Category Archives: Finance

How Cash-Strapped Chicago Snagged a Triple-A Rating for Its New Bonds

Since the city of Chicago is apparently no longer considered to be insurable by the bond insurance industry (cf., it has had to turn to other means for making its debt attractive to investors. The latest scheme involves issuing new bonds through a new (and separate) legal entity called the Sales Tax Securitization Corporation (STSC). The bonds offered by STSC are collateralized by a dedicated first claim to the city’s sales-tax revenue. Apparently similar strategies have been employed two years ago by the city of Detroit, throughout the past decade by Puerto Rico and 40-some years ago by New York City.

Interestingly, the bond rating agencies are somewhat split about the extent to which a so-called “dedicated first claim” to Chicago’s sales-tax revenues would obtain in the event of default; this divergence of opinion is reflected by the ratings given on these bonds; e.g., Fitch and Kroll gave STSC a AAA rating, whereas S&P scores it two grades lower.

Chicago has created a new company to sell the debt, offering a tempting pledge to investors: a dedicated first claim to the city’s sales taxes.

Is Da Vinci’s ‘Salvator Mundi’ Worth $450 Million or $454,680?

Fascinating geometric mean return calculation for Da Vinci’s ‘Salvator Mundi’; at its recent $450 million sales price, the annual rate of return for this artwork over the course of five centuries comes out to around 1.35%.

“Salvator Mundi” sold at Christie’s for more than $450 million. If we were to regard this work of art as an investment, has the Leonardo generated a good return since the master painted it?

A Federal Guarantee that is Sure to Go Broke

See the (November 2014) Wall Street Journal article entitled “A Federal Guarantee Is Sure to Go Broke” and related article from November 2015 entitled “Moody’s Predicts PBGC Premiums Will Become Unaffordable“.

Think of PBGC as essentially the FDIC of private pensions. Thus, the analysis the flowchart shown at the bottom of my “On the economics of financial guarantees” blog post concerning how FDIC guarantees bank deposits applies here; in the diagram from that posting, simply replace “FDIC” in the diagram with “PBGC”, and in place of “Bank” and “Depositors”, substitute “Company offering private pension to Workers” and “Workers”.

Quoting from the above referenced WSJ article:

How is the PBGC insurance program doing on its 40th anniversary? Well, it is dead broke. Its net worth is negative $62 billion as of the end of September. That is even more broke than it was a year ago, when its net worth was negative $36 billion… The PBGC has total assets of $90 billion but total liabilities of $152 billion. So its assets are a mere 59% of its liabilities. Put another way, its capital-to-asset ratio is negative 69%.

Why does the government have such a pathetic record at guaranteeing other people’s debts? It isn’t that Washington wasn’t warned. “My son, if you have become surety for your neighbor, have given your pledge for a stranger, you are snared in the utterance of your lips,” reads Proverbs 6: 1-2.

Merton: Applications of Option-Pricing Theory (shameless self-promotion alert)…

Now that we have begun our study of the famous Black-Scholes-Merton option pricing formula, it’s time for me to shamelessly plug a journal article that I published early in my academic career which Robert C. Merton cites in his Nobel Prize lecture (Merton shared the Nobel Prize in economics in 1997 with Myron Scholes “for a new method to determine the value of derivatives”).

Here’s the citation (and link) to Merton’s lecture:

Merton, Robert C., 1998, Applications of Option-Pricing Theory: Twenty-Five Years Later, The American Economic Review, Vol. 88, No. 3 (Jun. 1998), pp. 323-349.

See page 337, footnote 11 of Merton’s paper for the reference to Neil A. Doherty and James R. Garven (1986)… (Doherty and I “pioneered” the application of a somewhat modified version of the Black-Scholes-Merton model to the pricing of insurance; thus Merton’s reference to our Journal of Finance paper in his Nobel Prize lecture)…

On the economics of financial guarantees

On pp. 13-19 of the Derivatives Theory, part 3 lecture note and in Problem Set 9, we study how credit enhancement of risky debt works. Examples of credit enhancement in the real world include federal deposit insurance, public and private bond insurance, pension insurance, mortgage insurance, government loan guarantees, etc.; the list goes on.

Most credit enhancement schemes work in the fashion described below. Creditors loan money to “risky” borrowers who are at risk for defaulting on promised payments. Although borrowers promise to pay back $B at t=1, they may default (in whole or in part) and the shortfall to creditors resembles a put option with t=1 payoff of -Max[0, B-F]. Therefore, without credit enhancement, the value of risky debt at t=0 is

V(D) = B{e^{ - r}} - V(Max[0,B - F]).

However, when credit risk is intermediated by a guarantor (e.g., an insurance company or government agency), credit risk gets transferred to the guarantor, who receives an upfront “premium” worth V(Max[0,B - F]) at t=0 in exchange for having to cover a shortfall of Max[0,B - F] that may occur at t=1. If all credit risk is transferred to the guarantor (as shown in the graphic provided below), then from the creditors’ perspective it is as if the borrowers have issued riskless debt. Therefore, creditors charge borrowers the riskless rate of interest and are paid back what was promised from two sources: 1) borrowers pay D = B - Max[0,B - F], and 2) the guarantor pays Max[0,B - F].


Credit risk teaching note and spreadsheet

I’d like to call your attention to my credit risk teaching note @ and my credit risk spreadsheet @ This teaching note provides a brief synopsis of today’s presentation of the credit risk topic, and the spreadsheet provides the code required in order to produce the table on page 8 of the teaching note.

On the role of replicating portfolios in the pricing of financial derivatives

Replicating portfolios play a central role in terms of pricing financial derivatives. Here is a succinct summary from last Thursday’s class meeting:

  1. Buying forward is equivalent to buying the underlying on margin, and selling forward is equivalent to shorting the underlying and lending money. Like options, forwards and futures are priced by pricing the replicating portfolio and invoking the “no-arbitrage” condition. If the forward/futures price it too low, then one can earn positive returns with zero risk and zero net investment by buying forward, shorting the underlying and lending money. Similarly, if the forward futures price is too high, one can earn positive returns with zero risk and zero net investment by selling forward and buying the underlying with borrowed money. This is commonly referred to as “riskless arbitrage”; it’s riskless because you’re perfectly hedged, and it’s arbitrage because you are buying low and selling high.
  2. The replicating portfolio for a call option is a margined investment in the underlying. For example, in my teaching note entitled “A Simple Model of a Financial Market”, I provide a numerical example where the interest rate is zero, there are two states of the world, a bond which pays off $1 in both states is worth $1 today, and a stock that pays off $2 in one state and $.50 in the other state is also worth one dollar. In that example, the replicating portfolio for a European call option with an exercise price of $1 consists of 2/3 of 1 share of stock (costing $0.66) and a margin balance consisting of a short position in 1/3 of a bond (which is worth -$0.33). Thus, the value of the call option is $0.66 – $0.33 = $0.33.
  3. Since the replicating portfolio for a call option is a margined investment in the underlying, it should come as no surprise that the replicating portfolio for a put option consists of a short position in the underlying combined with lending. Thus, in order to price the put, you need to determine and price the components of the replicating portfolio; we will begin class on Tuesday, 10/24 by determining the the relative weightings (delta and beta) for the put’s replicating portfolio.

Insights gleaned from our coverage of portfolio and capital market theory

The topics covered during the course of the last couple of Finance 4335 class meetings (portfolio and capital market theory) rank among the most important finance topics; after all, the scientific foundations for these topics won Nobel prizes for Markowitz (portfolio theory) and Sharpe (capital market theory). The following outline pretty much summarizes what we covered in class on Thursday, October 12 and Tuesday, October 17:

  • Portfolio Theory
    1. Mean-variance efficiency
    2. Portfolio Mean-Variance calculations
    3. Minimum variance portfolio (n = 2 case)
    4. Efficient frontier (n = 2 case under various correlation assumptions)
  • Capital Market Theory
    1. Efficient frontiers with multiple number (“large” n) of risky assets (aka the “general” case)
    2. Portfolio allocation under the general case
      • degree of risk aversion/risk tolerance determines how steeply sloped indifference curves are
      • indifference curves for investors with high (low) degrees of risk tolerance (aversion) are less steeply sloped than indifference curves than for investors with low (high) degrees of risk tolerance (aversion)).
      • Optimal portfolios (i.e., portfolios that maximize expected utility) occur at points of tangency between indifference curves and efficient frontier.
    3. Introduction of a risk-free asset simplifies the portfolio selection problem since the efficient frontier is now a straight line rather than an ellipse in E({r_p}), {\sigma _p} space. The same selection principle holds as in the previous point (point 2); i.e., investors determine optimal portfolio by finding the tangency between highest indifference curve and the efficient frontier. The point of tangency occurs on the capital market line (CML) where the Sharpe ratio is maximized; everyone chooses some combination of the risk-free asset and the market portfolio, and risk tolerance determines whether the point of tangency involves either a lending (low risk tolerance) or borrowing (high risk tolerance) allocation strategy.
    4. The security market line (SML), aka the CAPM, is deduced by arbitrage arguments. Specifically, it must be the case that all risk-return trade-offs (as measured by the ratio of “excess” return (E({r_j}) - {r_f}) from investing in a risky rather than risk-free asset, divided by the risk taken on by the investor ({\sigma _{j,M}}) are the same. If not, then there will be excess demand for investments with more favorable risk-return trade-offs and excess supply for investments with less favorable risk-return trade-offs). “Equilibrium” occurs when markets clear; i.e., when there is neither excess demand or supply, which is characterized by risk-return ratios being the same for all possible investments. When this occurs, then the CAPM obtains: E({r_j}) = {r_f} + {\beta _j}(E({r_M}) - {r_f}).


A Summary of Portfolio and Capital Market Theory (source: The Royal Swedish Academy of Sciences)

During tomorrow’s Finance 4335 class meeting, we will complete our study of portfolio and capital market theory. The portfolio theory topic won Professor Harry Markowitz the Nobel Prize in Economics in 1990, and Professor William F. Sharpe shared the 1990 Nobel Prize with Markowitz for his work on capital market theory.

The very best summary of portfolio theory and capital market theory that I am aware of appears as part of an October 16, 1990 press release put out  by The Royal Swedish Academy of Sciences in commemoration of the prizes won by Markowitz and Sharpe (see  I have included an appropriately edited version of that press release below (it is important to also note that University of Chicago Finance Professor Merton Miller was cited that same year along with Markowitz and Sharpe for his work on the theory of corporate finance; I include below only the sections of the Royal Swedish Academy press release pertaining to the work by Messrs. Markowitz and Sharpe on the topics of portfolio and capital market theory):


Financial markets serve a key purpose in a modern market economy by allocating productive resources among various areas of production. It is to a large extent through financial markets that saving in different sectors of the economy is transferred to firms for investments in buildings and machines. Financial markets also reflect firms’ expected prospects and risks, which implies that risks can be spread and that savers and investors can acquire valuable information for their investment decisions.

The first pioneering contribution in the field of financial economics was made in the 1950s by Harry Markowitz who developed a theory for households’ and firms’ allocation of financial assets under uncertainty, the so-called theory of portfolio choice. This theory analyzes how wealth can be optimally invested in assets which differ in regard to their expected return and risk, and thereby also how risks can be reduced.

A second significant contribution to the theory of financial economics occurred during the 1960s when a number of researchers, among whom William Sharpe was the leading figure, used Markowitz’s portfolio theory as a basis for developing a theory of price formation for financial assets, the so-called Capital Asset Pricing Model, or CAPM.

Harrv M. Markowitz
The contribution for which Harry Markowitz now receives his award was first published in an essay entitled “Portfolio Selection” (1952), and later, more extensively, in his book, Portfolio Selection: Efficient Diversification (1959). The so-called theory of portfolio selection that was developed in this early work was originally a normative theory for investment managers, i.e., a theory for optimal investment of wealth in assets which differ in regard to their expected return and risk. On a general level, of course, investment managers and academic economists have long been aware of the necessity of taking returns as well as risk into account: “all the eggs should not be placed in the same basket”. Markowitz’s primary contribution consisted of developing a rigorously formulated, operational theory for portfolio selection under uncertainty – a theory which evolved into a foundation for further research in financial economics.

Markowitz showed that under certain given conditions, an investor’s portfolio choice can be reduced to balancing two dimensions, i.e., the expected return on the portfolio and its variance. Due to the possibility of reducing risk through diversification, the risk of the portfolio, measured as its variance, will depend not only on the individual variances of the return on different assets, but also on the pairwise covariances of all assets.

Hence, the essential aspect pertaining to the risk of an asset is not the risk of each asset in isolation, but the contribution of each asset to the risk of the aggregate portfolio. However, the “law of large numbers” is not wholly applicable to the diversification of risks in portfolio choice because the returns on different assets are correlated in practice. Thus, in general, risk cannot be totally eliminated, regardless of how many types of securities are represented in a portfolio.

In this way, the complicated and multidimensional problem of portfolio choice with respect to a large number of different assets, each with varying properties, is reduced to a conceptually simple two-dimensional problem – known as mean-variance analysis. In an essay in 1956, Markowitz also showed how the problem of actually calculating the optimal portfolio could be solved. (In technical terms, this means that the analysis is formulated as a quadratic programming problem; the building blocks are a quadratic utility function, expected returns on the different assets, the variance and covariance of the assets and the investor’s budget restrictions.) The model has won wide acclaim due to its algebraic simplicity and suitability for empirical applications.

Generally speaking, Markowitz’s work on portfolio theory may be regarded as having established financial micro analysis as a respectable research area in economic analysis.

William F. Sharpe

With the formulation of the so-called Capital Asset Pricing Model, or CAPM, which used Markowitz’s model as a “positive” (explanatory) theory, the step was taken from micro analysis to market analysis of price formation for financial assets. In the mid-1960s, several researchers – independently of one another – contributed to this development. William Sharpe’s pioneering achievement in this field was contained in his essay entitled, Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk (1964).

The basis of the CAPM is that an individual investor can choose exposure to risk through a combination of lending-borrowing and a suitably composed (optimal) portfolio of risky securities. According to the CAPM, the composition of this optimal risk portfolio depends on the investor’s assessment of the future prospects of different securities, and not on the investors’ own attitudes towards risk. The latter is reflected solely in the choice of a combination of a risk portfolio and risk-free investment (for instance treasury bills) or borrowing. In the case of an investor who does not have any special information, i.e., better information than other investors, there is no reason to hold a different portfolio of shares than other investors, i.e., a so-called market portfolio of shares.

What is known as the “beta value” of a specific share indicates its marginal contribution to the risk of the entire market portfolio of risky securities. Shares with a beta coefficient greater than 1 have an above-average effect on the risk of the aggregate portfolio, whereas shares with a beta coefficient of less than 1 have a lower than average effect on the risk of the aggregate portfolio. According to the CAPM, in an efficient capital market, the risk premium and thus also the expected return on an asset, will vary in direct proportion to the beta value. These relations are generated by equilibrium price formation on efficient capital markets.

An important result is that the expected return on an asset is determined by the beta coefficient on the asset, which also measures the covariance between the return on the asset and the return on the market portfolio. The CAPM shows that risks can be shifted to the capital market, where risks can be bought, sold and evaluated. In this way, the prices of risky assets are adjusted so that portfolio decisions become consistent.

The CAPM is considered the backbone of modern price theory for financial markets. It is also widely used in empirical analysis, so that the abundance of financial statistical data can be utilized systematically and efficiently. Moreover, the model is applied extensively in practical research and has thus become an important basis for decision-making in different areas. This is related to the fact that such studies require information about firms’ costs of capital, where the risk premium is an essential component. Risk premiums which are specific to an industry can thus be determined using information on the beta value of the industry in question.

Important examples of areas where the CAPM and its beta coefficients are used routinely, include calculations of costs of capital associated with investment and takeover decisions (in order to arrive at a discount factor); estimates of costs of capital as a basis for pricing in regulated public utilities; and judicial inquiries related to court decisions regarding compensation to expropriated firms whose shares are not listed on the stock market. The CAPM is also applied in comparative analyses of the success of different investors.

Along with Markowitz’ portfolio model, the CAPM has also become the framework in textbooks on financial economics throughout the world.

On the Determinants of Risk Aversion

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.