Arguelles, Amanda Marie Co, Darlene Angelini R. Assignment 2 I. Given:

U  W   e W

Where α is a parameter (1a) To prove that more is preferred than less, we need to show that this individual’s preferences are nonsatiated. This can be shown by satisfying the condition,

would show that for every one unit increase in wealth, satisfaction will increase as well. For this utility function, we will get:

U (W )  0 . This W

U (e W )  0  (e W )( )  0   e W  0 W
Because W > 0, we can then conclude that more is preferred to less consumption given the utility function. (1b) To prove that the consumer is risk averse we must show that this utility function has diminishing marginal utility wherein the condition

that the marginal utility from an extra increase of utility decreases as your payoff increases. Hence, this would represent risk-aversion since the bigger the amount of money that is at stake, the higher satisfaction from winning is offset by an even higher dissatisfaction from losing. For this utility function we get:

U 2 (e W )  0 must be satisfied. This means  2W

U 2 (e W ) U 2 (e W )  ( e W )( )   2e W  0 0  2 2 W W
Since the second order derivative of the function gives a negative value, we could say that the consumer is risk averse. (2a) There are two measures of risk aversion that was developed by Pratt and Arrow. The first measurement would be the absolute risk aversion coefficient/index. This measurement helps show how much wealth an individual is willing to expose to risk as a function of changes in wealth. In order to solve for this, we used the formula that was defined by Pratt and Arrow, A(W )   . For this utility function we get:

U "(W ) U '(W )

A(W )  

(2b) The second measurement of risk aversion is the relative risk aversion coefficient/index. This also helps measure how much wealth an individual is willing to expose to risk however in

 2e W Simplifying the equation, we get: A(W )    e  W

percentage form. Hence, it then helps us measure the percentage of an individual’s wealth he/she is willing to expose to risk. As defined, R(W )  

U "(W )  W . For this utility function, we get: U '(W )

R(W )  

 2e W  W  R(W )  W  e W

(3a) We could measure the effect of  on the absolute risk aversion by getting the partial derivative of A(W) with respect to  :

dA(W ) 1 d
This shows that for every unit increase in α, A(W) increases by 1. Hence, the greater the parameter, the more risk-averse this individual is. (3b) We could measure the effect of  on the relative risk aversion by getting the partial derivative of R(W) with respect to  :

This shows that for every unit increase in α, R(W) increases by the value of wealth. This then means that this individual’s risk aversion is dependent on the amount of wealth he/she has. (4a) To get the effect of the changes in W with the behavior of the degree of absolute risk aversion, we must get the partial derivative of A(W) with respect to W:

dR(W ) W d

dA(W ) 0 dW
This means that the individual has CARA or constant absolute risk aversion. That is, if his/her wealth increases, he/she will not change the amount of pesos invested in a risky assest. (4b) To get the effect of the changes in W with the behavior of the degree of relative risk aversion, we must get the partial derivative of R(W) with respect to W:

dR(W )  dW
This means that the individual has IRRA or increasesing relative risk aversion. That is, if his/her wealth increases, the percentage of his/her wealth invested in a risky asset decreases. Hence, the more wealth an individual receives, the larger the percentage of his wealth he would wish to keep safe.

II. Given:

U (W ) 

W

 1 G (50000;10000;0.5)
G 2 ( x;0;1)

where W>0 and α is a parameter

In order to determine the value of x, we must first show that the individual is indifferent to both gambles. In order for an individual to indifferent, the two gambles must give the same amount of satisfaction or expected utility according to the Expected Utility Theorem that states that an individual decides between gambles as if he/she were to maximize utility. Therefore, indifference ( ) would mean ( ) where represents a gamble with risk while represents a ( ) gamble which is sure. This can also be written as ( ). ( ) ∑ Because ( ) since a gamble has a random outcome and we assume then that it will occur on average we can then solve for it as: ( ) ( ( ( We end up with this equation: ( Then by substituting α with 0.25: ( Simplify, and you will get ) =( ( ) ( ) ) ( ) ( ) )) ( ) )( ( ( )) )

) and by transposition, we will come up with

By checking we will get:

0.5  (

49.91=49.91

500000.25 100000.25 24232.880.25 )  0.5  ( )( ) 0 0.25 0.25 0.25

By substituting α with 0.75: ( Simplify, and you will get ) =( ( ) ( )

) and by transposition, we will come up with

By checking we will get:

0.5  (

2,895.80=2,895.80

500000.75 100000.75 24232.880.75 )  0.5  ( )( ) 0 0.75 0.75 0.75

Since both sides of the equation are equal, we could say that the consumer is indifferent between the risky gamble G1 and the sure gamble of G 2 when the sure pay off of G 2 is equal to P24,232.88 when α = 0.25 and P 28,125.91 when α = 0.75. These values represent the amount of certain money the individual would receive in a sure gamble but has the same expected utility as if the individual would have entered into the risky gamble. This is also known as certainty equivalent of this individual’s wealth with the risky gamble. Although it may be less than what one can possibly get, this individual shows that he/she would be equally as pleased to have this certain wealth as the risky wealth. Being a risk-averse individual like most people, this certain equivalent would be preferred rather than the risky wealth. The parameter also has a lot to do with the risk aversion of this individual. As you can see, when the parameter increased, the certain equivalent of the risk gamble increased as well. This means that the higher the parameter, the less risk-averse the individual since he/she associates the same amount of risky wealth to higher certain wealth. Hence, if we were to place this in a scenario where the individual could get insured for this risky gamble, he would have to pay a smaller amount of insurance premium as compared to the individual with a parameter of 0.25. In relation, the more risk-averse you are, the more you demand to pay a higher premium to enter into a risky gamble. III. Given: W0 = ₱ 100,000

U (W )  ln W for W  0

As defined, W = W0 + Zs, hence to solve for the pay offs we just simply transpose. You can see in the table below what pay off each state of the world has. State, s 1 2 3 Worth (₱) W, Ws 1 50,000 100,000 Probability, Ps 0.02 0.04 0.94 Zs = W0 - Ws -99,999 -50,000 0

In order to find out the maximum amount he will pay for an insurance policy that pays in full any loss that he incurs, we must first fulfill the condition that EU(W)Insurance= EU(W)w/o insurance . It is at this point where you are as well off as you would be if forced to face the world without insurance. We must take note however that the expected utility of wealth with insurance is certain as you are covered even with a loss, which allows us to express it as: EU(W)Insurance = U(W0 - π) We subtract the premium, π, from our original wealth since you have to pay this in order to avail of the insurance. Through substitution, we will get: U(W0 - π) = ln(100,000 – π)

As defined previously, to solve for expected utility (which is a random outcome), we will assume that it will occur on average. This means we can calculate it by: EU(W)w/o Insurance = 0.2 (ln 1) + 0.04 (ln 50000) + 0.94 (ln 100,000) Hence, EU(W)Insurance= EU(W)w/o insurance through substitution would be: 0.2 (ln 1) + 0.04 (ln 50,000) + 0.94 (ln 100,000) = ln (100,000-π) 11.25494107= ln (100,000-π) ℮ 11. 25494107 = ℮ ln (100,000-π) Solving for the premium by transposition, we will get: π = 22,739.31279 The maximum amount that he is willing to pay for the insurance policy would be P 22,739.31. This would be mean that anything above this amount, the individual will not avail of the insurance policy anymore. IV. Given: Perfect capital markets: U (W) = W0.5 G1 = (36; 16; 0.5) in ₱ W0 = 0 When there are perfect capital markets, this means that transactions in the capital market (may it to borrow or to lend) can be made without payment of charges other than the prevailing market rate of interest since no agent is large enough to affect prices or interest rates. Also, there is symmetric information among market participants. This means that the individual is open to either borrowing or spending money throughout his decision of joining into the gamble or not. However, we will first assume that this person starts out with zero wealth hence, we can solve for the certainty equivalent of the gamble and maximum premium he is willing to pay. (1) The certainty equivalent, CE(W0,z) of the gamble is the peso amount of certain payoff at which the investor would be indifferent to the risky investment and this certain amount. By definition, EU(W0 + Z) = U(W0 + CE(W0,z)) = U(W0 + E[Z] – ∏) As defined previously, to solve for expected utility (which is a random outcome), we will assume that it will occur on average. This means we can calculate it by:

EU  W0  z   0.5  360.5  0.5  160.5 EU  W0  z   5

5  (CE(0,0.5))0.5 By raising both sides to 1 , we can solve for CE(0,0.5) : 0.5 CE(0,0.5) = 25 and Wc = W0 + CE = 25

This means that the individual is indifferent to receiving a sure pay off of P25.00 as compared to the risky pay off of receiving either P36.00 or P16.00. The P25.00 also represents this individual’s Wc or the amount of certain wealth that the individual is indifferent to when compared to receiving the risky wealth of either P36.00 or P16.00 since our starting wealth is 0. (2) The risk premium of an individual is the amount that the person is willing to pay in order to assure/insure a certain amount from his/her risky gamble. Hence, may the gamble go good or bad, the individual will still receive this certain amount which was solved earlier, CE. By definition: CE (W0, z) = E[z] - ∏ (W0, z) where ∏ is the premium In order to solve for the premium, we must solve for E[Z] first. To get this expected value of the pay offs 36 and 16, like explained many times earlier, we will assume that it occurs on average hence, the E[Z] would be equal to the mean of these two values or (0.5)(36) + (0.5)(16) = 26 Now that we have both CE (W0, z) and E[Z], we can now solve for ∏ (W0, z) by substitution: CE (W0, z) = E[z] - ∏ (W0, z) 25 = 26 - ∏ ∏ = 26 - 25 ∏=1 Hence, the consumer is willing to pay a maximum of P1.00 in order to insure an amount of P25.00 from the gamble. However we must consider the fact the individual has an initial wealth of 0. Hence, if ever he/she wishes to pay for this amount to acquire the insurance, he would have to borrow from the capital market. (3) Usually we try to prove that the individual will be willing to pay the premium because he/she is risk averse. In order for this to be proven we have to satisfy the condition that [ ]) ( )] [ ( provided that the investor is risk averse. This would also then imply that the investor will only be willing to pay for an uncertain payoff that would have to be priced less than its expected payoff, E[Z]. First, let’s prove the individual is risk averse through the second ordered derivative of his utility function in respect to W or U’’. This is because U’’ shows the shape of his utility function. As explained previously, if U’’ is negative that would mean that the shape of his/her utility function is strictly concave which means he is risk-averse. For this individual his U’’ is equal to: ( ) Hence, this individual is risk-averse.

Now, to solve for the probability premium, or the minimum increase in the probability of the high payoff state so that the individual will not be willing to pay any premium, we will have to then satisfy the condition that [ ]) ( )] [ ( because it is at this point where the individual will then be indifferent between the two utilities. Therefore, the individual will no longer be willing to avail of the premium at this point since there his/her risky payoff is priced just as much as his expected payoff. Therefore, to solve for the increase change in the higher payoff probability (denoted as p) we will add p to the probability of 36. Because there must be a balance between the two probabilities (since for an event to surely happen you only need a probability of 1 and nothing more than that), we must then subtract p from the probability of the lower payoff. This is illustrated in the solution below, after substitution: ( And when simplified, we get: Solving for p by transposition, we will arrive with: p = 0.0495097568 or 0.050 This would then mean that once the probability of 36 has increased by 0.050 or has reached 0.55, then the individual will no longer avail of the premium, as explained earlier. V. Given: )( ) ( )( )

5 2 W 0.2 W  10000 U (W )  
S 1 2 3 4 5 Ls (₱) 1000 2000 3000 5000 6000 Ps 0.10 0.20 0.35 0.20 0.15 Ws = W0 + Ls 10000 – 1000 = 9000 10000 – 2000 = 8000 10000 – 3000 = 7000 10000 – 5000 = 5000 10000 – 6000 = 4000

(1) The certainty equivalent of the risky gamble CE(W0,Z), as explained earlier, is the peso amount of certain payoff at which the investor would be indifferent to the risky investment and this certain amount. We must first solve for this before attaining the certainty equivalent of the risky wealth you can are willing to pay your risk premium for. By definition, ( ) ( ( ( ) ) ) ( [ ] ( ( )) ( )) ))

So that, ( )

(

[ ]

Where is the random payoff or the loss in this problem. Substituting the given values to our equations given we will arrive with:

( ( ( ( ( ( ) )

) ) ) ) (
(

( (

) )

(

)

(

)

1.130650242 (
( ))

))

By equating

(
(

)

(
( ))

) , we will get:

By solving for ( )

(

) through transposition method, we will arrive with:

This means that this is the amount that you are willing to pay a premium for to assure such a loss. Take note that the certainty equivalent for this situation is negative because in this situation, all payoffs are losses. This means that no matter what may happen, the individual will incur a loss and –P3,706.72 is the amount that the individual will be indifferent with the expected risky loss that he might incur . Hence, this individual’s Wc or his certainty equivalent of his risky wealth would be equal to: ( )  Wc = 10,000 + (-3,706.72)  Wc =P 6,293.28 W c = W0 + This is then the amount that the investor will be willing to pay for to avoid the risk of receiving anything less. So, if the insurance policy is availed of, then the individual is assured P6,293.28 no matter what may happen. (2) Similarly as part V of this assignment, we will solve for risk premium. As explained previously, the risk premium of an individual is the amount that the person is willing to pay in order to assure/insure a certain amount from his/her risky gamble. Hence, may the gamble go good or bad, the individual will still receive this certain amount which was solved earlier, CE. [ ] ( ) From (1) we have stated that ( )) . Hence, by transposing ( ) to the left side of the equation we could solve for the risk premium. So that: ( Where: ) [ ] ( )

[ ] ( )

(

(

)

)

(

)

(

)

By simplifying we will get: [ ] which was solved through the different rules of expectations for a random variable where we assume that it occurs “on average.” Hence, [ ] is the weighted mean of the losses given in the table which explains the sign of the value we came across with. Given the values, we could solve for the premium by substituting the values to the given formula: ( ( ( ) ) ) [ ] ( ( ) ) to avoid the risk in gambling that will allow him

The consumer is willing to pay to attain a sure wealth of P 6,293.28.