Empirical evidence of risk aversion in the real world: Small stocks, Large stocks, Government bonds, and Treasury bills, 1926-2017

Although there may be various social contexts in which people stray from risk averse behavior (e.g., the risk loving behavior which is on display whenever people place bets on gambles with unfair odds), in other (more economically consequential settings), it does appear that risk averse behavior is more the rule rather than the exception. Indeed, risk aversion is what motivates people to buy insurance and diversify risk in their asset holdings.

The financial markets provide us with a superb example of risk averse behavior writ large. Historically, here are the long run (1926-2017) compound annual returns on stocks, bonds, and bills that are traded in U.S. financial markets (source: page 9 of http://bit.ly/sbbi2018):

Risk for these various asset classes is lowest for Treasury bills, a bit higher for Government bonds, a bit higher yet for Large stocks, and highest for Small stocks. If you are risk averse, then if one asset has higher risk than another, you are not willing to invest in the riskier asset unless you can reasonably expect that on average, you’ll be compensated for bearing the extra risk in the form of a higher expected return, and it turns out that this is exactly what happens in the real world. If investors were to act in a risk neutral fashion, then the average returns wouldn’t be all that different from each other. Finally if investors were to act in a risk loving fashion, they’d pay more for risky assets than for safe; this would cause risky assets to be bid up in value relative to safe assets, which in turn would imply lower average returns for risky than for safe assets.

Midterm Exam 1 hints and formula sheet

I just posted the formula sheet for the exam at http://fin4335.garven.com/fall2018/formulas_part1.pdf. It is also linked as the first item on the formula sheets page on the course website.  This is identical to the formula sheet which will be attached to the exam booklet.

The exam consists of a total of 4 problems. The first problem 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 work all three of Problems #2-#4, in which case I will count the two problems with the highest scores toward your grade on this exam. Each of the graded problems is worth 32 points, so as a “bonus” I’ll add 4 points for including your name on the exam. Thus, the total number of points possible is 100.

Regarding content, the exam is all about stuff that we covered since the beginning of the semester; specifically, risk preferences, expected utility, certainty-equivalent of wealth, risk premiums, and stochastic dominance.

Risk Management: Premium loading Q&A

On 9/24/18, 1:17 PM, "Finance 4335 Student" <Finance4335student@baylor.edu> wrote:

Dr. Garven,

What does the term “premium loading” mean? I’m familiar with many of the concepts in the problem set, but was unable to find this term and how to calculate it in class notes.

– Finance 4335 Student

Here’s my response to "Finance 4335 Student" on the topic of premium loadings:

It corresponds to the markup in the premium relative to the expected value of the claim (also commonly referred to as the expected value of the "indemnity"). In percentage terms, % loading = (premiumE(indemnity))/E(indemnity), and in dollar terms, $ loading = premiumE(indemnity), where E(indemnity) corresponds to the expected value of the indemnity. As we showed in class last Thursday, the expected value of the indemnity corresponds to the expected value of the loss only when full coverage is offered. Under a deductible policy, the indemnity in state s, indemnity(s) = Min(L(s) – Min(L(s), d)), where L(s) corresponds to the state-contingent loss and d corresponds to the dollar value of the deductible. Under a coinsurance policy, the indemnity in state s, indemnity(s) = L(s) – (1-a)L(s), where a corresponds to the coinsurance rate. Under an upper limit policy, indemnity(s) = Min(L(s), u), where u corresponds to the dollar value of the deductible. If you look at the “Insurance Payment Calculations” worksheet tab in the Coinsurance, Deductibles, and Upper Limits Spreadsheet, you’ll see that these are the equations used in order to determine the indemnity schedules under the various contracts listed there.

Guidelines for completing parts B and D on Problem Set 5

For parts B and D on Problem Set 5, you may solve these problems via either calculus or a spreadsheet model.

If you decide to implement a spreadsheet model, then you must email your spreadsheet model to “risk@garven.com” prior to the start of class on Tuesday. In the problem set that you turn in at the beginning of class on Thursday, please reference your spreadsheet when you explain your answers for parts B and D problem. However, if you rely upon the calculus for maximizing expected utility, then no spreadsheet is necessary, although you might consider validating the result that you obtain via calculus with a spreadsheet model anyway. Or, you could validate your spreadsheet model with the calculus.

In order to solve this problem via spreadsheet, you’ll need to use the so-called Solver Add-in. The instructions for loading the Solver add-in into Excel are provided at the following webpage:

https://support.office.com/en-us/article/load-the-solver-add-in-in-excel-612926fc-d53b-46b4-872c-e24772f078ca

Assignments for class: Tuesday, September 25

Not surprisingly, the first midterm exam in Finance 4335 (scheduled for Thursday, September 27 in class) will be all about the "Decision Making under Risk and Uncertainty" and “Insurance Economics” topics.

In preparation for next Tuesday’s in-class review session, your assignments are to 1) review the two-page document entitled “Finance 4335 course synopsis for Midterm Exam 1 (also available from the Readings page on the course website as an assigned “optional” reading for Tuesday, September 25), and 2) review Problem Set 3, Problem Set 4, Problem Set 5, and the Sample Midterm 1 Exam Booklet (this is the exam that I gave in Finance 4335 during the Spring 2018 semester). Solutions for the third and fourth problem sets are available now (see http://risk.garven.com/category/problem-set-solutions/), and I will make the solutions for the fifth problem set and the sample Midterm 1 exam available after class on Tuesday.

In closing, I expect that y’all will prepare adequately so that we can have a productive review session next Tuesday. Be sure to come equipped with questions, and I will do my best to provide answers.

Recap of Analytic and Numerical Proofs of the Bernoulli Principle and the Mossin Theorem

Yesterday, we discussed three particularly important insurance economics concepts: 1) the Bernoulli Principle,  2) the Mossin Theorem, and 3) the Arrow Theorem.  The Bernoulli Principle implies that if an actuarially fair full coverage insurance policy is offered, then an arbitrarily risk averse individual will purchase it. The Mossin Theorem implies that if insurance is actuarially unfair, then an arbitrarily risk averse individual will prefer partial insurance coverage over full insurance coverage.  Finally, the Arrow Theorem implies an arbitrarily risk averse individual will select deductible insurance from the menu of partial insurance choices that were considered during class yesterday.

In what follows, I present a recap of the analytic and numerical proofs of the Bernoulli Principle and Mossin Theorem which took up most of our attention in class yesterday.

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).

Mark your calendars – yet another Finance 4335 extra credit opportunity!

I have decided to yet another extra credit opportunity for Finance 4335.   You can earn extra credit by attending and reporting on Evan Baehr‘s upcoming Free Enterprise Forum presentation which is scheduled for Tuesday, September 25 from 4-5:15 in Foster 240:If you decide to take advantage of this opportunity, I will use the grade you earn to replace your lowest quiz grade in Finance 4335 (assuming that your grade on the extra credit is higher than your lowest quiz grade). The report should be in the form of a 1-2 page executive summary in which you provide a critical analysis of Mr. Baehr’s lecture. In order to receive credit, the report must be submitted via email to risk@garven.com in either Word or PDF format by no later than Friday, September 28 at 5 p.m.

P.S.: This is the second extra credit opportunity that I have offered this semester for Finance 4335.  For information on Dr. Michael Munger’s  October 11 lecture entitled “Tomorrow 3.0: Transaction Costs and the Sharing Economy”, click here).  Also, see http://risk.garven.com/category/extracredit/ for an up-to-date list of extra credit opportunities in Finance 4335.

Finance 4335