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 *W*_{0 }= $100 to (risky) stock and (safe) bond investments for two investors who are identical in all respects except utility. Let represent the allocation to stock; then the plan is to invest $100 in the stock and $100(1-) in the bond. The key here is to find the value for 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*;*W*_{0 }= $100 for both investors;- Current bond and stock prices are
*B*_{0 }and*S*_{0}respectively; - End-of-period bond price is
*B*_{1 }=*B*_{0}(1.05) with probability 1.0; and - eEnd-of-period stock price is
*S*_{1}=*S*_{0}(1.3) with probability .6 and*S*_{1}=*S*_{0}(.7) with probability .4.

In order to compute expected utility of wealth for either investor, you must first determine state-contingent wealth (*W*_{s}). 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,
*W*_{s }=*W*_{0}(1.3) + (1-)*W*_{0}(1.05) = 100(1.30) + (1-)100(1.05) = 130 + (1-)105 = 105 + 25. - 40% of the time,
*W*_{s }=*W*_{0}(.7) + (1-)*W*_{0}(1.05) = 100(.7) + (1-)100(1.05) = 70 + (1-)105 = 105 – 35.

Therefore, expected utility for Investor A is: *E*(*U*(*W*)) = .6(105 + 25)^{.5} + .4(105 – 35)^{.5}, and expected utility for Investor B is *E*(*U*(*W*)) = .6ln (105 + 25) + .4ln(105 – 35). It is up to you to solve for the optimal value of 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.

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:

]]>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).

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.

Not only is Hurricane Florence a highly destructive natural catastrophe; it will also apparently be a financial catastrophe for the many thousands of households who, for whatever reason, lack flood insurance coverage.

… are available at the link listed below:

]]>… are available at http://fin4335.garven.com/fall2018/ps3solutions.pdf.

]]>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.

This *Atlantic* article is well worth reading!

The outrageous price of a U.S. degree is unique in the world.

This article from the *Wall Street Journal* provides an interesting follow-up to yesterday’s behavioral finance discussion. “Information avoidance” represents a particularly strong (and potentially deadly) form of confirmation bias!