# Rothschild-Stiglitz model and related numerical example from last Thursday’s class meeting

This past Thursday, we discussed the logic behind the Rothschild-Stiglitz “separating equilibrium” model (see pp. 22-24 of the Moral Hazard and Adverse Selection lecture note) and provided a numerical illustration of its inner workings .

From our initial study of adverse selection in insurance markets (see pp. 17-21 of the Moral Hazard and Adverse Selection lecture note), we find that low-risk insureds cross-subsidize high-risk insureds when pooled premiums (based upon the average of the expected costs for both risk-types) are charged. In the “dynamic” version where there are many different risk-types (see Adverse Selection Dynamics Class Problem), this results in the so-called “insurance death spiral”. The death spiral begins with the exit of the lowest risk members of the pool, because pooling makes insurance too expensive for them and they are better off self-insuring. Their exit causes premiums for remaining pool members to increase, which motivates even more lower risk members to also exit, further shrinking the risk pool and making it even more expensive. Unchecked, this dynamic ultimately results in the failure of the insurance market.

The purpose of the static (2 risk-types) Rothschild-Stiglitz model is to show how insurance contract design can mitigate the adverse selection problem described in the previous paragraph. By offering full coverage contracts (based on the high-risk loss probability) and partial coverage (based on the low-risk probability), the high-risk and low-risk types credibly confirm whether they are high-risk or low-risk by their contract choices.  Since the full coverage contract provides high-risk types with greater expected utility than the partial coverage contract, and the partial coverage contract provides low-risk types with greater expected utility than the full coverage contract,  voilà – the adverse selection problem goes away because the insurer now knows who’s who!

The problem that we worked on toward the end of Thursday’s class meeting provides a  numerical illustration of the Rothschild-Stiglitz model. Here is the problem description (from pg. 24 of the Moral Hazard and Adverse Selection lecture note):

Note that Policy A represents actuarially fair full coverage based on the high-risk probability, whereas Policy C represents actuarially fair partial coverage based on the high-risk probability.  Without any further calculation, the Bernoulli principle implies that high-risk types will prefer Policy A over Policy C, and that Policy A and Policy C are preferred to self-insurance.  Furthermore, Policy B will never be offered, since high-risk types prefer Policy B over A and the insurer would lose $19.50 ($65-$45.50) per high-risk type if it offered Policy B. Since we are interested in determining the policy pair which maximizes (expected value of) profit, it all boils down to whether the insurer offers Policy C or D. We already know that the high-risk types prefer A over C. We need to determine whether the low-risk types prefer C or D, and whether there’s any possibility that high-risk types might defect from A to D if D were offered (note that the choice of D over A by high-risk types loses money for the insurer, since the expected cost of 30% coverage of high-risk types costs$19.50, and policy D’s premium is only $13.65). Furthermore, while we know that high-risk types prefer A to C, we don’t yet know the preference ordering by low-risk types of self-insurance, Policy C, and Policy D. Under Policy C, the expected profit per low-risk type is$39 – .6(35) = $18, but it is only$13.65-$10.50 =$3.15 under Policy D.

The following spreadsheet provides with the answers that we need (clicking on the picture below brings up the spreadsheet from which this picture is obtained; see the worksheet labeled as “RS (Class Problem)”):

The various calculations in this worksheet confirm our intuition – the profit maximizing pair is A and C.  If A and C is offered, then the insurer earns expected profit of $0 on A per high-risk type (because A is purchased exclusively by high-risk types) and expected profit of$18 on C (because C is purchased exclusively by low-risk types).  If Policy D is offered instead of Policy C, then high-risk types still prefer A (and yield expected profit per high-risk type of $0), whereas low-risk types prefer D (and yield expected profit per low-risk type of$3.15)

# Moral Hazard and Adverse Selection Class Problems and Solutions

Here are the moral hazard and adverse selection class problems and solutions. These class problems were featured during the 10/15/2019 class meeting of Finance 4335.

Restrictions on Use: Section III.C.16 of Baylor’s Honor Code Policy and Procedures stipulates that using, uploading, or downloading any online resource derived from material pertaining to a Baylor course without the written permission of the professor constitutes an act of academic dishonesty.  Since Professor Garven gives no such permission for Finance 4335 course content, this means that the use and redistribution of Finance 4335-related documents involving any source other than Professor Garven are expressly forbidden.  For more information on the use restrictions of Finance 4335 course content, see http://bit.ly/4335honorcode.

# Change in Problem Set 5 due date

I have decided to change the Problem Set 5  due date from today to Thursday. This will give everyone more time to better understand (as well as demonstrate comprehension of) the logical underpinnings of this problem set.

# September 19th assignment (problem set 3)

Just wanted to give everyone a heads up that I have posted Problem Set 3 on the course website. It is due at the beginning of class on Thursday, September 19.

# Risk Pooling Class Problems and Solutions

Restrictions on Use: Section III.C.16 of Baylor’s Honor Code Policy and Procedures stipulates that using, uploading, or downloading any online resource derived from material pertaining to a Baylor course without the written permission of the professor constitutes an act of academic dishonesty.  Since Professor Garven gives no such permission for Finance 4335 course content, this means that the use and redistribution of Finance 4335-related documents involving any source other than Professor Garven are expressly forbidden.  For more information on the use restrictions of Finance 4335 course content, see http://bit.ly/4335honorcode.

# Risk Pooling Class Problem and plans for tomorrow’s meeting of Finance 4335

Here is the solution for the first question in the class problem that we worked on yesterday. Please check your work against this solution, and also solve problems 2 and 3.

In problem 2, we change the composition of the risk pool by adding an additional 5 risks.  While this change does not affect the average loss per policy (see equation 1a on page 4 of the lecture outline),  it lowers the average risk per policy (equation 2a on the same page).  The lowering of average risks causes the tail probability (i.e., the probability of an average loss exceeding \$1,500) to decline, which you will numerically confirm when you complete your work on problem 2.

In problem 3, we use the same number of risks (n = 10) as in problem 2, but introduce positive correlation ($\rho = .1$). This causes the average risk per policy to increase vis-a-vis the result shown in problem 2, which causes the tail probability to increase, which you will numerically confirm when  you complete your work on problem 3.

We’ll begin class tomorrow by finishing off the solutions to this risk pooling class problem and covering the following set of topics which appear on pp. 6-21 of the lecture outline @ http://fin4335.garven.com/fall2019/lecture5.pdf: (1) Making decisions under certainty, 2) Making decisions under uncertainty, 3) The expected value model, 4) The St. Petersburg Paradox, and 5) The expected utility model.

# Extra Credit Opportunity for Finance 4335 (repost)

(This is a repost that corrects various typos in the original; follow the instructions provided here instead!):

I have decided to offer the following extra credit opportunity for Finance 4335. You can earn extra credit by building an Excel spreadsheet which replicates the Standard Normal Distribution Function table that appears on the course website for z values ranging from 0 to 3 . This is actually not a particularly difficult exercise, since all it requires is building a 32 row by 11 column spreadsheet in which cells B2 through K32 reference sums of column A numbers (which begin at 0.0 in cell A2 and increase by .1 increments, ending at 3.0 in cell A32) with row 1 numbers (which begin at 0.00 in cell B2 and increase in .01 increments, ending at 0.09 in cell K1), using the “=NORMSDIST(z)” command.

This extra credit assignment must be emailed as a file attachment to fin4335@gmail.com by no later than Friday, September 13 at 5 pm. I will use your grade on this extra credit assignment to replace your lowest quiz grade in Finance 4335 (assuming that your grade on the extra credit is higher than your lowest grade).

Feel free to email me (James_Garven@baylor.edu) or call/text me at 254-307-1317 if you have any questions concerning this assignment.

# Problem Set 2 due tomorrow at the beginning of class

As y’all are well aware, Problem Set 2 is due tomorrow at the beginning of class.  Last Thursday, I posted some (what I hope you will find to be) helpful hints for Problem Set 2.

# Problem Set 2 helpful hints

Problem Set 2 is available from the course website at http://fin4335.garven.com/fall2019/ps2.pdf; its due date is Thursday, September 12.

Problem Set 2 consists of two problems. The first problem requires calculating expected value, standard deviation, and correlation, and using this information as inputs into determining expected returns and standard deviations for 2-asset portfolios; see pp. 15-18 of the http://fin4335.garven.com/fall2019/lecture3.pdf lecture note for coverage of this topic. The second problem involves using the standard normal probability distribution to calculate the probabilities of earning various levels of return by investing in risky securities and portfolios; see pp. 17-22 of the http://fin4335.garven.com/fall2019/lecture4.pdf lecture note for coverage of that topic.