An important clarification of the logical principles behind the stochastic dominance model

On problem set 4, part D, most of you had no apparent difficulty in correctly establishing that the sum of the differences between the cumulative distribution (CDF) for risk 2 and the CDF for risk 1 is positive.   However, many of you drew the wrong conclusion, claiming that since the sum of differences between F({W_{2,s}}) and F({W_{1,s}}) came out to a positive number, it followed that risk 2 second order stochastically dominates risk 1.  Actually, this result implies the opposite; i.e., that risk 1 second order stochastically dominates 2. The purpose of this blog posting is to clarify everyone’s understanding of the logic behind the stochastic dominance model.

The one-page exam formula sheet includes section 4 which explains that risk i dominates risk j, in both the first and second cases, when 1) the cumulative distribution function (CDF) of the ith risk is either less than or equal to the CDF of the jth risk for all states (first order dominance), or 2) the sum of the differences between the jth risk CDF and the ith risk CDF for all states is positive (second order dominance):

While the math behind first and second order stochastic dominance is summarized in my optional reading entitled “Technical Note on Stochastic Dominance and Expected Utility”, the intuition for first and second order stochastic dominance can be seen in the figures featured on pages 9 and 12 of my Decision-Making under Risk and Uncertainty, part 4 lecture note.

In the above figure from page 9 of my Decision-Making under Risk and Uncertainty, part 4 lecture note, the G risk has 50% of a $0 payoff, and 25% each of a $10 payoff and a $100 payoff.  The F risk involves removing 25 percentage points off the $0 payoff and adding 25 percentage points extra to the $100 payoff, and both F and G have a 25% probability of $10 payoffs.  Graphically, this ensures that F first order stochastically dominates G; i.e., G(Ws) is greater than or equal to F(Ws) for all s, which also implies that EF[U(W)] > EG[U(W)]. Intuitively, the picture which gets rendered by this analysis shows that most of the probability mass of the stochastically dominant risk (in this case, F) lies below the probability mass of the stochastically dominated risk (in this case, G). Furthermore, since risk F first order stochastically dominates risk G, risk F also second order stochastically dominates risk G because G(Ws) – F(Ws) > 0 for $0 and $10 payoffs, and G(Ws) – F(Ws) = 0 for the $100 payoff.

Next consider the figure from page 12 of my Decision-Making under Risk and Uncertainty, part 4 lecture note:

Here, G(Ws) – F(Ws) > 0 for payoffs ranging from 1-5, G(Ws) – F(Ws) < 0 for payoffs ranging from 5-8, and G(Ws) – F(Ws) = 0 payoffs ranging from 8-12.  Thus, there is no first order dominance.  However, since the positive difference between G(Ws) – F(Ws) for payoffs ranging from 1-5 exceeds the negative difference between G(Ws) – F(Ws) for payoffs ranging from 5-8, the sum of G(Ws) – F(Ws) over the entire range of payoffs comes out positive.  Thus, risk F second order stochastically dominates risk G, which also implies that EF[U(W)] > EG[U(W)].

 

Midterm Exam 1 Formula Sheet and Helpful Hints

The formula sheet for Midterm Exam 1 is now available for downloading at http://fin4335.garven.com/fall2023/formulas_part1.pdf.

Midterm Exam 1 comprises four problems, with a requirement to solve three out of the four. If all four problems are attempted, only the three highest-scoring problems will contribute to your Midterm Exam 1 grade. Each problem is valued at 32 points, and you will receive an additional 4 points for including your name on the exam booklet. Therefore, the maximum attainable score for Midterm Exam 1 is 100 points.

On the exam, it is important not only to neatly and concisely present your analysis, but also to provide thorough explanations that showcase your understanding of the concepts being examined. In other words, be sure to articulate your findings using clear and concise language.

In my estimation, the information I have shared in this blog post, coupled with the guidance provided in the study guide titled “Finance 4335 Midterm 1 Synopsis,” should prove very helpful as you prepare for the upcoming midterm exam.

Good luck!

Some comments and clarifications about solutions for problem set 3

Whenever you submit a problem set in Finance 4335, I highly recommend that you compare your work with the solutions that I post on the course blog.  In the case of problem set 3, students seemed to have the most difficulty with problem 1, part B, and also problem 2, parts B and C.

In a nutshell, in 1B, Ned is willing to pay the most for insuring risk, since after all, he is the only one of the three who is risk averse; Dusty is risk neutral, whereas Lucky is risk loving.  Indeed, as I show in the solution for 1B, Ned is willing to pay up to $12.64 more than the actuarially fair value of .25(100), which implies an insurance price of $37.64. Since Dusty is risk neutral, he is indifferent between having certain wealth of $115 and uncertain wealth with an expected value of $115; the most Dusty is willing to pay for insurance is its actuarially fair value of .25(100)  = $25.  Since Lucky is risk loving, he will only insure risk if it is available for a $4.46 discount from its actuarially fair value of $25, which comes to $20.54.

Also, in problem 2A, since you’re risk averse, the Bernoulli principle tells us that risk averters are happy to pay actuarially fair prices (in this case, $180).  Also, in problem 2B, a price of  $250 is acceptable because U($750) = 6.71 > expected utility of no insurance = 6.45.  Finally, in problem 2C, the maximum price for full coverage is equal to the sum of the actuarially fair price of $180 plus the risk premium (\lambda = 189.04), or $369.04.  Note that this also corresponds to the difference between the initial wealth of $1,000 and the certainty equivalent of wealth of $630.96 under the self-insurance option.

Finance 4335 policy for submitting completed problem sets via Canvas

At the risk of redundancy (e.g., see the screenshot below for Problem Set 4 on Section 2’s Canvas site and section “8. 2. Late Work” on page 3 of the course syllabus), let me remind everyone that Finance 4335 problem sets must be submitted in PDF format via Canvas; paper copies or file formats other than PDF are not accepted.  Furthermore, given how simple it is to merge multiple pages into a single PDF document (see the instructions provided here), I also expect students to submit their completed problem sets as single PDF documents.

Let me know whether any of you have any questions about the Finance 4335 Problem Set submission policy.

 

Finance 4335 student query about availability of Sample Midterm 1 Exam Booklet and Midterm Exam 1 Formula Sheet

From: Finance 4335 Student <Finance_4335_Student@baylor.edu>
Date: Wednesday, September 13, 2023 at 5:10 PM
To: Garven, James <James_Garven@baylor.edu>
Subject: FIN 4335 Midterm Exam 1

Good Morning, Dr. Garven,

I have a quick question regarding the Midterm Exam 1 coming up in the next few weeks. I was wondering when the Sample Midterm 1 Exam Booklet as well as the Midterm Exam 1 Formula Sheet would be available for us.

Also, would we be able to use this formula sheet on the exam?

Best,

Finance 4335 Student
_________________________________________________________
From: Garven, James <James_Garven@baylor.edu>
Sent: Wednesday, September 13, 2023 5:24 PM
To: Finance 4335 Student <Finance_4335_Student@baylor.edu>
Subject: Re: FIN 4335 Midterm Exam 1

Dear Finance 4335 Student,

My answer is yes to both of your questions. I intend to upload the sample Midterm Exam #1 to the course website on Tuesday, September 19. This is to ensure that students have ample time to review it before the Midterm Exam #1 Review Session, which is scheduled for our class meeting on Thursday, September 21.

Regarding the “real” Midterm Exam #1 Formula Sheet, I intend to upload this document to the course website in time for the Midterm Exam #1 Review Session on 9/21, and I will include this same document as an attachment to the Midterm Exam #1 booklet on the exam day itself, which is Tuesday, September 26.

Dr. Garven

Friendly reminder about assigned readings, Problem Set 3, and Quiz 4 due Thursday, September 14

Here’s a friendly reminder regarding the assigned readings, Problem Set 3, and Quiz 4 which are due prior to the start of class on Thursday, September 14th.

Here are the links for the two assigned readings:

1. Expected Utility, Mean-Variance, and Stochastic Dominance, by James R. Garven
2. Modeling Risk Preferences Using Taylor Series Expansions of Utility Functions, by James R. Garven

Here is the link for Problem Set 3:

Problem Set 3 (Expected Utility)