What is Certainty-Equivalent Cash Flows?

Certainty-Equivalent: It is a method in which uncertain cash flows are converted into certain cash flows by multiplying with probability of occurrence such cash flows .Certainty coefficient assumes value between 0 and 1. In this method risk free rate are used instead of risk-adjusted discount rate. We would use either IRR or NPV for evaluation of a project.

By using NPV method:

NPV > 0 project accepted

NPV < 0 project rejected

NPV = 0 project may be accepted or rejected

By using IRR method:

IRR > r project accepted

IRR < r project rejected

IRR = 0 project may be accepted or rejected

Formula of Certainty Coefficient:

Certainty Coefficient = Expected Cash Flows (Certain cash Flows) / Risky Cash Flows

= Expected Cash Flows (Certain cash Flows)/ (1+risk premium rate)

Example: In below table 5 years cash inflows and certainty coefficient are given which shows the probability of occurrence of cash flows.

Year	Cash inflows	Certainty Coefficient
1	28,000	0.8
2	32,000	0.6
3	46,000	0.4
4	58,000	0.2

The initial cost of investment is Rs.65, 000 and the discount rate is 8% annually. Find out the NPV with the help of certainty-equivalent method.

Solution:

Year	Cash inflows	Certainty Coefficient	Certain-Equivalent Cash flows	Discount Rate 8%	Present Value
1	28,000	0.8	22400	1.08	20,740.74
2	32,000	0.6	19200	1.17	16,410.26
3	46,000	0.4	18400	1.26	14,603.17
4	58,000	0.2	11600	1.36	8529.41
Total					60,283.58

NPV = Present cash inflows-Cash outflows

= Rs. (60,283.58-65, 000)

= - Rs.4,716.42

NPV is negative which means project is not acceptable.

By using IRR method:

Let’s assume r =10%

Year	Certain-Equivalent Cash flows	Discount rate 10%	Present Value
1	22400	1.1	20,363.63
2	19200	1.21	15,867.76
3	18400	1.33	13,834.58
4	11600	1.46	7,945.20
Total			58011.17

NPV = Present Value of cash inflows – Cash Outflows

= Rs. (58011.17-65000)

=- Rs.6988.83

Let’s assume r = 4%

Year	Certain-Equivalent Cash flows	Discount rate 10%	Present Value
1	22400	1.04	21,538.46
2	19200	1.08	17,777.77
3	18400	1.12	16,428.57
4	11600	1.16	10,000
Total			65744.8

NPV = Present Value of cash inflows – Cash Outflows

= Rs. (65744.8-65000)

= Rs. 744.8

IRR = lower rate+ (NPV at lower rate/ (NPV at lower rate-NPV at higher rate))*(higher rate-lower rate)

= 4+ (744.8/ (744.8- (6988.83)))*(10-4)

= 4.57%

So, this project is rejected due to IRR (4.57%) < risk free rate (8%).

Example:Company X wants to invest in project A . Find out the certainty equivalent cash flows and NPV  of that project if the discount rate is 6% and the expected cash flows for five year are as follows:

Cash flows	Certainty Coefficient
5, 000	08
8, 000	0.6
8, 900	0.7
12, 000	0.3
14, 400	0.4

Solution:

Year	Cash flows	Certainty Coefficient	Certain equivalent cash flows	Discount rate 6%	Present Value
1	5, 000	0.8	4, 000	1.06	3, 773.58
2	8, 000	0.6	4, 800	1.12	4, 285.71
3	8, 900	0.7	6, 230	1.19	5, 235.29
4	12, 000	0.3	3, 600	1.26	2,857.14
5	14, 400	0.4	5, 760	1.34	4, 298.51
Total					20, 450.23

NPV = Present value of cash flows – Initial investment

= 20, 450.23 – 20, 000

= Rs. 450.23

The NPV of this project is positive. So, the project A is accepted by the company X.