Problem set #4 (which is due on Thursday, February 14) consists of two problems: 1) an optimal (expected utility maximizing) portfolio problem, and 2) a stochastic dominance problem. We’ll discuss stochastic dominance next Tuesday, 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 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;
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
End-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 = W0(1.3) + (1-)W0(1.05) = 100(1.30) + (1-)100(1.05) = 130 + (1-)105 = 105 + 25.
40% of the time, Ws = W0(.7) + (1-)W0(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. However, at your option, you may rely solely on building your own spreadsheet model. If you do this, in order to receive full credit, you need to email your spreadsheet model to email@example.com along with turning in the completed problem set.
There is a section in the assigned “Optimization” reading due Thursday, 1/17 on pp. 74-76 entitled “Lagrangian Multipliers” which (as noted in footnote 9 of that reading) may be skipped without loss of continuity. The primary purpose of this chapter is to re-acquaint students with basic calculus and how to use the calculus to solve so-called optimization problems. Since the course only requires solving unconstrained optimization problems, there’s no need for Lagrangian multipliers.
A course blog has been established for Finance 4335 at the address http://risk.garven.com; it is also linked from the “Course Blog” button located on the course website. I recommend that you follow the risk management course blog regularly via email, Facebook, and/or Twitter.
The risk management course blog provides me with a convenient means for distributing important announcements to the class. Topics covered on the course blog typically include things like changes in the course schedule, clarifications and hints concerning problem sets, information about upcoming exams, announcements concerning extra credit opportunities, and short blurbs showing how current events relate to many of the topics which we cover in Finance 4335.
If you are either a Facebook or Twitter user, everything that is posted on the options, futures, and other derivatives course blog is automatically posted to Facebook and “tweeted”, so you can also subscribe by “liking” the Finance 4335 Facebook page or by “following” @fin4335 on Twitter. Finally, you can also subscribe via email. The remainder of this blog entry explains how to subscribe to the risk management course blog via email.
Email Subscription Instructions:
Email Subscription Instructions: If you would like to receive the risk management course blog via email, you can do this by going to http://risk.garven.com and entering your email address in the form provided on the left hand side of that webpage:
After clicking “Subscribe”, the following information will appear on the screen:
Next, check for an email from “Risk Management Blog <firstname.lastname@example.org> ”:
Next, simply click the “Confirm Follow” button. This will cause you to receive the following email:
From that point forward, whenever I post to the course blog, you will immediately receive a nicely formatted version of the blog posting via email. Also, you can opt to change your delivery preferences, or even cancel your subscription.
Since many of the topics covered in Finance 4335 require a basic knowledge and comfort level with algebra, differential calculus and probability & statistics, the second class meeting during the spring 2019 semester will include a mathematics tutorial, and the third and fourth class meetings will cover probability & statistics. I know of no better online resource for brushing up on (or learning for the first time) these topics than the Khan Academy.
So here are my suggestions for Khan Academy videos which cover these topics (unless otherwise noted, all sections included in the links which follow are recommended):
A subscription to the Wall Street Journal is required for Finance 4335. In order to subscribe to the Wall Street Journal (WSJ) for this semester, go to http://r.wsj.net/j73NM. Your WSJ subscription includes access to print, online, tablet and mobile editions, and only costs $1 for a 15 week subscription. At your option, you may choose to receive both the digital and paper versions of WSJ or only the WSJ digital version.
Throughout the semester, I will often reference specific WSJ articles on the course blog and in class. Finance 4335 topics (as well as topics in many of your other business school courses) come to life in the world outside the Baylor bubble when you read make a habit of reading the WSJ on a regular basis. Furthermore, if you expect to interview for jobs or internships anytime soon, reading the WSJ will give you a leg up on the competition, since you will be better informed and have more compelling ideas and insights to share with recruiters.
In closing, the following (2 minute) video provides a helpful introduction to the WSJ, providing time-saving tips to help you get the most from WSJ and succeed not only in Finance 4335, but also your other courses and careers: