Risk Sharing in Contingent-Claims Market

Risk Sharing between Consumers in a Contingent-Claims Market:

For instance Frank Knight is a dealer in a Casino. Michael Spence wants to play the card game with $100,000.

There are 3 unseen cards two of which have dandelions and the remaining one has a rose. If Spence selects a card with rose, he will win the amount that he bet. And if Spence picks a card with dandelion, he will lose the amount that he bet.

Let Cr as well as Cd denote the two possible commodity bundles that Spence will shop after the card game (contingent commodity).

Budget Line:

Spence will choose any amount within $100,000 to play the game.

687_budget line.jpg

F: initial endowment, FH¯ : Budget Set (or opportunity set) How about IF¯ ?

Iso-Expected Value Line:

Now on the basis of game framework we can calculate the expected value of the consumption that Spence will take.

Since the probabilities are 1/3 and 2/3 for rose and dandelion, respectively, if Spence bet $ 30K, then Cr = 130K and Cd = 70K.

Therefore the expected value if he bet $ 30K will be:

E = {(1/3)*130000}+{(2/3)*70000}=90000

Is this card game fair? Absolutely not. Let’s compute the expected value if Spence bet all the money he had ($100,000). How much is able to he expect to make?

{(1/3)*200000}+{(2/3)*0}=66667

Now, how can we design a new rule that is able to make this game fair a game that can guarantee the initial state regardless of how much money player bets. Net gain is forever zero? Simply because there is one card of rose along with two cards of dandelions the manner to make a fair game is to double the award when player picks a card of rose we got a new game rule! Can you draw a fresh budget line with the new framework? Sure you can.

1428_iso expected value line.jpg

 FJ¯ : iso-expected value line or else fair odds line

Meaning of Risk:

It is risk free not to play the game (45º line is called risk-free line).

The more cash the players bet the higher the risk s/he needs to take.

Moving farther from F which is risk free the riskiness will increase.

Point J represents the maximized stage of riskiness with given condition.

Preference and Choices of Decision-makers:

Risk neutral person will rely merely upon the expected value of the game. If the game is favorable based on the computed expected value, s/he will play.

Risk averse person will not play if not the game is sufficiently favourable. Therefore s/he will not play even if the game is fair with 0 expected value.

Risk loving person will play still though the game is disadvantageous based on the expected value.

Indifference Curves with Risk Averseness:

Convex indifference curve implies that people favour the average consumption bundles to the extreme ones. As well as the slope of indifference curve on 45º line (risk-free line) will have a same slope with the fair odds line. Why? (Elucidate with the green indifference curve Io)

5_risk averseness.jpg

therefore we can presume that the person with risk averseness will choose the point F as the optimal point if the game is fair (will play the game).

Now what if the game is beneficial to the player?

976_risk free line.jpg

Even if Frank tenders three times of the amount that Michael bet Michael will bet small amount of money (100K – R).

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