Author: drgarven

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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. This blog posting aims 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)].

 

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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: Monday, February 12, 2024 at 3:10 PM
To: Garven, James <James_Garven@baylor.edu>
Subject: FIN 4335 Midterm Exam 1

Good Morning, Dr. Garven,

I have a quick question about the Midterm Exam 1 next week. I would like to know when the Sample Midterm 1 Exam Booklet and the Midterm Exam 1 Formula Sheet will be available.

Also, can we use the Midterm Exam 1 Formula Sheet formula sheet on the exam?

Best,

Finance 4335 Student
_________________________________________________________
From: Garven, James <James_Garven@baylor.edu>
Sent: Monday, February 12, 2024 at 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 to both of your questions is yes. I have already uploaded the Sample Midterm Exam #1 to the course website so that students have ample time to review it before the Midterm Exam #1 Review Session, which is scheduled for our class meeting on Thursday, February 15.

Regarding the Midterm Exam #1 Formula Sheet, I have also uploaded this document to the course website (@ http://fin4335.garven.com/spring2024/formulas_part1.pdf), and I will include this same document as an attachment to the Midterm Exam #1 booklet on the exam day itself, which is Tuesday, February 20.

Dr. Garven

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On the Determinants of Risk Aversion

Several years ago, The Economist published a particularly interesting article about various behavioral 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 (somewhat dated, but still quite timely) 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, low tolerance for risk is linked to past emotional trauma.
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Some important intuitions from yesterday Finance 4335 class meeting…

  1. The most important concept covered in class yesterday is that people vary in terms of their preferences for bearing risk. Although we focused most of our attention on modeling risk-averse behavior, we also briefly considered examples of risk neutrality (where you only care about expected wealth and are indifferent about the riskiness of wealth) and risk loving (where you prefer to bear risk and are willing to pay money for the opportunity to do so).
  2. Related to point 1: irrespective of whether you are risk averse, risk neutral, or risk loving, the foundation for decision-making under conditions of risk and uncertainty is expected utility. Given a choice among various risky alternatives, one selects the choice that has the highest utility ranking.
  3. If you are risk averse, then E(W) > {W_{CE}} and the difference between E(W) and {W_{CE}} is equal to the risk premium \lambda. Some practical implications — if you are risk averse, then you are okay with buying “expensive” insurance at a price that exceeds the expected value of payment provided by the insurer, since (other things equal) you’d prefer to transfer risk to someone else if it’s not too expensive to do so. On the other hand, you are not willing to pay more than the certainty equivalent for a bet on a sporting event or a game of chance.
  4. If you are risk neutral, then E(W) = {W_{CE}} and \lambda = 0; risk is inconsequential and all you care about is maximizing the expected value of wealth.
  5. If you are risk loving, then E(W) < {W_{CE}} and \lambda < 0; you are quite willing to pay for the opportunity to (on average) lose money.