An Identity Crisis for the Variable ‘R’

By Christopher Faille

It might be the subject of a Sesame Street episode.  “R is an important letter.  It stands for Rate and Return and Risk-Free and lots of other words!”  Yet, like Oscar the Grouch if deprived of his garbage can, R has lost its fixed abode.

Much of the mathematics of finance over the last forty years has involved equations that require a variable “r,” the risk free rate of return.   The famous, or notorious, Black-Scholes model for the price of an option depends upon just five variables: the price of the underlying stock; the option’s exercise price; the time until expiration; the standard deviation of the stock price; and the risk-free rate of return.  Of these five, three can be known with certainty: the stock price is in the morning papers, the exercise price is defined by the terms of the contract, and the time until expiration requires only the possession of a calendar.

The critical two variables, the two that require some inference, are: the standard deviation (volatility), and the risk-free rate of interest.  The easy assumption is that the standard deviation is that dictated by the “normal distribution,” a Bell curve, though that has come under a good deal of critical fire of late.  None of that is our concern at the moment.

Our concern, rather, is the risk-free rate of interest.  It is after all a common sense notion that a decision to put one’s money into any specific investment, tying it up in call options for example, must be judged in the context of the other choices that are available – what else one might have done with it.  The use of a risk-free rate, an r, in such equations incorporates that intuition in mathematical form.  But how is r fixed when the calculation is run?  Generally, by the use of U.S. Treasury bills, which were regarded (until very recently) as risk free instruments.

Likewise, consider for a moment the Sharpe ratio, one of the most common means of measuring and evaluating the performance of hedge funds, or for that matter of diamonds.  The numerator of the Sharpe ratio is precisely the difference between the fund’s return and a risk-free instrument’s return (the denominator is the standard deviation, or volatility, of that numerator.)  William Sharpe himself used the yield on 10-year Treasury bonds as a proxy for the risk-free rate in the 1966 paper that gave rise to its use.  But since then, the shorter-term Treasury securities have become the standard here, too.

So: what next?  In the wake of an inane months-long political deadlock that resulted in widespread discussion of the possibility of default, even of the idea that the President should construe the U.S. Constitution in a previously unheard-of manner in order to ward off that danger; in the wake of a downgrade from Standard & Poor’s, and it may well be a looming downgrade from one of the other of the three major players on the small credit-rating stage; what if anything is now a reliable proxy for the risk-free rate?  Either all such formulae have to be abandoned or something has to fix r!

Asking this question produces the somewhat surprising answer that U.S. Treasuries may have some life left in them as a risk-free proxy.  Howard Marks, co-founder of Oaktree Capital and the author of some widely-admired memos to its clients over a period of years, in a recent e-mail exchange, said (“speaking as a non-quant”) that he doubts anyone believes there is any “riskless place in the world,” but that short T-bills remain “essentially” riskless.

Irene Aldridge, the author of High-Frequency Trading: A Practical Guide to Algorithmic Strategies and Trading Systems (2009), and one of the participants in last year’s “Battle of the Quants,” has a different take.  She said that she would “lean toward” using the bonds of the most secure U.S. municipalities as the new risk-free benchmark.  Some states in the U.S. have constitutional requirements that explicitly mandate procedures for the issuance and repayment of state and municipal obligations, making those bonds and the financial realities behind them more stable and transparent than the bonds of the U.S. government.  She also suggests the returns of still-triple-A sovereigns, especially United Kingdom and Swiss debt, as a possible proxy for r going forward.

Yet Aldridge is not very far from Marks’ view.  She, too, suggests that there may not be an immediate change in either of those directions.  She referred an inquirer to an article she wrote recently on this subject, which ends with the observation that TED, the spread between U.S. T-bills and the London Interbank Offering Rate (LIBOR), is about 18 times lower than it was at the worst of the crisis in the fall of 2008.  This indicates that the market still believes the possibility of a U.S. default is quite small, “and the perfect AAA rating may be restored in the not-so-distant future.”

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  1. Simon Connel
    August 16, 2011 at 3:53 am

    The Omega Ratio appears to be a far more effective method of quantifying hedge fund returns. It captures all the information in the returns distribution, including all the higher moments and does not assume normally distributed returns. There’s a strong argument, at the very least, for Omega to be given as much attention as Sharpe in the hedge fund space. See for more detail

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