EXPECTED UTILITY THEORY

[Sarthak GoelParticipant]

What do you mean by the utility and expected utility theory?

[Sarthak GoelParticipant]

<b>The utility</b> is the same as what we learned in Class XII Economics. It is the satisfaction derived or obtained from a particular course of action.

It is assumed that the numerical value of utility can be defined for different levels of investors’ wealth using a utility function.

It reflects the satisfaction that an investor would derive when having a particular amount of wealth.

Now, let’s see what expected utility theory is based upon.

It is based on 2 things, first being the ability to construct a utility function which represents the investors’ utility of wealth and the second that decisions are made taking into consideration the probabilities of different outcomes with an aim to maximize the expected utility of the investor.

Our main focus would be maximizing the expected utility of the investor.

The word expected is used here as the future is uncertain and calculating the utility of the wealth of a future date is not possible without taking probabilities or expectations into account.

Expected utility= E(U)= Ʃp_{i *}U(W_i)

Here p_i is the probability of an outcome i

W_i is the total wealth of the investor in case of an outcome i.

U(W_i) is the utility of wealth for the investor in case of outcome i

where i goes from 1 to n with n possible outcomes each having a probability of occurring.

So, we are calculating expected utility by multiplying the utility of a particular amount of wealth with the probability of achieving that amount of wealth and adding all the possible outcomes.

Let’s understand it with an example:

Suppose an investor is having a wealth of $100 which he decides to invest in an Asset A which yields following returns with the following probabilities.

5% with probability 1/5

10% with probability 2/5

-5% with probability 1/5

-10% with probability 1/5

The utility function for the investor is U(W)= log(W). Calculate the expected utility for the investor.

ANS:

Wealth= Investment amount + Return

So, the following amount of wealth is possible for the investor with the probabilities mentioned.

$105 with 1/5 probability

$110 with 2/5 probability

$95 with 1/5 probability

$90 with probability 1/5

E(U)= 1/5 *log (105) + 2/5 *log (110) +1/5 * log (95) + 1/5 * log (90)

= 2.0071

 

TRY THIS OUT!!

Now there is an asset B in which the above investor could invest which provides return with the mentioned probabilities below. Decide the asset in which the investor should invest provided the investor would have to invest the entire amount in the same asset.

20% with probability 1/5

-5% with probability 2/5

-10% with probability 1/5

-12% with probability 1/5

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