I am forwarding this to y’all just in case someone might be interested in this opportunity…
From: Khyltash, Kamran (BANK) [mailto:Kamran.Khyltash@capitalone.com]
Sent: Wednesday, September 28, 2016 5:38 PM
Subject: Capital One Information Session
Attached is a flier for a credit analyst role here at Capital One. We will be holding an information session on campus on October 4, from 5:30-7:30PM and would like to see a strong turn out. Please pass along to any students that are interested and feel free to put them in touch with myself.
Senior Credit Analyst | Commercial Real Estate
600 North Pearl Street, Suite 2500
Dallas, TX 75201
(214) 855-1614 – Office
(214) 855-1600 – Fax
I have finished writing the first midterm exam for Finance 4335.
The exam consists of a total of 4 problems. Problem #1 is required, and you are also required to work 2 out of the 3 remaining problems on the exam (i.e., select two problems from Problems #2-#4). At your option, you may select all three problems which comprise Problems #2-#4, in which case I will count the two problems with the highest scores toward your grade on this exam.
Not surprisingly, the exam is all about expected utility. I have posted a copy of the formula sheet that will appear as part of the exam booklet at http://fin4335.garven.com/fall2016/formulas_part1.pdf. I did not bother including the equations for either the risk premium or the certainty-equivalent of wealth – I assume by now that you understand that the risk premium is equal to the difference between expected wealth and the certainty-equivalent of wealth, and that you solve for the certainty-equivalent of wealth by calculating expected utility (E(U(W))) and equating E(U(W)) with the utility of the certainty equivalent of wealth (U(WCE)). One backs into WCE algebraically using the definition for the utility function; e.g., if utility is the square root of wealth, then WCE = E(U(W))2.
… are available at http://fin4335.garven.com/fall2016/ps5solutions.pdf…
1. Obviously the key concept centers around the notion that people vary in terms of their preferences for bearing risk. Although we focused most of our attention in upon modeling risk averse behavior, we also considered examples of risk neutrality (where you only care about expected wealth and are indifferent about riskiness of wealth) and risk loving (where you actually want to bear risk and are willing to pay money for the opportunity to do so).
2. Related to point 1: irrespective of whether one is risk averse, risk neutral, or risk loving, the foundation for decision-making under conditions of risk and uncertainty is expected utility. Given a choice amongst various risky alternatives, one selects the choice that has the highest utility ranking.
- If you are risk averse, then E(W) > Wce and the difference between E(W) and Wce is equal to the risk premium . Some practical applications – if you’re 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 expected value to place a bet on a sporting event or a game of chance such as rolling dice or tossing a coin.
- If you are risk neutral, then E(W) = Wce and .
- If you are risk loving, then E(W) < Wce and ; i.e., you pay for the opportunity to (on average) lose money.
3. We talked about “special cases” of expected utility – specifically, mean-variance and stochastic dominance analysis. If we impose various restrictive assumptions upon expected utility, these methods emerge.
- Of these two special cases, the mean-variance framework is quite a bit more restrictive than stochastic dominance. I numerically illustrated various circumstances under which the mean-variance framework is not even an appropriate method; particularly in cases where variance may not be a particularly good risk measure. We looked examples where there are so-called “fat tails” (which is a risk characteristic that lowers expected utility) and where there is skewness (positive skewness is “good”; negative skewness” is bad in an expected utility sense.
- Stochastic dominance appears to be more “robust” than the mean-variance framework because it can accommodate broader risk concepts such as skewness and fat tails.
- So long as various restrictive assumptions apply, we can be confident that if risk X “dominates” risk Y, then the expected utility for X is greater than the expected utility for Y.
4. We also discussed a couple of different methods for calculating .
- The exact method involves calculating expected utility (E(U(W))), setting expected utility equal to the certainty-equivalent of wealth (E(U(W)) = U(Wce), and solving for Wce directly; e.g., if E(U(W)) = U(Wce) = 10 and , then Wce = 100; if E(W) = $110, then the risk premium = $10.
- The approximate method involves evaluating the Arrow-Pratt coefficient at the expected value of wealth and multiplying it by half of the variance of wealth; i.e.,
5. Insurance economics focuses upon some examples of how risk aversion influences incentives for risk transfer to a counterparty (in this case, and insurer). The three major concepts include the so-called Bernoulli principle, Mossin’s theorem, and Arrow’s Theorem.
- Bernoulli principle – if insurance is actuarially fair, risk averters full insure.
- Mossin’s theorem – if insurance is actuarially unfair, risk averters partially insure.
- Arrow’s theorem – other things equal, the optimal partial insurance contract is the deductible contract.
A student asked me the following questions about problem set 5;
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.
During last Thursday’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.
We graphically illustrated the Bernoulli Principle by using the following figure (taken from p. 4 in the Insurance Economics lecture note):
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 α, where α represents the proportion of loss covered by the insurer. By definition, α is bounded from below at 0 and from above at 1. Thus, if the consumer selects α = 0, this means that she does not purchase any insurance (i.e., she self-insures). If she selects α = 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 (Pi) multiplied by α. 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 α which maximizes expected utility is α = 1. This insight may also be numerically validated with a spreadsheet model. I have posted an Excel spreadsheet called the “Bernoulli and Mossin Spreadsheet” which enables students to solve this same problem with 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 Pi = 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 (α) 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:
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 α = 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 α 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 α gets stored). Since insurance is actuarially fair (β = 0 in cell B4), the Bernoulli Principle implies that the optimal value for α is 1.0. You can confirm this by clicking on Solver’s “Solve” button:
Not only is α = 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 α = 1.0).
Next, let’s determine what the optimal coinsurance rate (α) 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 $α40. Reset α ’s value in cell B3 back to 0, β’s value in cell B4 equal to 0.60, and invoke Solver once again:
On p. 8 in the Insurance Economics lecture note, we showed analytically that the optimal coinsurance rate is 1/7, and this value for α is indicated by clicking on the “Solve” button:
Since α = 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., α = 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).
A Finance 4335 student sent the following email to me this morning:
From: Student, FIN4335 [mailto:FIN4335_Student@baylor.edu]
Sent: Monday, September 26, 2016 12:52 PM
To: Garven, James R. <James_Garven@baylor.edu>
Subject: Problem Set 4 and 5 Solutions
Hey Dr. Garven,
I am running through the problem sets and haven’t been able to locate the solutions for 4 and 5. Are the posted on the blog and I am just missing them?
Here’s my response to FIN4335 Student:
Dear FIN4335 Student,
Thanks for the reminder about problem set 4 solutions (I forgot to post them after class last Thursday). Anyway, I just posted the link on the course blog; see http://risk.garven.com/2016/09/26/the-solutions-for-problem-set-4/.
I have not yet posted solutions for problem set 5 since that problem set is due tomorrow at the beginning of class.
For future reference, my policy is to post solutions as quickly as possible after the problem sets have been turned in. The quickest way to find solutions (or anything else on the course blog) is to use the search tool denoted by magnifying glass at the top right portion of the course blog. For example, if you click on the magnifying glass and simply type in the word “solutions”, this will generate a webpage (located at http://risk.garven.com/?s=solutions) listing all blog postings that include the word “solutions”. I also categorize all blog postings about problem set solutions using the “Assignments” category, so if you click on the drop-down menu for blog postings (located on the left side of the site), you can create a page with links to all blog postings conforming to this category.
… are located at http://fin4335.garven.com/fall2016/ps4solutions.pdf…
The yield on the 10-year U.S. Treasury note hit an all-time low in July, as other bond yields across the developed world plunge. Why are we here and where are we going?