Author Archives: LTN

Student Loan Debt

Student Loan Debt by Age Group

Federal Reserve Bank of New York, March 29, 2013

First Official Three-Year Student Loan Default Rates Published

U.S. Department of Education, September 28, 2012

Too Big to Fail: Student debt hits a trillion

Consumer Financial Protection Bureau, Rohit Chopra, March 21, 2012

Grading Student Loans

Liberty Street Economics, Federal Reserve Bank of New York, Meta Brown, Andrew Haughwout, Donghoon Lee, Maricar Mabutas, and Wilbert van der Klaauw, March 5, 2012.

Degreeless in Debt: What Happens to Borrowers Who Drop Out

Education Sector, Mary Nguyen, February 23, 2012

Don’t Count on Settling Those Student Loans

The Atlantic, Megan McArdle, June 10, 2011.

Grodal on Macroeconomics

Birgit Grodal, Comment on L. H. Summers, The Scientific Illusion in Empirical Macroeconomics. Scandinavian Journal of Economics 93 (2), 155-159, 1991.

Summary

According to Grodal, the reason why sophisticated macro-econometrics has little impact is that the underlying macroeconomic theory is insufficiently developed.

Macroeconomic models are based on unwarranted simplifications. They use one or a few commodities, one or a few representative consumers and producers, perhaps a public sector, and simple institutional arrangements. Yet the conclusions drawn from these models are treated as if they hold in economies with many interacting agents.

There is no basis in economic theory for believing that these models should give a good description of the way an economy with many agents operates.

Only in economies where all consumers have identically homothetic preferences can the demand side can be represented by a representative consumer.

That is true if all income distributions are allowed. If only a fixed income distribution is considered, then it may be possible to represent the demand side by a representative consumer, but only under severe restrictions on the preferences of the individual agents. In this case, the welfare implications of economic policy for the representative consumer can be the opposite of the welfare implications for all the original consumers.

Thus it is not surprising that macroeconomic relations derived from theory are usually rejected empirically. The only way to obtain better empirical macroeconomic models is to develop better macroeconomic theory.

Grodal recommends studying the distribution of agents’ characteristics and using it to derive conclusions about aggregate behavior.

N(d1) and N(d2)

No, N(d1) is not the probability of exercise.

In one of my classes I derived the formulas for the values of standard options and various digital options in the Black-Scholes models, the point being to illustrate various concepts – the state price process, risk-adjusted probabilities, and the use of different numeraires.

We got into an argument about the meaning of N(d1) and N(d2).

I published a paper about this a number of years ago in Revue Finance, the journal of the French finance association: “Understanding N(d1) and N(d2): Risk-Adjusted Probabilities in the Black-Scholes Model,” Revue Finance (Journal of the French Finance Association) 14 (1993), 95-106. [Abstract][Abstract on the journal’s website][Paper (pdf)]

The paper explains N(d1) and N(d2) and relates them to the single-period and multi-period binomial models.

After so many years, people are still puzzling over this.

What is N(d1)? Well, N(d2) is the risk-adjusted probability of exercise. N(d1) is something else: It is the factor by which the present value of contingent receipt of the stock (contingent upon exercise) falls short of the current stock price.

Mathematical Genealogy

My Ph.D. is in economics, not in mathematics, but I did write a master’s thesis in differential topology, some of which was published in: Transversality and the Inverse Image of a Submanifold with Corners. Mathematica Scandinavica 49 (1981), 211-221.

My thesis advisor was Vagn Lundsgaard Hansen, then at the University of Copenhagen, now at the Technical University of Denmark. Through him, I trace my mathematical ancestry back to Birkhoff, and possibly all the way back to Poisson, Lagrange, Euler, Johan and Jacob Bernoulli, and, finally, Leibnitz.

Apart from being my advisor, Lundsgaard Hansen was also my teacher in first-year undergraduate mathematical analysis. The course started with an unforgettably vivid exposition of the topology of metric spaces — pulling back open and closed sets and mapping compact sets forward and so on. Later on I took his course in algebraic topology, where the main agenda was to use category theory to translate topological questions into easier algebraic ones.

Category theory was apparently invented by Samuel Eilenberg together with Saunders Mac Lane.

Samuel Eilenberg

Samuel Eilenberg

This bust of Samuel Eilenberg stands in the corner of the Faculty Lounge in the Columbia Mathematics Department. Eilenberg was a professor in the department from 1947 and until his retirement in 1982.

Instantaneous Arbitrage and the CAPM

This working paper from 2004–2006 studies the concept of instantaneous arbitrage in continuous time and its relation to the instantaneous CAPM. Absence of instantaneous arbitrage is equivalent to the existence of a trading strategy which satisfies the CAPM beta pricing relation in place of the market. Thus the difference between the arbitrage argument and the CAPM argument in Black and Scholes (1973) is this: the arbitrage argument assumes that there exists some portfolio satisfying the CAPM equation, whereas the CAPM argument assumes, in addition, that this portfolio is the market portfolio.

Instantaneous Arbitrage and the CAPM