Question

Consider the cash flow for projects A and B

Year: 0, 1, 2, 3, 4, 5

Project A: ($1000), 100, 600, 700, 900, 300

Project B: ($1000), 900, 700, 500, 300, 300

The cost of capital for both projects is 10%

1. Find NPV, IRR and profitability index (PI) of projects A and B.

2. If projects A and B are mutually exclusive, which project would you select

3. Find the crossover rate for projects A and B

Answer 1

1. Calculation of NPV, IRR and profitability index (PI) of projects A and B.

-Project A

a) Calculation of NPV

Year	Cashflow ($)	PVF@10%	Cashflow*PVF
0	(1,000)	1	(1,000.00)
1	100	0.909	90.90
2	600	0.826	495.60
3	700	0.751	525.70
4	900	0.683	614.70
5	300	0.621	186.30

Net Present Value = Present value of inflows -  Present value of outflows

= (90.9+495.6+525.7+614.7+186.3) - 1000

= 1913.20 - 1000

NPV = $913.20

*You can use the equation 1/(1+i)^n to find PVF using calculator

**Formula to calculate PV in excel is as follows

"=PV(interest rate,Year,0,cashflow)"

b) IRR

IRR is the discounting rate at which NPV is equal to zero. It is calculated by trial and error method. First find 2 discouting rates such that one will give positive NPV(herein after called as NPV1 and corresponding rate is R1 ) and other will give negative NPV((herein after called as NPV2 and corresponding rate is R2). Then use equation

IRR = R1 + ((NPV1 * (R2 - R1)) / (NPV1 - NPV2))

here I have taken 36% and 37%

Year	Cashflow ($)	PVF@36%	Cashflow*PVF
0	(1,000)	1	(1,000.00)
1	100	0.735	73.53
2	600	0.541	324.39
3	700	0.398	278.28
4	900	0.292	263.08
5	300	0.215	64.48

Net Present Value = Present value of inflows -  Present value of outflows

= (73.53+324.39+278.28+263.08+64.48) - 1000

= 1003.76 - 1000

= $3.76

Year	Cashflow ($)	PVF@37%	Cashflow*PVF
0	(1,000)	1	(1,000.00)
1	100	0.730	72.99
2	600	0.533	319.68
3	700	0.389	272.23
4	900	0.284	255.48
5	300	0.207	62.16

Net Present Value = Present value of inflows -  Present value of outflows

= (72.99+319.68+272.23+255.48+62.16)-1000

= 982.54 -1000

= -17.46

IRR = IRR = R1 + ((NPV1 * (R2 - R1)) / (NPV1 - NPV2))

= 36 + ((3.76 * (37-36)) / (3.76--17.46))

IRR = 36.177% (approximately)

NPV = -.04 at 36.177%.

c) profitability index (PI

It is also known as profit investment ratio and  value investment ratiowhich is calculated using the formula

PV of future cashflows / initial investment

= 1913.20 / 1000

= 1.9132

= 1.91

-Project B

a) NPV

Year	Cashflow ($)	PVF@10%	Cashflow*PVF
0	(1,000)	1	(1,000.00)
1	900	0.909	818.18
2	700	0.826	578.51
3	500	0.751	375.66
4	300	0.683	204.90
5	300	0.621	186.28

Net Present Value = Present value of inflows -  Present value of outflows

= (818.18+578.51+375.66+204.9+186.28) - 1000

= 2163.53 -1000

= 1163.53

b) IRR

I have taken 62% and 63%

Year	Cashflow ($)	PVF@62%	Cashflow*PVF
0	(1,000)	1	(1,000.00)
1	900	0.617	555.56
2	700	0.381	266.73
3	500	0.235	117.60
4	300	0.145	43.56
5	300	0.090	26.89

NPV = (555.56+266.73+117.6+43.56+26.89) - 1000

= 1010.33 - 1000

= 10.33

Year	Cashflow ($)	PVF@63%	Cashflow*PVF
0	(1,000)	1	(1,000.00)
1	900	0.613	552.15
2	700	0.376	263.46
3	500	0.231	115.45
4	300	0.142	42.50
5	300	0.087	26.07

NPV = (552.15+263.46+115.45+42.5+26.07) - 1000

= 999.64 - 1000

= -.36

IRR = R1 + ((NPV1 * (R2 - R1)) / (NPV1 - NPV2))

= 62 + ((10.33 * (63-62)) / (10.33+.36))

= 62.966%

c) profitability index (PI)

PV of future cashflows / initial investment

= 2163.53 /1000

= 2.16353

= 2.16

2. If projects A and B are mutually exclusive, which project would you select

Both projects have positive NPV, hence both are acceptable. But since the projects are mutually exclusive only one can be selected. So select Project B since it has higher NPV.

In IRR prospective, higher NPV is preferable, provided cost of investment is equal. Then also B is preferable

PI>1 itself is acceptable. So in a mutually exclusive project, select that which has higher PI.Then also B is preferable

3. Find the crossover rate for projects A and B

Crossover rate is the rate of return at which NPV of two projects are equal.

100/(1+r)^1 + 600/(1+r)^2 + 700/(1+r)^3 + 900/(1+r)^4 + 300/(1+r)^5 - 1000 = 900/(1+r)^1 + 700/(1+r)^2 + 500/(1+r)^3 + 300/(1+r)^4 + 300/(1+r)^5

-800/(1+r)^1 - 100/(1+r)^2 + 200/(1+r)^3 + 600/(1+r)^4 = 0

-800(1+r)^3 - 100(1+r)^2 + 200(1+r)^1 +600 = 0

This a cubic equation. We get the values .9564

1+r = .9564

r = .9564 - 1

= .0436

Crossover rate = -4.36%

It can also calculated as IRR of incremental cashflows.