All posts by jgarven

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.

Guidelines for completing the first problem on Problem Set 4

During class today, I elaborated further on my blog posting entitled “Hints for solving problem set #4 (Hint #1)”. Specifically, at your option, you may solve the first problem via either calculus or a spreadsheet model.

If you decide to solve the first problem by building your own spreadsheet model, then you must email your spreadsheet model to “risk@garven.com” prior to the start of class on Thursday. 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 this 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

Hints for solving problem set #4 (Hint #1)

Problem set #4 consists of two problems: 1) an optimal (expected utility maximizing) portfolio problem, and 2) a stochastic dominance problem.  We’ll discuss stochastic dominance tomorrow (and also (hopefully) work a class problem in connection with that concept), but in the meantime allow me to provide you with some hints for setting up  the first problem.

The first problem involves determining how to (optimally) allocate  initial wealth W0 = $100 to (risky) stock and (safe) bond investments for two investors who are identical in all respects except utility. Let \alpha represent the allocation to stock; then the plan is to invest $100\alpha in the stock and $100(1-\alpha) in the bond. The key here is to find the value for \alpha 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 = $100 for both investors;
  • Current bond and stock prices are B0 and S0 respectively;
  • End-of-period bond price is B1 = B0(1.05) with probability 1.0; and
  • eEnd-of-period stock price is S1 = S0(1.3) with probability .6 and S1 = S0(.7) 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 30%, a 40% chance that the stock decreases 30%, and a 100% chance that the bond increases in value by 5%, this implies the following:

  • 60% of the time, Ws = \alphaW0(1.3) + (1-\alpha)W0(1.05) = \alpha100(1.30) + (1-\alpha)100(1.05) = \alpha130 + (1-\alpha)105 = 105 + 25\alpha.
  • 40% of the time, Ws = \alphaW0(.7) + (1-\alpha)W0(1.05) = \alpha100(.7) + (1-\alpha)100(1.05) = \alpha70 + (1-\alpha)105 = 105 – 35\alpha.

Therefore, expected utility for Investor A is: E(U(W)) = .6(105 + 25\alpha).5 + .4(105 – 35\alpha).5, and expected utility for Investor B is E(U(W)) = .6ln (105 + 25\alpha) + .4ln(105 – 35\alpha). It is up to you to solve for the optimal value of \alpha for each investor.  There are two ways to do this – via calculus or a spreadsheet model.  Actually, I would encourage y’all to work this problem both ways if time permits because doing so will help you develop an even better grasp of the underlying principles and concepts in Finance 4335.

Problem Set #4 – new (replacement) version uploaded as of 6:15 p.m. this evening

Upon closer inspection, I noticed that my first problem on Problem Set 4 was identical to the Risk Aversion Class Problem that we worked in class last Thursday, so as of 6:15 p.m. this evening, I uploaded a new (replacement) version of Problem Set 4, which is due at the beginning of class on Thursday, 9/20. Therefore, if any of you downloaded this problem set  prior to 6:15 p.m. today, be sure to replace it with this new Problem Set 4, which looks like this:

Arrow-Pratt method vis–à–vis the “exact” method for calculating risk premiums

I received an email from a Finance 4335 student earlier today asking for further clarification of the two methods for calculating risk premiums which we covered in class last Thursday. Under the so-called “exact” method, one 1) calculates expected utility, 2) sets expected utility equal to the utility of the certainty-equivalent of wealth, 3) solves for the certainty-equivalent of wealth, and 4) obtains the risk premium by calculating the difference between expected wealth and the certainty-equivalent of wealth.  On the other hand, the Arrow-Pratt method is an alternative method for calculating the risk premium which is based upon Taylor series approximations of expected utility of wealth and the utility of the certainty equivalent of wealth (the derivation for which appears on pp. 16-18 of http://fin4335.garven.com/fall2018/lecture6.pdf). Both of these approaches for calculating risk premiums are perfectly acceptable for purposes of Finance 4335.

The value added of Arrow-Pratt is that it analytically demonstrates how risk premiums depend upon two factors: 1) the magnitude of the risk itself (as indicated by variance), and 2) the degree to which the decision-maker is risk averse. For example, we showed in class on Thursday that the Arrow-Pratt coefficient for the logarithmic investor (for whom U(W) = ln W) is twice as large as the Arrow-Pratt coefficient for the square root investor (for whom U(W) = W.5); 1/W for the logarithmic investor compared with .5/W for the square root investor. Thus, the logarithmic investor behaves in a more risk averse than the square root investor; other things equal, the logarithmic investor will prefer to allocate less of her wealth to risky assets and buy more insurance than the square root investor. Another important insight yielded by Arrow-Pratt (at least for the utility functions considered so far in Finance 4335) is the notion of decreasing absolute risk aversion (DARA). Other things equal,  an investor with DARA preferences become less (more) risk averse as wealth increases (decreases).  Furthermore, such an investor increases (reduces) the dollar amount that she is willing to put at risk as she becomes wealthier (poorer).

How Do Energy Companies Measure the Temperature? Not in Fahrenheit or Celsius

Instead of Fahrenheit or Celsius, a metric called “degree days” is used to capture variability in temperature. The risk management lesson here is that this metric makes it possible to create risk indices which companies can rely upon for pricing and hedging weather-related risks with weather derivatives.

How Hurricane Florence Could Move Insurance Markets

Hurricane Florence provides a particularly timely and compelling case study of the economic consequences of natural catastrophes; specifically, the nexus of direct and indirect effects upon property insurance markets, reinsurance markets, alternative risk markets (e.g., catastrophe bonds), and public policy.

Some hurricanes are worse than others — both for people in the way and the insurance industry that tries to understand storms and put a price on their risks.