Analytic and Numerical Proofs of the Bernoulli Principle and Mossin’s Theorem

During yesterday’s class meeting, we discussed two particularly important insurance economics concepts: 1) the Bernoulli Principle, and 2) Mossin’s Theorem. The Bernoulli Principle states that if an actuarially fair full coverage insurance policy is offered, then an arbitrarily risk averse individual will purchase it, and Mossin’s Theorem states that if insurance is actuarially unfair, then an arbitrarily risk averse individual will prefer partial insurance coverage over full insurance coverage.

The Bernoulli Principle is graphically illustrated in the following figure (taken from p. 4 in the Insurance Economics lecture note):

Capture

Here, the consumer has initial wealth of $120 and there is a 25% chance that a fire will occur which (in the absence of insurance) will reduce her wealth from $120 to $20. The expected loss is E(L) = .25(100) = $25, which is the actuarially fair price for a full coverage insurance policy. Thus if she remained uninsured, she would have E(W) = .25(20) + .75(120) = $95, and her utility would be EU. Note that the (95, EU) wealth/utility pair corresponds to point C in the figure above. Now suppose the consumer purchases an actuarially fair full coverage insurance policy. Since she is fully insured, she is no longer exposed to any risk; her net worth is $95 irrespective of whether the loss occurs, her utility is U(95), and her wealth/utility pair is located at point D in the figure above. Since U(95) > EU, she will fully insure. More generally, since this decision is optimal for an arbitrarily risk averse consumer, it is also optimal for all risk averse consumers, irrespective of the degree to which they are risk averse.

Next, we introduced coinsurance. A coinsurance contract calls for proportional risk sharing between the consumer and the insurer. The consumer selects a coinsurance rate \alpha , where \alpha represents the proportion of loss covered by the insurer. By definition, \alpha is bounded from below at 0 and from above at 1. Thus, if the consumer selects \alpha = 0, this means that she does not purchase any insurance (i.e., she self-insures). If she selects \alpha = 1, this implies that she obtains full coverage. Furthermore, the price of a coinsurance contract is equal to the price of a full coverage insurance contract (P) multiplied by \alpha . On pp. 6-7 of the Insurance Economics lecture note, I analytically (via the calculus) confirm the Bernoulli principle by showing that if the consumer’s utility is U = W.5 and insurance is actuarially fair, then the value for \alpha which maximizes expected utility is \alpha = 1. An Excel spreadsheet called the “Bernoulli and Mossin Spreadsheet” is available from the course website which enables students to work this same problem using Solver. I recommend that you download this spreadsheet and use Solver in order to validate the results for a premium loading percentage (β) equal to 0 (which implies a full coverage (actuarially fair) premium P = E(L)(1+β) =$25 x (1+0) = $25) and β = 0.6 (which implies an actuarially unfair premium of $40, the analytic solution for which is presented on p. 8 in the Insurance Economics lecture note).

Let’s use Solver to determine what the optimal coinsurance rate (\alpha ) is for U = W.5 when β = 0 (i.e., when insurance is actuarially fair and full coverage can be purchased for the actuarially fair premium of $25). To do this, open up the Bernoulli and Mossin Spreadsheet and invoke Solver by selecting “Data – Solver”; here’s how your screen will look at this point:

Solver1

This spreadsheet is based upon the so-called “power utility” function U = Wn, where is 0 < n < 1. Since we are interested in determining the optimal coinsurance rate for a consumer with U = W.5, we set cell B2 (labeled “exponent value”) equal to 0.500. With no insurance coverage (i.e., when \alpha = 0) , we find that E(W) = $95 and E(U(W)) = 9.334. Furthermore, the standard deviation (σ) of wealth is $43.30. This makes sense since the only source of risk in the model is the risk related to the potentially insurable loss.

Since we are interested in finding the value for \alpha which maximizes E(U(W)), Solver’s “Set Objective” field must be set to cell B1 (which is a cell in which the calculated value for E(U(W)) gets stored) and Solver’s “By Changing Variable Cells” field must be set to cell B3 (which is the cell in which the coinsurance rate \alpha gets stored). Since insurance is actuarially fair (β = 0 in cell B4), the Bernoulli Principle implies that the optimal value for \alpha is 1.0. You can confirm this by clicking on Solver’s “Solve” button:

Solver2

Not only is \alpha = 1.0, but we also find that E(W) = $95, σ = 0 and E(U(W)) = 9.75. Utility is higher because this risk averse individual receives the same expected value of wealth as before ($95) without having to bear any risk (since σ = 0 when \alpha = 1.0).

Next, let’s determine what the optimal coinsurance rate (\alpha ) is for U = W.5 when β = 0.60 (i.e., when insurance is actuarially unfair). As noted earlier, this implies that the insurance premium for a full coverage policy is $25(1.60) = $40. Furthermore, the insurance premium for partial coverage is $ \alpha 40. Reset \alpha ’s value in cell B3 back to 0, β’s value in cell B4 equal to 0.60, and invoke Solver once again:

Solver3

On p. 8 in the Insurance Economics lecture note, we showed analytically that the optimal coinsurance rate is 1/7, and this value for \alpha is indicated by clicking on the “Solve” button:

Solver4

Since \alpha = 1/7, this implies that the insurance premium is (1/7)40 = $5.71 and the uninsured loss in the Fire state is (6/7)(100) = $85.71. Thus in the Fire state, state contingent wealth is equal to initial wealth of $120 minus $5.71 for the insurance premium minus for $85.71 for the uninsured loss, or $28.57, and in the No Fire state, state contingent wealth is equal to initial wealth of $120 minus $5.71 for the insurance premium, or $114.29. Since insurance has become very expensive, this diminishes the benefit of insurance in a utility sense, so in this case only a very limited amount of coverage is demanded. Note that E(U(W)) = 9.354 compared with E(U(W)) = 9.334 if no insurance is purchased (as an exercise, try increasing β to 100%; you’ll find in that case that no insurance is demanded (i.e., \alpha = 0).

I highly recommend that students conduct sensitivity analysis by making the consumer poorer or richer (by reducing or increasing cell B5 from its initial value of 120) and more or less risk averse (by lowering or increasing cell B2 from its initial value of 0.500). Other obvious candidates for sensitivity analysis include changing the probability of Fire (note: I have coded the spreadsheet so that any changes in the probability of Fire are also automatically reflected by corresponding changes in the probability of No Fire) as well as experimenting with changes in loss severity (by changing cell C8).

Moral Hazard

During next Tuesday’s class meeting, we will discuss the concept of moral hazard. In finance, the moral hazard problem is commonly referred to as the “agency” problem. Many, if not most real-world contracts involve two parties – a “principal” and an “agent”. Contracts formed by principals and agents also usually have two key features: 1) the principal delegates some decision-making authority to the agent and 2) the principal and agent decide upon the extent to which they share risk.

The principal has good reason to be concerned whether the agent is likely to take actions that may not be in her best interests. Consequently, the principal has strong incentives to monitor the agent’s actions. However, since it is costly to closely monitor and enforce contracts, some actions can be “hidden” from the principal in the sense that she is not willing to expend the resources necessary to discover them since the costs of discovery may exceed the benefits of obtaining this information. Thus moral hazard is often described as a problem of “hidden action”.

Since it is not economically feasible to perfectly monitor all of the agent’s actions, the principal needs to be concerned about whether the agent’s incentives line up, or are compatible with the principal’s objectives. This concern quickly becomes reflected in the contract terms defining the formal relationship between the principal and the agent. A contract is said to be incentive compatible if it causes principal and agent incentives to coincide. In other words, actions taken by the agent usually also benefit the principal. In practice, contracts typically scale agent compensation to the benefit received by the principal. Thus in insurance markets, insurers are not willing to offer full coverage contracts; instead, they offer partial insurance coverage which exposes policyholders to some of the risk that they wish to transfer. In turn, partial coverage reinforces incentives for policyholders to prevent/mitigate loss.

Problem set 5 questions from a Finance 4335 student…

A student asked me the following questions about Problem Set 5 (due at the beginning of class on Tuesday, October 3):

Question 1: “I am having trouble with Problem Set 5. What exactly does Part A mean when it asks for premium loading? I cannot seem to recall in my notes what exactly that is and how it applies to this problem.”

My Answer to Question 1: In insurance, the premium loading corresponds to the “markup” from the actuarially fair value. Part A asks for the premium loading in dollar and percentage terms, so you need to figure out what the actuarially fair value is for the policy and compare that to the quoted price.

Question 2: “And for Part B, am I right to assume that the “optimal” level of insurance coverage is being calculated with the $240 insurance premium that is given in the problem?”

My Answer to Question 2: Yes.

Arrow-Pratt method vis–à–vis the “exact” method for calculating risk premiums

I received an email from a Finance 4335 student earlier today asking for some clarification regarding the Arrow-Pratt method vis–à–vis (what I like to refer to as) the “exact” method for calculating risk premiums. As I showed in class, the Arrow-Pratt method is an alternative method for calculating the risk premium. Thus, either approach (the “exact” method or the Arrow-Pratt method) is perfectly acceptable for calculating risk premiums.

The value added of Arrow-Pratt is (as I point out in my two page Finance 4335 synopsis) that it analytically demonstrates how risk premiums depend upon two factors: 1) the magnitude of the risk itself (as indicated by variance), and 2) the degree to which the decision-maker is risk averse. For example, the Arrow-Pratt coefficient for the logarithmic investor (for whom U(W) = ln W) is twice as large as the Arrow-Pratt coefficient for the square root investor (for whom U(W) = W.5); 1/W for the logarithmic investor compared with .5/W for the square root investor. Thus, the logarithmic investor behaves in a more risk averse than the square root investor; other things equal, the logarithmic investor will prefer to allocate less of her wealth to risky assets and buy more insurance than the the square root investor. Another important insight yielded by Arrow-Pratt (at least for the types of utility functions we have considered in Finance 4335; i.e., power and logarithmic utilities) is the notion of decreasing absolute risk aversion. Other things equal, investors become less (more) risk averse as wealth increases (decreases).

Finance 4335 Midterm Exam 1 information

I have just finished writing Midterm Exam 1 for Finance 4335. This test consists of 2 problems worth 32 points each and 1 problem worth 36 points. Thus, the maximum number of points possible for this exam is 100. This exam will be given during our scheduled class time on Tuesday, September 26, from 11 a.m. – 12:15 p.m. in Foster 402.

I also just uploaded the formula sheet that will be included as part of the exam booklet on Tuesday; I highly recommend that you download and review the “Formula Sheet for Midterm exam #1” (technical note: if you have previously clicked on this link, clear out your browser cache so as to ensure that you are able to access the current version of this document).

Not surprisingly, the exam is all about the “Decision Making under Risk and Uncertainty” topic. In my opinion, the best way to prepare for the exam is to review my updated two page synopsis of what we have covered in Finance 4335 since the start of this semester. Also, review the third and fourth problem sets, as well as the Sample Midterm 1 Exam Booklet and solutions.

Anyway, best of luck on the exam on Tuesday. If you have any questions or concerns, don’t hesitate to call me at my Baylor office number, which is 254-710-6207.

Finance 4335