Question

A county is considering using a piece of parkland for one of two alternative recreation projects. Project S would require construction costs of $2 million (year 0) and generate net benefits of $1 million per year for 10 years. (Assume the benefits are realized at the ends of years 1 through 10). Project L would require construction costs of $15.5 million and generate net benefits of $2 million per year for 20 years. (Assume the benefits are realized at the ends of years 1 through 20). If these figures are in real dollars, and the real discount rate is 8 percent: Find the NPV, IRR, PI, Discounted Payback Period, Regular Payback Period, and AEA for each of the two alternatives. Which project would the county select?

Answer 1

Answer : Calculation of NPV

NPV = Present value of Cash Inflow - Present Value of Cash Outflow

NPV of Project S

Year	Cash Inflow (in millions)	Present Value Factor @ 8%	Present value of cash inflow
1	1	0.925925926	0.925925926
2	1	0.85733882	0.85733882
3	1	0.793832241	0.793832241
4	1	0.735029853	0.735029853
5	1	0.680583197	0.680583197
6	1	0.630169627	0.630169627
7	1	0.583490395	0.583490395
8	1	0.540268885	0.540268885
9	1	0.500248967	0.500248967
10	1	0.463193488	0.463193488
		Total Present value of cash inflow	6.710081399 million
		Less : Cash outflow	2 million
		Net Present Value	4.7101 million

NPV of Project L

Year	Cash Inflow (million)	Present Value Factor @ 8%	Present value of cash inflow
1	2	0.925925926	1.851851852
2	2	0.85733882	1.714677641
3	2	0.793832241	1.587664482
4	2	0.735029853	1.470059706
5	2	0.680583197	1.361166394
6	2	0.630169627	1.260339254
7	2	0.583490395	1.166980791
8	2	0.540268885	1.080537769
9	2	0.500248967	1.000497934
10	2	0.463193488	0.926386976
11	2	0.428882859	0.857765719
12	2	0.397113759	0.794227517
13	2	0.367697925	0.735395849
14	2	0.340461041	0.680922083
15	2	0.315241705	0.63048341
16	2	0.291890468	0.583780935
17	2	0.270268951	0.540537903
18	2	0.250249029	0.500498058
19	2	0.231712064	0.463424128
20	2	0.214548207	0.429096415
		Total Present value of cash inflow	19.63629481
		Less : Cash outflow	15.5
		Net Present Value	4.1363 million

Calculation of IRR of Project S and Project L

Calculation of PI

PI = Present value of Cash Inflow / Present Value of Cash Outflow

Present values are calculated while calculating NPV

PI of Project S = 6.710081399 / 2

= 3.355 or 3.36

PI of Project L = 19.63629481 / 15.5

= 1.267 or 1.27

Calculation of Regular Payback Period

Regular Payback Period = Initial Investment / Annual Cash Flows

Regular Payback Period of Project S = 2 million / 1 million = 2 years

Regular Payback Period of Project L = 15.5 million / 2 million = 7.75 years

Calculation of Discounted Payback Period :

Project S

Year	Cash Flows	PVF @ 8%	Present value of Cash FlowsCash Flows	Cumulative Cash Flows
1	1	0.925925926	0.925925926	0.925925926
2	1	0.85733882	0.85733882	1.783264746
3	1	0.793832241	0.793832241	2.577096987
4	1	0.735029853	0.735029853	3.31212684
5	1	0.680583197	0.680583197	3.992710037
6	1	0.630169627	0.630169627	4.622879664
7	1	0.583490395	0.583490395	5.206370059
8	1	0.540268885	0.540268885	5.746638944
9	1	0.500248967	0.500248967	6.246887911
10	1	0.463193488	0.463193488	6.710081399

Discounted Paybavck Period =Complete years + [(Initial Investment - Cash Flow recovered) / Cash flow of the year]

= 2 years + [(2 - 1.783264746) / 0.793832241]

= 2.27 years

Project L

Year	Cash Flows	PVF @ 8%	Present value of Cash FlowsCash Flows	Cumulative Cash Flows
1	2	0.925925926	1.851851852	1.851851852
2	2	0.85733882	1.714677641	3.566529492
3	2	0.793832241	1.587664482	5.154193974
4	2	0.735029853	1.470059706	6.62425368
5	2	0.680583197	1.361166394	7.985420074
6	2	0.630169627	1.260339254	9.245759328
7	2	0.583490395	1.166980791	10.41274012
8	2	0.540268885	1.080537769	11.49327789
9	2	0.500248967	1.000497934	12.49377582
10	2	0.463193488	0.926386976	13.4201628
11	2	0.428882859	0.857765719	14.27792852
12	2	0.397113759	0.794227517	15.07215603
13	2	0.367697925	0.735395849	15.80755188
14	2	0.340461041	0.680922083	16.48847397
15	2	0.315241705	0.63048341	17.11895738
16	2	0.291890468	0.583780935	17.70273831
17	2	0.270268951	0.540537903	18.24327621
18	2	0.250249029	0.500498058	18.74377427
19	2	0.231712064	0.463424128	19.2071984
20	2	0.214548207	0.429096415	19.63629481

Discounted Paybavck Period =Complete years + [(Initial Investment - Cash Flow recovered) / Cash flow of the year]

= 12 years + [(15.5 - 15.07215603) / 0.735395849]

= 12.58 years

AEA of both projects

AEA (Annual Equivalent Approach ) of Project S = Net Present Value / Present value Annuity factor @8% for 10 years

= 4.7101 million / 6.71008139868

= 0.7019 million

AEA (Annual Equivalent Approach ) of Project L = Net Present Value / Present value Annuity factor @8% for 20 years

= 4.1363 million / 9.81814740671

= 0.4213 million

Project S shoul be selected as it is better in all aspects in project L