This document provides an overview of utility theory in actuarial science. It defines utility as the satisfaction obtained from consuming a commodity. Total utility is the sum of utility derived from consuming all units of a commodity, while marginal utility is the additional utility from consuming an extra unit. The utility of a capital amount depends on one's current wealth. Utility functions should be complete, transitive, and monotonic increasing. Examples of utility functions given include positive power functions and exponential utility functions. References on actuarial models and risk theory are also provided. Overall, the document outlines key concepts in utility theory as relevant to actuarial applications.

1. ACADEMIC WRITING
Topic:- UTILITY THEORY IN ACTURIAL SCIENCE
VINAY M
Application no:- a8bf718ceea611e9879cab44f7631e4a
Department of Mathematics and Statistics
Central University of Punjab, Bathinda

2. ACKNOWLEDGEMENT
I would like to express my special thanks of gratitude to
Swayam who gave this wonderful opportunity to take a
presentation on the topic Utility Theory in Actuarial
Science as well as professor Dr. Ajay Semalty who offer the
course Academic Writing.

3. UTILITY APPROACH
 Definition
 Utility means the satisfaction obtained from consuming a commodity
 Utility of Capital means Degree of satisfaction of having capital which depends of
the particular amount of capital.
eg.
if we give 1000 rupees to a person with a wealth of 1 crore and the same
1000 rupees to a person with zero capital, the first person will feel much less
satisfied than the second person.
 Utility of a capital x is denoted as U(x).

4. Features of UTILITY
 It is subjective(It differs from person to person)
 Utility is different from usefulness. Eg. Cigarette, Alcohol
 Utility is different from satisfaction.
Utility(Expected)
Satisfaction(actual)
When utility is actually used then it become satisfaction
 Utility is different from pleasure(it is not necessary that the thing which posses utility provide
pleasure) Eg. Injection
 Utility is independent of morality. Eg. Drunkard

5. Concepts of Utility
Total utility
Sum total of utility derived from the consumption
of all the units of the commodity.
TU= 𝑀𝑈
Eg. The person is consuming 2 ice creams, the
satisfaction from first one is 10 utils and
satisfaction from 2nd one is 9 utils. So total
satisfaction is 19 utils from both the ice creams.

6. Marginal Utility
 Additional utility derived from the consumption of an
additional unit of commodity.
Marginal utility of extra units become decline as more it
is consumed. Which is called Diminishing marginal utility.
 Eg. If 10 Units of commodity gives 100 utils of T.U and
11 units of commodity gives 105 utils of T.U
Then Marginal utility will be
MUn=TUn - TUn-1
MU11=TU11 – TU11-1
MU11=105-100=5 utils

7. Characteristics of Utility function
 Complete: U(x) is relevant for values of x. eg. 1/x, not exist when x=0.
 Transitivity: If I prefer A to B then U(A)>U(B). If I prefer B to C then
U(B)>U(C). That means U(A)>U(C).
 dU/dx >0: U is a monotonic increasing function.

8.  A utility function U(x) represents a preference relation if and only if:
 𝑋 ≽ 𝑌  E{U(X)} ≥ E{U(Y)}
 𝑋 ≼ 𝑌  E{U(X)} ≤ E{U(Y)}
 𝑋 ≃ 𝑌  E{U(X)} = E{U(Y)} X is equivalent to Y
 It means among two random variables, we prefer the random variable with
the larger expected utility.
 The investor who follows E(u(X)) ≥E(u(Y)) is called Expected Utility
Maximizer.

9. examples for utility functions
1. Positive power function
2. Exponential utility functions

10. The Positive power function
 Let u(x)=xα for all x≤0 and some α>0
 The expected utility in this case is considered only for positive random variables
 u(x) tends to 0 as x tends to 0
 E[u(X)]=E{Xα}, the moment of X of the order α.
 If α=1, then E[u(X)]=E{X}, that means EUM criterion coincides with the mean-value
criterion.
 If α<1 the function u(x) is concave(downward).
 If α>1 then function u(x) is convex(concave upward).
 Eg. Let X=b > 0 or 0 with equal probabilities. Then
c(X) = [(1/2)bα]1/α = 2-1/αb.
The smaller α is the smaller the certainty equivalent.

11. Exponential Utility functions
 Let u(x) = -e-βx, where β>0
 Here u(x) tends to 0 as x tends to ∞, faster than any power function, the
saturation effect in this case is stronger than negative power function.
 The expected utility E{u(X)} = -E{e-βX} = -M(-β), where M(z)=E{ezX}

12. Reference
 ACTUARIAL MODELS:The Mathematics of Insurance Second Edition, VLADIMIR I.
ROTAR
 Modern Actuarial Risk Theory by Rob Kaas , Marc Goovaerts , Jan Dhaene and
Michel Denuit.

13. Feedback of the course
 This course is really helpful for my rearch work which I will do after my M. sc
course. It helped to know about the important terms and facts, we have to
look before publishing a paper. The lectures are good and syllabus is
relevant. I am using this occasion to thanks for teaching me.