Category Archives: Helpful Hints

Information about tomorrow’s midterm exam in Finance 4335

Tomorrow afternoon’s midterm exam in Finance 4335 consists of four problems. Since your exam grade will be based upon the three
highest scoring problems of these four, feel free to either work all four problems or just three of the four problems. Each problem is worth 32 points; thus three problems times 32 points each totals 96
points. I will add an additional 4 points on your exam if you also legibly write your name on the cover page in the space provided. Thus, the maximum number of points possible on this exam is 100 points.  Furthermore, I have also posted the formula sheet for Midterm Exam 1, which will be included as part of the exam booklet. I recommend that y’all familiarize yourselves with this document sometime prior to tomorrow’s exam.

I’d like to make an important point about the formulas provided on the formula sheet.  What you’ll find there is not a complete census of all formulas used to date in Finance 4335.  For example, I don’t include any insurance pricing formulas.  By now, I assume that everyone knows that an actuarially fair price for an insurance policy is simply the expected value of the claim/indemnity under said policy.  On the other hand, an actuarially “unfair” policy has a “premium loading” which represents a “markup” from the actuarially fair price.  Also, I recommend when performing expected utility calculations, I recommend that you go out to no less than the 3rd digit to the right of the decimal point.

Unfortunately, I cannot be at the exam tomorrow due to an important medical issue affecting a close member of my family.  Professor Paul Anderson has graciously agreed to proctor tomorrow’s exam in my place.

See y’all next week!

Dr. Garven

Hints for solving problem set #4

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 \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
  • 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 = \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.  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 risk@garven.com along with turning in the completed problem set.

Problem Set 2 helpful hints

Problem Set 2 is now available from the course website at http://fin4335.garven.com/spring2019/ps2.pdf; its due date is Tuesday, January 29.

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. 17-22 of the http://fin4335.garven.com/spring2019/lecture3.pdf lecture note for coverage of this topic. 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.

Visualizing Taylor polynomial approximations

In his video lesson entitled “Visualizing Taylor polynomial approximations“, Sal Kahn essentially replicates the tail end of last Thursday’s Finance 4335 class meeting in which we approximated y = eˣ with a Taylor polynomial centered at x=0.  Sal approximates y = eˣ with a Taylor polynomial centered at x=3 instead of x=0, but the same insight obtains in both cases, which is that one can approximate functions using Taylor polynomials, and the accuracy of the approximation increases as the order of the polynomial increases (see pp. 19-25 in my Mathematics Tutorial lecture note if you wish to review what we did in class on Thursday).

Plans for next week’s Finance 4335 class meetings, along with a preview of future topics

Next week in Finance 4335 will be devoted to tutorials on probability and statistics. These tools are critically important in order to evaluate risk and develop appropriate risk management strategies for individuals and firms alike. Next Tuesday’s class meeting will be devoted to introducing discrete and continuous probability distributions, calculating parameters such as expected value, variance, standard deviation, covariance and correlation, and applying these concepts to measuring expected returns and risks for portfolios consisting of risky assets. Next Thursday will provide a deeper dive into discrete and continuous probability distributions, in which the binomial and normal distributions are showcased.

While I have your attention, let me briefly explain what the main “theme” will initially be in Finance 4335 (up to the first midterm exam, which is scheduled for Thursday, February 21). Starting on Tuesday, January 29, we will begin our discussion of decision theory. Decision theory addresses decision making under risk and uncertainty, and not surprisingly, risk management lies at the very heart of decision theory. Initially, we’ll focus attention on variance as our risk measure. Most basic finance models (e.g., portfolio theory, the capital asset pricing model (CAPM), and option pricing theory) implicitly or explicitly assume that risk = variance. We’ll learn that while this is not necessarily an unreasonable assumption, circumstances can arise where it is not an appropriate assumption. Furthermore, since individuals and firms are typically exposed to multiple sources of risk, we need to take into consideration the portfolio effects of risk. To the extent that risks are not perfectly positively correlated, this implies that risks often “manage” themselves by canceling each other out. Thus the risk of a portfolio is typically less than the sum of the individual risks which comprise the portfolio.

The decision theory provides a particularly useful framework for thinking about concepts such as risk aversion and risk tolerance. The calculus comes in handy by providing an analytic framework for determining how much risk to retain and how much risk to transfer to others. Such decisions occur regularly in daily life, encompassing practical problems such as deciding how to allocate assets in a 401-K or IRA account, determining the extent to which one insures health, life, and property risks, whether to work for a startup or an established business and so forth. There’s also quite a bit of ambiguity when we make decisions without complete information, but this course will at least help you think critically about costs, benefits, and trade-offs related to decision-making whenever you encounter risk and uncertainty.

After the first midterm, the remainder of the semester will be devoted to various other risk management topics, including the demand for insurance, asymmetric information, portfolio theory, capital market theory, option pricing theory, and corporate risk management.

Problem Set 1 hint…

Problem Set 1 is due at the beginning of class on Tuesday, January 22. Here is a hint for solving the 4th question on problem set 1.

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 total cost; if this were the case, there would be no hospital because marginal costs are positive, which implies that total cost is positively related to the number of patient-days.

The cost equation C = 4,700,000 + 0.00013X2 tells you the total cost as a function of the number of patient-days. This is why you are asked in part “a” of the 4th question to derive a formula for the relationship between cost per patient-day and the number of patient days. Once you have that equation, then that is what you minimize, and you’ll be able to answer the question concerning optimal hospital size.

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, 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:

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After clicking “Subscribe”, the following information will appear on the screen:Screen Shot 2016-06-16 at 3.45.08 PM
Next, check for an email from “Risk Management Blog <donotreply@wordpress.com> ”:

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Next, simply click the “Confirm Follow” button. This will cause you to receive the following email:

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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 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):

Finally, if your algebra skills are generally a bit on the rusty side, I would also recommend checking out the Khan Academy’s review of algebra.