Fair Price of Insurance Policy

A Finance 4335 student asked the following question earlier today:

Q: “I have a quick homework question. How do you find the fair price of an insurance policy?”

Here’s the answer I provided, which I now share with all Finance 4335 students:

A: “I believe you are referring to the “actuarially fair” price, or insurance premium. The actuarially fair premium corresponds to the expected value of the insurance indemnity; the indemnity is the amount of coverage offered by an insurance policy. Under “full coverage”, 100% of the loss is indemnified, and in such a case, the actuarially fair premium is equal to the expected value of the loss distribution.

For what it’s worth, the concept of “actuarially fair” insurance prices/premiums, along with implications for the demand for insurance, are explained in two of the September 8th assigned readings; e.g.,

  1. on page 4 of the Supply of Insurance reading (just prior to the section entitled “Example 2: Correlated Identically Distributed Losses), the following sentence appears, “A premium that is equal to the expected outcome is called an actuarially fair premium”;
  2. on page 30 of the Basic Economics: How Individuals Deal with Risk (Doherty, Chapter 2) reading, consider the following excerpt: “Ignoring transaction costs, an insurer charging a premium equal to expected loss would break even if it held a large portfolio of such policies. This premium could be called a fair premium or an actuarially fair premium, denoting that the premium is equal to the expected value of loss (sometimes called the actuarial value of the policy). The term fair is not construed in a normative sense; rather it is simply a reference point”; and
  3. on page 43 of Doherty, Chapter 2, in the first sentence of the first full paragraph: “We know from the Bernoulli principle that a risk averter will choose to fully insure at an actuarially fair premium.””

On the Determinants of Risk Aversion

This week, we began a series of five Finance 4335 class meetings (scheduled for September 8 – 22) devoted to decision-making under risk and uncertainty. We shall study how to measure risk, model consumer and investor risk preferences, and explore implications for the pricing and management of risk. We will focus especially on the concept of risk aversion. Other things equal, risk averse decision-makers prefer less risk to more risk. Risk aversion helps to explain some very basic facts of human behavior; e.g., why investors diversify, why consumers purchase insurance, etc.

A few years ago, The Economist published a particularly interesting article about various behavioral 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.

St. Petersburg Paradox…

During today’s class meeting, we will discuss (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 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 282 years ago in an obscure (presumably peer-reviewed) academic journal, it is fairly succinct and surprisingly easy to read.

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. The original numerical example proposed by Bernoulli focuses attention on Paul’s gamble per se and does not explicitly consider the effect of Paul’s initial wealth on his willingness to pay. However, the quote on page 31 of the article (“… that any reasonable man would sell his chance … for twenty ducats”) implies that Bernoulli may have assumed Paul to be a millionaire, since (as shown in the Wikipedia article) the certainty-equivalent value of this bet to a millionaire who has logarithmic utility comes out to 20.88 ducats.

On the ancient origin of the word “algorithm”

During class last week, I mentioned that the September 1 assigned reading entitled “The New Religion of Risk Management” (by Peter Bernstein, March-April 1996 issue of Harvard Business Review) is a synopsis of the same author’s 1996 book entitled “Against the Gods: The Remarkable Story of Risk”.  Here’s a fascinating quote from page 33 which explains the ancient origin of the word “algorithm”:

“The earliest known work in Arabic arithmetic was written by al­Khowarizmi, a mathematician who lived around 825, some four hun­dred years before Fibonacci. Although few beneficiaries of his work are likely to have heard of him, most of us know of him indirectly. Try saying “al­Khowarizmi” fast. That’s where we get the word “algo­rithm,” which means rules for computing.”

Note: The book cover shown above is a copy of a 1633 oil-on-canvas painting by the Dutch Golden Age painter Rembrandt van Rijn.

Also featured as one of “50 Things That Made the Modern Economy”: The Index Fund

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.”

Warren Buffett is one of the world’s great investors. His advice? Invest in an index fund

Insurance featured as one of “50 Things That Made the Modern Economy”

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 form 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.”

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

We will devote next week in Finance 4335 to tutorials on probability and statistics. These tools are critically important in the measurement of risk and development of 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 measure expected returns and risks for portfolios comprising risky assets. The following Thursday will provide a deeper dive into discrete and continuous probability distributions, in which we showcase the binomial and normal distributions.

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 on Tuesday, September 29). Starting on Tuesday, September 1, we will begin our discussion of decision theory. Decision theory addresses decision making under risk and uncertainty, which at the very heart of risk management. Initially, we’ll focus attention on variance as our risk measure. Most of the basic finance theories, including portfolio, capital market, and option pricing theories, define risk as variance. We’ll learn that while this is not necessarily an unreasonable assumption, circumstances may arise where it is not an appropriate assumption. Since individuals and firms encounter multiple sources of risk, we also need to take into consideration the portfolio effects of risk. Portfolio theory 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 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 ambiguity when we have incomplete information about risk. 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 rest 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.

On the relationship between the S&P 500 and the CBOE Volatility Index (VIX)

Besides going over the course syllabus during the first day of class on Tuesday, August 25, we will also discuss a particularly important “real world” example of financial risk. Specifically, we will look at the relationship between stock market returns (as indicated by daily percentage changes in the SP500 stock market index) and stock market volatility (as indicated by daily percentage changes in the CBOE Volatility Index (VIX)):

As indicated by this graph from page 21 of the lecture note for the first day of class, daily percentage changes on closing prices for VIX (which is the x-axis variable) and the SP500 (which is the y-axis variable) are strongly negatively correlated. The blue points represent 7,712 daily observations on these two variables, spanning the time period from January 3, 1990, through August 10, 2020. When we fit a regression line through this scatter diagram, we obtain the following equation:

{R_{SP500}} = 0.0621 - 0.1156{R_{VIX}},

where {R_{SP500}} corresponds to the daily return on the SP500 index and {R_{VIX}} corresponds to the daily return on the VIX index. The slope of this line (-0.1156) indicates that on average, daily VIX returns during this time period were inversely related to the contemporaneous daily return on the SP500; i.e., when volatility, as measured by VIX, went down (up), then the stock market return as indicated by SP500 typically went up (down). Nearly half of the variation in the stock market return during this time period (specifically, 48.75%) can be statistically “explained” by changes in volatility, and the correlation between {R_{SP500}} and {R_{VIX}} comes out to -0.698. While a correlation of -0.698 does not imply that {R_{SP500}} and {R_{VIX}} always move in opposite directions, it does suggest that this will be the case more often than not. Indeed, closing daily returns on {R_{SP500}} and {R_{VIX}} during this period moved inversely 78.62% of the time.

You can see how the relationship between the SP500 and VIX evolves prospectively by entering http://finance.yahoo.com/quotes/^GSPC,^VIX into your web browser’s address field.