Question

Cullumber Incorporated management is considering investing in two alternative production systems. The systems are mutually exclusive, and the cost of the new equipment and the resulting cash flows are shown in the accompanying table. The firm uses a 10 percent discount rate for production systems.

YearSystem 1System 2

0

-$14,240-$45,926

1

14,26 132,130

2

14,26132,130

3

14,26 132,130

Compute the IRR for both production system 1 and production system 2

Which has the higher IRR?

Which production system has the higher NPV?

Answer 1

The discount factor in the question = 10% or we can say .10

The discount factor for Year 1 will be = 1/(1+.10) = .909

The discount factor for Year 2 will be = 1/(1+.10)^2 = .826

The discount factor for Year 3 will be = 1/(1+.10)^3 = .751

Cash outflow for System 1 is given which is equals to -$14,240 and for the System it is -$45,926

Now converting the cash inflows to the present values:

Year	Cash Flow	Discount Factor	Value (Cash Flow* Discount Factor)
1	14,26132,130	.909	1296354106.17
2	14,26132,130	.826	1177985139.38
3	14,26132,130	.751	1071025229.63

The formula for Net Present Value = Sum of Present Values of cash Inflow - Cash Outflow

NPV for System 1 = 1296354106.17+1177985139.38+1071025229.63 - 14240

= 3545364475.18 - 14240

= $35,45350,235.18

NPV for System 2 = 1296354106.17+1177985139.38+1071025229.63 - 45926

= $35,45318,549.18

So clearly System 1 has higher NPV.

Now calculating the IRR for both the systems.

The formula of IRR = L_r + L_rNPV/(L_rNPV-H_rNPV)* (H_r-L_r)

where,

L_r = Lower Discount rate

H_r = Higher Discount rate

L_rNPV = Lower discount rate NPV

H_rNPV = Higher discount rate NPV

For calculation of IRR we will have to consider one more discount rate apart from the one company uses. Let's take 12% as the other discount rate

The discount factor at 12% for Year 1 = 1/(1+.12)^1 = .893

The discount factor at 12% for Year 2 = 1/(1+.12)^2 = .797

The discount factor at 12% for Year 3 = 1/(1+.12)^3 = .712

Year	Cash Flow	Discount Factor	Value (Cash Flow * Discount Factor)
1	14,26132,130	.893	1273535992.09
2	14,26132,130	.797	1136627307.61
3	14,26132,130	.712	1015406076.56

Now we will use the same process as we have used above to calculate the NPV at the new discount rate which is 12% in our case.

NPV System 1 (@12%) = Sum of present value of cash flows - Cash outflow

= 1273535992.09+1136627307.61+1015406076.56 - 14240

= $34,25555,136.26

NPV System 2 (@12%) = Sum of present value of cash flows - Cash outflow

= 1273535992.09+1136627307.61+1015406076.56 - 45926

= $34,25523,450.26

IRR System 1 = L_r + L_rNPV/(L_rNPV-H_rNPV)* (H_r-L_r)

the description of the formula has been explained above. Now we just have to put the values in the formula,

L_r = Lower discount rate which is 10, H_r = Higher discount rate which is 12

= 10 + 35,45350,235.18/(35,45350,235.18 - 34,25555,136.26) *(12-10)

= 69.19%

IRR System 2 = L_r + L_rNPV/(L_rNPV-H_rNPV)* (H_r-L_r)

= 10 + 35,45318,549.18/(35,45318,549.18-34,25523,450.26) * (12-10)

= 69.19%

The IRR for both the systems comes to be the same.