[Diebold, Et Al] Converting 1-Day Volatility to H-Day Volatility - Scaling by Root h is Worse Than...

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FinancialIns t i tu t ionsCenter

Converting 1-Day Volatility toh-Day Volatility: Scaling by

Is Worse than You Think

byFrancis X . DieboldA n d r e w Hi c k m a nAt sush i InoueTil Schuermann

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T HE WHARTON F INANCIAL INSTIT UT IONS CENT ER

T he Wharton Financial Institutions Center provides a multi-disciplinary research approach tothe problems and opportunities facing the financial services industry in its search forcompetitive excellence. T he Center's research focuses on the issues related to managing risk at the firm level as well as ways to improve productivity and performance.

T he Cen ter fosters the developmen t of a community of faculty, visiting scholars and Ph.D .candidates whose research interests complement and support the mission of the Center. T heCenter works closely with industry executives and practitioners to ensure that its research isinformed by the operating realities and compe titive demands facing industry participants asthey pursue competitive excellence.

Copies of the working papers summarized here are available from the Ce nte r. If you wouldlike to learn more about the Cen ter or become a member of our research community, please

let us know of your inte rest.

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The Working Paper Series is made possible by a generousgrant from the Alfred P. Sloan Foundation


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Converting 1-Day Volatility to h-Day Volatility:

Scaling by is Worse than You Think

Francis X. Diebold Andrew HickmanUniversity of Pennsylvania Oliver, Wyman and [email protected] [email protected]

Atsushi Inoue Til SchuermannUniversity of Pennsylvania Oliver, Wyman and [email protected] [email protected]

December 1996

This Print: July 3, 1997

Copyright 1997 F.X. Diebold, A. Hickman, A. Inoue, and T. Schuermann. This paper isavailable on the World Wide Web at http://www.ssc.upenn.edu/~diebold/ and may be freelyreproduced for educational and research purposes, so long as it is not altered, this copyrightnotice is reproduced with it, and it is not sold for profit.

Correspondence to:F.X. DieboldDepartment of EconomicsUniversity of Pennsylvania3718 Locust Walk Philadelphia, PA 19104-6297

Abstract: We show that the common practice of converting 1-day volatility estimates to h-day estimates by scaling by is inappropriate and produces overestimates of the variabilityof long-horizon volatility. We conclude that volatility models are best tailored to tasks: if interest centers on long-horizon volatility, then a long-horizon volatility model should beused. Economic considerations, however, confound even that prescription and point toimportant directions for future research.

Acknowledgments: The first author would like to thank the Oliver Wyman Institute for itshospitality and inspiration. Peter Christoffersen, Paul Kupiec, Jose Lopez and TonySantomero provided helpful discussion, as did participants at the Federal Reserve SystemConference on Financial Market Structure in a Global Environment, but all errors are oursalone. We gratefully acknowledge support from the National Science Foundation, the SloanFoundation, and the University of Pennsylvania Research Foundation.


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Chew (1994) provides insightful early discussion.1

A leading example is Bankers Trusts RAROC system; see Falloon (1995).2

See, for example, Smithson and Minton (1996a, b) and J.P. Morgan (1996).3

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1. Motivation and Background

What is the relevant horizon for risk management? This obvious question has no

obvious answer. Horizons such as 7 to 10 days for equity and foreign exchange, and longer1

horizons such as 30 days for interest rate instruments, are routinely discussed. In fact,

horizons as long as a year are not uncommon. Operationally, risk is often assessed at a 1-day2

horizon, and shorter (intra day) horizons have even been discussed. Short-horizon risk

measures are converted to other horizons, such as 10-day or 30-day, by scaling. For 3

example, to obtain a 10-day volatility we multiply the 1-day volatility by . Moreover, the

horizon conversion is often significantly longer than 10 days. Many banks, for example, link

trading volatility measurement to internal capital allocation and risk-adjusted performance

measurement schemes, which rely on annual volatility estimates. The temptation is to scale 1-

day volatility by .

The routine and uncritical use of scaling is also widely accepted by regulators. For

example, the Basle Committee's January 1996 "Amendment to the Capital Accord to

Incorporate Market Risks" features it prominently. Specifically, the amendment requires a

10-day holding period and advises conversion by scaling:

In calculating value at risk, an instantaneous price shock equivalent to a 10 daymovement in prices is to be used, i.e. the minimum "holding period" will be tentrading days. Banks may use value-at-risk numbers calculated according toshorter holding periods scaled up to ten days by the square root of time ...

(p. 44, section B.4, paragraph c)


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In this paper we sound an alarm: such scaling is inappropriate and misleading. We

show in section 2 that converting volatilities by scaling is statistically appropriate only under

strict conditions that are routinely violated by high-frequency (e.g., 1-day) asset returns. In

section 3, we provide a detailed illustrative example of the failure of scaling. We conclude in

section 4, in which we note that, even in the unlikely event that the conditions for its statistical

legitimacy are met, scaling is nevertheless problematic for economic reasons associated with

fluctuations in portfolio composition .

2. The Links Between Short-Horizon and Long-Horizon Risk: Statistical

Considerations

Scaling Works in iid Environments but Fails Otherwise

Here we describe the restrictive environment in which scaling is appropriate.

Let be a log price at time t, and suppose that changes in the log price are independently and

identically distributed,

Then the 1-day return is

with standard deviation . Similarly, the h-day return is

with variance and standard deviation . Hence the " rule": to convert a 1-day

standard deviation to an h-day standard deviation, simply scale by . For some


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