Category Archives: Finance

Arrow-Pratt method vis–à–vis the “exact” method for calculating risk premiums

I received an email from a Finance 4335 student asking for further clarification of the two methods for calculating risk premiums which we covered in class last Thursday. Under the so-called “exact” method, one 1) calculates expected utility, 2) sets expected utility equal to the utility of the certainty-equivalent of wealth, 3) solves for the certainty-equivalent of wealth, and 4) obtains the risk premium by calculating the difference between expected wealth and the certainty-equivalent of wealth. A numerical example of this approach is provided on page 3 of the lecture note. On the other hand, the Arrow-Pratt method is an alternative method for calculating the risk premium which is based upon Taylor series approximations of expected utility of wealth and the utility of the certainty equivalent of wealth (the derivation for which appears on pp. 16-18 of this same lecture note). Both of these approaches for calculating risk premiums are perfectly acceptable for purposes of Finance 4335.

The value added of Arrow-Pratt is that it analytically demonstrates how risk premiums depend upon two factors: 1) the magnitude of the risk itself (as indicated by variance), and 2) the degree to which the decision-maker is risk averse. For example, we showed in class on Thursday that the Arrow-Pratt coefficient for the logarithmic investor (for whom U(W) = ln W) is twice as large as the Arrow-Pratt coefficient for the square root investor (for whom U(W) = W.5); 1/W for the logarithmic investor compared with .5/W for the square root investor. Thus, the logarithmic investor behaves in a more risk averse than the square root investor; other things equal, the logarithmic investor will prefer to allocate less of her wealth to risky assets and buy more insurance than the the square root investor. Another important insight yielded by Arrow-Pratt (at least for the utility functions considered so far in Finance 4335) is the notion of decreasing absolute risk aversion. Other things equal, investors become less (more) risk averse as wealth increases (decreases).

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

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 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 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.”

It Has Been a Near-Perfect Investing Environment. But It May End Soon.

As this article from today’s WSJ points out, the “near-perfect” environment is in reference to a two decade-long financial market anomaly (dating back to the late 1990s) in which stock and bond  have tended to move in opposite directions.  Thus, investors have been able to (quite effectively) hedge risk by owning both asset classes.
For two decades, government bonds have provided what amounts to free insurance against stock-market struggles. But that’s a historical anomaly.

Stocks Weren’t Made for Social Climbing

Superb WSJ op-ed by (former hedge fund manager turned author) Andy Kessler about the corporate social responsibility “gospel” and the importance of profit; Kessler’s essay is essentially an homage to Milton Friedman’s famous 1970 New York Times Magazine article entitled “The Social Responsibility of Business Is to Increase Its Profits.”

Profits are the proper gauge of a company’s value to consumers—and to society.

More on Bitcoin specifically and cryptocurrencies generally…

For more information concerning Bitcoin (other than the Motley Fool articles that I posted a few minutes ago), I recommend reading the Wikipedia article at Technically, Bitcoin is an “application” of so-called “Blockchain” technology. Of course, I wish that I would have had the foresight to have purchased (for that matter, even “mined”) Bitcoin starting back in 2009, but such is the nature of uncertainty – what may seem  “obvious” now seemed borderline silly back then.

The level of volatility in Bitcoin spot and futures prices can be quite breathtaking at times.  Indeed, daily volatility of Bitcoin is roughly ten times the daily volatility of the SP 500 stock index (see WSJ Daily Shot, 11-January-2018). While there may be some entertainment value in buying and selling cryptocurrencies in the spot and futures markets, these instruments are clearly not suitable for most  investors.

For more on Blockchain, I recommend watching either NYU finance professor David Yermack (cf. or Duke finance professor Cam Harvey (cf. – they are the best of the best finance experts on this topic.

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

Besides going over the syllabus during the first day of class on Tuesday, January 9, we will also discuss a “real world” example of financial risk. Specifically, we will look at the relationship between short-term stock market volatility (as indicated by the CBOE Volatility Index (VIX)) and returns (as indicated by the SP500 stock market index).

As indicated by this graph from page 25 of next Tuesday’s lecture note, daily percentage changes on closing prices for VIX and the SP500 are strongly negatively correlated. In the graph above, the y-axis variable is the daily return on the SP500, whereas the x-axis variable is the daily return on the VIX. The blue points represent 7,056 daily observations on these two variables, spanning the time period from January 2, 1990 through December 29, 2017. When we fit a regression line through this scatter diagram, we obtain the following equation:

{R_{SP500}} = 0.00058 - 0.1187{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 (-0.1187) indicates that on average, daily VIX returns during this time period were inversely related to the 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, 49.2%) can be statistically “explained” by changes in volatility, and the correlation between {R_{SP500}} and {R_{VIX}} comes out to -0.7014. While a correlation of -0.7014 does not imply that {R_{SP500}} and {R_{VIX}} will always move in opposite directions, it does indicate 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% of the time.

You can see how the relationship between the SP500 and VIX evolves prospectively by entering^GSPC,^VIX into your web browser’s address field.