outflow today=$37/0.3909=$95

NPV today=Present value of cash inflow today-Present value of cash Outflow today

=$67-$95

=(-)$28

Lastly,If scanner is purchased after 10 years that is waiting time=10

No Inflow can be generated from next years.So,Present value of cash inflow=0

So it's present value today=0

Present value of cash Outflow at 10th year=$490*75%*75%*75%*75%*75%*75%*75%*75%*75%*75%=$28

present value of cash outflow today=$28/0.3522=$80

NPV today=Present value of cash inflow today-Present value of cash Outflow today

=$0-$80

=(-)$80

So,NPV is maximized when the scanner is purchased after a waiting time of 5 years from today and so 5th year is the best time to purchase the scanner and in this case the NPV is highest that is $83 in terms of today's value.