# 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 = $\alpha$W0(1.3) + (1-$\alpha$)W0(1.05) = $\alpha$100(1.30) + (1-$\alpha$)100(1.05) = $\alpha$130 + (1-$\alpha$)105 = 105 + 25$\alpha$.
• 40% of the time, Ws = $\alpha$W0(.7) + (1-$\alpha$)W0(1.05) = $\alpha$100(.7) + (1-$\alpha$)100(1.05) = $\alpha$70 + (1-$\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 2 helpful hints

Problem Set 2 is now available from the course website at http://fin4335.garven.com/fall2018/ps2.pdf; its due date is Tuesday, September 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; see pp. 17-23 of the http://fin4335.garven.com/fall2018/lecture3.pdf lecture note for coverage of this topic. The second problem involves using the standard normal probability distribution to calculate probabilities of earning various levels of return by investing in risky securities and portfolios. We will devote tomorrow’s class meeting to these and related topics.

# Problem Set 1 Hint…

Problem Set 1 is due at the beginning of class tomorrow. Here is a hint for solving the 4th question on this problem set.

The objective is to determine how big a hospital must be so that the cost per patient-day is minimized. We are not interested in minimizing the total cost of operating a hospital; a sure-fire way to minimize total cost would be to not even have a hospital in the first place. Indeed, if you were to differentiate the total cost function given by C = 4,700,000 + 0.00013X2 with respect to X, this is what the math would tell you.

In part “a” of the 4th question, you are asked to “derive” a formula for the relationship between cost per patient-day and the number of patient days; in other words, what you are interested in determining is what is the most cost-efficient way to scale a hospital facility such that the cost per patient-day is minimized. Once you obtain that equation, then you’ll be able to answer the question concerning optimal hospital size.

# Lagrangian Multipliers

There is a section in the assigned “Optimization” reading tomorrow on pp. 74-76 entitled “Lagrangian Multipliers” which (as noted in footnote 9) 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.

Besides reading the articles entitled “Optimization” and “How long does it take to double (triple/quadruple/n-tuple) your money?” in preparation for tomorrow’s meeting of Finance 4335, make sure that you fill out and email the student information form as a file attachment to risk@garven.com prior to the beginning of tomorrow’s class.  As I explained during yesterday’s class meeting, this assignment counts as a problem set, and your grade is 100 if you turn this assignment in on time (i.e., sometime prior to tomorrow’s class meeting) and 0 otherwise.

# How to know whether you are on track with Finance 4335 assignments

At any given point in time during the upcoming semester, you can ensure that you are on track with Finance 4335 assignments by monitoring due dates which are published on the course website. See http://fin4335.garven.com/readings/ for due dates pertaining to reading assignments, and http://fin4335.garven.com/problem-sets/ for due dates pertaining to problem sets. Also keep in mind that short quizzes will be administered in class on each of the dates indicated for required readings. As a case in point, since the required readings entitled “Optimization” and ” How long does it take to double (triple/quadruple/n-tuple) your money?” are listed for Thursday, August 23, this means that a quiz based upon these readings will be given in class on that day.

Important assignments due on the second class meeting of Finance 4335 (scheduled for Thursday, August 23) include: 1) filling out and emailing the student information form as a file attachment to risk@garven.com, 2) subscribing to the Wall Street Journal, and 3) subscribing to the course blog. A completed student information form is graded as a problem set and receives 100 points; if you fail to turn in a student information form, then you will receive a 0 for this “problem set”. Furthermore, tasks 2 and 3 listed above count toward your class participation grade in Finance 4335.

Regarding the student information form, I prefer that you complete this form (by either typing or writing) and email it to risk@garven.com prior to the beginning of class on Thursday.  However, if you prefer, you may turn in a hard copy instead at the beginning of class on Thursday.

# Instructions for subscribing to the Risk Management Course Blog

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, RSS, 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 already are familiar with RSS, this is a great way to subscribe to the options, futures, and other derivatives course blog. By going to the http://risk.garven.com/feed webpage, you can subscribe by using Firefox’s Live Bookmarks feature, Internet Explorer’s RSS feed subscription feature, or an RSS reader. 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 <donotreply@wordpress.com> ”:

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.

# Calculus and Probability & Statistics recommendations…

Since many of the topics covered in Finance 4335 require a basic knowledge and comfort level with differential calculus and probability & statistics, the second class meeting (August 23) will include a mathematics tutorial, and the third and fourth class meetings (August 28-30) 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):

Finally, if your algebra is a bit rusty, I would also recommend checking out the Khan Academy’s review of algebra.