# 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 #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:

# Gamma Iota Sigma Interest Meeting: Tuesday, August 28 at 5:30 pm in Foster 226

(Note from Dr. Garven: I am posting this announcement on behalf of Gamma Iota Sigma, which is a professional business fraternity for which I am a faculty advisor.)

You’re invited to an interest meeting for Gamma Iota Sigma (GIS), the international risk management and insurance (RMI) business fraternity.  The purpose of GIS is to promote, encourage, and sustain student interest in insurance, risk management, and actuarial science as professions and to facilitate interaction with the business community.

Members of GIS have the opportunity to:

– meet executives from insurance and risk management companies;
– learn about internship and job opportunities in insurance and other risk management related companies;
– attend and network at industry conferences in places like Chicago, Atlanta, Phoenix and elsewhere;
– meet and enjoy fun and fellowship with students of like interests.

The first meeting of GIS for the 2018-2019 academic year is Tuesday, August 28 at 5:30 pm in Foster 226. Anyone who wants to learn more about risk management and insurance is welcome, regardless of major.

# 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, September 27). Starting on Tuesday, September 4, 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.