Category Archives: Risk

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.

Confirmation bias in the form of “information avoidance”

This article from the Wall Street Journal provides an interesting followup to yesterday’s behavioral finance discussion. “Information avoidance” represents a particularly strong (and potentially deadly) form of confirmation bias!

Getting past information avoidance to deal with health issues, financial difficulties and other worries.
wsj.com|By Elizabeth Bernstein

On the Determinants of Risk Aversion

In January 2014, The Economist published a particularly interesting article about the determinants of risk aversion, entitled “Risk off: Why some people are more cautious with their finances than others”. Here are some key takeaways from this article:

1. Economists have long known that people are risk-averse; yet the willingness to run risks varies enormously among individuals and over time.

2. Genetics explains a third of the difference in risk-taking; e.g., a Swedish study of twins finds that identical twins had “… a closer propensity to invest in shares” than fraternal ones.

3. Upbringing, environment and experience also matter; e.g., . “…the educated and the rich are more daring financially. So are men, but apparently not for genetic reasons”.

4. People’s financial history has a strong impact on their taste for risk; e.g., “… people who experienced high (low) returns on the stock market earlier in life were, years later, likelier to report a higher (lower) tolerance for risk, to own (not own) shares and to invest a bigger (smaller) slice of their assets in shares.”

5. “Exposure to economic turmoil appears to dampen people’s appetite for risk irrespective of their personal financial losses.” Furthermore, a low tolerance for risk is linked to past emotional trauma.

More on the St. Petersburg Paradox…

During last Thursday’s class meeting, we discussed (among other things) the famous St. Petersburg Paradox. The source for this is Daniel Bernoulli’s famous article entitled “Exposition of a new theory on the measurement of risk”. As was the standard academic practice in academia at the time, Bernoulli’s article was originally published in Latin in 1738. It was subsequently translated into English in 1954 and published a second time that same year in Econometrica (Volume 22, No. 1): pp. 22–36. Considering that this article was published 277 years ago in an obscure (presumably peer reviewed) academic journal, it is fairly succinct and surprisingly easy to read. The direct link to the PDF of this article is here: http://www.jstor.org/stable/1909829 (P.S.: the article is gated but you can access it from anywhere on campus when connected to Airbear).

Also, the Wikipedia article about Bernoulli’s article is worth reading. It provides the mathematics for determining the price at which the apostle Paul would have been indifferent about taking the apostle Peter up on this bet. Specifically, if Paul were a millionaire, he should be willing to pay up to $20.88, On the other hand, if Paul were a thousandaire (total wealth = $1000), he should be willing to pay up to $10.95. Finally, if Paul was quite poor (e.g., total wealth = $2), then he might consider borrowing $1.35 and paying up to $3.35 to place this bet…

Interview With Meir Statman (extra credit opportunity)

Meir Statman has very important things to say about decision-making under risk and uncertainty; I introduced Professor Statman to you in my previous blog posting entitled “Your Tolerance for Investment Risk Is Probably Not What You Think.”  Here is an extra credit opportunity for Finance 4335 based upon a 1 hour, 25 minute podcast (recorded in July 2017) hosted by Barry Ritholtz’s Masters of Business podcast (link provided below) entitled “Interview with Meir Statman.”

You may earn extra credit by listening to and reporting on Mr. Ritholtz’s interview with Meir Statman about behavioral finance.   In order to receive extra credit for this assignment, you must submit (via email sent to risk@garven.com) a 1-2 page executive summary of what you learned from this podcast; it is due by no later than 5 p.m. on Monday, September 18.  This extra credit assignment will replace your lowest quiz grade in Finance 4335 (assuming the extra credit grade is higher).

Bloomberg View columnist Barry Ritholtz interviews Meir Statman, the Glenn Klimek Professor of Finance at Santa Clara University. His research focuses on behavioral finance. He attempts to understand how investors and managers make financial decisions and how these decisions are reflected in financial markets. His most recent book is “Finance for Normal People: How Investors and Markets Behave,” published by Oxford University Press. This commentary aired on Bloomberg Radio.

Your Tolerance for Investment Risk Is Probably Not What You Think

This WSJ article is authored by Professor Meir Statman,  the Glenn Klimek Professor of Finance at Santa Clara University.  Professor Statman’s research focuses on behavioral finance, which is a very important topic in decision theory that I hope to cover during tomorrow’s meeting of Finance 4335.
The questions financial advisers ask clients to get at the answer actually measure something completely different—often leading to misguided investment strategies.

Q3 & PS2

Hello Class,

Overall, very well on the Problem Set with almost 70% of the class achieving a perfect score. While almost everyone missed at least one question on the quiz, the majority of the class scored 80% or higher. It appears that some of you may need to brush up on your statistics, and it is always prudent to thoroughly check your work to make sure you answer the question asked. Also, always be sure to carefully read the assigned readings in order to perform optimally on the quiz, as there may be some nuances where certain concepts sound similar but are distinctly different.

Thank you for your time and considerations.

Sincerely,

Alexander Law

 

How government policy exacerbates hurricanes like Harvey

Here’s the (very timely) cover story of the latest issue of The Economist. Quoting from the article, “Underpricing (of flood insurance) encourages the building of new houses and discourages existing owners from renovating or moving out. According to the Federal Emergency Management Agency, houses that repeatedly flood account for 1% of NFIP’s properties but 25-30% of its claims. Five states, Texas among them, have more than 10,000 such households and, nationwide, their number has been going up by around 5,000 each year. Insurance is meant to provide a signal about risk; in this case, it stifles it.”

As if global warming were not enough of a threat, poor planning and unwise subsidies make floods worse.

Harvey’s Test: Businesses Struggle With Flawed Insurance as Floods Multiply

This WSJ article provides a fairly comprehensive look at the financial implications for #Harvey for small business. What’s particularly disconcerting is that NFIP is already for all intents and purposes technically insolvent (current debt to the US Treasury stands at around $25 billion) and Congress is supposed to reauthorize funding for the program’s next five years by September 30. On the lighter side of things, it’s fun to see a couple of academic colleagues’ names in print in this article; specifically, Erwann Michel-Kerjan of the Organization for Economic Cooperation and Development Board on Financial Management of Catastrophes and Ben Collier, who is a faculty member at Temple University’s Fox School of Business.

Hurricane will strain a National Flood Insurance Program out of step with needs of small businesses in era of extreme weather.