I just posted a new (1 page) reading entitled "Mean and Variance of a Two-Asset Portfolio" which (not surprisingly) explains the origins of the formulas for mean and variance of a two-asset portfolio. I expect you’ll find such knowledge very helpful in understanding and completing the 4th and 5th sections of the Statistics Class Problem as well as parts D, E, and F of Problem 1 and part B of Problem 2 in Problem Set 2.
As of 2:15 p.m. central standard time today, the Predictit.org prediction market put the odds of President Trump being removed from office at 8%. Specifically, Predict.org currently offers for sale a “share” which pays $1 if the answer to the question, “Will the Senate convict Donald Trump in his first term?, turns out to be “yes”.
Allow me to provide further context for this “prediction”. PredictIt.org is a New Zealand-based prediction market that offers “shares” on political and financial events. The idea behind PredictIt.org shares (technically, these are binary options, but I digress) is quite simple – you can buy and sell “yes” and “no” shares which pay off $1 if the answer to the contract question ends up being “yes” or “no”. If you buy yes (no) but no (yes) is the answer, then your share expires worthless and you have lost the full value of your original “investment”. However, if you sell yes (no) and no (yes) is the answer, then you don’t owe your counterparty any money and you get to pocket the price received (net of transactions costs) as profit.
Since the payoffs on PredictIt.org shares feature binary payoffs (i.e., $1 if yes and $0 if no), these shares are canonical examples of Arrow-Debreu, or “pure” securities. Arrow-Debreu securities pay $1 if a particular state (in this case, either “yes” or “no”) occurs at a particular time in the future. Thus, the current price for a given PredictIt.org share is the “state price”, which corresponds to the value today of $1 received when a particular future state of the world is realized. Breaking the state price down further, its components include 1) the probability of a particular future state of the world, 2) the rate of interest (to compensate for the time value of money), and 3) a further discount (to compensate for risk averse behavior by the bettor) or premium (to compensate for risk loving behavior by the bettor).
Prediction market prices are frequently referred to in the news media as probabilities for future state-contingent events; if prediction market participants are risk neutral and interest rates are negligible, then this is technically appropriate and roughly correct. What’s fascinating about prediction markets is that they showcase, in very pure form, how market prices reflect the statistical odds of some future event happening. Similarly, prices of speculative assets generally (e.g., corporate securities such as stocks and bonds and derivative securities such as options and futures) also reflect probabilistic beliefs about future states of the world, albeit in more of an opaque fashion.
… are available at http://fin4366.garven.com/spring2020/ps1solutions.pdf.
Besides insurance, Tim Harford also features the index fund in his “Fifty Things That Made the Modern Economy” radio and podcast series. This 9 minute long podcast lays out the history of the development of the index fund in particular and the evolution of so-called of passive portfolio strategies in general. Much of the content of this podcast is sourced from Vanguard founder Jack Bogle’s September 2011 WSJ article entitled “How the Index Fund Was Born” (available at https://www.wsj.com/articles/SB10001424053111904583204576544681577401622). Here’s the description of this podcast:
“Warren Buffett is the world’s most successful investor. In a letter he wrote to his wife, advising her how to invest after he dies, he offers some clear advice: put almost everything into “a very low-cost S&P 500 index fund”. Index funds passively track the market as a whole by buying a little of everything, rather than trying to beat the market with clever stock picks – the kind of clever stock picks that Warren Buffett himself has been making for more than half a century. Index funds now seem completely natural. But as recently as 1976 they didn’t exist. And, as Tim Harford explains, they have become very important indeed – and not only to Mrs Buffett.”
From November 2016 through October 2017, Financial Times writer Tim Harford presented an economic history documentary radio and podcast series called 50 Things That Made the Modern Economy. This same information is available in book under the title “Fifty Inventions That Shaped the Modern Economy“. While I recommend listening to the entire series of podcasts (as well as reading the book), I would like to call your attention to Mr. Harford’s episode on the topic of insurance, which I link below. This 9-minute long podcast lays out the history of the development of the various institutions which exist today for the sharing and trading of risk, including markets for financial derivatives as well as for insurance.
“Legally and culturally, there’s a clear distinction between gambling and insurance. Economically, the difference is not so easy to see. Both the gambler and the insurer agree that money will change hands depending on what transpires in some unknowable future. Today the biggest insurance market of all – financial derivatives – blurs the line between insuring and gambling more than ever. Tim Harford tells the story of insurance; an idea as old as gambling but one which is fundamental to the way the modern economy works.”
Not only are the Khan Academy Calculus and Statistics videos that I referenced in a previous posting quite useful; I am also a big fan of the Khan Academy “Finance and capital markets” videos; these videos do a great job of effectively presenting some of the most important concepts which are typically covered in undergraduate and MBA level finance curricula. Note also that the “Options, swaps, futures, MBSs, CDOs, and other derivatives” subsection of the “Finance and capital markets” page covers many of the same topics that we will cover this semester in Finance 4366!
Problem Set 1 is due at the beginning of tomorrow’s Finance 4366 meeting. We will start class with a brief quiz based upon the assigned readings, which include “The New Religion of Risk Management”, by Peter Bernstein and “Normal and standard normal distribution”, by yours truly.
Going forward, I will typically not post reminders like this concerning Finance 4366 assignment deadlines; however, you’ll be “good to go” in Finance 4366 if you faithfully follow the guidelines listed in my “How to know whether you are on track with Finance 4366 assignments” posting.
At any point in time this semester, you can ensure that you are on track with Finance 4366 assignments by monitoring due dates on the course website. See http://fin4366.garven.com/readings/ for the reading assignment due dates, and http://fin4366.garven.com/problem-sets/ for the problem set due dates. Also, keep in mind that I will administer short quizzes in class on each of the dates shown for required readings.
In his video lesson entitled “Visualizing Taylor polynomial approximations“, Sal Kahn essentially replicates the tail end of last Thursday’s Finance 4366 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. 18-23 in my Mathematics Tutorial lecture note if you wish to review what we did in class last Thursday).