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How to use Utility Theory to measure risk?


Utility Theory: It is based on economic theory of marginal utility which means a satisfaction we gain from consuming an additional unit of a product. Law of diminishing marginal utility is also applied here but the difference is money can never be zero. The utility increasing with increase in money (so you can start fulfilling your basic needs and also your future needs) after reaching a certain point the utility of money start decreasing (at this point your all needs are fulfill) now money is less valuable for you in comparison to earlier stage. This function is useful for those investors who consider more risk. It is not used by risk averse investors who protect his investment from high risk.

Example: According to below information of a 3 projects A, B and C and its cash inflows. Find out the without utility of cash flows and with utility cash flows.
PROJECT A
Year
Cash inflow
Probability
1
30000
0.3
2
20000
0.3
3
50000
0.4
Year
PROJECT B
Probability
1
32000
0.4
2
38000
0.3
3
48000
0.3
Year
PROJECT C
Probability
1
43000
0.3
2
29000
0.6
3
40000
0.2


Utilities of first Rs.10000 are 5 next Rs.10000 are 4 then 3, 2 and 1 respectively.

Solution:
 Calculation of without considering utility of 3 projects:
Project A:

Year
Cash Inflows
Probability
Expected cash flows
1
30000
0.3
9000
2
20000
0.3
6000
3
50000
0.4
20000
Total
1,00,000
1
35000

Project B:
Year
Cash Inflows
Probability
Expected cash flows
1
32000
0.4
12800
2
38000
0.3
11400
3
48000
0.3
14400
Total
118000
1
38600


Project C:
Year
Cash Inflows
Probability
Expected cash flows
1
43000
0.3
12900
2
29000
0.6
17400
3
40000
0.2
8000
Total
112000
1
38300


Without considering the utility function Project B has higher expected net cash flow in comparison to Project B and C. So, Project B is best option.

Calculation of the utility Cash Flows:

Project A:

Year
Cash Inflows
Probability
Utility with each cash flows
Total Utility
1
30000
0.3
5+4+3=12
3.6
2
20000
0.3
5+4=9
2.7
3
50000
0.4
5+4+3+2+1=15
6
Total
1,00,000
1
36
12.3

Project B:

Year
Cash Inflows
Probability
Utility with each cash flows
Total Utility
1
32000
0.4
5+4+3+*0.25=12.25
4.9
2
38000
0.3
5+4+3+*1.6=13.6
4.08
3
48000
0.3
5+4+3+2+*0.8=14.8
4.44
Total
118000
1
40.65
13.42

Project C:

Year
Cash Inflows
Probability
Utility with each cash flows
Total Utility
1
43000
0.3
5+4+3+2+0.3=14.3
4.29
2
29000
0.6
5+4+2.7=11.7
7.02
3
40000
0.2
5+4+3+2=14
2.8
Total
112000
1
40
14.11

*0.25= (2/10000)*2000
*1.6= (2/10000)*8000
*0.8= (1/10000)*8000

With considering the utility function Project C has higher total utility in comparison to A and B. So, Project C is best option.

Example: Company Y has two projects A and B both the project has same risk but different in investment value that is Rs.60, 000 and Rs.45, 000 respectively. The probability of gaining full amount of investment is 0.70 of project A and 0.65 of project B. The utility of those investment are 2.30 and 4.40 respectively. Find out which investment is better for company Y.

Solution :
According to the utility function:
Project A = 0.70* 2.30 = 1.61 utils
Project B = 0.65 * 4.40 = 2.86 utils
Project B has a higher utils in comparison to project A. So, project B is better for company Y.



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