Mea Culpa concerning today’s Arrow-Pratt numeric illustration

I reexamined my analysis of the ad hoc exact and approximation methods from the review session today, and here is a screenshot from Excel of this problem:The fair coin toss involved payoffs of $25 and $100; the expected value of wealth under this coin toss is $62.50, and the standard deviation is half of the dispersion between $25 and $100, or $37.50.  Applying the “exact” method, we get a risk premium of $6.25.  In calculating the risk premium under the Arrow-Pratt (approximation) method, I incorrectly calculated variance as 75^2 = 5,625.  Since variance is actually 37.5^2 = 1,406.25, the Arrow-Pratt equation produces a risk premium of $5.63, not $22.50.  I knew that right away that one of the model inputs had to be wrong, and it was the variance input, which was 4x the actual variance of the coin toss.

Midterm 1 formula sheet, helpful hints, pre-exam office hours

I just posted the formula sheet for the Midterm 1 exam, which will be administered during class on Thursday, February 25. Actually, the formula sheet consists of two pages; the first page is a formula sheet, and the second page is a standard normal distribution table.

The exam consists of 3 problems worth 32 points each; I add 4 points to everyone’s scores so that that the maximum number of points possible is 100. On Thursday, plan on allocating no more than 80 minutes to complete the exam, and no more than 10 minutes to upload a single PDF of your written work which clearly demonstrates your conceptual grasp and ability to clearly explain, in plain English, how you arrived at all of your answers on the exam.

You may use either a calculator or a spreadsheet for any computations that are required for the exam; please keep in mind the Official Finance 4335 Course Policy Concerning the Use of Excel for Problem Sets and Exams.

Here are some (what I think are) helpful hints:

1. Review definitions for risk aversion, risk neutrality, and risk-loving behavior; see especially page 2 of http://fin4335.garven.com/spring2021/lecture6.pdf about these topics:

  • Risk-averse utility functions are characterized by diminishing marginal utility; thus, E(U(W)) < U(E(W));
  • Risk-neutral utility functions are characterized by constant marginal utility; thus, E(U(W)) = U(E(W)); and
  • Risk-loving utility functions are characterized by increasing marginal utility; thus, E(U(W)) > U(E(W)).

2. Full coverage insurance: Under a “full coverage” policy, the insured pays a premium that transfers all risk to the insurer; if the premium charged for such coverage is actuarially fair, then the optimal choice for all arbitrarily risk-averse decision-makers is to purchase a full-coverage policy; this result is commonly referred to as “Bernoulli Principle” (see http://risk.garven.com/2021/02/22/actuarially-fair-price-of-insurance-policy/).

3. Degree of risk aversion. For logarithmic and power utilities, we saw that all such utilities feature decreasing absolute risk aversion, which means that as one’s initial wealth increases, the one’s degree of aversion to a given risk declines; resulting in a lower risk premium at higher levels of initial wealth (see pp. 9-12 of http://fin4335.garven.com/spring2021/lecture6.pdf).

On Wednesday, I plan to be available in my virtual Zoom office from 3-5 pm CT in case if any students would like to stop by for a pre-exam chat.

Good luck!

 

 

 

 

Actuarially Fair Price of Insurance Policy

A Finance 4335 student asked the following question earlier today:

Q: “How do you find the actuarially fair price (premium) for an insurance policy?”

Here’s the answer I provided, which I now share with all Finance 4335 students:

A: “The actuarially fair price (premium) corresponds to the expected value of the insurance indemnity; the indemnity is the amount of coverage offered by an insurance policy. Under “full coverage”, 100% of the loss is indemnified, and in such a case, the actuarially fair premium is equal to the expected value of the loss distribution.

For what it’s worth, the concept of “actuarially fair” insurance prices/premiums, along with implications for the demand for insurance, is explained in two previously assigned readings; e.g.,

  1. on page 4 of the Supply of Insurance reading (just prior to the section entitled “Example 2: Correlated Identically Distributed Losses), the following sentence appears, “A premium that is equal to the expected outcome is called an actuarially fair premium”;
  2. on page 30 of the Basic Economics: How Individuals Deal with Risk (Doherty, Chapter 2) reading, consider the following excerpt: “Ignoring transaction costs, an insurer charging a premium equal to expected loss would break even if it held a large portfolio of such policies. This premium could be called a fair premium or an actuarially fair premium, denoting that the premium is equal to the expected value of loss (sometimes called the actuarial value of the policy). The term fair is not construed in a normative sense; rather it is simply a reference point”; and
  3. on page 43 of Doherty, Chapter 2, in the first sentence of the first full paragraph: “We know from the Bernoulli principle that a risk averter will choose to fully insure at an actuarially fair premium.””

Hints for solving problem set #4 (spreadsheet hint)

An acceptable alternative way (either in lieu of or in addition to) for solving problem 2 in Problem Set #4 would be to build a spreadsheet model in which you use Solver to determine the optimal exposure to risk for both investors. If you decide to build your own spreadsheet model, upload it in addition to the problem set itself in order to get credit for working problem 2 this way.

An example of how to use Solver for figuring out an optimal decision is provided in my “Optimal insurance demand” spreadsheet. There, alpha corresponds to the percent of the risk to be insured and beta corresponds to the percentage markup from the actuarially fair price; when the beta is zero (as currently coded), then insurance is actuarially fair and Solver returns an alpha value of 1 (i.e., full coverage; as currently coded, this spreadsheet corroborates the Bernoulli principle for a consumer with a square root utility function.

Hints for solving problem set #4 (calculus hint)

In a previous “helpful” hint pertaining to the second problem in Problem Set 4, I noted (among other things) that expected utility for Investor A is E(U(W)) = .6(1,020 + 100x).5 + .4(1,020 – 140x).5, where x corresponds to the proportion of the portfolio that is to be allocated to the risky asset. “Optimal” exposure to risk is determined by maximizing E(U(W)); this is accomplished by differentiating E(U(W)) with respect to x, setting that result equal to 0, and solving for x. The mathematical logic applied here is the same as the approach shown on pp. 14-15 in the Mathematics Tutorial for determining the profit-maximizing production decision of a firm. Here, since the square root utility function is itself a function of x; i.e., U(W(x)), this means that we must apply the chain rule in order to differentiate E(U(W)) with respect to x:

\displaystyle\frac{{dE(U(W))}}{{dx}} = \frac{{dE(U(W))}}{{dW}}\frac{{dW}}{{dx}} = .5W{(x)^{ - .5}}\frac{{dW}}{{dx}}.

Since there are two states, this means that the are two state-contingent values for W(x); specifically, 60% of the time, W(x) = 1,020 + 100x , and 40% of the time, W(x) = 1,020 -140x. Once the chain rule has been applied to differentiating both terms on the right-hand side of the E(U(W)) equation for Investor A, set that result (also known as the “first-order condition”) equal to 0 and solve for Investor A’s optimal exposure to the risky asset. Once that proportion has been determined, the allocation to the safe asset is equal to 1-x.

Rinse and repeat to determine Investor B’s optimal exposure to risk. Since Investor B’s utility function is E(U(W)) = .6ln (1,020 + 100x) + .4ln(1,020 – 140x), it follows that the first-order condition is \displaystyle \frac{{dE(U(W))}}{{dx}} = \frac{{dE(U(W))}}{{dW}}\frac{{dW}}{{dx}} = \frac{1}{{W(x)}}\frac{{dW}}{{dx}}. Set Investor B’s first-order condition equal to 0, solve for x and 1-x, and you’re good to go!

Hints for solving problem set #4 (due on Thursday, February 18)

Problem set #4 (which is due on Thursday, September 24) consists of two problems: 1) a problem that incorporates both expected utility and stochastic dominance, and 2) an optimal (expected utility-maximizing) portfolio problem. We’ll discuss stochastic dominance in class on Tuesday, but in the meantime, here are some hints for setting up the second problem.

The second problem involves determining how to (optimally) allocate initial wealth W0 = $1,000 to (risky) stock and (safe) bond investments for two investors who are identical in all respects except utility. Let x represent the allocation to stock; then the plan is to invest $1,000x in the stock and $1,000(1-x) in the bond. The key here is to find the value for x which maximizes expected utility. The problem is based on the following facts:

  • U(W) = W.5; for Investor A and U(W) = ln W for Investor B;
  • W0 = $1,000 for both investors;
  • Current bond and stock prices are B0 and S0 respectively;
  • End-of-period bond price is B1 = B0(1.02) with probability 1.0; and
  • End-of-period stock price is S1 = S0(1.12) with probability .6 and S1 = S0(.88) with probability .4.

In order to compute expected utility of wealth for either investor, you must first determine state-contingent wealth (Ws). Since there is a 60% chance that the stock increases in value by 12%, a 40% chance that the stock decreases in value by 12%, and a 100% chance that the bond increases in value by 2%, this implies the following:

  • 60% of the time, Ws = xW0(1.12) + (1-x)W0(1.02) = x1,000(1.12) + (1-x)1,000(1.02) = x1,120 + (1-x)1,020 = 1,020 + 100x.
  • 40% of the time, Ws = xW0(.88) + (1-x)W0(1.02) = x1,000(.88) + (1-x)1,000(1.02) = x880 + (1-x)1,020 = 1,020 – 140x.

Therefore, expected utility for Investor A is: E(U(W)) = .6(1,020 + 100x).5 + .4(1,020 – 140x).5, and expected utility for Investor B is E(U(W)) = .6ln (1,020 + 100x) + .4ln(1,020 – 140x). There are two ways to solve for the optimal value of x for each investor – via calculus (applying the power rule and the chain rule) or a spreadsheet model; either approach suffices.

Problem Set 2 helpful hints

Problem Set 2 is available from the course website at http://fin4335.garven.com/spring2021/ps2.pdf; its due date is Thursday, February 4.

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. My one-page teaching note entitled “Mean and Variance of a Two-Asset Portfolio” (assigned reading from last Tuesday) provides simple and succinct explanations concerning how to calculate the expected return (equations 1-2) and variance (equations 3-4) of a two-asset portfolio. 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. 17-23 of the http://fin4335.garven.com/spring2021/lecture4.pdf lecture note for coverage of that topic.