Option valuation a first course in financial mathematics

Finance/Mathematics

A First Course in Financial Mathematics
Option Valuation: A First Course in Financial Mathematics
provides a straightforward introduction to the mathematics and
models used in the valuation of financial derivatives. It examines
the principles of option pricing in detail via standard binomial and
stochastic calculus models. Developing the requisite mathematical
background as needed, the text introduces probability theory and
stochastic calculus at an undergraduate level.

Hugo D. Junghenn

Option Valuation
A First Course in
Financial Mathematics

Junghenn

Largely self-contained, this classroom-tested text offers a sound
introduction to applied probability through a mathematical finance
perspective. Numerous examples and exercises help readers
gain expertise with financial calculus methods and increase their
general mathematical sophistication. The exercises range from
routine applications to spreadsheet projects to the pricing of a
variety of complex financial instruments. Hints and solutions to
odd-numbered problems are given in an appendix.

A First Course in
Financial Mathematics

The first nine chapters of the book describe option valuation
techniques in discrete time, focusing on the binomial model. The
author shows how the binomial model offers a practical method
for pricing options using relatively elementary mathematical tools.
The binomial model also enables a clear, concrete exposition of
fundamental principles of finance, such as arbitrage and hedging,
without the distraction of complex mathematical constructs. The
remaining chapters illustrate the theory in continuous time, with
an emphasis on the more mathematically sophisticated Black–
Scholes–Merton model.

Option Valuation

Option Valuation

K14090

K14090_Cover.indd 1

10/7/11 11:23 AM


Option Valuation
A First Course in
Financial Mathematics


CHAPMAN & HALL/CRC
Financial Mathematics Series
Aims and scope:

The field of financial mathematics forms an ever-expanding slice of the financial sector. This series
aims to capture new developments and summarize what is known over the whole spectrum of this
field. It will include a broad range of textbooks, reference works and handbooks that are meant to
appeal to both academics and practitioners. The inclusion of numerical code and concrete realworld examples is highly encouraged.

Series Editors
M.A.H. Dempster

Dilip B. Madan

Rama Cont

Centre for Financial Research
Department of Pure
Mathematics and Statistics
University of Cambridge

Robert H. Smith School
of Business
University of Maryland

Center for Financial
Engineering
Columbia University
New York

Published Titles
American-Style Derivatives; Valuation and Computation, Jerome Detemple
Analysis, Geometry, and Modeling in Finance: Advanced Methods in Option Pricing,
 Pierre Henry-Labordère

Credit Risk: Models, Derivatives, and Management, Niklas Wagner
Engineering BGM, Alan Brace
Financial Modelling with Jump Processes, Rama Cont and Peter Tankov
Interest Rate Modeling: Theory and Practice, Lixin Wu
Introduction to Credit Risk Modeling, Second Edition, Christian Bluhm, Ludger Overbeck, and
 Christoph Wagner
Introduction to Stochastic Calculus Applied to Finance, Second Edition,
 Damien Lamberton and Bernard Lapeyre
Monte Carlo Methods and Models in Finance and Insurance, Ralf Korn, Elke Korn,
 and Gerald Kroisandt
Numerical Methods for Finance, John A. D. Appleby, David C. Edelman, and John J. H. Miller
Option Valuation: A First Course in Financial Mathematics, Hugo D. Junghenn
Portfolio Optimization and Performance Analysis, Jean-Luc Prigent
Quantitative Fund Management, M. A. H. Dempster, Georg Pflug, and Gautam Mitra
Risk Analysis in Finance and Insurance, Second Edition, Alexander Melnikov
Robust Libor Modelling and Pricing of Derivative Products, John Schoenmakers
Stochastic Finance: A Numeraire Approach, Jan Vecer
Stochastic Financial Models, Douglas Kennedy
Structured Credit Portfolio Analysis, Baskets & CDOs, Christian Bluhm and Ludger Overbeck
Understanding Risk: The Theory and Practice of Financial Risk Management, David Murphy
Unravelling the Credit Crunch, David Murphy
Proposals for the series should be submitted to one of the series editors above or directly to:
CRC Press, Taylor & Francis Group
4th, Floor, Albert House
1-4 Singer Street
London EC2A 4BQ
UK


Option Valuation

A First Course in
Financial Mathematics

Hugo D. Junghenn


CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2011 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Version Date: 20150312
International Standard Book Number-13: 978-1-4398-8912-1 (eBook - PDF)
This book contains information obtained from authentic and highly regarded sources. Reasonable
efforts have been made to publish reliable data and information, but the author and publisher cannot
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❚❖ ▼❨ ❋❆▼■▲❨

▼❛r②✱

❑❛t✐❡✱

P❛tr✐❝❦✱

❙❛❞✐❡

✈


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❈♦♥t❡♥ts

①✐

Pr❡❢❛❝❡

✶ ■♥t❡r❡st ❛♥❞ Pr❡s❡♥t ❱❛❧✉❡
✶✳✶

❈♦♠♣♦✉♥❞ ■♥t❡r❡st

✶✳✷

❆♥♥✉✐t✐❡s

✶✳✸

❇♦♥❞s

✶✳✹

❘❛t❡ ♦❢ ❘❡t✉r♥

✶✳✺

❊①❡r❝✐s❡s

✶

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✻

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✾

✷ Pr♦❜❛❜✐❧✐t② ❙♣❛❝❡s

✶✸

✷✳✶

❙❛♠♣❧❡ ❙♣❛❝❡s ❛♥❞ ❊✈❡♥ts

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✸

✷✳✷

❉✐s❝r❡t❡ Pr♦❜❛❜✐❧✐t② ❙♣❛❝❡s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✹

✷✳✸

●❡♥❡r❛❧ Pr♦❜❛❜✐❧✐t② ❙♣❛❝❡s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✻

✷✳✹

❈♦♥❞✐t✐♦♥❛❧ Pr♦❜❛❜✐❧✐t②

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✵

✷✳✺

■♥❞❡♣❡♥❞❡♥❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✷

✷✳✻

❊①❡r❝✐s❡s

✷✹

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸ ❘❛♥❞♦♠ ❱❛r✐❛❜❧❡s

✷✼

✸✳✶

❉❡✜♥✐t✐♦♥ ❛♥❞ ●❡♥❡r❛❧ Pr♦♣❡rt✐❡s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✳✷

❉✐s❝r❡t❡ ❘❛♥❞♦♠ ❱❛r✐❛❜❧❡s

✸✳✸

❈♦♥t✐♥✉♦✉s ❘❛♥❞♦♠ ❱❛r✐❛❜❧❡s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✷

✸✳✹

❏♦✐♥t ❉✐str✐❜✉t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✹

✸✳✺

■♥❞❡♣❡♥❞❡♥t ❘❛♥❞♦♠ ❱❛r✐❛❜❧❡s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✺

✸✳✻

❙✉♠s ♦❢ ■♥❞❡♣❡♥❞❡♥t ❘❛♥❞♦♠ ❱❛r✐❛❜❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✽

✸✳✼

❊①❡r❝✐s❡s

✹✶

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹ ❖♣t✐♦♥s ❛♥❞ ❆r❜✐tr❛❣❡

✷✼
✷✾

✹✸

✹✳✶

❆r❜✐tr❛❣❡

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✳✷

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❉❡r✐✈❛t✐✈❡s

✹✹

✹✳✸

❋♦r✇❛r❞s

✹✳✹

❈✉rr❡♥❝② ❋♦r✇❛r❞s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✽

✹✳✺

❋✉t✉r❡s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✾

✹✳✻

❖♣t✐♦♥s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✳✼

Pr♦♣❡rt✐❡s ♦❢ ❖♣t✐♦♥s

✹✳✽

❉✐✈✐❞❡♥❞✲P❛②✐♥❣ ❙t♦❝❦s

✹✳✾

❊①❡r❝✐s❡s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✻

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✻

✺✵

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✸

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✺

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✼

✈✐✐


✈✐✐✐

✺ ❉✐s❝r❡t❡✲❚✐♠❡ P♦rt❢♦❧✐♦ Pr♦❝❡ss❡s

✺✾

✺✳✶

❉✐s❝r❡t❡✲❚✐♠❡ ❙t♦❝❤❛st✐❝ Pr♦❝❡ss❡s✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✳✷

❙❡❧❢✲❋✐♥❛♥❝✐♥❣ P♦rt❢♦❧✐♦s

✺✾

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✳✸

❖♣t✐♦♥ ❱❛❧✉❛t✐♦♥ ❜② P♦rt❢♦❧✐♦s

✻✶

✺✳✹

❊①❡r❝✐s❡s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✻✹

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✻✻

✻ ❊①♣❡❝t❛t✐♦♥ ♦❢ ❛ ❘❛♥❞♦♠ ❱❛r✐❛❜❧❡

✻✼

✻✳✶

❉✐s❝r❡t❡ ❈❛s❡✿ ❉❡✜♥✐t✐♦♥ ❛♥❞ ❊①❛♠♣❧❡s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✻✳✷

❈♦♥t✐♥✉♦✉s ❈❛s❡✿ ❉❡✜♥✐t✐♦♥ ❛♥❞ ❊①❛♠♣❧❡s

✻✳✸

Pr♦♣❡rt✐❡s ♦❢ ❊①♣❡❝t❛t✐♦♥

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✻✾

✻✳✹

❱❛r✐❛♥❝❡ ♦❢ ❛ ❘❛♥❞♦♠ ❱❛r✐❛❜❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼✶

✻✳✺

❚❤❡ ❈❡♥tr❛❧ ▲✐♠✐t ❚❤❡♦r❡♠

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼✸

✻✳✻

❊①❡r❝✐s❡s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼✺

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼ ❚❤❡ ❇✐♥♦♠✐❛❧ ▼♦❞❡❧

✻✼
✻✽

✼✼

✼✳✶

❈♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ❇✐♥♦♠✐❛❧ ▼♦❞❡❧

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼✳✷

Pr✐❝✐♥❣ ❛ ❈❧❛✐♠ ✐♥ t❤❡ ❇✐♥♦♠✐❛❧ ▼♦❞❡❧

✼✳✸

❚❤❡ ❈♦①✲❘♦ss✲❘✉❜✐♥st❡✐♥ ❋♦r♠✉❧❛

✼✳✹

❊①❡r❝✐s❡s

✼✼

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽✵

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽✸

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽✻

✽ ❈♦♥❞✐t✐♦♥❛❧ ❊①♣❡❝t❛t✐♦♥ ❛♥❞ ❉✐s❝r❡t❡✲❚✐♠❡ ▼❛rt✐♥❣❛❧❡s

✽✾

✽✳✶

❉❡✜♥✐t✐♦♥ ♦❢ ❈♦♥❞✐t✐♦♥❛❧ ❊①♣❡❝t❛t✐♦♥

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽✳✷

❊①❛♠♣❧❡s ♦❢ ❈♦♥❞✐t✐♦♥❛❧ ❊①♣❡❝t❛t✐♦♥

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✾✷

✽✳✸

Pr♦♣❡rt✐❡s ♦❢ ❈♦♥❞✐t✐♦♥❛❧ ❊①♣❡❝t❛t✐♦♥

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✾✹

✽✳✹

❉✐s❝r❡t❡✲❚✐♠❡ ▼❛rt✐♥❣❛❧❡s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✾✻

✽✳✺

❊①❡r❝✐s❡s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✾✽

✾ ❚❤❡ ❇✐♥♦♠✐❛❧ ▼♦❞❡❧ ❘❡✈✐s✐t❡❞

✽✾

✶✵✶

✾✳✶

▼❛rt✐♥❣❛❧❡s ✐♥ t❤❡ ❇✐♥♦♠✐❛❧ ▼♦❞❡❧

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✾✳✷

❈❤❛♥❣❡ ♦❢ Pr♦❜❛❜✐❧✐t②

✾✳✸

❆♠❡r✐❝❛♥ ❈❧❛✐♠s ✐♥ t❤❡ ❇✐♥♦♠✐❛❧ ▼♦❞❡❧

✾✳✹

❙t♦♣♣✐♥❣ ❚✐♠❡s

✾✳✺

❖♣t✐♠❛❧ ❊①❡r❝✐s❡ ♦❢ ❛♥ ❆♠❡r✐❝❛♥ ❈❧❛✐♠

✾✳✻

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✵✶
✶✵✸

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✵✺

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✵✽

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✶✶

❉✐✈✐❞❡♥❞s ✐♥ t❤❡ ❇✐♥♦♠✐❛❧ ▼♦❞❡❧

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✶✹

✾✳✼

❚❤❡ ●❡♥❡r❛❧ ❋✐♥✐t❡ ▼❛r❦❡t ▼♦❞❡❧

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✶✺

✾✳✽

❊①❡r❝✐s❡s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✶✼

✶✵ ❙t♦❝❤❛st✐❝ ❈❛❧❝✉❧✉s

✶✶✾

✶✵✳✶ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✶✾

✶✵✳✷ ❈♦♥t✐♥✉♦✉s✲❚✐♠❡ ❙t♦❝❤❛st✐❝ Pr♦❝❡ss❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✷✵

✶✵✳✸ ❇r♦✇♥✐❛♥ ▼♦t✐♦♥

✶✷✷

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✵✳✹ ❱❛r✐❛t✐♦♥ ♦❢ ❇r♦✇♥✐❛♥ P❛t❤s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✷✸

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✷✻

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✷✻

✶✵✳✺ ❘✐❡♠❛♥♥✲❙t✐❡❧t❥❡s ■♥t❡❣r❛❧s
✶✵✳✻ ❙t♦❝❤❛st✐❝ ■♥t❡❣r❛❧s

✶✵✳✼ ❚❤❡ ■t♦✲❉♦❡❜❧✐♥ ❋♦r♠✉❧❛

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✵✳✽ ❙t♦❝❤❛st✐❝ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✸✶
✶✸✻


✐①
✶✵✳✾ ❊①❡r❝✐s❡s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✶ ❚❤❡ ❇❧❛❝❦✲❙❝❤♦❧❡s✲▼❡rt♦♥ ▼♦❞❡❧
✶✶✳✶ ❚❤❡ ❙t♦❝❦ Pr✐❝❡ ❙❉❊

✶✹✶

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✶✳✷ ❈♦♥t✐♥✉♦✉s✲❚✐♠❡ P♦rt❢♦❧✐♦s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✶✳✸ ❚❤❡ ❇❧❛❝❦✲❙❝❤♦❧❡s✲▼❡rt♦♥ P❉❊

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✶✳✹ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❇❙▼ ❈❛❧❧ ❋✉♥❝t✐♦♥
✶✶✳✺ ❊①❡r❝✐s❡s

✶✸✾

✶✹✶
✶✹✷
✶✹✸

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✹✻

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✹✾

✶✷ ❈♦♥t✐♥✉♦✉s✲❚✐♠❡ ▼❛rt✐♥❣❛❧❡s
✶✷✳✶ ❈♦♥❞✐t✐♦♥❛❧ ❊①♣❡❝t❛t✐♦♥

✶✺✶

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✷✳✷ ▼❛rt✐♥❣❛❧❡s✿ ❉❡✜♥✐t✐♦♥ ❛♥❞ ❊①❛♠♣❧❡s

✶✺✷

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✺✹

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✺✻

✶✷✳✸ ▼❛rt✐♥❣❛❧❡ ❘❡♣r❡s❡♥t❛t✐♦♥ ❚❤❡♦r❡♠
✶✷✳✹ ▼♦♠❡♥t ●❡♥❡r❛t✐♥❣ ❋✉♥❝t✐♦♥s

✶✺✶

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✷✳✺ ❈❤❛♥❣❡ ♦❢ Pr♦❜❛❜✐❧✐t② ❛♥❞ ●✐rs❛♥♦✈✬s ❚❤❡♦r❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✺✽

✶✷✳✻ ❊①❡r❝✐s❡s

✶✻✶

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✸ ❚❤❡ ❇❙▼ ▼♦❞❡❧ ❘❡✈✐s✐t❡❞

✶✻✸

✶✸✳✶ ❘✐s❦✲◆❡✉tr❛❧ ❱❛❧✉❛t✐♦♥ ♦❢ ❛ ❉❡r✐✈❛t✐✈❡
✶✸✳✷ Pr♦♦❢s ♦❢ t❤❡ ❱❛❧✉❛t✐♦♥ ❋♦r♠✉❧❛s
✶✸✳✸ ❱❛❧✉❛t✐♦♥ ✉♥❞❡r

P

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✻✺

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✻✼

✶✸✳✹ ❚❤❡ ❋❡②♥♠❛♥✲❑❛❝ ❘❡♣r❡s❡♥t❛t✐♦♥ ❚❤❡♦r❡♠
✶✸✳✺ ❊①❡r❝✐s❡s

✶✻✸

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✻✽

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✼✶

✶✹ ❖t❤❡r ❖♣t✐♦♥s

✶✼✸

✶✹✳✶ ❈✉rr❡♥❝② ❖♣t✐♦♥s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✹✳✷ ❋♦r✇❛r❞ ❙t❛rt ❖♣t✐♦♥s
✶✹✳✸ ❈❤♦♦s❡r ❖♣t✐♦♥s

✶✼✸

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✼✺

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✼✻

✶✹✳✹ ❈♦♠♣♦✉♥❞ ❖♣t✐♦♥s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✹✳✺ P❛t❤✲❉❡♣❡♥❞❡♥t ❉❡r✐✈❛t✐✈❡s

✶✼✼

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✼✽

✶✹✳✺✳✶ ❇❛rr✐❡r ❖♣t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✼✾

✶✹✳✺✳✷ ▲♦♦❦❜❛❝❦ ❖♣t✐♦♥s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✽✺

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✾✶

✶✹✳✺✳✸ ❆s✐❛♥ ❖♣t✐♦♥s

✶✹✳✻ ◗✉❛♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✾✺

✶✹✳✼ ❖♣t✐♦♥s ♦♥ ❉✐✈✐❞❡♥❞✲P❛②✐♥❣ ❙t♦❝❦s

✶✾✼

✶✹✳✼✳✶ ❈♦♥t✐♥✉♦✉s ❉✐✈✐❞❡♥❞ ❙tr❡❛♠

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✾✼

✶✹✳✼✳✷ ❉✐s❝r❡t❡ ❉✐✈✐❞❡♥❞ ❙tr❡❛♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✾✽

✶✹✳✽ ❆♠❡r✐❝❛♥ ❈❧❛✐♠s ✐♥ t❤❡ ❇❙▼ ▼♦❞❡❧
✶✹✳✾ ❊①❡r❝✐s❡s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✵✵

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✵✸

❆ ❙❡ts ❛♥❞ ❈♦✉♥t✐♥❣

✷✵✾

❇ ❙♦❧✉t✐♦♥ ♦❢ t❤❡ ❇❙▼ P❉❊

✷✶✺

❈ ❆♥❛❧②t✐❝❛❧ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❇❙▼ ❈❛❧❧ ❋✉♥❝t✐♦♥

✷✶✾


①

❉ ❍✐♥ts ❛♥❞ ❙♦❧✉t✐♦♥s t♦ ❖❞❞✲◆✉♠❜❡r❡❞ Pr♦❜❧❡♠s

✷✷✺

❇✐❜❧✐♦❣r❛♣❤②

✷✹✼

■♥❞❡①

✷✹✾


①✐

Pr❡❢❛❝❡
❚❤✐s t❡①t ✐s ✐♥t❡♥❞❡❞ ❛s ❛♥ ✐♥tr♦❞✉❝t✐♦♥ t♦ t❤❡ ♠❛t❤❡♠❛t✐❝s ❛♥❞ ♠♦❞❡❧s
✉s❡❞ ✐♥ t❤❡ ✈❛❧✉❛t✐♦♥ ♦❢ ✜♥❛♥❝✐❛❧ ❞❡r✐✈❛t✐✈❡s✳ ■t ✐s ❞❡s✐❣♥❡❞ ❢♦r ❛♥ ❛✉❞✐❡♥❝❡
✇✐t❤ ❛ ❜❛❝❦❣r♦✉♥❞ ✐♥ st❛♥❞❛r❞ ♠✉❧t✐✈❛r✐❛❜❧❡ ❝❛❧❝✉❧✉s✳ ❖t❤❡r✇✐s❡✱ t❤❡ ❜♦♦❦ ✐s
❡ss❡♥t✐❛❧❧② s❡❧❢✲❝♦♥t❛✐♥❡❞✿ ❚❤❡ r❡q✉✐s✐t❡ ♣r♦❜❛❜✐❧✐t② t❤❡♦r② ✐s ❞❡✈❡❧♦♣❡❞ ❢r♦♠
✜rst ♣r✐♥❝✐♣❧❡s ❛♥❞ ✐♥tr♦❞✉❝❡❞ ❛s ♥❡❡❞❡❞✱ ❛♥❞ ✜♥❛♥❝❡ t❤❡♦r② ✐s ❡①♣❧❛✐♥❡❞ ✐♥
❞❡t❛✐❧ ✉♥❞❡r t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t t❤❡ r❡❛❞❡r ❤❛s ♥♦ ❜❛❝❦❣r♦✉♥❞ ✐♥ t❤❡ s✉❜❥❡❝t✳
❚❤❡ ❜♦♦❦ ✐s ❛♥ ♦✉t❣r♦✇t❤ ♦❢ ❛ s❡t ♦❢ ♥♦t❡s ❞❡✈❡❧♦♣❡❞ ❢♦r ❛♥ ✉♥❞❡r❣r❛❞✉❛t❡
❝♦✉rs❡ ✐♥ ✜♥❛♥❝✐❛❧ ♠❛t❤❡♠❛t✐❝s ♦✛❡r❡❞ ❛t ❚❤❡ ●❡♦r❣❡ ❲❛s❤✐♥❣t♦♥ ❯♥✐✈❡rs✐t②✳
❚❤❡ ❝♦✉rs❡ s❡r✈❡s ♠❛✐♥❧② ♠❛❥♦rs ✐♥ ♠❛t❤❡♠❛t✐❝s✱ ❡❝♦♥♦♠✐❝s✱ ♦r ✜♥❛♥❝❡ ❛♥❞
✐s ✐♥t❡♥❞❡❞ t♦ ♣r♦✈✐❞❡ ❛ str❛✐❣❤t❢♦r✇❛r❞ ❛❝❝♦✉♥t ♦❢ t❤❡ ♣r✐♥❝✐♣❧❡s ♦❢ ♦♣t✐♦♥
♣r✐❝✐♥❣✳ ❚❤❡ ♣r✐♠❛r② ❣♦❛❧ ♦❢ t❤❡ t❡①t ✐s t♦ ❡①❛♠✐♥❡ t❤❡s❡ ♣r✐♥❝✐♣❧❡s ✐♥ ❞❡t❛✐❧ ✈✐❛
t❤❡ st❛♥❞❛r❞ ❜✐♥♦♠✐❛❧ ❛♥❞ st♦❝❤❛st✐❝ ❝❛❧❝✉❧✉s ♠♦❞❡❧s✳ ❖❢ ❝♦✉rs❡✱ ❛ r✐❣♦r♦✉s
❡①♣♦s✐t✐♦♥ ♦❢ s✉❝❤ ♠♦❞❡❧s r❡q✉✐r❡s ❛ ❝♦❤❡r❡♥t ❞❡✈❡❧♦♣♠❡♥t ♦❢ t❤❡ r❡q✉✐s✐t❡
♠❛t❤❡♠❛t✐❝❛❧ ❜❛❝❦❣r♦✉♥❞✱ ❛♥❞ ✐t ✐s ❛♥ ❡q✉❛❧❧② ✐♠♣♦rt❛♥t ❣♦❛❧ t♦ ♣r♦✈✐❞❡ t❤✐s
❜❛❝❦❣r♦✉♥❞ ✐♥ ❛ ❝❛r❡❢✉❧ ♠❛♥♥❡r ❝♦♥s✐st❡♥t ✇✐t❤ t❤❡ s❝♦♣❡ ♦❢ t❤❡ t❡①t✳ ■♥❞❡❡❞✱
✐t ✐s ❤♦♣❡❞ t❤❛t t❤❡ t❡①t ♠❛② s❡r✈❡ ❛s ❛♥ ✐♥tr♦❞✉❝t✐♦♥ t♦ ❛♣♣❧✐❡❞ ♣r♦❜❛❜✐❧✐t②
✭t❤r♦✉❣❤ t❤❡ ❧❡♥s ♦❢ ♠❛t❤❡♠❛t✐❝❛❧ ✜♥❛♥❝❡✮✳
❚❤❡ ❜♦♦❦ ❝♦♥s✐sts ♦❢ ❢♦✉rt❡❡♥ ❝❤❛♣t❡rs✱ t❤❡ ✜rst ♥✐♥❡ ♦❢ ✇❤✐❝❤ ❞❡✈❡❧♦♣
♦♣t✐♦♥ ✈❛❧✉❛t✐♦♥ t❡❝❤♥✐q✉❡s ✐♥ ❞✐s❝r❡t❡ t✐♠❡✱ t❤❡ ❧❛st ✜✈❡ ❞❡s❝r✐❜✐♥❣ t❤❡ t❤❡✲
♦r② ✐♥ ❝♦♥t✐♥✉♦✉s t✐♠❡✳ ❚❤❡ ❡♠♣❤❛s✐s ✐s ♦♥ t✇♦ ♠♦❞❡❧s✱ t❤❡ ✭❞✐s❝r❡t❡ t✐♠❡✮

❜✐♥♦♠✐❛❧

♠♦❞❡❧ ❛♥❞ t❤❡ ✭❝♦♥t✐♥✉♦✉s t✐♠❡✮

❇❧❛❝❦✲❙❝❤♦❧❡s✲▼❡rt♦♥

♠♦❞❡❧✳ ❚❤❡

❜✐♥♦♠✐❛❧ ♠♦❞❡❧ s❡r✈❡s t✇♦ ♣✉r♣♦s❡s✿ ❋✐rst✱ ✐t ♣r♦✈✐❞❡s ❛ ♣r❛❝t✐❝❛❧ ✇❛② t♦ ♣r✐❝❡
♦♣t✐♦♥s ✉s✐♥❣ r❡❧❛t✐✈❡❧② ❡❧❡♠❡♥t❛r② ♠❛t❤❡♠❛t✐❝❛❧ t♦♦❧s✳ ❙❡❝♦♥❞✱ ✐t ❛❧❧♦✇s ❛
str❛✐❣❤t❢♦r✇❛r❞ ❛♥❞ ❝♦♥❝r❡t❡ ❡①♣♦s✐t✐♦♥ ♦❢ ❢✉♥❞❛♠❡♥t❛❧ ♣r✐♥❝✐♣❧❡s ♦❢ ✜♥❛♥❝❡✱
s✉❝❤ ❛s ❛r❜✐tr❛❣❡ ❛♥❞ ❤❡❞❣✐♥❣✱ ✇✐t❤♦✉t t❤❡ ♣♦ss✐❜❧❡ ❞✐str❛❝t✐♦♥ ♦❢ ❝♦♠♣❧❡①
♠❛t❤❡♠❛t✐❝❛❧ ❝♦♥str✉❝ts✳ ▼❛♥② ♦❢ t❤❡ ✐❞❡❛s t❤❛t ❛r✐s❡ ✐♥ t❤❡ ❜✐♥♦♠✐❛❧ ♠♦❞❡❧
❢♦r❡s❤❛❞♦✇ ♥♦t✐♦♥s ✐♥❤❡r❡♥t ✐♥ t❤❡ ♠♦r❡ ♠❛t❤❡♠❛t✐❝❛❧❧② s♦♣❤✐st✐❝❛t❡❞ ❇❧❛❝❦✲
❙❝❤♦❧❡s✲▼❡rt♦♥ ♠♦❞❡❧✳
❈❤❛♣t❡r ✶ ❣✐✈❡s ❛♥ ❡❧❡♠❡♥t❛r② ❛❝❝♦✉♥t ♦❢ ♣r❡s❡♥t ✈❛❧✉❡✳ ❍❡r❡ t❤❡ ❢♦❝✉s
✐s ♦♥ r✐s❦✲❢r❡❡ ✐♥✈❡st♠❡♥ts✱ s✉❝❤ ♠♦♥❡② ♠❛r❦❡t ❛❝❝♦✉♥ts ❛♥❞ ❜♦♥❞s✱ ✇❤♦s❡
✈❛❧✉❡s ❛r❡ ❞❡t❡r♠✐♥❡❞ ❜② ❛♥ ✐♥t❡r❡st r❛t❡✳ ■♥✈❡st♠❡♥ts ♦❢ t❤✐s t②♣❡ ♣r♦✈✐❞❡ ❛
✇❛② t♦ ♠❡❛s✉r❡ t❤❡ ✈❛❧✉❡ ♦❢ ❛ r✐s❦② ❛ss❡t✱ s✉❝❤ ❛s ❛ st♦❝❦ ♦r ❝♦♠♠♦❞✐t②✱ ❛♥❞
♠❛t❤❡♠❛t✐❝❛❧ ❞❡s❝r✐♣t✐♦♥s ♦❢ s✉❝❤ ✐♥✈❡st♠❡♥ts ❢♦r♠ ❛♥ ✐♠♣♦rt❛♥t ❝♦♠♣♦♥❡♥t
♦❢ ♦♣t✐♦♥ ♣r✐❝✐♥❣ t❡❝❤♥✐q✉❡s✳
❈❤❛♣t❡rs ✷✱ ✸✱ ❛♥❞ ✻ ❢♦r♠ t❤❡ ❝♦r❡ ♦❢ t❤❡ ❣❡♥❡r❛❧ ♣r♦❜❛❜✐❧✐t② ♣♦rt✐♦♥ ♦❢
t❤❡ t❡①t✳ ❚❤❡ ❡①♣♦s✐t✐♦♥ ✐s s❡❧❢✲❝♦♥t❛✐♥❡❞ ❛♥❞ ✉s❡s ♦♥❧② ❜❛s✐❝ ❝♦♠❜✐♥❛t♦r✐❝s
❛♥❞ ❡❧❡♠❡♥t❛r② ❝❛❧❝✉❧✉s✳ ❆♣♣❡♥❞✐① ❆ ♣r♦✈✐❞❡s ❛ ❜r✐❡❢ ♦✈❡r✈✐❡✇ ♦❢ t❤❡ ❡❧❡✲
♠❡♥t❛r② s❡t t❤❡♦r② ❛♥❞ ❝♦♠❜✐♥❛t♦r✐❝s ✉s❡❞ ✐♥ t❤❡s❡ ❝❤❛♣t❡rs✳ ❘❡❛❞❡rs ✇✐t❤ ❛
❣♦♦❞ ❜❛❝❦❣r♦✉♥❞ ✐♥ ♣r♦❜❛❜✐❧✐t② ♠❛② s❛❢❡❧② ❣✐✈❡ t❤✐s ♣❛rt ♦❢ t❤❡ t❡①t ❛ ❝✉rs♦r②
r❡❛❞✐♥❣✳ ❲❤✐❧❡ ♦✉r ❛♣♣r♦❛❝❤ ✐s ❧❛r❣❡❧② st❛♥❞❛r❞✱ t❤❡ ♠♦r❡ s♦♣❤✐st✐❝❛t❡❞ ♥♦✲
t✐♦♥s ♦❢ ❡✈❡♥t

σ ✲✜❡❧❞

❛♥❞ ✜❧tr❛t✐♦♥ ❛r❡ ✐♥tr♦❞✉❝❡❞ ❡❛r❧② t♦ ♣r❡♣❛r❡ t❤❡ r❡❛❞❡r


①✐✐
❢♦r t❤❡ ♠❛rt✐♥❣❛❧❡ t❤❡♦r② ❞❡✈❡❧♦♣❡❞ ✐♥ ❧❛t❡r ❝❤❛♣t❡rs✳ ❲❡ ❤❛✈❡ ❛✈♦✐❞❡❞ ✉s✲
✐♥❣ ▲❡❜❡s❣✉❡ ✐♥t❡❣r❛t✐♦♥ ❜② ❝♦♥s✐❞❡r✐♥❣ ♦♥❧② ❞✐s❝r❡t❡ ❛♥❞ ❝♦♥t✐♥✉♦✉s r❛♥❞♦♠

✈❛r✐❛❜❧❡s✳
❈❤❛♣t❡r ✹ ❞❡s❝r✐❜❡s t❤❡ ♠♦st ❝♦♠♠♦♥ t②♣❡s ♦❢ ✜♥❛♥❝✐❛❧ ❞❡r✐✈❛t✐✈❡s ❛♥❞
❡♠♣❤❛s✐③❡s t❤❡ r♦❧❡ ♦❢ ❛r❜✐tr❛❣❡ ✐♥ ✜♥❛♥❝❡ t❤❡♦r②✳ ❚❤❡ ❛ss✉♠♣t✐♦♥ ♦❢ ❛♥
❛r❜✐tr❛❣❡✲❢r❡❡ ♠❛r❦❡t✱ t❤❛t ✐s✱ ♦♥❡ t❤❛t ❛❧❧♦✇s ♥♦ ✏❢r❡❡ ❧✉♥❝❤✱✑ ✐s ❝r✉❝✐❛❧ ✐♥
❞❡✈❡❧♦♣✐♥❣ ✉s❡❢✉❧ ♣r✐❝✐♥❣ ♠♦❞❡❧s✳ ❆♥ ✐♠♣♦rt❛♥t ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤✐s ❛ss✉♠♣✲
t✐♦♥ ✐s t❤❡ ♣✉t✲❝❛❧❧ ♣❛r✐t② ❢♦r♠✉❧❛✱ ✇❤✐❝❤ r❡❧❛t❡s t❤❡ ❝♦st ♦❢ ❛ st❛♥❞❛r❞ ❝❛❧❧
♦♣t✐♦♥ t♦ t❤❛t ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♣✉t✳
❉✐s❝r❡t❡✲t✐♠❡ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ❛r❡ ✐♥tr♦❞✉❝❡❞ ✐♥ ❈❤❛♣t❡r ✺ t♦ ♣r♦✈✐❞❡
❛ r✐❣♦r♦✉s ♠❛t❤❡♠❛t✐❝❛❧ ❢r❛♠❡✇♦r❦ ❢♦r t❤❡ ♥♦t✐♦♥ ♦❢ ❛ s❡❧❢✲✜♥❛♥❝✐♥❣ ♣♦rt❢♦❧✐♦✳
❚❤❡ ❝❤❛♣t❡r ❞❡s❝r✐❜❡s ❤♦✇ s✉❝❤ ♣♦rt❢♦❧✐♦s ♠❛② ❜❡ ✉s❡❞ t♦ r❡♣❧✐❝❛t❡ ♦♣t✐♦♥s ✐♥
❛♥ ❛r❜✐tr❛❣❡✲❢r❡❡ ♠❛r❦❡t✳
❈❤❛♣t❡r ✼ ✐♥tr♦❞✉❝❡s t❤❡ r❡❛❞❡r t♦ t❤❡ ❜✐♥♦♠✐❛❧ ♠♦❞❡❧✳ ❚❤❡ ♠❛✐♥ r❡s✉❧t ✐s
t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ❛ r❡♣❧✐❝❛t✐♥❣✱ s❡❧❢✲✜♥❛♥❝✐♥❣ ♣♦rt❢♦❧✐♦ ❢♦r ❛ ❣❡♥❡r❛❧ ❊✉r♦♣❡❛♥
❝❧❛✐♠✳ ❚❤❡ ♠♦st ✐♠♣♦rt❛♥t ❝♦♥s❡q✉❡♥❝❡ ✐s t❤❡ ❈♦①✲❘♦ss✲❘✉❜✐♥st❡✐♥ ❢♦r♠✉❧❛
❢♦r t❤❡ ♣r✐❝❡ ♦❢ ❛ ❝❛❧❧ ♦♣t✐♦♥✳ ❈❤❛♣t❡r ✾ ❝♦♥s✐❞❡rs t❤❡ ❜✐♥♦♠✐❛❧ ♠♦❞❡❧ ❢r♦♠
t❤❡ ✈❛♥t❛❣❡ ♣♦✐♥t ♦❢ ❞✐s❝r❡t❡✲t✐♠❡ ♠❛rt✐♥❣❛❧❡ t❤❡♦r②✱ ✇❤✐❝❤ ✐s ❞❡✈❡❧♦♣❡❞ ✐♥
❈❤❛♣t❡r ✽✱ ❛♥❞ t❛❦❡s ✉♣ t❤❡ t❤❡ ♠♦r❡ ❞✐✣❝✉❧t ♣r♦❜❧❡♠ ♦❢ ♣r✐❝✐♥❣ ❛♥❞ ❤❡❞❣✐♥❣
❛♥ ❆♠❡r✐❝❛♥ ❝❧❛✐♠✳
❈❤❛♣t❡r ✶✵ ❣✐✈❡s ❛♥ ♦✈❡r✈✐❡✇ ♦❢ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✱ ❝♦♥str✉❝ts t❤❡ ■t♦ ✐♥✲
t❡❣r❛❧ ❢♦r ♣r♦❝❡ss❡s ✇✐t❤ ❝♦♥t✐♥✉♦✉s ♣❛t❤s✱ ❛♥❞ ✉s❡s ■t♦✬s ❢♦r♠✉❧❛ t♦ s♦❧✈❡
✈❛r✐♦✉s st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✳ ❖✉r ❛♣♣r♦❛❝❤ t♦ st♦❝❤❛st✐❝ ❝❛❧❝✉❧✉s
❜✉✐❧❞s ♦♥ t❤❡ r❡❛❞❡r✬s ❦♥♦✇❧❡❞❣❡ ♦❢ ❝❧❛ss✐❝❛❧ ❝❛❧❝✉❧✉s ❛♥❞ ❡♠♣❤❛s✐③❡s t❤❡ s✐♠✲
✐❧❛r✐t✐❡s ❛♥❞ ❞✐✛❡r❡♥❝❡s ❜❡t✇❡❡♥ t❤❡ t✇♦ t❤❡♦r✐❡s ✈✐❛ t❤❡ ♥♦t✐♦♥ ♦❢ ✈❛r✐❛t✐♦♥
♦❢ ❛ ❢✉♥❝t✐♦♥✳
❈❤❛♣t❡r ✶✶ ✉s❡s t❤❡ t♦♦❧s ❞❡✈❡❧♦♣❡❞ ✐♥ ❈❤❛♣t❡r ✶✵ t♦ ❝♦♥str✉❝t t❤❡ ❇❧❛❝❦✲
❙❝❤♦❧❡s✲▼❡rt♦♥ P❉❊✱ t❤❡ s♦❧✉t✐♦♥ ♦❢ ✇❤✐❝❤ ❧❡❛❞s t♦ t❤❡ ❝❡❧❡❜r❛t❡❞ ❇❧❛❝❦✲
❙❝❤♦❧❡s ❢♦r♠✉❧❛ ❢♦r t❤❡ ♣r✐❝❡ ♦❢ ❛ ❝❛❧❧ ♦♣t✐♦♥✳ ❆ ❞❡t❛✐❧❡❞ ❛♥❛❧②s✐s ♦❢ t❤❡ ❛♥✲
❛❧②t✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❢♦r♠✉❧❛ ✐s ❣✐✈❡♥ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥ ♦❢ t❤❡ ❝❤❛♣t❡r✳
❚❤❡ ♠♦r❡ t❡❝❤♥✐❝❛❧ ♣r♦♦❢s ❛r❡ r❡❧❡❣❛t❡❞ t♦ ❛♣♣❡♥❞✐❝❡s s♦ ❛s ♥♦t t♦ ✐♥t❡rr✉♣t
t❤❡ ♠❛✐♥ ✢♦✇ ♦❢ ✐❞❡❛s✳

❈❤❛♣t❡r ✶✷ ❣✐✈❡s ❛ ❜r✐❡❢ ♦✈❡r✈✐❡✇ ♦❢ t❤♦s❡ ❛s♣❡❝ts ♦❢ ❝♦♥t✐♥✉♦✉s✲t✐♠❡ ♠❛r✲
t✐♥❣❛❧❡s ♥❡❡❞❡❞ ❢♦r r✐s❦✲♥❡✉tr❛❧ ♣r✐❝✐♥❣✳ ❚❤❡ ♣r✐♠❛r② r❡s✉❧t ✐s ●✐rs❛♥♦✈✬s ❚❤❡✲
♦r❡♠✱ ✇❤✐❝❤ ❣✉❛r❛♥t❡❡s t❤❡ ❡①✐st❡♥❝❡ ♦❢ r✐s❦✲♥❡✉tr❛❧ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡s✳
❈❤❛♣t❡rs ✶✸ ❛♥❞ ✶✹ ♣r♦✈✐❞❡ ❛ ♠❛rt✐♥❣❛❧❡ ❛♣♣r♦❛❝❤ t♦ ♦♣t✐♦♥ ♣r✐❝✐♥❣✱ ✉s✐♥❣
r✐s❦✲♥❡✉tr❛❧ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡s t♦ ✜♥❞ t❤❡ ✈❛❧✉❡ ♦❢ ❛ ✈❛r✐❡t② ♦❢ ❞❡r✐✈❛t✐✈❡s✱
✐♥❝❧✉❞✐♥❣ ♣❛t❤✲❞❡♣❡♥❞❡♥t ♦♣t✐♦♥s✳ ❘❛t❤❡r t❤❛♥ ❜❡✐♥❣ ❡♥❝②❝❧♦♣❡❞✐❝✱ t❤❡ ♠❛✲
t❡r✐❛❧ ✐s ✐♥t❡♥❞❡❞ t♦ ❝♦♥✈❡② t❤❡ ❡ss❡♥t✐❛❧ ✐❞❡❛s ♦❢ ❞❡r✐✈❛t✐✈❡ ♣r✐❝✐♥❣ ❛♥❞ t♦
❞❡♠♦♥str❛t❡ t❤❡ ✉t✐❧✐t② ❛♥❞ ❡❧❡❣❛♥❝❡ ♦❢ ♠❛rt✐♥❣❛❧❡ t❡❝❤♥✐q✉❡s ✐♥ t❤✐s ❡♥❞❡❛✈♦r✳
❚❤❡ t❡①t ❝♦♥t❛✐♥s ♥✉♠❡r♦✉s ❡①❛♠♣❧❡s ❛♥❞ ✷✵✵ ❡①❡r❝✐s❡s ❞❡s✐❣♥❡❞ t♦ ❤❡❧♣
t❤❡ r❡❛❞❡r ❣❛✐♥ ❡①♣❡rt✐s❡ ✐♥ t❤❡ ♠❡t❤♦❞s ♦❢ ✜♥❛♥❝✐❛❧ ❝❛❧❝✉❧✉s ❛♥❞✱ ♥♦t ✐♥❝✐✲
❞❡♥t❛❧❧②✱ t♦ ✐♥❝r❡❛s❡ ❤✐s ♦r ❤❡r ❧❡✈❡❧ ♦❢ ❣❡♥❡r❛❧ ♠❛t❤❡♠❛t✐❝❛❧ s♦♣❤✐st✐❝❛t✐♦♥✳
❚❤❡ ❡①❡r❝✐s❡s r❛♥❣❡ ❢r♦♠ r♦✉t✐♥❡ ❝❛❧❝✉❧❛t✐♦♥s t♦ s♣r❡❛❞s❤❡❡t ♣r♦❥❡❝ts t♦ t❤❡


①✐✐✐
♣r✐❝✐♥❣ ♦❢ ❛ ✈❛r✐❡t② ♦❢ ❝♦♠♣❧❡① ✜♥❛♥❝✐❛❧ ✐♥str✉♠❡♥ts✳ ❍✐♥ts ❛♥❞ s♦❧✉t✐♦♥s t♦
t❤❡ ♦❞❞✲♥✉♠❜❡r❡❞ ♣r♦❜❧❡♠s ❛r❡ ❣✐✈❡♥ ✐♥ ❆♣♣❡♥❞✐① ❉✳
❋♦r ❣r❡❛t❡r ❝❧❛r✐t② ❛♥❞ ❡❛s❡ ♦❢ ❡①♣♦s✐t✐♦♥ ✭❛♥❞ t♦ r❡♠❛✐♥ ✇✐t❤✐♥ t❤❡ ✐♥✲
t❡♥❞❡❞ s❝♦♣❡ ♦❢ t❤❡ t❡①t✮✱ ✇❡ ❤❛✈❡ ❛✈♦✐❞❡❞ st❛t✐♥❣ r❡s✉❧ts ✐♥ t❤❡✐r ♠♦st ❣❡♥❡r❛❧
❢♦r♠✳ ❚❤✉s✱ ✐♥t❡r❡st r❛t❡s ❛r❡ ❛ss✉♠❡❞ t♦ ❜❡ ❝♦♥st❛♥t✱ ♣❛t❤s ♦❢ st♦❝❤❛st✐❝ ♣r♦✲
❝❡ss❡s ❛r❡ r❡q✉✐r❡❞ t♦ ❜❡ ❝♦♥t✐♥✉♦✉s✱ ❛♥❞ ✜♥❛♥❝✐❛❧ ♠❛r❦❡ts tr❛❞❡ ✐♥ ❛ s✐♥❣❧❡
r✐s❦② ❛ss❡t✳ ❲❤✐❧❡ t❤❡s❡ ❛ss✉♠♣t✐♦♥s ♠❛② ❜❡ ✉♥r❡❛❧✐st✐❝✱ ✐t ✐s ♦✉r ❜❡❧✐❡❢ t❤❛t
t❤❡ r❡❛❞❡r ✇❤♦ ❤❛s ♦❜t❛✐♥❡❞ ❛ s♦❧✐❞ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ t❤❡ t❤❡♦r② ✐♥ t❤✐s s✐♠♣❧✐✲
✜❡❞ s❡tt✐♥❣ ✇✐❧❧ ❤❛✈❡ ❧✐tt❧❡ ❞✐✣❝✉❧t② ✐♥ ♠❛❦✐♥❣ t❤❡ tr❛♥s✐t✐♦♥ t♦ ♠♦r❡ ❣❡♥❡r❛❧
❝♦♥t❡①ts✳
❲❤✐❧❡ t❤❡ t❡①t ❝♦♥t❛✐♥s ♥✉♠❡r♦✉s ❡①❛♠♣❧❡s ❛♥❞ ♣r♦❜❧❡♠s ✐♥✈♦❧✈✐♥❣ t❤❡
✉s❡ ♦❢ s♣r❡❛❞s❤❡❡ts✱ ✇❡ ❤❛✈❡ ♥♦t ✐♥❝❧✉❞❡❞ ❛♥② ❞✐s❝✉ss✐♦♥ ♦❢ ❣❡♥❡r❛❧ ♥✉♠❡r✐❝❛❧
t❡❝❤♥✐q✉❡s✱ ❛s t❤❡r❡ ❛r❡ s❡✈❡r❛❧ ❡①❝❡❧❧❡♥t t❡①ts ❞❡✈♦t❡❞ t♦ t❤✐s s✉❜❥❡❝t✳ ■♥❞❡❡❞✱
s✉❝❤ ❛ t❡①t ❝♦✉❧❞ ❜❡ ✉s❡❞ t♦ ❣♦♦❞ ❡✛❡❝t ✐♥ ❝♦♥❥✉♥❝t✐♦♥ ✇✐t❤ t❤❡ ♣r❡s❡♥t ♦♥❡✳
■t ✐s ✐♥❡✈✐t❛❜❧❡ t❤❛t ❛♥② s❡r✐♦✉s ❞❡✈❡❧♦♣♠❡♥t ♦❢ ♦♣t✐♦♥ ♣r✐❝✐♥❣ ♠❡t❤♦❞s ❛t

t❤❡ ✐♥t❡♥❞❡❞ ❧❡✈❡❧ ♦❢ t❤✐s ❜♦♦❦ ♠✉st ♦❝❝❛s✐♦♥❛❧❧② r❡s♦rt t♦ ✐♥✈♦❦✐♥❣ ❛ r❡s✉❧t
t❤❛t ❢❛❧❧s ♦✉ts✐❞❡ t❤❡ s❝♦♣❡ ♦❢ t❤❡ t❡①t✳ ❋♦r t❤❡ ❢❡✇ t✐♠❡s t❤❛t t❤✐s ❤❛s ♦❝✲
❝✉rr❡❞✱ ✇❡ ❤❛✈❡ tr✐❡❞ ❡✐t❤❡r t♦ ❣✐✈❡ ❛ s❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢ ♦r✱ ❢❛✐❧✐♥❣ t❤❛t✱ t♦
❣✐✈❡ r❡❢❡r❡♥❝❡s✱ ❣❡♥❡r❛❧ ♦r s♣❡❝✐✜❝✱ ✇❤❡r❡ t❤❡ r❡❛❞❡r ♠❛② ✜♥❞ ❛ r❡❛s♦♥❛❜❧②
❛❝❝❡ss✐❜❧❡ ♣r♦♦❢✳
❚❤❡ t❡①t ✐s ♦r❣❛♥✐③❡❞ t♦ ❛❧❧♦✇ ❛s ✢❡①✐❜❧❡ ✉s❡ ❛s ♣♦ss✐❜❧❡✳ ❚❤❡ ♣r❡❝✉rs♦r
t♦ t❤❡ ❜♦♦❦✱ ✐♥ t❤❡ ❢♦r♠ ♦❢ ❛ s❡t ♦❢ ♥♦t❡s✱ ❤❛s ❜❡❡♥ s✉❝❝❡ss❢✉❧❧② t❡st❡❞ ✐♥ t❤❡
❝❧❛ssr♦♦♠ ❛s ❛ s✐♥❣❧❡ s❡♠❡st❡r ❝♦✉rs❡ ✐♥ ❞✐s❝r❡t❡✲t✐♠❡ t❤❡♦r② ♦♥❧② ✭❈❤❛♣t❡rs
✶✕✾✮ ❛♥❞ ❛s ❛ ♦♥❡✲s❡♠❡st❡r ❝♦✉rs❡ ❣✐✈✐♥❣ ❛♥ ♦✈❡r✈✐❡✇ ♦❢ ❜♦t❤ ❞✐s❝r❡t❡✲t✐♠❡ ❛♥❞
❝♦♥t✐♥✉♦✉s✲t✐♠❡ ♠♦❞❡❧s ✭❈❤❛♣t❡rs ✶✕✼✱ ✶✵✱ ❛♥❞ ✶✶✮✳ ■t ♠❛② ❛❧s♦ ❡❛s✐❧② s❡r✈❡
❛s ❛ t✇♦✲s❡♠❡st❡r ❝♦✉rs❡✱ ✇✐t❤ ❈❤❛♣t❡rs ✶✕✶✸ ❢♦r♠✐♥❣ t❤❡ ❝♦r❡ ❛♥❞ s❡❧❡❝t✐♦♥s
❢r♦♠ ❈❤❛♣t❡r ✶✹✳
❚♦ t❤❡ st✉❞❡♥ts ✇❤♦s❡ s❤❛r♣ ❡②❡ ❝❛✉❣❤t t②♣♦s✱ ✐♥❝♦♥s✐st❡♥❝✐❡s✱ ❛♥❞ ❞♦✇♥✲
r✐❣❤t ❡rr♦rs ✐♥ t❤❡ ♥♦t❡s ❧❡❛❞✐♥❣ ✉♣ t♦ t❤❡ ❜♦♦❦✿ t❤❛♥❦ ②♦✉✳ ❚♦ t❤❡ r❡❛❞❡rs ♦❢
t❤✐s t❡①t✿ t❤❡ ❛✉t❤♦r ✇♦✉❧❞ ❜❡ ❣r❛t❡❢✉❧ ✐♥❞❡❡❞ ❢♦r s✐♠✐❧❛r ♦❜s❡r✈❛t✐♦♥s✱ s❤♦✉❧❞
t❤❡ ♦♣♣♦rt✉♥✐t② ❛r✐s❡✱ ❛s ✇❡❧❧ ❛s ❢♦r s✉❣❣❡st✐♦♥s ❢♦r ✐♠♣r♦✈❡♠❡♥ts✳

❍✉❣♦ ❉✳ ❏✉♥❣❤❡♥♥
❲❛s❤✐♥❣t♦♥✱ ❉✳❈✳✱ ❯❙❆


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❈❤❛♣t❡r ✶
■♥t❡r❡st ❛♥❞ Pr❡s❡♥t ❱❛❧✉❡

■♥ t❤✐s ❝❤❛♣t❡r✱ ✇❡ ❝♦♥s✐❞❡r ❛ss❡ts ✇❤♦s❡ ✈❛❧✉❡ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ❛♥ ✐♥t❡r❡st
r❛t❡✳ ■❢ t❤❡ ❛ss❡t ✐s ❣✉❛r❛♥t❡❡❞✱ ❛s ✐♥ t❤❡ ❝❛s❡ ♦❢ ❛♥ ✐♥s✉r❡❞ s❛✈✐♥❣s ❛❝❝♦✉♥t ♦r ❛
❣♦✈❡r♥♠❡♥t ❜♦♥❞ ✭✇❤✐❝❤✱ t②♣✐❝❛❧❧②✱ ❤❛s ♦♥❧② ❛ s♠❛❧❧ ❧✐❦❡❧✐❤♦♦❞ ♦❢ ❞❡❢❛✉❧t✮✱ t❤❡

❛ss❡t ✐s s❛✐❞ t♦ ❜❡

r✐s❦✲❢r❡❡ ✳ ❙✉❝❤ ❛♥ ❛ss❡t st❛♥❞s ✐♥ ❝♦♥tr❛st t♦ ❛ r✐s❦② ❛ss❡t✱

❢♦r ❡①❛♠♣❧❡✱ ❛ st♦❝❦ ♦r ❝♦♠♠♦❞✐t②✱ ✇❤♦s❡ ❢✉t✉r❡ ✈❛❧✉❡s ❝❛♥♥♦t ❜❡ ❞❡t❡r♠✐♥❡❞
✇✐t❤ ❝❡rt❛✐♥t②✳ ❆s ✇❡ s❤❛❧❧ s❡❡ ✐♥ ❧❛t❡r ❝❤❛♣t❡rs✱ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧s t❤❛t
❞❡s❝r✐❜❡ t❤❡ ✈❛❧✉❡ ♦❢ ❛ r✐s❦② ❛ss❡t t②♣✐❝❛❧❧② ✐♥❝❧✉❞❡ ❛ ❝♦♠♣♦♥❡♥t ✐♥✈♦❧✈✐♥❣ ❛
r✐s❦✲❢r❡❡ ❛ss❡t✳ ❚❤❡r❡❢♦r❡✱ ♦✉r ✜rst ❣♦❛❧ ✐s t♦ ❞❡s❝r✐❜❡ ❤♦✇ r✐s❦✲❢r❡❡ ❛ss❡ts ❛r❡
✈❛❧✉❡❞✳

✶✳✶ ❈♦♠♣♦✉♥❞ ■♥t❡r❡st
■♥t❡r❡st

✐s ❛ ❢❡❡ ♣❛✐❞ ❜② ♦♥❡ ♣❛rt② ❢♦r t❤❡ ✉s❡ ♦❢ ❝❛s❤ ❛ss❡ts ♦❢ ❛♥♦t❤❡r✳

❚❤❡ ❛♠♦✉♥t ♦❢ ✐♥t❡r❡st ✐s ❣❡♥❡r❛❧❧② t✐♠❡ ❞❡♣❡♥❞❡♥t✿ t❤❡ ❧♦♥❣❡r t❤❡ ♦✉tst❛♥❞✐♥❣
❜❛❧❛♥❝❡✱ t❤❡ ♠♦r❡ ✐♥t❡r❡st ✐s ❛❝❝r✉❡❞✳ ❆ ❢❛♠✐❧✐❛r ❡①❛♠♣❧❡ ✐s t❤❡ ✐♥t❡r❡st ❣❡♥✲
❡r❛t❡❞ ❜② ❛ ♠♦♥❡② ♠❛r❦❡t ❛❝❝♦✉♥t✳ ❚❤❡ ❜❛♥❦ ♣❛②s t❤❡ ❞❡♣♦s✐t♦r ❛♥ ❛♠♦✉♥t
t❤❛t ✐s ✉s✉❛❧❧② ❛ ❢r❛❝t✐♦♥ ♦❢ t❤❡ ❜❛❧❛♥❝❡ ✐♥ t❤❡ ❛❝❝♦✉♥t✱ t❤❛t ❢r❛❝t✐♦♥ ❣✐✈❡♥ ✐♥
t❡r♠s ♦❢ ❛ ♣r♦r❛t❡❞ ❛♥♥✉❛❧ ♣❡r❝❡♥t❛❣❡ ❝❛❧❧❡❞ t❤❡

♥♦♠✐♥❛❧ r❛t❡✳

n =
1, 2, . . .✳ ❙✉♣♣♦s❡ t❤❡ ✐♥✐t✐❛❧ ❞❡♣♦s✐t ✐s A0 ❛♥❞ t❤❡ ✐♥t❡r❡st r❛t❡ ♣❡r ♣❡r✐♦❞
✐s i✳ ■❢ ✐♥t❡r❡st ✐s ❝♦♠♣♦✉♥❞❡❞ ✱ t❤❡♥✱ ❛❢t❡r t❤❡ ✜rst ♣❡r✐♦❞✱ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡
❛❝❝♦✉♥t ✐s A1 = A0 + iA0 = A0 (1 + i)✱ ❛❢t❡r t❤❡ s❡❝♦♥❞ ♣❡r✐♦❞ t❤❡ ✈❛❧✉❡ ✐s
A2 = A1 + iA1 = A1 (1 + i) = A0 (1 + i)2 ✱ ❛♥❞ s♦ ♦♥✳ ■♥ ❣❡♥❡r❛❧✱ t❤❡ ✈❛❧✉❡ ♦❢
t❤❡ ❛❝❝♦✉♥t ❛t t✐♠❡ n ✐s
❈♦♥s✐❞❡r ✜rst ❛♥ ❛❝❝♦✉♥t t❤❛t ♣❛②s ✐♥t❡r❡st ❛t t❤❡ ❞✐s❝r❡t❡ t✐♠❡s

An = A0 (1 + i)n ,
A0

✐s ❝❛❧❧❡❞ t❤❡

❢✉t✉r❡ ✈❛❧✉❡✳

♣r❡s❡♥t ✈❛❧✉❡

♦r

n = 0, 1, 2, . . . .

❞✐s❝♦✉♥t❡❞ ✈❛❧✉❡

◆♦✇ s✉♣♣♦s❡ t❤❛t t❤❡ ♥♦♠✐♥❛❧ r❛t❡ ✐s
t✐♠❡s ❛ ②❡❛r✳ ❚❤❡♥

i = r/m

r

✭✶✳✶✮

♦❢ t❤❡ ❛❝❝♦✉♥t ❛♥❞

An

❛♥❞ ✐♥t❡r❡st ✐s ❝♦♠♣♦✉♥❞❡❞

❤❡♥❝❡ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❛❝❝♦✉♥t ❛❢t❡r

At = A0 (1 + r/m)mt .

t

❛

m

②❡❛rs ✐s
✭✶✳✷✮

✶


❖♣t✐♦♥ ❱❛❧✉❛t✐♦♥✿ ❆ ❋✐rst ❈♦✉rs❡ ✐♥ ❋✐♥❛♥❝✐❛❧ ▼❛t❤❡♠❛t✐❝s

✷

❚❤❡ ❞✐st✐♥❝t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❢♦r♠✉❧❛s ✭✶✳✶✮ ❛♥❞ ✭✶✳✷✮ ✐s t❤❛t t❤❡ ❢♦r♠❡r ❡①✲
♣r❡ss❡s t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❛❝❝♦✉♥t ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ❝♦♠♣♦✉♥❞✐♥❣
✐♥t❡r✈❛❧s ✭t❤❛t ✐s✱ ❛t t❤❡ ❞✐s❝r❡t❡ t✐♠❡s
❛ ❢✉♥❝t✐♦♥ ♦❢ ❝♦♥t✐♥✉♦✉s t✐♠❡

t

n✮✱

✇❤✐❧❡ t❤❡ ❧❛tt❡r ❣✐✈❡s t❤❡ ✈❛❧✉❡ ❛s

✭✐♥ ②❡❛rs✮✳

■♥ ❝♦♥tr❛st t♦ ❛♥ ❛❝❝♦✉♥t ❡❛r♥✐♥❣ ❝♦♠♣♦✉♥❞ ✐♥t❡r❡st✱ ❛♥ ❛❝❝♦✉♥t ❞r❛✇✐♥❣

s✐♠♣❧❡ ✐♥t❡r❡st

❤❛s t✐♠❡✲t ✈❛❧✉❡

At = A0 (1 + tr).
■♥ t❤✐s ❝❛s❡✱ ✐♥t❡r❡st ✐s ❝❛❧❝✉❧❛t❡❞ ♦♥❧② ♦♥ t❤❡ ✐♥✐t✐❛❧ ❞❡♣♦s✐t

A0

❛♥❞ ♥♦t ♦♥

t❤❡ ♣r❡❝❡❞✐♥❣ ❛❝❝♦✉♥t ✈❛❧✉❡✳

❊①❛♠♣❧❡ ✶✳✶✳✶✳

❚❛❜❧❡ ✶✳✷ ❣✐✈❡s t❤❡ ✈❛❧✉❡ ❛❢t❡r t✇♦ ②❡❛rs ♦❢ ❛♥ ❛❝❝♦✉♥t ✇✐t❤

♣r❡s❡♥t ✈❛❧✉❡ ✩✽✵✵✳ ❚❤❡ ❛❝❝♦✉♥t ✐s ❛ss✉♠❡❞ t♦ ❡❛r♥ ✐♥t❡r❡st ❛t ❛♥ ❛♥♥✉❛❧ r❛t❡
♦❢ ✶✷✪✳

❱❛❧✉❡

800(1.12)2
800(1.06)4

800(1.03)8
800(1.01)24
800(1.0003)730

❈♦♠♣♦✉♥❞ ▼❡t❤♦❞

❂

✩✶✱✵✵✸✳✺✷

❛♥♥✉❛❧❧②

❂

✩✶✱✵✵✾✳✾✽

s❡♠✐❛♥♥✉❛❧❧②

❂

✩✶✱✵✶✸✳✹✷

q✉❛rt❡r❧②

❂

✩✶✱✵✶✺✳✼✾

♠♦♥t❤❧②

❂

✩✶✱✵✶✻✳✾✻

❞❛✐❧②

❚❆❇▲❊ ✶✳✶✿ ❆❝❝♦✉♥t ❱❛❧✉❡ ✐♥ ❚✇♦ ❨❡❛rs
◆♦t❡ t❤❛t ❢♦r s✐♠♣❧❡ ✐♥t❡r❡st t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❛❝❝♦✉♥t ❛❢t❡r t✇♦ ②❡❛rs ✐s

800(1.24) = $992.00✳
❚❤❡ ❛❜♦✈❡ ❡①❛♠♣❧❡ s✉❣❣❡sts t❤❛t ❝♦♠♣♦✉♥❞✐♥❣ ♠♦r❡ ❢r❡q✉❡♥t❧② r❡s✉❧ts ✐♥

(1+r/m)m
x = m/r ✐♥ ✭✶✳✷✮

❛ ❣r❡❛t❡r r❡t✉r♥✳ ❚❤✐s ✐s ❝❛♥ ❜❡ s❡❡♥ ❢r♦♠ t❤❡ ❢❛❝t t❤❛t t❤❡ s❡q✉❡♥❝❡
✐s ✐♥❝r❡❛s✐♥❣ ✐♥

m✳

❚♦ s❡❡ ✇❤❛t ❤❛♣♣❡♥s ✇❤❡♥

s♦ t❤❛t

m → ∞✱

s❡t

rt

At = A0 [(1 + 1/x)x ] .
❆s

m → ∞✱ ❧✬❍♦s♣✐t❛❧✬s r✉❧❡ s❤♦✇s t❤❛t (1 + 1/x)x → e✳ ■♥ t❤✐s ✇❛②✱ ✇❡ ❛rr✐✈❡

❛t t❤❡ ❢♦r♠✉❧❛ ❢♦r

❝♦♥t✐♥✉♦✉s❧② ❝♦♠♣♦✉♥❞❡❞ ✐♥t❡r❡st ✿
At = A0 ert .

✭✶✳✸✮

❘❡t✉r♥✐♥❣ t♦ ❊①❛♠♣❧❡ ✶✳✶✳✶ ✇❡ s❡❡ t❤❛t✱ ✐❢ ✐♥t❡r❡st ✐s ❝♦♠♣♦✉♥❞❡❞ ❝♦♥t✐♥✉✲
♦✉s❧②✱ t❤❡♥ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❛❝❝♦✉♥t ❛❢t❡r t✇♦ ②❡❛rs ✐s

800e(.12)2 = $1, 016.99✱

♥♦t s✐❣♥✐✜❝❛♥t❧② ♠♦r❡ t❤❛♥ ❢♦r ❞❛✐❧② ❝♦♠♣♦✉♥❞✐♥❣✳
❚❤❡

❡✛❡❝t✐✈❡ ✐♥t❡r❡st r❛t❡ re

✐s t❤❡ s✐♠♣❧❡ ✐♥t❡r❡st r❛t❡ t❤❛t ♣r♦❞✉❝❡s t❤❡


■♥t❡r❡st ❛♥❞ Pr❡s❡♥t ❱❛❧✉❡

✸

s❛♠❡ ②✐❡❧❞ ✐♥ ♦♥❡ ②❡❛r ❛s ❝♦♠♣♦✉♥❞ ✐♥t❡r❡st✳ ■❢ ✐♥t❡r❡st ✐s ❝♦♠♣♦✉♥❞❡❞

t✐♠❡s ❛ ②❡❛r✱ t❤✐s ♠❡❛♥s t❤❛t

A0 (1 + r/m)m = A0 (1 + re )

m

❤❡♥❝❡

re = (1 + r/m)m − 1.
■❢ ✐♥t❡r❡st ✐s ❝♦♠♣♦✉♥❞❡❞ ❝♦♥t✐♥✉♦✉s❧②✱ t❤❡♥

A0 er = A0 (1 + re )

s♦ t❤❛t

re = er − 1.

❊①❛♠♣❧❡ ✶✳✶✳✷✳

❨♦✉ ❥✉st ✐♥❤❡r✐t❡❞ ✩✶✵✱✵✵✵✱ ✇❤✐❝❤ ②♦✉ ❞❡❝✐❞❡ t♦ ❞❡♣♦s✐t ✐♥

♦♥❡ ♦❢ t❤r❡❡ ❜❛♥❦s✱ ❆✱ ❇✱ ♦r ❈✳ ❇❛♥❦ ❆ ♣❛②s ✶✶✪ ❝♦♠♣♦✉♥❞❡❞ s❡♠✐❛♥♥✉✲
❛❧❧②✱ ❜❛♥❦ ❇ ♣❛②s ✶✵✳✼✻✪ ❝♦♠♣♦✉♥❞❡❞ ♠♦♥t❤❧②✱ ❛♥❞ ❜❛♥❦ ❈ ♣❛②s ✶✵✳✼✷ ✪
❝♦♠♣♦✉♥❞❡❞ ❝♦♥t✐♥✉♦✉s❧②✳ ❲❤✐❝❤ ❜❛♥❦ s❤♦✉❧❞ ②♦✉ ❝❤♦♦s❡❄

❙♦❧✉t✐♦♥✿

❲❡ ❝♦♠♣✉t❡ t❤❡ ❡✛❡❝t✐✈❡ r❛t❡

re

❢♦r ❡❛❝❤ ❣✐✈❡♥ ✐♥t❡r❡st r❛t❡✳

❘♦✉♥❞✐♥❣✱ ✇❡ ❤❛✈❡

= (1 + .11/2)2 − 1
= 0.113025
= (1 + .1076/12)12 − 1 = 0.113068
= e.1072 − 1
= 0.113156

re
re
re

❢♦r ❇❛♥❦ ❆,
❢♦r ❇❛♥❦ ❇✱
❢♦r ❇❛♥❦ ❈✳

❇❛♥❦ ❈ ❤❛s t❤❡ ❤✐❣❤❡st ❡✛❡❝t✐✈❡ r❛t❡ ❛♥❞ ✐s t❤❡r❡❢♦r❡ t❤❡ ❜❡st ❝❤♦✐❝❡✳

✶✳✷ ❆♥♥✉✐t✐❡s
❆♥

❛♥♥✉✐t②

✐s ❛ s❡q✉❡♥❝❡ ♦❢ ♣❡r✐♦❞✐❝ ♣❛②♠❡♥ts ♦❢ ❛ ✜①❡❞ ❛♠♦✉♥t✱ s❛②✱

P✳

❚❤❡ ♣❛②♠❡♥ts ♠❛② t❛❦❡ t❤❡ ❢♦r♠ ♦❢ ❞❡♣♦s✐ts ✐♥t♦ ❛♥ ❛❝❝♦✉♥t✱ s✉❝❤ ❛s ❛ ♣❡♥s✐♦♥
❢✉♥❞ ♦r ❧❛②❛✇❛② ♣❧❛♥✱ ♦r ✇✐t❤❞r❛✇❛❧s ❢r♦♠ ❛♥ ❛❝❝♦✉♥t✱ ❢♦r ❡①❛♠♣❧❡✱ ❛ tr✉st
❢✉♥❞ ♦r r❡t✐r❡♠❡♥t ❛❝❝♦✉♥t✳
❛♥♥✉❛❧ r❛t❡

r

❝♦♠♣♦✉♥❞❡❞

✶ ❙✉♣♣♦s❡ t❤❛t t❤❡ ❛❝❝♦✉♥t ♣❛②s ✐♥t❡r❡st ❛t ❛♥

m

t✐♠❡s ♣❡r ②❡❛r ❛♥❞ t❤❛t ❛ ❞❡♣♦s✐t ✭✇✐t❤❞r❛✇❛❧✮

✐s ♠❛❞❡ ❛t t❤❡ ❡♥❞ ♦❢ ❡❛❝❤ ❝♦♠♣♦✉♥❞✐♥❣ ✐♥t❡r✈❛❧✳ ❲❡ s❡❡❦ t❤❡ ✈❛❧✉❡
❛❝❝♦✉♥t ❛t t✐♠❡

n✱

t❤❛t ✐s✱ ✐♠♠❡❞✐❛t❡❧② ❛❢t❡r t❤❡

nt❤

An

♦❢ t❤❡

♣❛②♠❡♥t✳

An ✐s t❤❡ s✉♠ ♦❢ t❤❡ t✐♠❡✲n ✈❛❧✉❡s ♦❢ ♣❛②♠❡♥ts ✶
j ❛❝❝r✉❡s ✐♥t❡r❡st ♦✈❡r n − j ♣❛②♠❡♥t ♣❡r✐♦❞s✱ ✐ts
P (1 + r/m)n−j ✳ ❚❤✉s✱

■♥ t❤❡ ❝❛s❡ ♦❢ ❞❡♣♦s✐ts✱
t❤r♦✉❣❤

n✳

❙✐♥❝❡ ♣❛②♠❡♥t

t✐♠❡✲n ✈❛❧✉❡ ✐s

An = P (1 + x + x2 + · · · + xn−1 ),
❚❤❡ ❣❡♦♠❡tr✐❝ s❡r✐❡s s✉♠s t♦

(xn − 1)/(x − 1)✱

An = P
✶ ❆♥

(1 + i)n − 1
,
i

x := 1 +

r
.
m

❤❡♥❝❡

i :=

r
.
m

✭✶✳✹✮

❛❝❝♦✉♥t ✐♥t♦ ✇❤✐❝❤ ♣❡r✐♦❞✐❝ ❞❡♣♦s✐ts ❛r❡ ♠❛❞❡ ❢♦r t❤❡ ♣✉r♣♦s❡ ♦❢ r❡t✐r✐♥❣ ❛ ❞❡❜t ♦r

♣✉r❝❤❛s✐♥❣ ❛♥ ❛ss❡t ✐s s♦♠❡t✐♠❡s ❝❛❧❧❡❞ ❛

s✐♥❦✐♥❣ ❢✉♥❞✳


❖♣t✐♦♥ ❱❛❧✉❛t✐♦♥✿ ❆ ❋✐rst ❈♦✉rs❡ ✐♥ ❋✐♥❛♥❝✐❛❧ ▼❛t❤❡♠❛t✐❝s

✹

❋♦r ✇✐t❤❞r❛✇❛❧s ✇❡ ❛r❣✉❡ ❛s ❢♦❧❧♦✇s✿ ▲❡t
❛❝❝♦✉♥t✳ ❚❤❡ ✈❛❧✉❡ ❛t t❤❡ ❡♥❞ ♦❢ ♣❡r✐♦❞

nt❤

♣❛②♠❡♥t✱ ✐s

An−1

♣❧✉s t❤❡ ✐♥t❡r❡st

✇✐t❤❞r❛✇❛❧ r❡❞✉❝❡s t❤❛t ✈❛❧✉❡ ❜②

P

n✱

iAn−1

A0

❜❡ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♦❢ t❤❡

❥✉st ❜❡❢♦r❡ ✇✐t❤❞r❛✇❛❧ ♦❢ t❤❡
♦✈❡r t❤❛t ♣❡r✐♦❞✳ ▼❛❦✐♥❣ t❤❡

s♦

An = aAn−1 − P,

a := 1 + i.

■t❡r❛t✐♥❣✱ ✇❡ ♦❜t❛✐♥

An = a2 An−2 − (1 + a)P = · · · = an A0 − (1 + a + a2 + · · · + an−1 )P.
❚❤✉s✱

1 − (1 + i)n

i
(1 + i)n (iA0 − P ) + P
.
=
i

An = (1 + i)n A0 + P

✭✶✳✺✮

◆♦✇ ❛ss✉♠❡ t❤❛t t❤❡ ❛❝❝♦✉♥t ✐s ❞r❛✇♥ ❞♦✇♥ t♦ ③❡r♦ ❛❢t❡r
❙❡tt✐♥❣

n=N

❛♥❞

AN = 0

✐♥ ✭✶✳✺✮ ❛♥❞ s♦❧✈✐♥❣ ❢♦r

A0 = P

A0

1 − (1 + i)−N
.
i

❚❤✐s ✐s t❤❡ ✐♥✐t✐❛❧ ❞❡♣♦s✐t r❡q✉✐r❡❞ t♦ s✉♣♣♦rt ❡①❛❝t❧②

P

N

✇✐t❤❞r❛✇❛❧s✳

②✐❡❧❞s

✭✶✳✻✮

N

✇✐t❤❞r❛✇❛❧s ♦❢ ❛♠♦✉♥t

❢r♦♠✱ s❛②✱ ❛ r❡t✐r❡♠❡♥t ❛❝❝♦✉♥t ♦r tr✉st ❢✉♥❞✳ ■t ♠❛② ❜❡ s❡❡♥ ❛s t❤❡ s✉♠ ♦❢

t❤❡ ♣r❡s❡♥t ✈❛❧✉❡s ♦❢ t❤❡
❙♦❧✈✐♥❣ ❢♦r

P

N

✇✐t❤❞r❛✇❛❧s✳

✐♥ ✭✶✳✻✮ ✇❡ ♦❜t❛✐♥

P = A0

i
,
1 − (1 + i)−N

✭✶✳✼✮

✇❤✐❝❤ ♠❛② ❜❡ ✉s❡❞✱ ❢♦r ❡①❛♠♣❧❡✱ t♦ ❝❛❧❝✉❧❛t❡ t❤❡ ♠♦rt❣❛❣❡ ♣❛②♠❡♥t ❢♦r ❛

A0

♠♦rt❣❛❣❡ ♦❢ s✐③❡

✭s❡❡ ❊①❛♠♣❧❡ ✶✳✷✳✷✱ ❜❡❧♦✇✮✳ ❙✉❜st✐t✉t✐♥❣ ✭✶✳✼✮ ✐♥t♦ ✭✶✳✺✮

✇❡ ♦❜t❛✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛ ❢♦r t❤❡ t✐♠❡✲n ✈❛❧✉❡ ♦❢ ❛♥ ❛♥♥✉✐t② s✉♣♣♦rt✐♥❣
❡①❛❝t❧②

N

✇✐t❤❞r❛✇❛❧s✿

An = A0

❊①❛♠♣❧❡ ✶✳✷✳✶✳
s✐③❡

P

❆❢t❡r
❢♦r

1 − (1 + i)n−N
,
1 − (1 + i)−N

n = 0, 1, . . . , N.

✭✶✳✽✮

✭❘❡t✐r❡♠❡♥t ♣❧❛♥✮✳ ❙✉♣♣♦s❡ ②♦✉ ♠❛❦❡ ♠♦♥t❤❧② ❞❡♣♦s✐ts ♦❢

r✱ ❝♦♠♣♦✉♥❞❡❞ ♠♦♥t❤❧②✳
t ②❡❛rs ②♦✉ ✇✐s❤ t♦ ♠❛❦❡ ♠♦♥t❤❧② ✇✐t❤❞r❛✇❛❧s ♦❢ s✐③❡ Q ❢r♦♠ t❤❡ ❛❝❝♦✉♥t
✐♥t♦ ❛ r❡t✐r❡♠❡♥t ❛❝❝♦✉♥t ✇✐t❤ ❛♥ ❛♥♥✉❛❧ r❛t❡

s ②❡❛rs✱ ❞r❛✇✐♥❣ ❞♦✇♥ t❤❡ ❛❝❝♦✉♥t t♦ ③❡r♦✳ ❇② ✭✶✳✹✮ ❛♥❞ ✭✶✳✻✮ ✐t ♠✉st t❤❡♥

❜❡ t❤❡ ❝❛s❡ t❤❛t

P

1 − (1 + i)−12s
(1 + i)12t − 1
=Q
,
i
i

i :=

r
,
12


■♥t❡r❡st ❛♥❞ Pr❡s❡♥t ❱❛❧✉❡
♦r

P
1 − (1 + i)−12s
=
.
Q
(1 + i)12t − 1

❋♦r ❛ ♥✉♠❡r✐❝❛❧ ❡①❛♠♣❧❡✱ s✉♣♣♦s❡ t❤❛t

t = 40✱ s = 30✱

✺

✭✶✳✾✮
❛♥❞

r = .06✳

❚❤❡♥

P
1 − (1.005)−360

=
≈ .084,
Q
(1.005)480 − 1
s♦ t❤❛t ❛ ✇✐t❤❞r❛✇❛❧ ♦❢✱ s❛②✱

Q = $5000

❞✉r✐♥❣ r❡t✐r❡♠❡♥t ✇♦✉❧❞ r❡q✉✐r❡

♠♦♥t❤❧② ❞❡♣♦s✐ts ♦❢

P = (.084)5000 ≈ $419.
❆ ♠♦r❡ r❡❛❧✐st✐❝ ❛♥❛❧②s✐s t❛❦❡s ✐♥t♦ ❛❝❝♦✉♥t t❤❡ r❡❞✉❝t✐♦♥ ♦❢ ♣✉r❝❤❛s✐♥❣ ♣♦✇❡r
❞✉❡ t♦ ✐♥✢❛t✐♦♥✳ ❙✉♣♣♦s❡ t❤❛t ✐♥✢❛t✐♦♥ ✐s r✉♥♥✐♥❣ ❛t ✸✪ ♣❡r ②❡❛r ♦r

.25✪

♣❡r ♠♦♥t❤✳ ❚❤✐s ♠❡❛♥s t❤❛t ❣♦♦❞s ❛♥❞ s❡r✈✐❝❡s t❤❛t ❝♦st ✩✶ ♥♦✇ ✇✐❧❧ ❝♦st
✩(1.0025)

n

n

♠♦♥t❤s ❢r♦♠ ♥♦✇✳ ❚❤❡ ♣r❡s❡♥t ✈❛❧✉❡ ♣✉r❝❤❛s✐♥❣ ♣♦✇❡r ♦❢ t❤❡

✜rst ✇✐t❤❞r❛✇❛❧ ✐s t❤❡♥

5000(1.0025)−481 ≈ $1504,

✇❤✐❧❡ t❤❛t ♦❢ t❤❡ ❧❛st ✇✐t❤❞r❛✇❛❧ ✐s ♦♥❧②

5000(1.0025)−840 ≈ $614.
❋♦r t❤❡ ✜rst ✇✐t❤❞r❛✇❛❧ t♦ ❤❛✈❡ t❤❡ ❝✉rr❡♥t ♣✉r❝❤❛s✐♥❣ ♣♦✇❡r ♦❢ ✩✺✵✵✵✱

Q

✇♦✉❧❞ ❤❛✈❡ t♦ ❜❡

5000(1.0025)481 ≈ $16, 617,
✇❤✐❝❤ ✇♦✉❧❞ r❡q✉✐r❡ ♠♦♥t❤❧② ❞❡♣♦s✐ts ♦❢

P = (.084)16, 617 ≈ $1396.
❋♦r t❤❡ ❧❛st ✇✐t❤❞r❛✇❛❧ t♦ ❤❛✈❡ t❤❡ ❝✉rr❡♥t ♣✉r❝❤❛s✐♥❣ ♣♦✇❡r ♦❢ ✩✺✵✵✵✱

Q

✇♦✉❧❞ ❤❛✈❡ t♦ ❜❡

5000(1.0025)840 ≈ $40, 724,
r❡q✉✐r✐♥❣ ♠♦♥t❤❧② ❞❡♣♦s✐ts ♦❢

P = (.084)40, 724 ≈ $3421,
♠♦r❡ t❤❛♥ ❡✐❣❤t t✐♠❡s t❤❡ ❛♠♦✉♥t ❝❛❧❝✉❧❛t❡❞ ✇✐t❤♦✉t ❝♦♥s✐❞❡r✐♥❣ ✐♥✢❛t✐♦♥✦

❊①❛♠♣❧❡ ✶✳✷✳✷✳

✭❆♠♦rt✐③❛t✐♦♥✮✳ ❙✉♣♣♦s❡ ②♦✉ t❛❦❡ ♦✉t ❛ ✷✵✲②❡❛r✱ ✩✷✵✵✱✵✵✵

♠♦rt❣❛❣❡ ❛t ❛♥ ❛♥♥✉❛❧ r❛t❡ ♦❢ ✽✪ ❝♦♠♣♦✉♥❞❡❞ ♠♦♥t❤❧②✳ ❨♦✉r ♠♦♥t❤❧② ♠♦rt✲

P ❝♦♥st✐t✉t❡ ❛♥
N = 240✳ ❍❡r❡ An ✐s

A0 = $200, 000✱ i = .08/12 =
n✳ ❇②

❣❛❣❡ ♣❛②♠❡♥ts

❛♥♥✉✐t② ✇✐t❤

.0067✱

t❤❡ ❛♠♦✉♥t ♦✇❡❞ ❛t t❤❡ ❡♥❞ ♦❢ ♠♦♥t❤

❛♥❞

✭✶✳✼✮✱ t❤❡ ♠♦rt❣❛❣❡ ♣❛②♠❡♥ts ❛r❡

P = 200, 000

.0067
= $1677.85.
1 − (1.0067)−240


❖♣t✐♦♥ ❱❛❧✉❛t✐♦♥✿ ❆ ❋✐rst ❈♦✉rs❡ ✐♥ ❋✐♥❛♥❝✐❛❧ ▼❛t❤❡♠❛t✐❝s

✻

◆♦✇ ❧❡t

In

❛♥❞

Pn

❞❡♥♦t❡✱ r❡s♣❡❝t✐✈❡❧②✱ t❤❡ ♣♦rt✐♦♥s ♦❢ t❤❡

t❤❛t ❛r❡ ✐♥t❡r❡st ❛♥❞ ♣r✐♥❝✐♣❧❡✳ ❙✐♥❝❡

n − 1✱

An−1

nt❤

♣❛②♠❡♥t

✇❛s ♦✇❡❞ ❛t t❤❡ ❡♥❞ ♦❢ ♠♦♥t❤

✭✶✳✽✮ s❤♦✇s t❤❛t

In = iAn−1 = iA0

1 − (1 + i)n−1−N
,
1 − (1 + i)−N

❛♥❞ t❤❡r❡❢♦r❡

Pn = P − In = iA0
■♥ ♣❛rt✐❝✉❧❛r✱ ❢r♦♠ ✭✶✳✶✵✮ ✇❡ ❤❛✈❡

(1 + i)n−1−N
.
1 − (1 + i)−N

P1 = $337.86

❛♥❞

✭✶✳✶✵✮

P240 = $1666.70✳

❚❤✉s

♦♥❧② ❛❜♦✉t ✷✵✪ ♦❢ t❤❡ ✜rst ♣❛②♠❡♥t ❣♦❡s t♦ r❡❞✉❝✐♥❣ t❤❡ ♣r✐♥❝✐♣❧❡✱ ✇❤✐❧❡
❛❧♠♦st ✶✵✵✪ ♦❢ t❤❡ ❧❛st ♣❛②♠❡♥t ❞♦❡s s♦✳
❚❤❡ s❡q✉❡♥❝❡s

(Pn )✱ (In )✱

❛♠♦rt✐③❛t✐♦♥ s❝❤❡❞✉❧❡

❛♥❞

(An )

❢♦r♠ t❤❡ ❜❛s✐s ♦❢ ✇❤❛t ✐s ❝❛❧❧❡❞ t❤❡

♦❢ t❤❡ ♠♦rt❣❛❣❡✳

■♥ t❤❡ ❛❜♦✈❡ ❛♥♥✉✐t② ❢♦r♠✉❧❛s✱ t❤❡ ❝♦♠♣♦✉♥❞✐♥❣ ✐♥t❡r✈❛❧ ❛♥❞ t❤❡ ♣❛②♠❡♥t

❡♥❞ ♦❢ t❤❡ ❝♦♠♣♦✉♥❞✐♥❣
♦r❞✐♥❛r② ❛♥♥✉✐t②✳ ■❢ ♣❛②♠❡♥t ✐s ♠❛❞❡ ❛t

✐♥t❡r✈❛❧ ❛r❡ t❤❡ s❛♠❡✱ ❛♥❞ ♣❛②♠❡♥t ✐s ♠❛❞❡ ❛t t❤❡
✐♥t❡r✈❛❧✱ ❞❡s❝r✐❜✐♥❣ ✇❤❛t ✐s ❝❛❧❧❡❞ ❛♥
t❤❡

❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ♣❡r✐♦❞✱ ❛s ✐s t❤❡ ❝❛s❡ ❢♦r✱ s❛②✱ r❡♥ts ❛♥❞
❛♥♥✉✐t② ❞✉❡✱ ❛♥❞ t❤❡ ❢♦r♠✉❧❛s ❝❤❛♥❣❡ ❛❝❝♦r❞✐♥❣❧②✳

✐♥s✉r❛♥❝❡✱ ♦♥❡

♦❜t❛✐♥s ❛♥

✶✳✸ ❇♦♥❞s
❇♦♥❞s ❛r❡ ✜♥❛♥❝✐❛❧ ❝♦♥tr❛❝ts ✐ss✉❡❞ ❜② ❣♦✈❡r♥♠❡♥ts✱ ❝♦r♣♦r❛t✐♦♥s✱ ❛♥❞
♦t❤❡r ✐♥st✐t✉t✐♦♥s✳ ❚❤❡ s✐♠♣❧❡st t②♣❡ ♦❢ ❜♦♥❞ ✐s t❤❡

③❡r♦ ❝♦✉♣♦♥ ❜♦♥❞✳

❯✳❙✳

❚r❡❛s✉r② ❜✐❧❧s ❛♥❞ ❯✳❙✳ s❛✈✐♥❣s ❜♦♥❞s ❛r❡ ❝♦♠♠♦♥ ❡①❛♠♣❧❡s✳ ❚❤❡ ♣✉r❝❤❛s❡r

B0 ✭✇❤✐❝❤ ♠❛② ❜❡ ❞❡t❡r♠✐♥❡❞ ❜② ❜✐❞s✮ ❛♥❞ r❡❝❡✐✈❡s
F ✱ t❤❡ ❢❛❝❡ ✈❛❧✉❡ ♦❢ t❤❡ ❜♦♥❞✱ ❛t ❛ ♣r❡s❝r✐❜❡❞ t✐♠❡ T ✱ t❤❡
♠❛t✉r✐t② ❞❛t❡✳ ❚❤❡ ✈❛❧✉❡ Bt ♦❢ t❤❡ ❜♦♥❞ ❛t t✐♠❡ t ♠❛② ❜❡ ❡①♣r❡ss❡❞ ✐♥ t❡r♠s
♦❢ ❛ ❝♦♥t✐♥✉♦✉s❧② ❝♦♠♣♦✉♥❞❡❞ ✐♥t❡r❡st r❛t❡ r ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ❡q✉❛t✐♦♥
♦❢ ❛ ❜♦♥❞ ♣❛②s ❛♥ ❛♠♦✉♥t
❛ ♣r❡s❝r✐❜❡❞ ❛♠♦✉♥t

B0 = F e−rT .
Bt

✐s t❤❡♥ t❤❡ ❢❛❝❡ ✈❛❧✉❡ ♦❢ t❤❡ ❜♦♥❞ ❞✐s❝♦✉♥t❡❞ t♦ t✐♠❡

Bt = F e−r(T −t) = B0 ert ,
❚❤✉s✱ ❞✉r✐♥❣ t❤❡ t✐♠❡ ✐♥t❡r✈❛❧

[0, T ]✱

t✿

0 ≤ t ≤ T.

t❤❡ ❜♦♥❞ ❛❝ts ❧✐❦❡ ❛ ♠♦♥❡② ♠❛r❦❡t

❛❝❝♦✉♥t ✇✐t❤ ❝♦♥t✐♥✉♦✉s❧② ❝♦♠♣♦✉♥❞❡❞ ✐♥t❡r❡st✳ ❚❤❡ t✐♠❡ r❡str✐❝t✐♦♥ ♠❛② ❜❡
t❤❡♦r❡t✐❝❛❧❧② r❡♠♦✈❡❞ ❛s ❢♦❧❧♦✇s✿ ❆t t✐♠❡
t❤❡ ❜♦♥❞ ❜② ❜✉②✐♥❣

F/B0

T✱

r❡✐♥✈❡st t❤❡ ♣r♦❝❡❡❞s

❜♦♥❞s✱ ❡❛❝❤ ❢♦r t❤❡ ❛♠♦✉♥t

B0

F

❢r♦♠

❛♥❞ ❡❛❝❤ ✇✐t❤ t❤❡


■♥t❡r❡st ❛♥❞ Pr❡s❡♥t ❱❛❧✉❡

✼

F ❛♥❞ ♠❛t✉r✐t② ❞❛t❡ 2T ✳ ❆t t✐♠❡ t ∈ [T, 2T ] ❡❛❝❤
F e−r(2T −t) = B0 e−rT ert ✱ s♦ t❤❡ ❜♦♥❞ ❛❝❝♦✉♥t ❤❛s ✈❛❧✉❡
❢❛❝❡ ✈❛❧✉❡

❜♦♥❞ ❤❛s ✈❛❧✉❡

Bt = (F/B0 )B0 e−rT ert = F e−rT ert = B0 ert , (T ≤ t ≤ 2T ).
❈♦♥t✐♥✉✐♥❣ t❤✐s ♣r♦❝❡ss ✇❡ s❡❡ t❤❛t t❤❡ ❢♦r♠✉❧❛

t≥0

♦✈❡r ✇❤✐❝❤ t❤❡ r❛t❡

r✱

Bt = B0 ert

❤♦❧❞s ❢♦r ❛❧❧ t✐♠❡s

❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ❢❛❝❡ ✈❛❧✉❡ ♦❢ t❤❡ ❜♦♥❞ ❛♥❞ t❤❡

❜✐❞✱ ✐s ❝♦♥st❛♥t✳
❲✐t❤ ❛

❝♦✉♣♦♥ ❜♦♥❞✱

♦♥❡ r❡❝❡✐✈❡s ♥♦t ♦♥❧② t❤❡ ❛♠♦✉♥t

F

❛t t✐♠❡

T

❜✉t

❛❧s♦ ❛ s❡q✉❡♥❝❡ ♦❢ ♣❛②♠❡♥ts ❞✉r✐♥❣ t❤❡ ❧✐❢❡ ♦❢ t❤❡ ❜♦♥❞✳ ❚❤✉s✱ ❛t ♣r❡s❝r✐❜❡❞

t1 < t2 < · · · < tN ✱ t❤❡ ❜♦♥❞ ♣❛②s ❛♥ ❛♠♦✉♥t Cn ✱ ❝❛❧❧❡❞ ❛ ❝♦✉♣♦♥✱ ❛♥❞
T ♦♥❡ r❡❝❡✐✈❡s t❤❡ ❢❛❝❡ ✈❛❧✉❡ F ✳ ❚❤❡ ♣r✐❝❡ ♦❢ t❤❡ ❜♦♥❞ ✐s t❤❡ t♦t❛❧

t✐♠❡s

❛t ♠❛t✉r✐t②

♣r❡s❡♥t ✈❛❧✉❡

N

e−rtn Cn + F e−rT .

B0 =

✭✶✳✶✶✮

n=1
◆♦t❡ t❤❛t t❤✐s ✐s t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♦❢ ❛ ♣♦rt❢♦❧✐♦ ❝♦♥s✐st✐♥❣ ♦❢
❜♦♥❞s ♠❛t✉r✐♥❣ ❛t t✐♠❡s

t1 ✱ t2 ✱ . . .✱ tN ✱

❛♥❞

N +1 ③❡r♦✲❝♦✉♣♦♥

T✳

✶✳✹ ❘❛t❡ ♦❢ ❘❡t✉r♥
❈♦♥s✐❞❡r ❛♥ ✐♥✈❡st♠❡♥t t❤❛t r❡t✉r♥s✱ ❢♦r ❛♥ ✐♥✐t✐❛❧ ♣❛②♠❡♥t ♦❢
❛♠♦✉♥t

An > 0

❛t t❤❡ ❡♥❞ ♦❢ ♣❡r✐♦❞

n✱ n = 1, 2, . . . , N ✳

❚❤❡

♦❢ t❤❡ ✐♥✈❡st♠❡♥t ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❛t ♣❡r✐♦❞✐❝ ✐♥t❡r❡st r❛t❡

i

P > 0✱

❛♥

r❛t❡ ♦❢ r❡t✉r♥

❢♦r ✇❤✐❝❤ t❤❡

♣r❡s❡♥t ✈❛❧✉❡ ♦❢ t❤❡ s❡q✉❡♥❝❡ ♦❢ r❡t✉r♥s ❡q✉❛❧s t❤❡ ✐♥✐t✐❛❧ ♣❛②♠❡♥t

P✱

t❤❛t ✐s✱

N

An (1 + i)−n .

P =

✭✶✳✶✷✮

n=1
❊①❛♠♣❧❡s ♦❢ s✉❝❤ ✐♥✈❡st♠❡♥ts ❛r❡ ❛♥♥✉✐t✐❡s ❛♥❞ ❝♦✉♣♦♥ ❜♦♥❞s✳ ❋♦r ❛ ❝♦✉♣♦♥
❜♦♥❞ t❤❛t ♣❛②s t❤❡ ❛♠♦✉♥t
♣❛②s t❤❡ ❢❛❝❡ ✈❛❧✉❡

F

P =C
✇❤❡r❡

P = B0

C ❛t ❡❛❝❤ ♦❢ t❤❡ t✐♠❡s n = 1, 2, . . . , N − 1
N ✱ ❊q✉❛t✐♦♥ ✭✶✳✶✷✮ r❡❞✉❝❡s t♦

1 − (1 + i)−N
+ F (1 + i)−N ,
i

✐s t❤❡ ♣r✐❝❡ ♦❢ t❤❡ ❜♦♥❞✳

❚♦ s❡❡ t❤❛t ❊q✉❛t✐♦♥ ✭✶✳✶✷✮ ❤❛s ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥
s✐❞❡ ❜②

f (i) ❛♥❞ ♥♦t❡ t❤❛t f

❙✐♥❝❡

i > −1✱ ❞❡♥♦t❡ t❤❡ r✐❣❤t
(−1, ∞) ❛♥❞ s❛t✐s✜❡s

✐s ❝♦♥t✐♥✉♦✉s ♦♥ t❤❡ ✐♥t❡r✈❛❧

lim f (i) = 0

i→∞

P

❛♥❞

❛t t✐♠❡

❛♥❞

lim f (i) = ∞.

i→−1+

P > 0✱ t❤❡ ■♥t❡r♠❡❞✐❛t❡ ❱❛❧✉❡ ❚❤❡♦r❡♠ ✐♠♣❧✐❡s t❤❛t t❤❡ ❡q✉❛t✐♦♥ f (i) =
i > −1✳ ❇❡❝❛✉s❡ f ✐s str✐❝t❧② ❞❡❝r❡❛s✐♥❣✱ t❤❡ s♦❧✉t✐♦♥ ✐s ✉♥✐q✉❡✳

❤❛s ❛ s♦❧✉t✐♦♥


✽

❖♣t✐♦♥ ❱❛❧✉❛t✐♦♥✿ ❆ ❋✐rst ❈♦✉rs❡ ✐♥ ❋✐♥❛♥❝✐❛❧ ▼❛t❤❡♠❛t✐❝s
❆ r❛t❡ ♦❢ r❡t✉r♥

i

♠❛② ❜❡ ♣♦s✐t✐✈❡✱ ③❡r♦✱ ♦r ♥❡❣❛t✐✈❡✳ ■❢

f (0) > P ✱

t❤❛t ✐s✱

t❤❡ s✉♠ ♦❢ t❤❡ ♣❛②♦✛s ✐s ❣r❡❛t❡r t❤❛♥ t❤❡ ✐♥✐t✐❛❧ ✐♥✈❡st♠❡♥t✱ t❤❡♥✱ ❜❡❝❛✉s❡
✐s ❞❡❝r❡❛s✐♥❣✱

i > 0✳

❧❡ss t❤❛♥ t❤❡ ✐♥✐t✐❛❧ ✐♥✈❡st♠❡♥t✱

❊①❛♠♣❧❡ ✶✳✹✳✶✳

f (0) < P ✱
t❤❡♥ i < 0✳

❙✐♠✐❧❛r❧②✱ ✐❢

f

t❤❛t ✐s✱ t❤❡ s✉♠ ♦❢ t❤❡ ♣❛②♦✛s ✐s

❙✉♣♣♦s❡ ②♦✉ ❧♦❛♥ ❛ ❢r✐❡♥❞ ✩✶✵✵ ❛♥❞ ❤❡ ❛❣r❡❡s t♦ ♣❛② ②♦✉

✩✸✺ ❛t t❤❡ ❡♥❞ ♦❢ t❤❡ ✜rst ②❡❛r✱ ✩✸✼ ❛t t❤❡ ❡♥❞ ♦❢ t❤❡ s❡❝♦♥❞ ②❡❛r✱ ❛♥❞ ✩✸✾ ❛t
t❤❡ ❡♥❞ ♦❢ t❤❡ t❤✐r❞ ②❡❛r✱ ❛t ✇❤✐❝❤ t✐♠❡ t❤❡ ❧♦❛♥ ✐s ❝♦♥s✐❞❡r❡❞ t♦ ❜❡ ♣❛✐❞ ♦✛✳
❚❤❡ s✉♠ ♦❢ t❤❡ ♣❛②♦✛s ✐s ❣r❡❛t❡r t❤❛♥ ✶✵✵✱ s♦ t❤❡ ❡q✉❛t✐♦♥

39
37
35
+
= 100
+
2
(1 + i) (1 + i)
(1 + i)3
❤❛s ❛ ✉♥✐q✉❡ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥

i✱

i✳

❖♥❡ ❝❛♥ ✉s❡ ◆❡✇t♦♥✬s ♠❡t❤♦❞ t♦ ❞❡t❡r♠✐♥❡

♦r ♦♥❡ ❝❛♥ s✐♠♣❧② s♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥ ❜② tr✐❛❧ ❛♥❞ ❡rr♦r ✉s✐♥❣ ❛ s♣r❡❛❞s❤❡❡t✳

❚❤❡ ❧❛tt❡r ❛♣♣r♦❛❝❤ ❣✐✈❡s

i ≈ 0.053✱ t❤❛t ✐s✱ ❛♥ ❛♥♥✉❛❧ r❛t❡ ♦❢ ❛❜♦✉t ✺✳✸✪✳

■♥t❡r❡st ❛♥❞ Pr❡s❡♥t ❱❛❧✉❡

✾

✶✳✺ ❊①❡r❝✐s❡s
✶✳ ❙✉♣♣♦s❡ ②♦✉ ❞❡♣♦s✐t ✩✶✺✵✵ ✐♥ ❛♥ ❛❝❝♦✉♥t ♣❛②✐♥❣ ❛♥ ❛♥♥✉❛❧ r❛t❡ ♦❢ ✻✪✳
❋✐♥❞ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❛❝❝♦✉♥t ✐♥ t❤r❡❡ ②❡❛rs ✐❢ ✐♥t❡r❡st ✐s ❝♦♠♣♦✉♥❞❡❞
✭❛✮ ②❡❛r❧②❀ ✭❜✮ q✉❛rt❡r❧②❀ ✭❝✮ ♠♦♥t❤❧②❀ ✭❞✮ ❞❛✐❧②❀ ✭❡✮ ❝♦♥t✐♥✉♦✉s❧②✳
✷✳ ❲❤❛t ❛♥♥✉❛❧ ✐♥t❡r❡st r❛t❡

r

✇♦✉❧❞ ❛❧❧♦✇ ②♦✉ t♦ ❞♦✉❜❧❡ ②♦✉r ✐♥✐t✐❛❧ ❞❡✲

♣♦s✐t ✐♥ ✻ ②❡❛rs ✐❢ ✐♥t❡r❡st ✐s ❝♦♠♣♦✉♥❞❡❞ q✉❛rt❡r❧②❄ ❈♦♥t✐♥✉♦✉s❧②❄
✸✳ ❋✐♥❞ t❤❡ ❡✛❡❝t✐✈❡ ✐♥t❡r❡st r❛t❡ ✐❢ ❛ ♥♦♠✐♥❛❧ r❛t❡ ♦❢ ✶✷✪ ✐s ❝♦♠♣♦✉♥❞❡❞
✭❛✮ q✉❛rt❡r❧②❀ ✭❜✮ ♠♦♥t❤❧②❀ ✭❝✮ ❝♦♥t✐♥✉♦✉s❧②✳
✹✳ ■❢ ②♦✉ r❡❝❡✐✈❡ ✻✪ ✐♥t❡r❡st ❝♦♠♣♦✉♥❞❡❞ ♠♦♥t❤❧②✱ ❛❜♦✉t ❤♦✇ ♠❛♥② ②❡❛rs
✇✐❧❧ ✐t t❛❦❡ ❢♦r ②♦✉r ✐♥✈❡st♠❡♥t t♦ tr✐♣❧❡❄
✺✳ ■❢ ②♦✉ ❞❡♣♦s✐t ✩✹✵✵ ❛t t❤❡ ❡♥❞ ♦❢ ❡❛❝❤ ♠♦♥t❤ ✐♥t♦ ❛♥ ❛❝❝♦✉♥t ❡❛r♥✐♥❣
✽✪ ✐♥t❡r❡st ❝♦♠♣♦✉♥❞❡❞ ♠♦♥t❤❧②✱ ✇❤❛t ✐s t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❛❝❝♦✉♥t ❛t
t❤❡ ❡♥❞ ♦❢ ✺ ②❡❛rs❄ ✶✵ ②❡❛rs❄
✻✳ ❨♦✉ ❞❡♣♦s✐t ✩✼✵✵ ❛t t❤❡ ❡♥❞ ♦❢ ❡❛❝❤ ♠♦♥t❤ ✐♥t♦ ❛♥ ❛❝❝♦✉♥t ❡❛r♥✐♥❣
✐♥t❡r❡st ❛t ❛♥ ❛♥♥✉❛❧ r❛t❡ ♦❢
t♦ ✜♥❞ t❤❡ ✈❛❧✉❡ ♦❢

r

r

❝♦♠♣♦✉♥❞❡❞ ♠♦♥t❤❧②✳ ❯s❡ ❛ s♣r❡❛❞s❤❡❡t

t❤❛t ♣r♦❞✉❝❡s ❛♥ ❛❝❝♦✉♥t ✈❛❧✉❡ ♦❢ ✩✺✵✱✵✵✵ ✐♥ ✺

②❡❛rs✳
✼✳ ❨♦✉ ❞❡♣♦s✐t ✩✹✵✵ ❛t t❤❡ ❡♥❞ ♦❢ ❡❛❝❤ ♠♦♥t❤ ✐♥t♦ ❛♥ ❛❝❝♦✉♥t ✇✐t❤ ❛♥
❛♥♥✉❛❧ r❛t❡ ♦❢ ✻✪ ❝♦♠♣♦✉♥❞❡❞ ♠♦♥t❤❧②✳ ❯s❡ ❛ s♣r❡❛❞s❤❡❡t t♦ ❞❡t❡r♠✐♥❡
t❤❡ ♠✐♥✐♠✉♠ ♥✉♠❜❡r ♦❢ ♣❛②♠❡♥ts r❡q✉✐r❡❞ ❢♦r t❤❡ ❛❝❝♦✉♥t t♦ ❤❛✈❡ ❛
✈❛❧✉❡ ♦❢ ❛t ❧❡❛st ✩✸✵✱✵✵✵✳
✽✳ ❙✉♣♣♦s❡ ❛♥ ❛❝❝♦✉♥t ♦✛❡rs ❝♦♥t✐♥✉♦✉s❧② ❝♦♠♣♦✉♥❞❡❞ ✐♥t❡r❡st ❛t ❛♥ ❛♥✲
♥✉❛❧ r❛t❡

r

❛♥❞ t❤❛t ❛ ❞❡♣♦s✐t ♦❢ s✐③❡

P

✐s ♠❛❞❡ ❛t t❤❡ ❡♥❞ ♦❢ ❡❛❝❤

♠♦♥t❤✳ ❙❤♦✇ t❤❛t t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❛❝❝♦✉♥t ❛❢t❡r

An = P

n

❞❡♣♦s✐ts ✐s

ern/12 − 1
.

er/12 − 1

✾✳ ❨♦✉ ♠❛❦❡ ❛♥ ✐♥✐t✐❛❧ ❞❡♣♦s✐t ♦❢ ✩✷✵✵✱✵✵✵ ✐♥t♦ ❛♥ ❛❝❝♦✉♥t ♣❛②✐♥❣ ✻✪
❝♦♠♣♦✉♥❞❡❞ ♠♦♥t❤❧②✳ ■❢ ②♦✉ ✇✐t❤❞r❛✇ ✩✷✵✵✵ ❡❛❝❤ ♠♦♥t❤✱ ❤♦✇ ♠✉❝❤
✇✐❧❧ ❜❡ ❧❡❢t ✐♥ t❤❡ ❛❝❝♦✉♥t ❛❢t❡r ✺ ②❡❛rs❄ ✶✵ ②❡❛rs❄ ❲❤❡♥ ✇✐❧❧ t❤❡ ❛❝❝♦✉♥t
❜❡ ❞r❛✇♥ ❞♦✇♥ t♦ ③❡r♦❄
✶✵✳ ❆♥ ❛❝❝♦✉♥t ♣❛②s ❛♥ ❛♥♥✉❛❧ r❛t❡ ♦❢ ✽✪ ♣❡r❝❡♥t ❝♦♠♣♦✉♥❞❡❞ ♠♦♥t❤❧②✳
❲❤❛t ❧✉♠♣ s✉♠ ♠✉st ②♦✉ ❞❡♣♦s✐t ✐♥t♦ t❤❡ ❛❝❝♦✉♥t ♥♦✇ s♦ t❤❛t ✐♥ ✶✵
②❡❛rs ②♦✉ ❝❛♥ ❜❡❣✐♥ t♦ ✇✐t❤❞r❛✇ ✩✹✵✵✵ ❡❛❝❤ ♠♦♥t❤ ❢♦r t❤❡ ♥❡①t ✷✵ ②❡❛rs✱
❞r❛✇✐♥❣ ❞♦✇♥ t❤❡ ❛❝❝♦✉♥t t♦ ③❡r♦❄


✶✵

❖♣t✐♦♥ ❱❛❧✉❛t✐♦♥✿ ❆ ❋✐rst ❈♦✉rs❡ ✐♥ ❋✐♥❛♥❝✐❛❧ ▼❛t❤❡♠❛t✐❝s

✶✶✳ ❆ tr✉st ❢✉♥❞ ❤❛s ❛♥ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♦❢ ✩✸✵✵✱✵✵✵ ❛♥❞ ❡❛r♥s ✐♥t❡r❡st ❛t ❛♥
❛♥♥✉❛❧ r❛t❡ ♦❢ ✻✪✱ ❝♦♠♣♦✉♥❞❡❞ ♠♦♥t❤❧②✳ ■❢ ❛ ✇✐t❤❞r❛✇❛❧ ♦❢ ✩✺✵✵✵ ✐s
♠❛❞❡ ❛t t❤❡ ❡♥❞ ♦❢ ❡❛❝❤ ♠♦♥t❤✱ ✇❤❡♥ ✇✐❧❧ t❤❡ ❛❝❝♦✉♥t ✇✐❧❧ ❢❛❧❧ ❜❡❧♦✇
✩✶✺✵✱✵✵✵❄ ✭❯s❡ ❛ s♣r❡❛❞s❤❡❡t✳✮

A0 ✐♥ t❡r♠s ♦❢ P
i t❤❛t ✇✐❧❧ ❢✉♥❞ ❛ ♣❡r♣❡t✉❛❧ ❛♥♥✉✐t②✱ t❤❛t ✐s✱ ❛♥ ❛♥♥✉✐t② ❢♦r ✇❤✐❝❤
An > 0 ❢♦r ❛❧❧ n✳ ❲❤❛t ✐s t❤❡ ✈❛❧✉❡ ♦❢ An ✐♥ t❤✐s ❝❛s❡❄

✶✷✳ ❘❡❢❡rr✐♥❣ t♦ ❊q✉❛t✐♦♥ ✭✶✳✺✮✱ ✜♥❞ t❤❡ s♠❛❧❧❡st ✈❛❧✉❡ ♦❢
❛♥❞

✶✸✳ ❙✉♣♣♦s❡ t❤❛t ❛♥ ❛❝❝♦✉♥t ♦✛❡rs ❝♦♥t✐♥✉♦✉s❧② ❝♦♠♣♦✉♥❞❡❞ ✐♥t❡r❡st ❛t ❛♥
❛♥♥✉❛❧ r❛t❡

r

❛♥❞ t❤❛t ✇✐t❤❞r❛✇❛❧s ♦❢ s✐③❡

❡❛❝❤ ♠♦♥t❤✳ ■❢ t❤❡ ✐♥✐t✐❛❧ ❞❡♣♦s✐t ✐s
t♦ ③❡r♦ ❛❢t❡r

N

A0

P

❛r❡ ♠❛❞❡ ❛t t❤❡ ❡♥❞ ♦❢

❛♥❞ t❤❡ ❛❝❝♦✉♥t ✐s ❞r❛✇♥ ❞♦✇♥

✇✐t❤❞r❛✇❛❧s✱ s❤♦✇ t❤❛t t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❛❝❝♦✉♥t ❛❢t❡r

✇✐t❤❞r❛✇❛❧s ✐s

An = P

n

1 − e−r(N −n)/12
.
er/12 − 1

t = 30✱ s = 20✱ ❛♥❞ r = .12✳ ❋✐♥❞ t❤❡
Q ♦❢ ✩✸✵✵✵ ♣❡r ♠♦♥t❤✳ ■❢ ✐♥✢❛t✐♦♥
✐s r✉♥♥✐♥❣ ❛t ✷✪ ♣❡r ②❡❛r✱ ✇❤❛t ✈❛❧✉❡ ♦❢ P ✇✐❧❧ ❣✐✈❡ t❤❡ ✜rst ✇✐t❤❞r❛✇❛❧

✶✹✳ ■♥ ❊①❛♠♣❧❡ ✶✳✷✳✶✱ s✉♣♣♦s❡ t❤❛t
♣❛②♠❡♥t ❛♠♦✉♥t

P

❢♦r ✇✐t❤❞r❛✇❛❧s

t❤❡ ❝✉rr❡♥t ♣✉r❝❤❛s✐♥❣ ♣♦✇❡r ♦❢ ✩✸✵✵✵❄ ❚❤❡ ❧❛st ✇✐t❤❞r❛✇❛❧❄
✶✺✳ ❋♦r ❛ ✸✵✲②❡❛r✱ ✩✸✵✵✱✵✵✵ ♠♦rt❣❛❣❡✱ ❞❡t❡r♠✐♥❡ t❤❡ ❛♥♥✉❛❧ r❛t❡

r

②♦✉ ✇✐❧❧

❤❛✈❡ t♦ ❧♦❝❦ ✐♥ t♦ ❤❛✈❡ ♣❛②♠❡♥ts ♦❢ ✩✶✽✵✵ ♣❡r ♠♦♥t❤❄
✶✻✳ ■♥ ❊①❛♠♣❧❡ ✶✳✷✳✷✱ s✉♣♣♦s❡ t❤❛t ②♦✉ ♠✉st ♣❛② ❛♥ ✐♥s♣❡❝t✐♦♥ ❢❡❡ ♦❢ ✩✶✵✵✵✱
❛ ❧♦❛♥ ✐♥✐t✐❛t✐♦♥ ❢❡❡ ♦❢ ✩✶✵✵✵✱ ❛♥❞ ✷

♣♦✐♥ts✱

t❤❛t ✐s✱ ✷✪ ♦❢ t❤❡ ♥♦♠✐♥❛❧

❧♦❛♥ ♦❢ ✩✷✵✵✱✵✵✵✳ ❊✛❡❝t✐✈❡❧②✱ t❤❡♥✱ ②♦✉ ❛r❡ r❡❝❡✐✈✐♥❣ ♦♥❧② ✩✶✾✹✱✵✵✵ ❢r♦♠
t❤❡ ❧❡♥❞✐♥❣ ✐♥st✐t✉t✐♦♥✳ ❈❛❧❝✉❧❛t❡ t❤❡ ❛♥♥✉❛❧ ✐♥t❡r❡st r❛t❡

r

②♦✉ ✇✐❧❧

♥♦✇ ❜❡ ♣❛②✐♥❣✱ ❣✐✈❡♥ t❤❡ ❛❣r❡❡❞ ✉♣♦♥ ♠♦♥t❤❧② ♣❛②♠❡♥ts ♦❢ ✩✶✻✻✼✳✽✺✳
✶✼✳ ❍♦✇ ❧❛r❣❡ ❛ ❧♦❛♥ ❝❛♥ ②♦✉ t❛❦❡ ♦✉t ❛t ❛♥ ❛♥♥✉❛❧ r❛t❡ ♦❢ ✶✺✪ ✐❢ ②♦✉ ❝❛♥
❛✛♦r❞ t♦ ♣❛② ❜❛❝❦ ✩✶✵✵✵ ❛t t❤❡ ❡♥❞ ♦❢ ❡❛❝❤ ♠♦♥t❤ ❛♥❞ ②♦✉ ✇❛♥t t♦
r❡t✐r❡ t❤❡ ❧♦❛♥ ❛❢t❡r ✺ ②❡❛rs❄
✶✽✳ ❙✉♣♣♦s❡ ②♦✉ t❛❦❡ ♦✉t ❛ ✷✵✲②❡❛r✱ ✩✸✵✵✱✵✵✵ ♠♦rt❣❛❣❡ ❛t ✼✪ ❛♥❞ ❞❡❝✐❞❡
❛❢t❡r ✶✺ ②❡❛rs t♦ ♣❛② ♦✛ t❤❡ ♠♦rt❣❛❣❡✳ ❍♦✇ ♠✉❝❤ ✇✐❧❧ ②♦✉ ❤❛✈❡ t♦ ♣❛②❄
✶✾✳ ❨♦✉ ❝❛♥ r❡t✐r❡ ❛ ❧♦❛♥ ❡✐t❤❡r ❜② ♣❛②✐♥❣ ♦✛ t❤❡ ❡♥t✐r❡ ❛♠♦✉♥t ✩✽✵✵✵ ♥♦✇✱
♦r ❜② ♣❛②✐♥❣ ✩✻✵✵✵ ♥♦✇ ❛♥❞ ✩✻✵✵✵ ❛t t❤❡ ❡♥❞ ♦❢ ✶✵ ②❡❛rs✳ ❋✐♥❞ ❛ ❝✉t♦✛
✈❛❧✉❡

r0

s✉❝❤ t❤❛t ✐❢ t❤❡ ♥♦♠✐♥❛❧ r❛t❡

t❤❡ ❡♥t✐r❡ ❧♦❛♥ ♥♦✇✱ ❜✉t ✐❢

r > r0 ✱

r

✐s

< r0 ✱

t❤❡♥ ②♦✉ s❤♦✉❧❞ ♣❛② ♦✛

t❤❡♥ ✐t ✐s ♣r❡❢❡r❛❜❧❡ t♦ ✇❛✐t✳ ❆ss✉♠❡

t❤❛t ✐♥t❡r❡st ✐s ❝♦♠♣♦✉♥❞❡❞ ❝♦♥t✐♥✉♦✉s❧②✳
✷✵✳ ❨♦✉ ❝❛♥ r❡t✐r❡ ❛ ❧♦❛♥ ❡✐t❤❡r ❜② ♣❛②✐♥❣ ♦✛ t❤❡ ❡♥t✐r❡ ❛♠♦✉♥t ✩✽✵✵✵ ♥♦✇✱
♦r ❜② ♣❛②✐♥❣ ✩✻✵✵✵ ♥♦✇✱ ✩✷✵✵✵ ❛t t❤❡ ❡♥❞ ♦❢ ✺ ②❡❛rs✱ ❛♥❞ ❛♥ ❛❞❞✐t✐♦♥❛❧
✩✷✵✵✵ ❛t t❤❡ ❡♥❞ ♦❢ ✶✵ ②❡❛rs✳ ❋✐♥❞ ❛ ❝✉t♦✛ ✈❛❧✉❡
♥♦♠✐♥❛❧ r❛t❡

r

✐s

< r0 ✱

r0

s✉❝❤ t❤❛t ✐❢ t❤❡

t❤❡♥ ②♦✉ s❤♦✉❧❞ ♣❛② ♦✛ t❤❡ ❡♥t✐r❡ ❧♦❛♥ ♥♦✇✱