Class on Thursday, September 21 will be devoted to 1) completing the “Decision-Making under Risk and Uncertainty” topic and 2) a review session for Midterm Exam 1, which is scheduled for Tuesday, September 26 (in class). Prior to coming to class on Thursday, you’ll want to review your notes about the “Decision-Making under Risk and Uncertainty” topic, review problem sets 3 and 4, and try solving the Finance 4335 Midterm 1 Exam that I gave during the Spring 2017 semester (solutions are available at http://fin4335.garven.com/fall2017/midterm1_spring2017_solutions.pdf). This way we can have a very productive review session that will enable you to be better prepared for the first midterm in Finance 4335!
See the one-page PDF document: Finance 4335 Synopsis (September 17, 2017).
A student asked me whether it is possible for the value of α in problem set 4 to be negative. Here, α represents the optimal level of exposure to the risky asset; 1-α represents the optimal level of exposure to the safe bond. While it is certainly theoretically possible for α to have a negative value, for this particular problem it turns out that α > 0. The reason α is positive in this case is because the 60/40 probabilities imply a positive expected return on the risky asset which exceeds the expected return on the bond. Thus, even though the stock is risky, a square root utility investor is willing to invest some of her money in the stock because in an expected utility sense, some positive exposure to risk is worthwhile.
If you obtain a negative value for α, this means that you must have made a math error somewhere. Since E(U(W)) = .6(105 + 25α ).5 + .4(105 – 35α ).5, then one can find the optimal value for a by applying the chain rule individually to both the .6(105 + 25α ).5 and the .4(105 – 35α ).5 terms, setting the resulting equation equal to zero (this is the so-called “first order condition”) and solving for α .
If you get stuck on the math at all, you might consider inputting the data into an Excel spreadsheet and use Solver to find the optimal value for α . For what it’s worth, I just did this a few moments ago and confirmed that the (positive) α value which I obtained using the calculus is identical to the α value indicated by Excel (obviously I was not surprised, since in both cases I knew a priori that my calculus and Excel code were correct :-)).
This raises an interesting question; specifically, what would have to be different about this problem in order to obtain a negative value for α ? If this problem were reparameterized such that the risk/return characteristics of the stock vis-a-vis the bond were sufficiently worsened, and/or if the investor was inclined to act in a more risk averse fashion (e.g., if initial wealth declined and/or the investor’s utility function were different), then a negative value for α is a possibility. For example, suppose that everything stayed the same, but that the state probabilities for the stock were 55/45 rather than 60/40. If this occurred, then you would find that the investor’s optimal α is -48.33%. Note that with 55/45 state probabilities, the stock has an expected return of .55(.3) + .45 (-.3) = 3% and a standard deviation of 14.92% (check this calculation for yourself). If short selling were prohibited, then this investor would optimally invest all of her initial wealth in the bond. However, if short selling were allowed, then at date 0, she would sell short $48.33 of stock and invest her initial wealth of $100 plus the $48.33 in proceeds from the short sale in the bond. From date 0 to date 1, she would earn 5%, or $7.42 on her $148.33 bond investment. At date 1, she would close out her short position by buying the stock back at either $48.33 x (1.30) = $62.83 (in which case she would lose $7.08 on her $100 net investment) or at $48.33 x (.7) = $33.83 (in which case she would gain $21.92 on her $100 net investment). Thus the expected return on her portfolio is .55(-7.08%) + .45(21.92%) = 5.97%, and the standard deviation is 14.43% (short selling is risky because you might get stuck having to close out the short position at a high price; this is why the standard deviation is so high).
I have posted my spreadsheet for this problem at http://risk.garven.com/wp-content/uploads/2017/09/assetallocation.xls. You are welcome to use this spreadsheet if you are interested in numerically validating your calculus-based solution procedure.
Problem set #4 involves determining how to (optimally) allocate your initial wealth W0 = $100 to (risky) stock and (safe) bond investments. Let α represent the allocation to stock; then the plan is to invest $100α in the stock and $100(1-α) in the bond. The key here is to find the value for a which maximizes expected utility. The problem is based on the following facts:
- U(W) = W.5;
- W0 = $100;
- 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, 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 = αW0(1.3) + (1-α)W0(1.05) = α100(1.30) + (1-α)100(1.05) = α130 + (1-α)105 = 105 + 25α.
- 40% of the time, Ws = αW0(.7) + (1-α)W0(1.05) = α100(.7) + (1-α)100(1.05)] = α70 + (1-α)105 = 105 – 35α.
Therefore, expected utility is: E(U(W)) = .6(105 + 25α).5 + .4(105 – 35α).5. It is up to you to solve for the optimal value of α. This requires solving the first order condition, which involves differentiating E(U(W)) with respect to α, setting the result equal to 0 and solving for α.
… are available at http://fin4335.garven.com/fall2017/ps3solutions.pdf…
This article from the Wall Street Journal provides an interesting followup to yesterday’s behavioral finance discussion. “Information avoidance” represents a particularly strong (and potentially deadly) form of confirmation bias!
In January 2014, The Economist published a particularly interesting article about the determinants of risk aversion, entitled “Risk off: Why some people are more cautious with their finances than others”. Here are some key takeaways from this article:
1. Economists have long known that people are risk-averse; yet the willingness to run risks varies enormously among individuals and over time.
2. Genetics explains a third of the difference in risk-taking; e.g., a Swedish study of twins finds that identical twins had “… a closer propensity to invest in shares” than fraternal ones.
3. Upbringing, environment and experience also matter; e.g., . “…the educated and the rich are more daring financially. So are men, but apparently not for genetic reasons”.
4. People’s financial history has a strong impact on their taste for risk; e.g., “… people who experienced high (low) returns on the stock market earlier in life were, years later, likelier to report a higher (lower) tolerance for risk, to own (not own) shares and to invest a bigger (smaller) slice of their assets in shares.”
5. “Exposure to economic turmoil appears to dampen people’s appetite for risk irrespective of their personal financial losses.” Furthermore, a low tolerance for risk is linked to past emotional trauma.
During last Thursday’s class meeting, we discussed (among other things) the famous St. Petersburg Paradox. The source for this is Daniel Bernoulli’s famous article entitled “Exposition of a new theory on the measurement of risk”. As was the standard academic practice in academia at the time, Bernoulli’s article was originally published in Latin in 1738. It was subsequently translated into English in 1954 and published a second time that same year in Econometrica (Volume 22, No. 1): pp. 22–36. Considering that this article was published 277 years ago in an obscure (presumably peer reviewed) academic journal, it is fairly succinct and surprisingly easy to read. The direct link to the PDF of this article is here: http://www.jstor.org/stable/1909829 (P.S.: the article is gated but you can access it from anywhere on campus when connected to Airbear).
Also, the Wikipedia article about Bernoulli’s article is worth reading. It provides the mathematics for determining the price at which the apostle Paul would have been indifferent about taking the apostle Peter up on this bet. Specifically, if Paul were a millionaire, he should be willing to pay up to $20.88, On the other hand, if Paul were a thousandaire (total wealth = $1000), he should be willing to pay up to $10.95. Finally, if Paul was quite poor (e.g., total wealth = $2), then he might consider borrowing $1.35 and paying up to $3.35 to place this bet…
Meir Statman has very important things to say about decision-making under risk and uncertainty; I introduced Professor Statman to you in my previous blog posting entitled “Your Tolerance for Investment Risk Is Probably Not What You Think.” Here is an extra credit opportunity for Finance 4335 based upon a 1 hour, 25 minute podcast (recorded in July 2017) hosted by Barry Ritholtz’s Masters of Business podcast (link provided below) entitled “Interview with Meir Statman.”
You may earn extra credit by listening to and reporting on Mr. Ritholtz’s interview with Meir Statman about behavioral finance. In order to receive extra credit for this assignment, you must submit (via email sent to email@example.com) a 1-2 page executive summary of what you learned from this podcast; it is due by no later than 5 p.m. on Monday, September 18. This extra credit assignment will replace your lowest quiz grade in Finance 4335 (assuming the extra credit grade is higher).
Bloomberg View columnist Barry Ritholtz interviews Meir Statman, the Glenn Klimek Professor of Finance at Santa Clara University. His research focuses on behavioral finance. He attempts to understand how investors and managers make financial decisions and how these decisions are reflected in financial markets. His most recent book is “Finance for Normal People: How Investors and Markets Behave,” published by Oxford University Press. This commentary aired on Bloomberg Radio.
The questions financial advisers ask clients to get at the answer actually measure something completely different—often leading to misguided investment strategies.