Forward und Zukunftsverträge Copyright 1997 von Campbell R. Harvey und Stephen Gray. Alle Rechte vorbehalten. Kein Teil dieser Vorlesung darf ohne Zustimmung der Autoren reproduziert werden. Letzte Änderung: 15. Februar 1997 4.0 Übersicht: Diese Klasse bietet einen Überblick über Terminkontrakte und Terminkontrakte. Forwards und Futures gehören zur Klasse der Derivate, da ihr Wert aus dem Wert eines anderen Wertpapiers abgeleitet wird. Der Preis eines Devisentermingeschäftes hängt zum Beispiel von dem Preis der zugrunde liegenden Währung ab, und der Preis eines Schweinebauch-Futures-Kontraktes hängt von dem Preis von Schweinebauch ab. Derivate handeln sowohl an Börsen (wo Verträge standardisiert sind) als auch außerbörslich (wo die Kontraktspezifikation angepasst werden kann). Der Fokus dieser Klasse liegt auf (1) Definitionen und Kontraktspezifikationen der wichtigsten börsengehandelten Derivate, (2) der Mechanismen des Kaufs, des Verkaufs, der Ausübung und der Abwicklung von Terminkontrakten, (3) Derivatgeschäften einschließlich Hedging, Und (4) die Beziehungen zwischen Derivaten, den zugrunde liegenden Sicherheiten und risikolosen Anleihen. Insbesondere ist es möglich, Kombinationen von Derivaten und dem zugrundeliegenden Wert zu bilden, die risikolos sind und ein Mittel zur Bewertung von Derivaten darstellen. 4.1 Ziele Nach Abschluss dieser Klasse sollten Sie in der Lage sein, die möglichen Auszahlungen von Termingeschäften und Terminkontrakten zu ermitteln. Verstehen Sie die Mechanik von Kauf, Verkauf, Ausübung und Abwicklung von Terminkontrakten. Bestimmen Sie die möglichen Auszahlungen für Futures, Forwards und den zugrunde liegenden Vermögenswert. Untersuchen Sie die Marktpreise, um festzustellen, ob Arbitragegrenzen in Termin - und Terminmärkten verletzt werden. Die Richtwirkungen der relevanten Variablen auf den Wert der derivativen Wertpapiere verstehen. Zur Ermittlung des Preises von Termingeschäften und Terminkontrakten werden Standardbewertungsmethoden verwendet. 4.2 Einleitung Trotz der jüngsten negativen Presse haben die derivativen Wertpapiere eine Reihe nützlicher Funktionen in den Bereichen Risikomanagement und Investitionen. Tatsächlich wurden Derivate ursprünglich entworfen, um es den Marktteilnehmern zu ermöglichen, das Risiko zu eliminieren. Ein Weizenbauer kann zum Beispiel einen Preis für seine Ernte bereits vor der Bepflanzung festlegen, wodurch das Preisrisiko beseitigt wird. Ein Exporteur kann einen Wechselkurs schon vor Beginn der Herstellung des Produkts zu beheben, wodurch Wechselkursrisiken. Bei missbräuchlicher Verwendung sind jedoch auch derivative Wertpapiere in der Lage, das Risiko drastisch zu erhöhen. Dieses Modul konzentriert sich auf die Mechanik von Terminkontrakten und Terminkontrakten. Besonderes Augenmerk liegt auf der Wechselbeziehung zwischen den einzelnen Kontrakten und dem Kassakurs des Basiswerts. Der Spotpreis ist der Preis eines Vermögenswertes, bei dem der Verkauf und die Abwicklung sofort erfolgen sollen. 4.3 Terminkontrakte Der Mechanismus eines Terminkontrakts Ein Terminkontrakt ist ein Vertrag, der heute für die Lieferung eines Vermögenswerts zu einem vorgegebenen Zeitpunkt in der Zukunft zu einem heute vereinbarten Preis abgeschlossen wird. Der Käufer eines Terminkontraktes verpflichtet sich, die Lieferung eines Basiswertes zu einem zukünftigen Zeitpunkt zu übernehmen. Zu einem heute vereinbarten Preis. Kein Geld ändert sich bis zur Zeit T. Der Verkäufer ist bereit, den zugrunde liegenden Vermögenswert zu einem späteren Zeitpunkt zu liefern. Zu einem heute vereinbarten Preis. Wieder wechselt kein Geld bis zur Zeit T. Ein Terminkontrakt bedeutet also einfach, einen Preis für einen Handel festzulegen, der in Zukunft erfolgen wird. Beispiel 4.4 zeigt die Mechanik eines Terminkontraktes. Da Terminkontrakte im Freiverkehr gehandelt werden und nicht an Börsen, zeigt das Beispiel einen Vertrag zwischen einem Benutzer und einem Produzenten der zugrunde liegenden Ware. Beispiel 4.4: Kontraktmechanik. Ein Weizenbauer hat gerade eine Ernte gepflanzt, die erwartet wird, um 5000 Scheffel zu ergeben. Um das Risiko einer Verringerung der Preise für Weizen vor der Ernte zu beseitigen, kann der Bauer die 5000 Bushels von Weizen nach vorne verkaufen. Ein Müller kann bereit sein, die andere Seite des Vertrages zu nehmen. Die beiden Parteien vereinbaren heute zu einem Terminkurs von 550 Cent pro Scheffel, für die Lieferung fünf Monate ab jetzt, wenn die Ernte geerntet wird. Kein Geld wechselt jetzt die Hände. In fünf Monaten liefert der Bauer die 5000 Bushels an den Müller im Austausch für 27.500. Beachten Sie, dass dieser Preis festgelegt ist und nicht abhängt von der Spot-Preis von Weizen zum Zeitpunkt der Lieferung und Zahlung. 4.5 Bewertung von Terminkontrakten Termingeschäfte können bewertet werden, indem anerkannt werden, dass die Terminmärkte in vielen Fällen überflüssig sind. Dies geschieht, wenn die Auszahlung aus einem Terminkontrakt durch eine Position in (1) dem Basiswert und (2) risikolose Anleihen repliziert werden kann. Bevor wir dieses Konzept veranschaulichen, definieren wir die Transportkosten der zugrunde liegenden Ware. Das sind die Kosten, die mit dem Halten einer physischen Menge der Ware verbunden sind. Für Weizen sind die Transportkosten Lagerkosten, für lebende Schweine bestehen sie aus Lager - und Futterkosten, und für Gold bestehen sie aus Lager - und Sicherheitskosten. Einige Waren haben eine negative Kosten von tragen. Beispielsweise bietet das Halten eines Aktienindex den Vorteil, Dividenden zu erhalten. In Terminmärkten ist es üblich, die Kosten von Carry als kontinuierlich zusammengesetzte Jahresrate auszudrücken, die zu Beginn fällig ist. Wenn zum Beispiel die Transportkosten für Weizen 5 sind, bedeutet dies, dass die Kosten für die Lagerung von 100 Weizen für sechs Monate 100 (e 0,05 (0,5) -1) 2,53 betragen. Sofort fällig. Wir verwenden die folgende Notation, die in Terminmärkten üblich ist: Der Spot-Kurs der zugrunde liegenden Ware zum Zeitpunkt t Da dieses Portfolio keinen anfänglichen Cash-Aufwand erfordert, wird die Abwesenheit von Arbitrage-Chancen sicherstellen, dass die Terminal-Auszahlung gleich Null ist. Daher kann der Futures-Kontrakt wie folgt bewertet werden. Das folgende Beispiel zeigt, wie Arbitrage möglich ist, wenn diese Preisrelation verletzt wird. Beispiel 4.6: Forward Arbitrage. Angenommen, der Spotpreis für Weizen beträgt 550 Cent pro Scheffel, der Sechsmonats-Terminkurs 600, der risikoloser Zinssatz 5 p. a. Und die Kosten für den Transport beträgt 6 p. a. Um eine Arbitrage auszuführen, leihen Sie Geld, kaufen Sie einen Scheffel Weizen, zahlen, um es zu speichern, und verkaufen es nach vorn. Die Cash-Flows sind: Anfangs-Cash Flow Terminal Cash-Flow Kaufen Sie eine Einheit der Ware Pay Cost of Carry Borrow Geben Sie 6-Monats-Terminkäufe Net Portfolio-Wert Das ist, ist es möglich, in einem sicheren Gewinn, der keinen anfänglichen Barausgaben erfordert zu sperren. 4.7 Absicherung von Terminkontrakten Das primäre Motiv für den Einsatz von Terminkontrakten ist das Risikomanagement. Der Weizenbauer in Beispiel 4.4 konnte das Preisrisiko beseitigen, indem er seine Ernte vorwarf. Beispiel 4.8 enthält ein umfassenderes Beispiel zum Devisenrisikomanagement. Beispiel 4.8: Kontakte weiterleiten und Risikomanagement. XYZ ist ein multinationales Unternehmen mit Sitz in den USA. Seine Produktionsstätten befinden sich in Pittsburgh und damit sind seine Arbeits - und Herstellungskosten in US-Dollar (USD) anfallen. Ein Großteil des Umsatzes erfolgt jedoch an deutsche Kunden, die die Waren in Deutschemarks (GDM) bezahlen. Es ist eine sechsmonatige Vorlaufzeit zwischen der Platzierung eines Kundenauftrags und der Lieferung des Produkts. XYZs Kosten der Produktion ist 80 des Verkaufspreises. Angenommen, XYZ erhält einen 1MM GDM-Auftrag, und der aktuelle USDGDM-Wechselkurs beträgt 0,60 (d. h. 1 GDM 0,60 USD). Die Herstellungskosten dieser Bestellung betragen 480.000 (0.60 x 1MM x 0.80). Der Wechselkurs von sechs Monaten ist natürlich ungewiss, in welchem Fall XYZ dem Wechselkursrisiko ausgesetzt ist. Wenn der Wechselkurs bei 0,60 bleibt, konvertiert XYZ die 1MM GDM auf 600.000 und verdient einen Gewinn von 25% bei den 480.000 Herstellungskosten. Wenn jedoch der Wechselkurs auf 0,40 sechs Monate ab dem Zeitpunkt, XYZ wird die 1MM GDM auf nur 400.000 zu konvertieren, registrieren einen Verlust auf den Verkauf. Umgekehrt, wenn der Wechselkurs steigt auf 0,80 sechs Monaten ab jetzt wird XYZ konvertieren die 1MM GDM auf 800.000, Registrierung ein sehr großer Gewinn auf den Verkauf. Während XYZ sehr gut in der Herstellung und Vermarktung ihrer Produkte sind, haben sie keine Expertise bei der Prognose von Wechselkursschwankungen. Daher wollen sie das mit dieser Transaktion verbundene Wechselkursrisiko vermeiden (d. H. Das Risiko, dass sie alles richtig machen und dann Geld für den Verkauf verlieren, nur weil die Wechselkurse sich gegen sie richten). Sie können dies durch den Verkauf vorwärts 1MM GDM tun. Dies beinhaltet, einen Vertrag heute mit, sagen wir, eine Investmentbank, unter denen XYZ stimmt zu 1MM GDM sechs Monate ab jetzt im Austausch für eine feste Anzahl von US-Dollar zu liefern. Dieser Wechselkurs ist der Sechs-Monats-Terminkurs. Angenommen, der Sechs-Monats-Terminkurs beträgt 0,62 (der entsprechend den Markterwartungen und den relativen Zinssätzen wie nachfolgend beschrieben festgelegt wird). Dann, wenn XYZ erhält 1m GDM von seinem Kunden, liefern sie es an die Investmentbank im Austausch für 620.000 (Verriegelung in einem Gewinn) unabhängig davon, ob der Wechselkurs zu 0,40 oder 0,80 zu diesem Zeitpunkt geschieht. 4.9 Futures-Kontrakte Die Mechanik von Futures-Kontrakten Ein Futures-Kontrakt ähnelt einem Terminkontrakt, abgesehen von zwei wichtigen Differenzen. Erstens werden Zwischengewinne oder Verluste täglich während der Laufzeit des Futures-Kontraktes gebucht. Dieses Merkmal ist bekannt als Markierung auf den Markt. Die Zwischengewinne oder - verluste ergeben sich aus der Differenz zwischen dem heutigen Futures-Preis und dem gestern Futures-Kurs. Zweitens werden Futures-Kontrakte an organisierten Börsen mit standardisierten Bedingungen gehandelt, während Termingeschäfte im Freiverkehr gehandelt werden (maßgeschneiderte Einmalgeschäfte zwischen einem Käufer und einem Verkäufer). Beispiel 4.10 veranschaulicht die Markierung, um die Mechanik des Futures-Kontrakts der All Ordinaries Aktienkursindex (SPI) an der Sydney Futures Exchange zu vermarkten. Der SPI-Vertrag ist ähnlich wie der Chicago Mercantile Exchange (CME) SampP 500 Vertrag und der London International Financial Futures Exchange (LIFFE) FTSE 100 Vertrag. Die Mechanik ist für alle diese Verträge gleich. Aktienindex-Futures wurden in Australien im Jahr 1983 in Form von Aktienkursindex (SPI) Futures, die auf der australischen Börsen (ASX) All Ordinaries Index, die die Benchmark-Indikator der australischen Börse basiert basiert eingeführt. Zu den Anwendern von SPI-Futures zählen bedeutende internationale und australische Banken, Fondsmanager und andere große Investmentgesellschaften. SFE-Einheimische und private Investoren sind auch aktive Teilnehmer am Markt. SPI-Futures haben einen Basiswert des A25 x Index (dh ein SPI-Futures-Kontrakt mit einem Kurs von 2000,00 wird einen Vertragswert von A50.000 haben). Der All-Ordinaries-Aktienkursindex (AOI) ist ein kapitalisierungsgewichteter Index und wird unter Verwendung der Marktpreise von etwa 318 der größten an der Australischen Börse (ASX) notierten Unternehmen berechnet. Der Gesamtmarktwert dieser Gesellschaften beläuft sich auf über 95 des Wertes der 1.186 inländischen Aktien. Beispiel 4.10: Markierung auf den Markt. Angenommen, ein australischer Futures-Spekulant kauft einen SPI-Futures-Kontrakt an der Sydney Futures Exchange (SFE) um 11:00 Uhr am 6. Juni. Zu diesem Zeitpunkt ist der Futures-Preis 2300. Am Ende des Handels am 6. Juni ist der Futures-Preis gefallen Auf 2290 (was dazu führt, dass sich die Futurespreise bewegen, wird unten diskutiert). Der zugrunde liegende Futures-Kontrakt beträgt 25 x Index, so dass sich die Käuferposition um 25 (2290-2300) -250 geändert hat. Da der Käufer den Futures-Kontrakt gekauft hat und der Kurs gesunken ist, hat er am Tag Geld verloren und sein Makler wird sofort 250 aus seinem Konto nehmen. Diese unmittelbare Reflexion der Gewinn oder Verlust ist bekannt als Markierung auf den Markt. Wo die 250 gehen Auf der gegenüberliegenden Seite der Käufer Kauf Bestellung gab es einen Verkäufer, der einen Gewinn von 250 (beachten Sie, dass Futures-Handel ist ein Null-Summen-Spiel - was eine Partei verliert, die Gegenpartei gewinnt). Die 250 wird dem Verkäuferkonto gutgeschrieben. Angenommen, am Ende des Handels am nächsten Tag ist der Futures-Preis 2310. Da der Käufer die Futures gekauft hat und der Preis gestiegen ist, macht er Geld. Insbesondere wird 25 (2310-2290) 500 auf sein Konto gutgeschrieben. Dieses Geld, natürlich, kommt aus dem Verkäufer-Konto. Dieses Konzept der Markierung auf den Markt ist Standard für alle großen Futures-Kontrakte. Die Kontrakte werden bis zum Ende des Vertrags am Tag des Handelsschlusses auf den Markt gebracht. Beim Verfall gibt es zwei verschiedene Mechanismen für die Abrechnung. Die meisten Finanzterminkontrakte (wie Aktienindex, Devisen und Zinsfutures) werden in bar abgerechnet, während die meisten physischen Futures (landwirtschaftliche, Metall - und Energie-Futures) durch Lieferung der physischen Ware abgewickelt werden. Beispiel 4.11 zeigt den Barausgleich. Beispiel 4.11: Barausgleich. Angenommen, der Kurs der SPI-Futures-Kontrakte lag am Handelstag am Tag vor dem Verfallsdatum bei 2350 und am Verfallstag bei 2360 am Ende des Handels. Die Abwicklung beinhaltet lediglich eine Zahlung von 25 (2360-2350) 250 vom Verkäuferkonto auf das Käuferkonto. Der Verfalltag wird wie jeder andere Tag im Hinblick auf die marktübliche Markierung behandelt. Eine Alternative zur Barauszahlung ist die physische Lieferung. Betrachten Sie die SFE Wolle Futures-Vertrag, der Lieferung von 2500 kg Wolle erfordert, wenn der Vertrag reift. Natürlich gibt es verschiedene Arten von Wolle, so dass eine Reihe von Regeln für die Lieferung Qualität erforderlich ist. Hierbei handelt es sich um detaillierte Regeln, die die Standardqualität der zugrunde liegenden Ware und einen Zeitplan für Rabatte und Prämien für die Lieferung niedrigerer und höherer Qualität regeln. Beispiel 4.12 veranschaulicht die Regeln für die lieferbare Qualität für den SFE-Schmierwoll-Futures-Kontrakt. Beispiel 4.12: Lieferbare Qualität: Fette Wolle Futures. Die Lieferung erfolgt in zugelassenen Lagerhäusern in den großen Wollhandelszentren in ganz Australien. Für die Lieferung von Wolle muss sie die entsprechenden von der australischen Wollteststelle (AWTA) ausgestellten Gutachten und die von der Australian Wool Exchange Limited (AWEX) ausgestellten Gutachterzeugnisse besitzen. Insbesondere muss es ein gutes Topmacherminerinvlies mit einem durchschnittlichen Faserdurchmesser von 21,0 Mikrometern mit einer gemessenen mittleren Stapelfestigkeit von 35 nktx, einer mittleren Stapellänge von 90 mm, einer guten Farbe mit weniger als 1,0 pflanzlichen Stoffen, sein. Da ein bestimmter Ballen aus Wolle nicht exakt mit diesen Spezifikationen übereinstimmt, ist Wolle innerhalb einer vorgegebenen Toleranz lieferbar. Insbesondere wurden 2.400 bis 2.600 Reingewichte Kilogramm Merinofleece Wolle, gute Topmaking-Stil oder besser, gute Farbe, mit durchschnittlichen Mikrometer zwischen 19,6 und 22,5 Mikron, gemessen Stapellänge zwischen 80mm und 100mm, gemessen Stapelstärke größer als 30 nktx, weniger Als 2.0 Gemüse ist lieferbar. Prämien und Rabatte für Lieferung, die nicht mit den genauen Spezifikationen des zugrunde liegenden Vertrages übereinstimmen, werden am Freitag vor dem letzten Handelstag für alle lieferbaren Wollen oberhalb und unterhalb des Standards festgelegt, in Cent pro Kilogramm sauber angegeben. Beispiel 4.13 veranschaulicht den Prozess der physischen Lieferung für den SFE-Schmierwoll-Futures-Kontrakt. Das Verfahren ist für die meisten Rohstoff-Futures-Kontrakte ähnlich. Beispiel 4.13: Physische Lieferung. Angenommen, der fettige Wolle-Futures-Kontaktpreis betrug 700 Cents am Ende des Handels am Verfalltag. Die Abwicklung beinhaltet die physische Lieferung der zugrunde liegenden Menge an Wolle (2.500 Kilogramm) vom Verkäufer des Futures-Kontrakts an den Käufer am Geschäftstag nach dem Verfalltag. Die Lieferung umfasst daher den Verkäufer, der dem Käufer 2500 kg Wolle gegen eine Zahlung von A 17 500 zugeführt wird. Die Wolle muss innerhalb der oben beschriebenen Toleranz sein. Wenn die Wolle von besserer Qualität ist als im Vertrag angegeben, muss eine Prämie gezahlt werden. Umgekehrt bedeutet Wolle von geringerer Qualität einen Rabatt. Es ist der Verkäufer der Futures, die die Lieferung der Wolle zu machen und er hat die Wahl zu wählen, welche Qualität er liefern wird, vorbehaltlich der Zeitplan der Rabatte und Prämien. 4.14 Marge Obwohl Futures-Kontrakte keine Anfangsinvestitionen erfordern, verlangen Futuresbörsen sowohl den Käufer als auch den Verkäufer, eine als Margin bekannte Kaution zu hinterlegen. Die Margin wird typischerweise auf einen Betrag festgelegt, der größer ist als ein gewöhnlicher Tageskurs im Futures-Preis. Damit wird sichergestellt, dass beide Parteien über genügend verfügbare Mittel verfügen, um sie auf den Markt zu bringen. Das restliche Kreditrisiko besteht nur insoweit, als (1) die Futures-Preise so dramatisch verschoben werden, dass der zur Markierung erforderliche Marktbetrag größer ist als der Saldo eines Margin-Kontos für Privatpersonen und (2) die einzelnen Zahlungsausfälle für die Restzahlung. In diesem Fall trägt der Austausch den Verlust, so dass die Teilnehmer an Futures-Märkten im Wesentlichen kein Kreditrisiko tragen. Die Margin-Regeln werden in Bezug auf die anfängliche Marge (die bei Vertragsabschluss gebucht werden muss) und die Instandhaltungsspanne (das ist der minimale annehmbare Saldo im Margin-Konto) angegeben. Wenn der Saldo des Kontos unterhalb des Wartungsniveaus sinkt, führt die Börse eine Margin-Aufforderung für die Einzelperson durch, die dann das Konto auf die Höhe der Anfangsspanne vor Beginn des Handels am darauffolgenden Tag wiederherstellen muss. Beispiel 4.15 zeigt das Margining-Verfahren. Beispiel 4.15: Margin Angenommen, ein Vertrag erfordert eine Anfangsmarge von 7.000 und eine Wartungsmarge von 5.000. Die folgende Tabelle zeigt das Marginverfahren und die Cashflows, die für den Käufer eines Futures-Kontrakts erforderlich sind. Beachten Sie, dass, wenn die Margin Balance unter die Wartungs-Marge fällt, muss es auf die ursprüngliche Ebene wiederhergestellt werden. Beachten Sie auch, dass, wenn die Terminkontrakte günstiger (wie zum Zeitpunkt 3) die Markierung zu vermarkten Geldzufluss kann sofort zurückgenommen werden - es muss nicht im Margin-Konto bleiben. 4.16 Bewertung von Futures-Kontrakten Während die Bewertung von Termingeschäften relativ einfach ist, erschwert die Markierung des Marktes die Bewertung von Futures-Kontrakten. Die Cashflows aus Forward - und Futures-Kontrakten sind in der folgenden Tabelle dargestellt. Cash Flows aus Forward - und Futures-Kontrakten Bei beiden Verträgen wechselt kein Geld zum Zeitpunkt des Vertragsbeginns (Zeitpunkt 0). Für den Terminkontrakt ändert sich kein Geld, bis der Kontrakt fällig wird (Zeitpunkt T). Für den Futures-Vertrag ändert sich das Geld täglich je nach Bewegungen des Futures-Preises. Unter bestimmten Umständen ist jedoch ein Futures-Kontrakt vollkommen gleichbedeutend mit einem Terminkontrakt, in welchem Fall die beiden Kontrakte denselben Wert haben müssen. Da Terminkontrakte mit einem No-Arbitrage-Argument relativ einfach zu bewerten sind, bietet dies eine bequeme Möglichkeit, einen Futures-Kontrakt zu bewerten. Insbesondere wenn die Zinssätze während der Laufzeit des Kontraktes konstant sind (bei einer kontinuierlich zusammengesetzten Jahresrate von r), sind die Preise des Terminkontrakts und des Terminkontraktes identisch. Diese Äquivalenz kann durch eine Roll-Over-Strategie ermittelt werden, wobei zum Zeitpunkt 0 ein Anleger E r-Futures-Kontrakte kauft und FU 0 in einer risikolosen Anleihe investiert, bei der FU den Futures-Preis repräsentiert). Zum Zeitpunkt 1 ist der Gewinn (ggf. negativ) an der Futures-Position e r (FU 1 - FU 0). Die er bis zur Fälligkeit investiert (oder leiht). Bei Fälligkeit ist dies auf e r (T-1) e r FU 1 - FU 0) e rT (FU 1 - FU 0) gewachsen. Zur Zeit 1 erhöht er seine Beteiligung auf e 2r Verträge. Zum Zeitpunkt 2 ist der Gewinn (möglicherweise negativ) an dieser Position e 2r (FU 2 - FU 1). Die er bis zur Fälligkeit investiert (oder leiht). Bei Reife ist dies auf e r (T-2) e 2r (FU 2 - FU 1) e rT (FU 2 - FU 1) gewachsen. Zur Zeit 2 erhöht er seinen Betrieb auf e 3r und so weiter. Bei Fälligkeit beträgt der Gesamtauszahlungsbetrag an der Futures-Position: wobei wir anmerken, dass FU T S T. Die Auszahlung der Anleihe ist FU 0 e rT. Daher ist die gesamte Anfangsinvestition, die für diese Strategie erforderlich ist, FU 0 und die Gesamtauszahlung zum Zeitpunkt T ist S 0 e rT. Betrachten wir nun die Strategie, e rT Terminkontrakte am Tag o zu kaufen und FO 0 in eine risikolose Anleihe zu investieren (wobei FO den Preis eines Terminkontrakts darstellt). Die gesamte anfängliche Investition, die für diese Strategie erforderlich ist, ist FO 0 und die Gesamtauszahlung zum Zeitpunkt T ist: Da beide Strategien dieselbe Auszahlung haben, müssen sie dasselbe kosten. Das ist FO 0 FU 0. Die folgende Tabelle zeigt die Cashflows der beiden Strategien. 4.17 Arbitrage-Beziehungen Für den Rest dieses Moduls gehen wir davon aus, dass die Zinssätze während der Vertragslaufzeit tatsächlich konstant sind und somit der Futures-Preis dem Terminkurs entspricht. Das heißt, wir können den Preis und die Auszahlungen eines Futures-Kontraktes mit denen eines Terminkontraktes identifizieren. Dies vereinfacht die Dinge, weil ein Terminkontrakt nur eine einzige Auszahlung bei Fälligkeit hat. Betrachten wir beispielsweise die Bewertung eines Futures-Kontrakts auf den SampP 500-Aktienindex. Dieser Vertrag, der an der Chicago Mercantile Exchange (CME) abschließt, berechtigt den Käufer, den Barwert des SampP 500-Aktienindex am Ende der Vertragslaufzeit zu erhalten. Im März, Juni, September und Dezember sind immer vier Verträge gültig. Im Gegensatz zu den bisherigen Beispielen, bei denen es sich um Transportkosten handelt, ergibt sich für den SampP 500-Index ein Vorteil, in Form von erhaltenen Dividenden und nicht für Kosten. Das Ergebnis ist, dass der Wert eines SampP 500-Futures-Kontrakts ausgedrückt werden kann als Der Futures-Preis Der aktuelle Wert des SampP 500-Aktienindex Der Zinssatz (jährlicher kontinuierlich zusammengesetzter T-Bill-Satz) Die Dividendenrendite auf dem Index (kontinuierlich zusammengefasst jährlich Rate) Die Laufzeit des Kontraktes Dies ist die Gleichung (1) mit der Ausnahme, dass der q-Quotient durch "-d quot ersetzt worden ist, da die Kosten für die Beförderung (Lagerung von Weizen) durch eine Leistung (Dividenden) ersetzt wurden. Um zu sehen, warum diese Beziehung gilt, betrachten wir die Strategie von (1) Entleihen von e - dT S 0 durch die Zeit T. (2) mit dem Kauf von e - dT-Einheiten des Index und der Wiederanlage aller Dividenden wieder in den Index und (3) Verkauf eines Futures-Kontraktes, der zum Zeitpunkt T reift. Wenn die Zinssätze konstant sind, entspricht der Futures-Kontrakt einem Terminkontrakt, was die Analyse vereinfacht. Insbesondere die mit dieser Strategie verbundenen (äquivalenten) Cashflows sind in der folgenden Tabelle dargestellt. Beachten Sie, dass die Wiederanlage der Dividenden dazu geführt hat, dass die e - dT-Einheiten des Index mit einer Rate von d um eine Einheit bis zur Fälligkeit ansteigen. Die Beispiele 4.18 und 4.19 veranschaulichen, wie eine risikolose Arbitrage ausgeführt werden kann, wenn diese Gleichheit nicht gilt. Arbitrage-Beziehung zwischen Spot - und Futures-Kontrakt Noch einmal, da diese Strategie keinen anfänglichen Cash-Aufwand erfordert, muss der Cashflow bei Fälligkeit ebenfalls Null sein oder eine Arbitrage-Chance bestehen. Insbesondere wenn F gt S 0 e (r-dT) die Strategie des Kaufs des Index und des Verkaufs der Futures einen Arbitragegewinn erzeugt. Umgekehrt, wenn F lt S 0 e (r-dT) die Strategie des Verkaufens des Index und der Kauf der Futures einen Arbitragegewinn erzeugt. Die Beispiele 4.18 und 4.19 veranschaulichen, wie eine risikolose Arbitrage ausgeführt werden kann, wenn diese Gleichheit nicht gilt. Beispiel 4.18: Futures-Arbitrage: Index kaufen - Futures verkaufen Angenommen, der SampP 500-Aktienindex liegt bei 295 und der Sechsmonats-Futures-Kontrakt auf diesem Index bei 300. Wenn der aktuelle T-Bill-Satz 7 ist und die Dividendenquote 5 beträgt. Eine Arbitrage-Gelegenheit besteht, weil F300 gt S e (r-d) T 297,96. Das Arbitrage kann durch den Kauf von niedrigen und hohen Verkauf ausgeführt werden. In diesem Fall ist der Futures-Kontrakt relativ überbewertet, so dass wir die Futures verkaufen und den Index kaufen. Die Strategie besteht insbesondere darin, e - dT S 0 287,72 auf 7 in 6 Monaten zurückzuzahlen. Verwenden Sie diese 287.72, um e - dT 0.975 Einheiten des SampP-Index zu kaufen, und reinvestieren Sie alle Dividenden im Index. Verkaufen Sie einen Futures-Kontrakt für die Lieferung des Index in sechs Monaten. Hierdurch entstehen folgende Cashflows: 4.20 Absicherung mit Futures In diesem Abschnitt wird untersucht, wie drei gemeinsame Geschäftsrisiken - Zinsänderungsrisiken, Börsenrisiken und Währungsrisiken - praktisch abgesichert werden können. In jedem Fall beschreiben wir die Art des Risikos und veranschaulichen anhand einer Reihe von praktischen Beispielen, wie das Risiko verwaltet werden kann. 4.21 Absicherung von Zinsrisiken Es gibt zwei primäre Zinsfutures, die an US-Börsen handeln. Der Eurodollar-Futures-Kontrakt handelt von der Chicago Mercantile Exchange und den US-T-Bill Futures-Kontrakten auf dem Chicago Board of Trade. Der Eurodollar-Vertrag ist der erfolgreichere und schwer gehandelte Vertrag. Zu jedem Zeitpunkt beträgt der fiktive Darlehensbetrag für ausstehende Eurodollar-Futures-Kontrakte mehr als 4 Billionen. Dieser Vertrag basiert auf der LIBOR (London Interbank Offer Rate), einem Zinssatz für Eurodollar-Termineinlagen. Dieser Satz ist der Maßstab für viele US-Kreditnehmer und Kreditgeber. Zum Beispiel kann ein Unternehmen Kreditnehmer eine Rate von LIBOR200 Basispunkte auf einem kurzfristigen Darlehen. Eurodollar-Festgelder sind nicht verhandelbare, festverzinsliche US-Dollar-Einlagen bei Banken, die nicht den US-Bankvorschriften unterliegen. Diese Banken können sich in Europa, der Karibik, Asien oder Südamerika befinden. US-Banken können auf unregulierter Basis über ihre internationalen Bankinstitute Einlagen tätigen. LIBOR ist die Rate, mit der die großen Geldzentrumsbanken bereit sind, Eurodollar-Termineinlagen bei anderen großen Geldzentrumsbanken zu platzieren. Unternehmen leihen in der Regel auf einer Spread über LIBOR, da ein Unternehmen Kreditrisiko ist größer als die einer großen Geld-Center-Bank. Nach Übereinkommen wird der LIBOR als jährlicher Zinssatz auf der Grundlage eines tatsächlichen 60-Tage-Jahres (d. h. Zinsen für jeden Tag zum Jahreszins360) berechnet. Beispiel 4.22 zeigt, wie die Zinsen auf ein LIBOR-Darlehen gemäß den Konventionen berechnet werden. Beispiel 4.22: LIBOR-Konventionen. Wenn 3-Monats - (90 Ist-Tage) LIBOR mit 8 notiert ist, sind die Zinsen, die am Ende des dreimonatigen Leihungszeitraums auf ein 1-Million-Darlehen gezahlt werden, (0,08) (90360) 1.000.000 ((.08) 4) 1.000.000 20.000 Der Eurodollar-Futures-Kontrakt basiert auf einer 3-Monats-1-Million-Eurodollar-Termineinlage. Es ist bar abgerechnet, so dass keine tatsächliche Ablieferung der Anzahlung erfolgt, wenn der Vertrag ausläuft. Liefermonate sind März, Juni, September und Dezember. Die Mindestpreisverschiebung beträgt 25 pro Vertrag, die 1 Basispunkt entspricht: (.00014) 1.000.00025. Der Futures-Preis bei Verfall (Zeitpunkt T) wird als F T 100-LIBOR ermittelt. Vor dem Verfall der Futures-Preis impliziert den Zinssatz, der effektiv für ein 3-Monats-Darlehen, die am Tag des Kontrakts fällig gesperrt werden kann. Die Abwicklung des Eurodollar-Futures-Kontraktes ist in Beispiel 4.23 dargestellt. Beispiel 4.23: Abwicklung des Eurodollar-Futures-Kontrakts. Angenommen, Sie kauften 1 Dezember Eurodollar Futures-Vertrag am 15. November, wenn der Preis 94,86 war. Wenn der Zinssatz zwischen dem 15. November und dem Ablauf des Futures-Kontraktes im Dezember um 100 Basispunkte sinkt, was ist Ihr Gesamtgewinn oder - verlust aus dem Vertrag bei der Abwicklung? Zuerst beachten Sie, dass kein Geld die Hände zum Zeitpunkt des Kaufs des Vertrages wechselt. Dies ist die Art aller Futures-Kontrakte. Der 15. November Preis von 94,86 bedeutet, dass der LIBOR-Zinssatz 100-94.865,14 zu diesem Zeitpunkt war. Wenn der LIBOR bis zum Ablauf des Dezember-Vertrags 100 Basispunkte unterschreitet, wird LIBOR dann 4,14. Daher wird der Auslauf-Futures-Preis 100-4.1495.86 sein. Der Gesamtgewinn ist also: Das heißt, um den Vertrag zu begleichen, wird Ihre Gegenpartei Ihnen 2.500 geben. Beispiel 4.24 enthält eine detaillierte Darstellung, wie der Eurodollar-Futures-Kontrakt zur Absicherung des Zinsrisikos eingesetzt werden kann. Beispiel 4.24: Absicherung mit dem Eurodollar-Futures-Kontrakt. Es ist derzeit 15. November und Ihr Unternehmen ist sich bewusst, dass es braucht, um 1 Million zu leihen am 16. Dezember eine Haftung, die fällig an diesem Tag zu zahlen. Das Darlehen kann am 16. März zurückbezahlt werden, wenn eine Forderung erhoben wird. Die aktuelle LIBOR-Rate ist 5.14. Ihr Unternehmen ist besorgt, dass die Zinsen steigen wird zwischen jetzt und 16. Dezember, in diesem Fall werden Sie zahlen einen höheren Zinssatz für Ihr Darlehen. Wie kann Ihr Unternehmen sperren in der aktuellen Rate von 5,14 Ihr Unternehmen steht zu verlieren, wenn die Zinsen zu erhöhen. Daher müssen Sie eine Futures-Position eintragen, die im Wert steigt, wenn die Zinsen steigen. Dann, wenn die Zinsen steigen, verliert Ihr Unternehmen durch die Zahlung höherer Zinsen auf das Darlehen, aber Ihr Unternehmen Gewinne durch Profitieren auf der Futures-Position. Umgekehrt, wenn die Zinsen fallen, Ihr Unternehmen gewinnt durch die Zahlung niedrigere Zinsen Gebühren für das Darlehen, aber Ihr Unternehmen verliert auf der Futures-Position. Idealerweise würde der Verlust und der Gewinn genau abbrechen, ob die Zinsen steigen oder fallen. Aus dem Bau des Eurodollar-Futures-Kontraktes wissen wir, dass, wenn der Zinssatz steigt, der Futures-Preis sinkt. Daher werden Sie 1 Dezember Eurodollar Futures-Kontrakt auf 94,86 zu verkaufen. Grundlage für diesen Vertrag ist ein fiktives 3-Monats-Darlehen in Höhe von 1 Million Dollar, das am 16. Dezember (am Tag des Vertragsabschlusses) abgeschlossen wird. Wenn wir den Satz von 5,14 verriegeln könnten, wäre der Gesamtzins für das Darlehen 0,0514 (1 Million) 4 12,850. Zuerst nehmen wir an, dass am 16. Dezember der LIBOR 6.14 ist. Der Zinssatz für das Darlehen beträgt 0,0614 (1 Million) 4 15.350, und der Gewinn auf der Futures-Position wird -10000 sein (93.86-94.86) 4 2.500. Daraus ergibt sich ein Nettomittelabfluss von -15.3502.500 -12.850, der gleich dem 3-Monats-Zins auf 1 Million bei 5,14 ist. Angenommen, am 16. Dezember ist der LIBOR 4.14. Der Zinssatz für das Darlehen beträgt 0,0414 (1 Million) 4 10.350 und der Gewinn auf der Futures-Position wird -10000 (95.86-94.86) 4 -2.500 betragen. Daraus ergibt sich ein Nettomittelabfluss von -10.350-2.500 -12.850, der gleich dem 3-Monats-Zins auf 1 Million bei 5,14 ist. 4.25 Hedging-Marktrisiko Eine weitere Risikoquelle, die eine Einzelperson oder eine Organisation wünschen kann, ist das Aktienmarktrisiko. Beispielsweise kann eine Person, die sich dem Ruhestand nähert, wünschen, den Wert der Aktienkomponente seines Rentenfonds gegen einen Börsencrash abzusichern, bevor er sich zurückzieht. Ein Fondsmanager, der glaubt, er könne Gewinner unter den einzelnen Aktien auswählen, könnte wünschte, die Marktbewegungen abzusichern. Der dominierende Börsenindex-Futures-Kontrakt ist der SampP 500 Futures-Kontrakt. Dieser Vertrag arbeitet an der Chicago Mercantile Exchange und hat Liefermonate März, Juni, September und Dezember. Die zugrunde liegende Menge beträgt das 500-fache des SampP 500-Index. Die minimale Kursbewegung beträgt 0,05 Indexpunkte, was 25 pro Kontrakt entspricht. Beispiel 4.26 zeigt die Abwicklungsmechanik für den SampP 500-Vertrag. Beispiel 4.26: Abrechnung des SampP 500 Futures-Kontrakts. Es ist derzeit 15. November und der SampP 500 Index ist bei 382,62. Der Dezember SampP 500 Futures Preis ist 383,50. Wenn Sie 1 Dezember SampP 500 Futures-Kontrakt kaufen, wie viel werden Sie gewinnen, wenn der Futures-Preis bei Verfall 393,50 Der Gewinn auf Ihrer Futures-Position ist 500 (F t - F 0) 500 (393.50-383.50) 5.000. Das heißt, um den Vertrag zu begleichen, wird Ihr Kontrahent Ihnen 5.000 geben. Beispiel 4.27 enthält eine detaillierte Darstellung, wie der SampP 500 Futures-Kontrakt zur Absicherung des Aktienmarktrisikos eingesetzt werden kann. Beispiel 4.27: Absicherung mit dem SampP 500 Futures-Kontrakt. Ein Portfolio-Manager hält ein Portfolio, das den SampP 500 Index nachbildet. Der SampP 500 Index startete das Jahr bei 306,8 und ist derzeit bei 382,62. The December SampP 500 futures price is currently 383.50. The managers fund was valued at 76.7 million at the beginning of the year. Since the fund has already generated a handsome return for the year, the manager wishes to lock in its current value. That is, he is willing to give up potential increases in order to ensure that the value of the fund does not decrease. How does he lock in the current value of the fund First note that at the December futures price of 383.50, the return on the index, since the beginning of the year, is 383.5306.80-1 25. If the manager is able to lock in this return on his fund, the value of the fund will be 1.25(76.7 million) 95.875 million . Since the notional amount underlying an SampP 500 futures contract is 500(383.50) 191,750 . the manager can lock in the 25 return by selling 95,875,000191,750500 contracts. To illustrate that this position does indeed form a perfect hedge, we examine the net value of the hedged position under two scenarios. First, suppose the value of the SampP 500 index is 303.50 at the end of December. In this case, the value of the fund will be (303.50383.50)95.875 million 75.875 million . The gain on the futures position will be -500(500)(303.50-383.50) 20 million. Hence the total value of the hedged position is 75.875 20 95.875 million, locking in a 25 return for the year. Now suppose that the value of the SampP 500 index is 403.50 at the end of December. In this case, the value of the fund will be 403.50383.50(95.875 million) 100.875 million . The gain on the futures position will be -500(500)(403.50-383.50) -5 million. Hence the total value of the hedged position is 100.875-595.875 million, again locking in a 25 return for the year. 4.28 Hedging Foreign Exchange Risk Another source of risk that an individual or organization may wish to hedge is foreign exchange risk. For example, a person who will be travelling overseas in the coming months may wish to hedge the value of the amount of money he intends to spend abroad against a devaluation of his domestic currency relative to the foreign currency. An exporter who sells goods overseas on credit may wish to hedge against a devaluation of the foreign currency in which payment occurs. A number of foreign currency futures contracts trade on the International Monetary Market division of the Chicago Mercantile Exchange. The currencies on which contracts are based, and the underlying notional amount are listed in the following Table. Delivery months for all contracts are March, June, September, and December. Prices are quoted as US dollars per unit of foreign currency. For example, if one Swiss franc buys 69.15 US cents, the price will be quoted as 0.6915. Denomination of Foreign Currency Futures Contracts Example 4.29 contains a detailed illustration of hedging exchange risk. Example 4.29: Hedging with the Swiss Franc Futures Contract. Your company sells 10 machines to a Swiss company. The sale price is 100,000 Swiss Francs each and payment is to be made at the end of the calendar year. The December futures price for Swiss Francs is 0.6915. You are worried that the Swiss Franc will depreciate against the US Dollar between now and the end of the year. How can you hedge this exchange rate risk Note that since (1) the total exposure is one million Swiss Francs and (2) each futures contract is for 125, 000 Francs, eight contracts are required to hedge the exposure. Further, since (1) the company stands to lose if the Swiss Franc depreciates (each Swiss Franc can be converted back into a smaller number of Dollars) and (2) the futures contracts decrease in value if the Swiss Franc depreciates (since the basis of the contract is Swiss Francs per Dollar), the contracts should be sold. To illustrate that selling eight futures contracts provides an adequate hedge, first suppose that the value of the Swiss Franc is 0.30 at the end of December. In this case, the US Dollar value of the payment for the machines will be 0.30(10)(100,000) 300,000. The gain on the futures position will be -8(125,000)(0.30-0.6915) 391,500. Hence the total income is 691,500, which equals the unhedged income in dollars if the exchange rate does not fluctuate. 4.30 Basis Risk There is no such thing as a perfect hedge. You can never completely eliminate a cash positions risk. Consider a holder of Q Treasury bonds maturing in 2004 with a coupon rate of 8. Assume that the holder of bonds believes that bond prices are going to fall. To hedge his risk, the person shorts an equivalent amount of futures contracts for Treasury bonds. At a later date, the person will close out both its bond and futures positions. At the close, the firm will receive B T per bond sold in the regular spot or cash market. The futures price is F 0 at the time the futures are sold short, and its price at the closeout is F T . Prior to the closeout, both B T and F T are uncertain, although F 0 is known. The usual computation of the funds that the person will have at closeout is: From the above equation, the net revenue from the hedge position is composed of (1) a certain component that depends upon the futures price at the time of the hedge ( F 0 ) and (2) an uncertain component that depends upon the difference between the price received for bonds in the spot market and the futures price at closeout ( B T - F T ). The difference between the spot and the futures price is called the basis . Thus, uncertainty about the net hedged revenue arises if there is uncertainty about the basis. To quote Holbrook Working, quothedging is speculation in the basisquot. There are many reasons for the basis to be uncertain. First, the good or instrument being hedged may be different from the good or instrument for which there is a futures contract. This would be the case if a corporate bond offering is hedged with Treasury bond futures basis risk arises due to the uncertainty of the yield differential at the time the hedge is lifted. Second, in commodity futures, there is basis risk due to locational differentials. For example, a cattle farmer in Texas who hedges with a cattle futures contract that calls for delivery in Omaha has the uncertainty of the closeout differential between the Texas steer price and the Omaha steer price. This is called locational basis risk . This is usually an important factor in agricultural contracts. The risk is compounded by the fact that the seller usually has the option of where delivery is made. The third type of basis risk arises because the seller of the futures contract often has the option to choose the quality of the goods or financial instrument delivered. For example, the Treasury bond futures market calls for delivery of any U. S. Treasury bond that is not callable within 15 years. Since there are many instruments that are candidates for delivery, the hedge has the risk of fluctuations in the yield spread between the instrument hedged and the instrument ultimately delivered. Fourthly, with most futures contracts, the seller has the choice of the date of delivery within the delivery month. This choice is an uncertain value and thus contributes to basis risk. Finally, the mark to market aspect of futures results in hedging risk. The uncertainty is about the amount of interest earned or forfeited due to the daily transfers of profits and losses. In fact, the equations for net revenue are not exactly right due to the omission of interest earned (lost) on futures profits (losses). 4.31 The Volatility of Futures A common mistake made is to assume that futures are much more volatile than stocks. Percentage changes of futures prices are generally less volatile than the percentage changes of a typical stock. Annualized standard deviations for most futures contracts are in the 15-20 range whereas a typical stocks is about 30. There is no reason that the futures should be played in a high risk manner by a large investor. Of course, if the futures investor does not have enough capital (5-8 times margin), then he is required to play with considerable leverage or not at all. Before taking great leverage, the small investor should consider looking at a smaller contract (grain on CBT is 5,000 bushels whereas Mid-America contract is 1,000 bushels). The effect of leverage is to increase volatility. Borrowing to meet the margin requirements will increase gains but also increase losses. Setting aside larger amounts of capital which are invested in a safe asset will decrease the volatility. 4.32 Trading the Contracts The futures exchanges provide detailed information on each of their contracts. You should obtain this information before trading. It is especially important to know the details of the delivery procedure (if you are going to make delivery or take delivery). WWWFinance contains contract specifications and general information on over 200 futures and options contracts traded worldwide. In addition, a serious trader would be in contact with the exchange to confirm the most recent set or specifications. 4.33 Risk in the Futures Markets As we have already seem, one the most important applications of the futures is for hedging. Futures contracts were initially introduced to help farmers that did not want to bear the risk of price fluctuations. The farmer could short hedge in March (agree to sell his crop) for a September delivery. This effectively locks in the price that the farmer receives. On the other side, a cereal company may want to guarantee in March the price that it will pay for grain in September. The cereal company will enter into a long hedge . There are a number of important insights that should be reviewed. The first is that we should be careful about what we consider the investment in a futures contract. It is unlikely that the margin is the investment for most traders. It is rare that somebody plays the futures with a total equity equal to the margin. It is more common to invest some of your capital in a money market fund and draw money out of that account as you need it for margin and add to that account as you gain on the futures contract. It is also uncommon to put the full value of the underlying contract in the money market fund. It is more likely that the futures investor will put a portion of the value of the futures contract into a money market fund. The ratio of the value of the underlying contract to the equity invested in the money market fund is known as the leverage. The leverage is a key determinant of both the return on investment and on the volatility of the investment. The higher the leverage -- the more volatile are the returns on your portfolio of money market funds and futures. The most extreme leverage is to include no money in the money market fund -- only commit your margin. The second important insight had to do with hedging with futures contracts. The concept of basis risk was introduced. It is extremely unlikely that you can create a perfect hedge. A perfect hedge is when the loss on your cash position is exactly offset by the gain in the futures position. We suggested some reasons why it is unlikely that we can construct a perfect hedge. The most obvious case is when you are trying to hedge a cash position with futures positions in different instruments. This is the case that we introduced in one of the first lectures when we hold the Ginnie Mae security and want to hedge this security with a combination of T-Bonds and Euros. It is unlikely, however, that at the expiration of the futures contract, the cash price of the T-Bond and Euros will equal the Ginnie Mae. This is the basis risk. A second type of basis risk arises out of the quality option . We discussed this in terms of food and financial instruments. If you are a farmer and want to lock in the price for your crop of wheat, you may use a futures contract that may call for delivery of a number of different types of wheat. Similarly, in the T-Bond and T-Note contracts, there are a whole range of instruments that are available for delivery. This difference will induce basis risk. Third, there is a timing option. The futures contract is different from an options contract. Most futures call for delivery within the contract month. It is unclear when the short will deliver the goods. This uncertainty leads to basis risk. The fourth type of risk is locational basis risk. This is mainly applicable to agricultural commodities. There could be a difference the cash price of the good that you are selling (cattle) and the futures price at a different location. The last type of uncertainty is linked to the uncertain interest rate flows from the money you make in excess of the margin. 4.34 More on Hedging Since hedging is such an important application of futures contracts, we have provided more examples of hedging Some textbook examples which do the hedging incorrectly are included, to show you some of the common pitfalls involved with hedging. Summary of Important Formulas The price of a forward contract when there is a cost of carry q . When interest rates are constant, the same relationship holds for a futures contract. The price of a forward contract when there is a dividend benefit d . When interest rates are constant, the same relationship holds for a futures contract. W hat is V aluation Knowing what an asset is worth and what determines that value is a pre-requisite for intelligent decision making -- in choosing investments for a portfolio, in deciding on the appropriate price to pay or receive in a takeover and in making investment, financing and dividend choices when running a business. The premise of valuation is that we can make reasonable estimates of value for most assets, and that the same fundamental principles determine the values of all types of assets, real as well as financial. Some assets are easier to value than others, the details of valuation vary from asset to asset, and the uncertainty associated with value estimates is different for different assets, but the core principles remain the same. This introduction lays out some general insights about the valuation process and outlines the role that valuation plays in portfolio management, acquisition analysis and in corporate finance. It also examines the three basic approaches that can be used to value an asset. A philosophical basis for valuation A postulate of sound investing is that an investor does not pay more for an asset than it is worth. This statement may seem logical and obvious, but it is forgotten and rediscovered at some time in every generation and in every market. There are those who are disingenuous enough to argue that value is in the eyes of the beholder, and that any price can be justified if there are other investors willing to pay that price. That is patently absurd. Perceptions may be all that matter when the asset is a painting or a sculpture, but we do not and should not buy most assets for aesthetic or emotional reasons we buy financial assets for the cashflows we expect to receive from them. Consequently, perceptions of value have to be backed up by reality, which implies that the price we pay for any asset should reflect the cashflows it is expected to generate. Valuation models attempt to relate value to the level of, uncertainty about and expected growth in these cashflows. There are many aspects of valuation where we can agree to disagree, including estimates of true value and how long it will take for prices to adjust to that true value. But there is one point on which there can be no disagreement. Asset prices cannot be justified by merely using the argument that there will be other investors around who will pay a higher price in the future. That is the equivalent of playing a very expensive game of musical chairs, where every investor has to answer the question, quotWhere will I be when the music stops before playing. The problem with investing with the expectation that there will be a bigger fool around to sell an asset to, when the time comes, is that you might end up being the biggest fool of all. Inside the Valuation Process There are two extreme views of the valuation process. At one end are those who believe that valuation, done right, is a hard science, where there is little room for analyst views or human error. At the other are those who feel that valuation is more of an art, where savvy analysts can manipulate the numbers to generate whatever result they want. The truth does lies somewhere in the middle and we will use this section to consider three components of the valuation process that do not get the attention they deserve 8211 the bias that analysts bring to the process, the uncertainty that they have to grapple with and the complexity that modern technology and easy access to information have introduced into valuation. Value first, Valuation to follow: Bias in Valuation We almost never start valuing a company with a blank slate. All too often, our views on a company are formed before we start inputting the numbers into the models that we use and not surprisingly, our conclusions tend to reflect our biases. We will begin by considering the sources of bias in valuation and then move on to evaluate how bias manifests itself in most valuations. We will close with a discussion of how best to minimize or at least deal with bias in valuations. Sources of Bias The bias in valuation starts with the companies we choose to value . These choices are almost never random, and how we make them can start laying the foundation for bias. It may be that we have read something in the press (good or bad) about the company or heard from an expert that it was under or over valued. Thus, we already begin with a perception about the company that we are about to value. We add to the bias when we collect the information we need to value the firm. The annual report and other financial statements include not only the accounting numbers but also management discussions of performance, often putting the best possible spin on the numbers. With many larger companies, it is easy to access what other analysts following the stock think about these companies . Zacks, IBES and First Call, to name three services among many, provide summaries of how many analysts are bullish and bearish about the stock, and we can often access their complete valuations. Finally, we have the markets own estimate of the value of the company - the market price 8211 adding to the mix. Valuations that stray too far from this number make analysts uncomfortable, since they may reflect large valuation errors (rather than market mistakes). In many valuations, there are institutional factors that add to this already substantial bias. For instance, it is an acknowledged fact that equity research analysts are more likely to issue buy rather than sell recommendations, i. e. that they are more likely to find firms to be undervalued than overvalued. 1 This can be traced partly to the difficulties analysts face in obtaining access and collecting information on firms that they have issued sell recommendations on, and partly to pressure that they face from portfolio managers, some of whom might have large positions in the stock, and from their own firms investment banking arms which have other profitable relationships with the firms in question. The reward and punishment structure associated with finding companies to be under and over valued is also a contributor to bias. An analyst whose compensation is dependent upon whether she finds a firm is under or over valued will be biased in her conclusions. This should explain why acquisition valuations are so often biased upwards. The analysis of the deal, which is usually done by the acquiring firms investment banker, who also happens to be responsible for carrying the deal to its successful conclusion, can come to one of two conclusions. One is to find that the deal is seriously over priced and recommend rejection, in which case the analyst receives the eternal gratitude of the stockholders of the acquiring firm but little else. The other is to find that the deal makes sense (no matter what the price) and to reap the ample financial windfall from getting the deal done. Manifestations of Bias There are three ways in which our views on a company (and the biases we have) can manifest themselves in value. The first is in the inputs that we use in the valuation . When we value companies, we constantly come to forks in the road where we have to make assumptions to move on. These assumptions can be optimistic or pessimistic. For a company with high operating margins now, we can either assume that competition will drive the margins down to industry averages very quickly (pessimistic) or that the company will be able to maintain its margins for an extended period (optimistic). The path we choose will reflect our prior biases. It should come as no surprise then that the end value that we arrive at is reflective of the optimistic or pessimistic choices we made along the way. The second is in what we will call post-valuation tinkering . where analysts revisit assumptions after a valuation in an attempt to get a value closer to what they had expected to obtain starting off. Thus, an analyst who values a company at 15 per share, when the market price is 25, may revise his growth rates upwards and his risk downwards to come up a higher value, if she believed that the company was under valued to begin with. The third is to leave the value as is but attribute the difference between the value we estimate and the value we think is the right one to a qualitative factor such as synergy or strategic considerations . This is a common device in acquisition valuation where analysts are often called upon to justify the unjustifiable. In fact, the use of premiums and discounts, where we augment or reduce estimated value, provides a window on the bias in the process. The use of premiums 8211 control and synergy are good examples 8211 is commonplace in acquisition valuations, where the bias is towards pushing value upwards (to justify high acquisition prices). The use of discounts 8211 illiquidity and minority discounts, for instance 8211 are more typical in private company valuations for tax and divorce court, where the objective is often to report as low a value as possible for a company. What to do about bias Bias cannot be regulated or legislated out of existence. Analysts are human and bring their biases to the table. However, there are ways in which we can mitigate the effects of bias on valuation: 1. Reduce institutional pressures . As we noted earlier, a significant portion of bias can be attributed to institutional factors. Equity research analysts in the 1990s, for instance, in addition to dealing with all of the standard sources of bias had to grapple with the demand from their employers that they bring in investment banking business. Institutions that want honest sell-side equity research should protect their equity research analysts who issue sell recommendations on companies, not only from irate companies but also from their own sales people and portfolio managers. 2. De-link valuations from rewardpunishment . Any valuation process where the reward or punishment is conditioned on the outcome of the valuation will result in biased valuations. In other words, if we want acquisition valuations to be unbiased, we have to separate the deal analysis from the deal making to reduce bias. 3. No pre-commitments . Decision makers should avoid taking strong public positions on the value of a firm before the valuation is complete. An acquiring firm that comes up with a price prior to the valuation of a target firm has put analysts in an untenable position, where they are called upon to justify this price. In far too many cases, the decision on whether a firm is under or over valued precedes the actual valuation, leading to seriously biased analyses. 4. Self-Awareness . The best antidote to bias is awareness. An analyst who is aware of the biases he or she brings to the valuation process can either actively try to confront these biases when making input choices or open the process up to more objective points of view about a companys future. 5. Honest reporting . In Bayesian statistics, analysts are required to reveal their priors (biases) before they present their results from an analysis. Thus, an environmentalist will have to reveal that he or she strongly believes that there is a hole in the ozone layer before presenting empirical evidence to that effect. The person reviewing the study can then factor that bias in while looking at the conclusions. Valuations would be much more useful if analysts revealed their biases up front. While we cannot eliminate bias in valuations, we can try to minimize its impact by designing valuation processes that are more protected from overt outside influences and by report our biases with our estimated values. It is only an estimate: Imprecision and Uncertainty in Valuation Starting early in life, we are taught that if we do things right, we will get the right answers. In other words, the precision of the answer is used as a measure of the quality of the process that yielded the answer. While this may be appropriate in mathematics or physics, it is a poor measure of quality in valuation. Barring a very small subset of assets, there will always be uncertainty associated with valuations, and even the best valuations come with a substantial margin for error. In this section, we examine the sources of uncertainty and the consequences for valuation. Sources of Uncertainty Uncertainty is part and parcel of the valuation process, both at the point in time that we value a business and in how that value evolves over time as we get new information that impacts the valuation. That information can be specific to the firm being valued, more generally about the sector in which the firm operates or even be general market information (about interest rates and the economy). When valuing an asset at any point in time . we make forecasts for the future. Since none of us possess crystal balls, we have to make our best estimates, given the information that we have at the time of the valuation. Our estimates of value can be wrong for a number of reasons, and we can categorize these reasons into three groups. ein. Estimation Uncertainty . Even if our information sources are impeccable, we have to convert raw information into inputs and use these inputs in models. Any mistakes or mis-assessments that we make at either stage of this process will cause estimation error. B. Firm-specific Uncertainty . The path that we envision for a firm can prove to be hopelessly wrong. The firm may do much better or much worse than we expected it to perform, and the resulting earnings and cash flows will be very different from our estimates. C. Macroeconomic Uncertainty . Even if a firm evolves exactly the way we expected it to, the macro economic environment can change in unpredictable ways. Interest rates can go up or down and the economy can do much better or worse than expected. These macro economic changes will affect value. The contribution of each type of uncertainty to the overall uncertainty associated with a valuation can vary across companies. When valuing a mature cyclical or commodity company, it may be macroeconomic uncertainty that is the biggest factor causing actual numbers to deviate from expectations. Valuing a young technology company can expose analysts to far more estimation and firm-specific uncertainty. Note that the only source of uncertainty that can be clearly laid at the feet of the analyst is estimation uncertainty. Even if we feel comfortable with our estimates of an assets values at any point in time, that value itself will change over time . as a consequence of new information that comes out both about the firm and about the overall market. Given the constant flow of information into financial markets, a valuation done on a firm ages quickly, and has to be updated to reflect current information. Thus, technology companies that were valued highly in late 1999, on the assumption that the high growth from the nineties would continue into the future, would have been valued much less in early 2001, as the prospects of future growth dimmed. With the benefit of hindsight, the valuations of these companies (and the analyst recommendations) made in 1999 can be criticized, but they may well have been reasonable, given the information available at that time. Responses of Uncertainty Analysts who value companies confront uncertainty at every turn in a valuation and they respond to it in both healthy and unhealthy ways. Among the healthy responses are the following: 1. Better Valuation Models . Building better valuation models that use more of the information that is available at the time of the valuation is one way of attacking the uncertainty problem. It should be noted, though, that even the best-constructed models may reduce estimation uncertainty but they cannot reduce or eliminate the very real uncertainties associated with the future 2. Valuation Ranges . A few analysts recognize that the value that they obtain for a business is an estimate and try to quantify a range on the estimate. Some use simulations and others derive expected, best-case and worst-case estimates of value. The output that they provide therefore yields both their estimates of value and their uncertainty about that value. 3. Probabilistic Statements . Some analysts couch their valuations in probabilistic terms to reflect the uncertainty that they feel. Thus, an analyst who estimates a value of 30 for a stock which is trading at 25 will state that there is a 60 or 70 probability that the stock is under valued rather than make the categorical statement that it is under valued. Here again, the probabilities that accompany the statements provide insight into the uncertainty that the analyst perceives in the valuation. In general, healthy responses to uncertainty are open about its existence and provide information on its magnitude to those using the valuation. These users can then decide how much caution they should exhibit while acting on the valuation. Unfortunately, not all analysts deal with uncertainty in ways that lead to better decisions. The unhealthy responses to uncertainty include: 1. Passing the buck . Some analysts try to pass on responsibility for the estimates by using other peoples numbers in the valuation. For instance, analysts will often use the growth rate estimated by other analysts valuing a company as their estimate of growth. If the valuation turns out to be right, they can claim credit for it, and if it turns out wrong, they can blame other analysts for leading them down the garden path. 2. Giving up on fundamentals . A significant number of analysts give up, especially on full-fledged valuation models, unable to confront uncertainty and deal with it. All too often, they fall back on more simplistic ways of valuing companies (multiples and comparables, for example) that do not require explicit assumptions about the future. A few decide that valuation itself is pointless and resort to reading charts and gauging market perception. In closing, it is natural to feel uncomfortable when valuing equity in a company. We are after all trying to make our best judgments about an uncertain future. The discomfort will increase as we move from valuing stable companies to growth companies, from valuing mature companies to young companies and from valuing developed market companies to emerging market companies. What to do about uncertainty The advantage of breaking uncertainty down into estimation uncertainty, firm-specific and macroeconomic uncertainty is that it gives us a window on what we can manage, what we can control and what we should just let pass through into the valuation. Building better models and accessing superior information will reduce estimation uncertainty but will do little to reduce exposure to firm-specific or macro-economic risk. Even the best-constructed model will be susceptible to these uncertainties. In general, analysts should try to focus on making their best estimates of firm-specific information 8211 how long will the firm be able to maintain high growth How fast will earnings grow during that period What type of excess returns will the firm earn8211 and steer away from bringing in their views on macro economic variables. To see why, assume that you believe that interest rates today are too low and that they will go up by about 1.5 over the next year. If you build in the expected rise in interest rates into your discounted cash flow valuations, they will all yield low values for the companies that you are analyzing. A person using these valuations will be faced with a conundrum because she will have no way of knowing how much of this over valuation is attributable to your macroeconomic views and how much to your views of the company. In summary, analysts should concentrate on building the best models they can with as much information as they can legally access, trying to make their best estimates of firm-specific components and being as neutral as they can on macro economic variables. As new information comes in, they should update their valuations to reflect the new information. There is no place for false pride in this process. Valuations can change dramatically over time and they should if the information warrants such a change. The Payoff to Valuation Even at the end of the most careful and detailed valuation, there will be uncertainty about the final numbers, colored as they are by assumptions that we make about the future of the company and the economy in which it operates. It is unrealistic to expect or demand absolute certainty in valuation, since the inputs are estimated with error. This also means that analysts have to give themselves reasonable margins for error in making recommendations on the basis of valuations. The corollary to this statement is that a valuation cannot be judged by its precision. Some companies can be valued more precisely than others simply because there is less uncertainty about the future. We can value a mature company with relatively few assumptions and be reasonably comfortable with the estimated value. Valuing a technology firm will require far more assumptions, as will valuing an emerging market company. A scientist looking at the valuations of these companies (and the associated estimation errors) may very well consider the mature company valuation the better one, since it is the most precise, and the technology firms and emerging market company valuations to be inferior because there is most uncertainty associated with the estimated values. The irony is that the payoff to valuation will actually be highest when you are most uncertain about the numbers. After all, it is not how precise a valuation is that determines its usefulness but how precise the value is relative to the estimates of other investors trying to value the same company. Any one can value a zero-coupon default-free bond with absolute precision. Valuing a young technology firm or an emerging market firm requires a blend of forecasting skills, tolerance for ambiguity and willingness to make mistakes that many analysts do not have. Since most analysts tend to give up in the face of such uncertainty, the analyst who perseveres and makes her best estimates (error-prone though they might be) will have a differential edge. We do not want to leave the impression that we are completely helpless in the face of uncertainty. Simulations, decision trees and sensitivity analyses are tools that help us deal with uncertainty but not eliminate it. Are bigger models better Valuation Complexity Valuation models have become more and more complex over the last two decades, as a consequence of two developments. On the one side, computers and calculators have become far more powerful and accessible in the last few decades. With technology as our ally, tasks that would have taken us days in the pre-computer days can be accomplished in minutes. On the other side, information is both more plentiful, and easier to access and use. We can download detailed historical data on thousands of companies and use them as we see fit. The complexity, though, has come at a cost. In this section, we will consider the trade off on complexity and how analysts can decide how much to build into models. More detail or less detail A fundamental question that we all face when doing valuations is how much detail we should break a valuation down into. There are some who believe that more detail is always better than less detail and that the resulting valuations are more precise. We disagree. The trade off on adding detail is a simple one. On the one hand, more detail gives analysts a chance to use specific information to make better forecasts on each individual item. On the other hand, more detail creates the need for more inputs, with the potential for error on each one, and generates more complicated models. Thus, breaking working capital down into its individual components 8211 accounts receivable, inventory, accounts payable, supplier credit etc. 8211 gives an analyst the discretion to make different assumptions about each item, but this discretion has value only if the analyst has the capacity to differentiate between the items. The Cost of Complexity A parallel and related question to how much detail there should be in a valuation is the one of how complex a valuation model should be. There are clear costs that we pay as models become more complex and require more information. 1. Information Overload . More information does not always lead to better valuations. In fact, analysts can become overwhelmed when faced with vast amounts of conflicting information and this can lead to poor input choices. The problem is exacerbated by the fact that analysts often operate under time pressure when valuing companies. Models that require dozens of inputs to value a single company often get short shrift from users. A models output is only as good as the inputs that go into it it is garbage in, garbage out. 2. Black Box Syndrome . The models become so complicated that the analysts using them no longer understand their inner workings. They feed inputs into the models black box and the box spits out a value. In effect, the refrain from analysts becomes The model valued the company at 30 a share rather than We valued the company at 30 a share. Of particular concern should be models where portions of the models are proprietary and cannot be accessed (or modified) by analysts. This is often the case with commercial valuation models, where vendors have to keep a part of the model out of bounds to make their services indispensable. 3. Big versus Small Assumptions . Complex models often generate voluminous and detailed output and it becomes very difficult to separate the big assumptions from the small assumptions. In other words, the assumption that pre-tax operating margins will stay at 20 (a big assumption that doubles the value of the company) has to compete with the assumption that accounts receivable will decline from 5 of revenues to 4 of revenues over the next 10 years (a small assumption that has almost no impact on value). The Principle of Parsimony In the physical sciences, the principle of parsimony dictates that we try the simplest possible explanation for a phenomenon before we move on to more complicated ones. We would be well served adopting a similar principle in valuation. When valuing an asset, we want to use the simplest model we can get away with. In other words, if we can value an asset with three inputs, we should not be using five. If we can value a company with 3 years of cashflow forecasts, forecasting ten years of cash flows is asking for trouble. The problem with all-in-one models that are designed to value all companies is that they have to be set up to value the most complicated companies that we will face and not the least complicated. Thus, we are forced to enter inputs and forecast values for simpler companies that we really do not need to estimate. In the process, we can mangle the values of assets that should be easy to value. Consider, for instance, the cash and marketable securities held by firms as part of their assets. The simplest way to value this cash is to take it at face value. Analysts who try to build discounted cash flow or relative valuation models to value cash often mis-value it, either by using the wrong discount rate for the cash income or by using the wrong multiple for cash earnings. 2 Approaches to Valuation Analysts use a wide spectrum of models, ranging from the simple to the sophisticated. These models often make very different assumptions about the fundamentals that determine value, but they do share some common characteristics and can be classified in broader terms. There are several advantages to such a classification -- it makes it is easier to understand where individual models fit in to the big picture, why they provide different results and when they have fundamental errors in logic. In general terms, there are three approaches to valuation. The first, discounted cashflow valuation, relates the value of an asset to the present value of expected future cashflows on that asset. The second, relative valuation, estimates the value of an asset by looking at the pricing of comparable assets relative to a common variable like earnings, cashflows, book value or sales. The third, contingent claim valuation, uses option pricing models to measure the value of assets that share option characteristics. While they can yield different estimates of value, one of the objectives of discussing valuation models is to explain the reasons for such differences, and to help in picking the right model to use for a specific task. Discounted Cashflow Valuation In discounted cashflows valuation, the value of an asset is the present value of the expected cashflows on the asset, discounted back at a rate that reflects the riskiness of these cashflows. This approach gets the most play in classrooms and comes with the best theoretical credentials. In this section, we will look at the foundations of the approach and some of the preliminary details on how we estimate its inputs. Basis for Approach We buy most assets because we expect them to generate cash flows for us in the future. In discounted cash flow valuation, we begin with a simple proposition. The value of an asset is not what someone perceives it to be worth but it is a function of the expected cash flows on that asset. Put simply, assets with high and predictable cash flows should have higher values than assets with low and volatile cash flows. In discounted cash flow valuation, we estimate the value of an asset as the present value of the expected cash flows on it. n Life of the asset E(CF t) Expected cashflow in period t r Discount rate reflecting the riskiness of the estimated cashflows The cashflows will vary from asset to asset -- dividends for stocks, coupons (interest) and the face value for bonds and after-tax cashflows for a business. The discount rate will be a function of the riskiness of the estimated cashflows, with higher rates for riskier assets and lower rates for safer ones. Using discounted cash flow models is in some sense an act of faith. We believe that every asset has an intrinsic value and we try to estimate that intrinsic value by looking at an assets fundamentals. What is intrinsic value Consider it the value that would be attached to an asset by an all-knowing analyst with access to all information available right now and a perfect valuation model. No such analyst exists, of course, but we all aspire to be as close as we can to this perfect analyst. The problem lies in the fact that none of us ever gets to see what the true intrinsic value of an asset is and we therefore have no way of knowing whether our discounted cash flow valuations are close to the mark or not. Classifying Discounted Cash Flow Models There are three distinct ways in which we can categorize discounted cash flow models. In the first, we differentiate between valuing a business as a going concern as opposed to a collection of assets. In the second, we draw a distinction between valuing the equity in a business and valuing the business itself. In the third, we lay out three different and equivalent ways of doing discounted cash flow valuation 8211 the expected cash flow approach, a value based upon excess returns and adjusted present value. ein. Going Concern versus Asset Valuation The value of an asset in the discounted cash flow framework is the present value of the expected cash flows on that asset. Extending this proposition to valuing a business, it can be argued that the value of a business is the sum of the values of the individual assets owned by the business. While this may be technically right, there is a key difference between valuing a collection of assets and a business. A business or a company is an on-going entity with assets that it already owns and assets it expects to invest in the future. This can be best seen when we look at the financial balance sheet (as opposed to an accounting balance sheet) for an ongoing company in figure 1.1: Note that investments that have already been made are categorized as assets in place, but investments that we expect the business to make in the future are growth assets. A financial balance sheet provides a good framework to draw out the differences between valuing a business as a going concern and valuing it as a collection of assets. In a going concern valuation, we have to make our best judgments not only on existing investments but also on expected future investments and their profitability. While this may seem to be foolhardy, a large proportion of the market value of growth companies comes from their growth assets. In an asset-based valuation, we focus primarily on the assets in place and estimate the value of each asset separately. Adding the asset values together yields the value of the business. For companies with lucrative growth opportunities, asset-based valuations will yield lower values than going concern valuations. One special case of asset-based valuation is liquidation valuation, where we value assets based upon the presumption that they have to be sold now. In theory, this should be equal to the value obtained from discounted cash flow valuations of individual assets but the urgency associated with liquidating assets quickly may result in a discount on the value. How large the discount will be will depend upon the number of potential buyers for the assets, the asset characteristics and the state of the economy. B. Equity Valuation versus Firm Valuation There are two ways in which we can approach discounted cash flow valuation. The first is to value the entire business, with both assets-in-place and growth assets this is often termed firm or enterprise valuation. The cash flows before debt payments and after reinvestment needs are called free cash flows to the firm . and the discount rate that reflects the composite cost of financing from all sources of capital is called the cost of capital . The second way is to just value the equity stake in the business, and this is called equity valuation. The cash flows after debt payments and reinvestment needs are called free cash flows to equity, and the discount rate that reflects just the cost of equity financing is the cost of equity. Note also that we can always get from the former (firm value) to the latter (equity value) by netting out the value of all non-equity claims from firm value. Done right, the value of equity should be the same whether it is valued directly (by discounting cash flows to equity a the cost of equity) or indirectly (by valuing the firm and subtracting out the value of all non-equity claims). C. Variations on DCF Models The model that we have presented in this section, where expected cash flows are discounted back at a risk-adjusted discount rate, is the most commonly used discounted cash flow approach but there are two widely used variants. In the first, we separate the cash flows into excess return cash flows and normal return cash flows. Earning the risk-adjusted required return (cost of capital or equity) is considered a normal return cash flow but any cash flows above or below this number are categorized as excess returns excess returns can therefore be either positive or negative. With the excess return valuation framework, the value of a business can be written as the sum of two components: Value of business Capital Invested in firm today Present value of excess return cash flows from both existing and future projects If we make the assumption that the accounting measure of capital invested (book value of capital) is a good measure of capital invested in assets today, this approach implies that firms that earn positive excess return cash flows will trade at market values higher than their book values and that the reverse will be true for firms that earn negative excess return cash flows. In the second variation, called the adjusted present value (APV) approach . we separate the effects on value of debt financing from the value of the assets of a business. In general, using debt to fund a firms operations creates tax benefits (because interest expenses are tax deductible) on the plus side and increases bankruptcy risk (and expected bankruptcy costs) on the minus side. In the APV approach, the value of a firm can be written as follows: Value of business Value of business with 100 equity financing Present value of Expected Tax Benefits of Debt 8211 Expected Bankruptcy Costs In contrast to the conventional approach, where the effects of debt financing are captured in the discount rate, the APV approach attempts to estimate the expected dollar value of debt benefits and costs separately from the value of the operating assets. While proponents of each approach like to claim that their approach is the best and most precise, we will argue that the three approaches yield the same estimates of value, if we make consistent assumptions. Inputs to Discounted Cash Flow Models There are three inputs that are required to value any asset in this model - the expected cash flow . the timing of the cash flow and the discount rate that is appropriate given the riskiness of these cash flows. ein. Discount Rates In valuation, we begin with the fundamental notion that the discount rate used on a cash flow should reflect its riskiness, with higher risk cash flows having higher discount rates. There are two ways of viewing risk. The first is purely in terms of the likelihood that an entity will default on a commitment to make a payment, such as interest or principal due, and this is called default risk . When looking at debt, the cost of debt is the rate that reflects this default risk. The second way of viewing risk is in terms of the variation of actual returns around expected returns . The actual returns on a risky investment can be very different from expected returns the greater the variation, the greater the risk. When looking at equity, we tend to use measures of risk based upon return variance. While the discussion of risk and return models elsewhere in this site will look at the different models that attempt to do this in far more detail, there are some basic points on which these models agree. The first is that risk in an investment has to perceived through the eyes of the marginal investor in that investment, and this marginal investor is assumed to be well diversified across multiple investments. Therefore, the risk in an investment that should determine discount rates is the non-diversifiable or market risk of that investment. The second is that the expected return on any investment can be obtained starting with the expected return on a riskless investment, and adding to it a premium to reflect the amount of market risk in that investment. This expected return yields the cost of equity . The cost of capital can be obtained by taking an average of the cost of equity, estimated as above, and the after-tax cost of borrowing, based upon default risk, and weighting by the proportions used by each. We will argue that the weights used, when valuing an on-going business, should be based upon the market values of debt and equity. While there are some analysts who use book value weights, doing so violates a basic principle of valuation, which is that at a fair value 3. one should be indifferent between buying and selling an asset. B. Expected Cash Flows In the strictest sense, the only cash flow an equity investor gets out of a publicly traded firm is the dividend models that use the dividends as cash flows are called dividend discount models . A broader definition of cash flows to equity would be the cash flows left over after the cash flow claims of non-equity investors in the firm have been met (interest and principal payments to debt holders and preferred dividends) and after enough of these cash flows has been reinvested into the firm to sustain the projected growth in cash flows. This is the free cash flow to equity (FCFE), and models that use these cash flows are called FCFE discount models . The cashflow to the firm is the cumulated cash flow to all claimholders in the firm. One way to obtain this cashflow is to add the free cash flows to equity to the cash flows to lenders (debt) and preferred stockholders. A far simpler way of obtaining the same number is to estimate the cash flows prior to debt and preferred dividend payments, by subtracting from the after-tax operating income the net investment needs to sustain growth. This cash flow is called the free cash flow to the firm (FCFF) and the models that use these cash flows are called FCFF models . C. Expected Growth It is while estimating the expected growth in cash flows in the future that analysts confront uncertainty most directly. There are three generic ways of estimating growth. One is to look at a companys past and use the historical growth rate posted by that company. The peril is that past growth may provide little indication of future growth. The second is to obtain estimates of growth from more informed sources. For some analysts, this translates into using the estimates provided by a companys management whereas for others it takes the form of using consensus estimates of growth made by others who follow the firm. The bias associated with both these sources should raise questions about the resulting valuations. We will promote a third way, where the expected growth rate is tied to two variables that are determined by the firm being valued - how much of the earnings are reinvested back into the firm and how well those earnings are reinvested. In the equity valuation model, this expected growth rate is a product of the retention ratio, i. e. the proportion of net income not paid out to stockholders, and the return on equity on the projects taken with that money. In the firm valuation model, the expected growth rate is a product of the reinvestment rate, which is the proportion of after-tax operating income that goes into net new investments and the return on capital earned on these investments. The advantages of using these fundamental growth rates are two fold. The first is that the resulting valuations will be internally consistent and companies that are assumed to have high growth are required to pay for the growth with more reinvestment. The second is that it lays the foundation for considering how firms can make themselves more valuable to their investors. DCF Valuation: Pluses and Minuses To true believers, discounted cash flow valuation is the only way to approach valuation, but the benefits may be more nuanced that they are willing to admit. On the plus side, discounted cash flow valuation, done right, requires analysts to understand the businesses that they are valuing and ask searching questions about the sustainability of cash flows and risk. Discounted cash flow valuation is tailor made for those who buy into the Warren Buffett adage that what we are buying are not stocks but the underlying businesses. In addition, discounted cash flow valuations is inherently contrarian in the sense that it forces analysts to look for the fundamentals that drive value rather than what market perceptions are. Consequently, if stock prices rise (fall) disproportionately relative to the underlying earnings and cash flows, discounted cash flows models are likely to find stocks to be over valued (under valued). There are, however, limitations with discounted cash flow valuation. In the hands of sloppy analysts, discounted cash flow valuations can be manipulated to generate estimates of value that have no relationship to intrinsic value. We also need substantially more information to value a company with discounted cash flow models, since we have to estimate cashflows, growth rates and discount rates. Finally, discounted cash flow models may very well find every stock in a sector or even a market to be over valued, if market perceptions have run ahead of fundamentals. For portfolio managers and equity research analysts, who are required to find equities to buy even in the most over valued markets, this creates a conundrum. They can go with their discounted cash flow valuations and conclude that everything is overvalued, which may put them out of business, or they can find an alternate approach that is more sensitive to market moods. It should come as no surprise that many choose the latter. Relative Valuation While the focus in classrooms and academic discussions remains on discounted cash flow valuation, the reality is that most assets are valued on a relative basis. In relative valuation, we value an asset by looking at how the market prices similar assets. Thus, when determining what to pay for a house, we look at what similar houses in the neighborhood sold for rather than doing an intrinsic valuation. Extending this analogy to stocks, investors often decide whether a stock is cheap or expensive by comparing its pricing to that of similar stocks (usually in its peer group). In this section, we will consider the basis for relative valuation, ways in which it can be used and its advantages and disadvantages. Basis for approach In relative valuation, the value of an asset is derived from the pricing of comparable assets, standardized using a common variable. Included in this description are two key components of relative valuation. The first is the notion of comparable or similar assets. From a valuation standpoint, this would imply assets with similar cash flows, risk and growth potential. In practice, it is usually taken to mean other companies that are in the same business as the company being valued. The other is a standardized price . After all, the price per share of a company is in some sense arbitrary since it is a function of the number of shares outstanding a two for one stock split would halve the price. Dividing the price or market value by some measure that is related to that value will yield a standardized price. When valuing stocks, this essentially translates into using multiples where we divide the market value by earnings, book value or revenues to arrive at an estimate of standardized value. We can then compare these numbers across companies. The simplest and most direct applications of relative valuations are with real assets where it is easy to find similar assets or even identical ones. The asking price for a Mickey Mantle rookie baseball card or a 1965 Ford Mustang is relatively easy to estimate given that there are other Mickey Mantle cards and 1965 Ford Mustangs out there and that the prices at which they have been bought and sold can be obtained. With equity valuation, relative valuation becomes more complicated by two realities. The first is the absence of similar assets, requiring us to stretch the definition of comparable to include companies that are different from the one that we are valuing. After all, what company in the world is remotely similar to Microsoft or GE The other is that different ways of standardizing prices (different multiples) can yield different values for the same company. Harking back to our earlier discussion of discounted cash flow valuation, we argued that discounted cash flow valuation was a search (albeit unfulfilled) for intrinsic value. In relative valuation, we have given up on estimating intrinsic value and essentially put our trust in markets getting it right, at least on average. Variations on Relative Valuation In relative valuation, the value of an asset is based upon how similar assets are priced. In practice, there are three variations on relative valuation, with the differences primarily in how we define comparable firms and control for differences across firms: a. Direct comparison . In this approach, analysts try to find one or two companies that look almost exactly like the company they are trying to value and estimate the value based upon how these similar companies are priced. The key part in this analysis is identifying these similar companies and getting their market values. B. Peer Group Average . In the second, analysts compare how their company is priced (using a multiple) with how the peer group is priced (using the average for that multiple). Thus, a stock is considered cheap if it trade at 12 times earnings and the average price earnings ratio for the sector is 15. Implicit in this approach is the assumption that while companies may vary widely across a sector, the average for the sector is representative for a typical company. C. Peer group average adjusted for differences . Recognizing that there can be wide differences between the company being valued and other companies in the comparable firm group, analysts sometimes try to control for differences between companies. In many cases, the control is subjective: a company with higher expected growth than the industry will trade at a higher multiple of earnings than the industry average but how much higher is left unspecified. In a few cases, analysts explicitly try to control for differences between companies by either adjusting the multiple being used or by using statistical techniques. As an example of the former, consider PEG ratios. These ratios are computed by dividing PE ratios by expected growth rates, thus controlling (at least in theory) for differences in growth and allowing analysts to compare companies with different growth rates. For statistical controls, we can use a multiple regression where we can regress the multiple that we are using against the fundamentals that we believe cause that multiple to vary across companies. The resulting regression can be used to estimate the value of an individual company. In fact, we will argue that statistical techniques are powerful enough to allow us to expand the comparable firm sample to include the entire market. Applicability of multiples and limitations The allure of multiples is that they are simple and easy to relate to. They can be used to obtain estimates of value quickly for firms and assets, and are particularly useful when there are a large number of comparable firms being traded on financial markets, and the market is, on average, pricing these firms correctly. In fact, relative valuation is tailor made for analysts and portfolio managers who not only have to find under valued equities in any market, no matter how overvalued, but also get judged on a relative basis. An analyst who picks stocks based upon their PE ratios, relative to the sectors they operate in, will always find under valued stocks in any market if entire sectors are over valued and his stocks decline, he will still look good on a relative basis since his stocks will decline less than comparable stocks (assuming the relative valuation is right). By the same token, they are also easy to misuse and manipulate, especially when comparable firms are used. Given that no two firms are exactly similar in terms of risk and growth, the definition of comparable firms is a subjective one. Consequently, a biased analyst can choose a group of comparable firms to confirm his or her biases about a firms value. While this potential for bias exists with discounted cashflow valuation as well, the analyst in DCF valuation is forced to be much more explicit about the assumptions which determine the final value. With multiples, these assumptions are often left unstated. The other problem with using multiples based upon comparable firms is that it builds in errors (over valuation or under valuation) that the market might be making in valuing these firms. If, for instance, we find a company to be under valued because it trades at 15 times earnings and comparable companies trade at 25 times earnings, we may still lose on the investment if the entire sector is over valued. In relative valuation, all that we can claim is that a stock looks cheap or expensive relative to the group we compared it to, rather than make an absolute judgment about value. Ultimately, relative valuation judgments depend upon how well we have picked the comparable companies and how how good a job the market has done in pricing them. Contingent Claim Valuation There is little in either discounted cashflow or relative valuation that can be considered new and revolutionary. In recent years, though, analysts have increasingly used option-pricing models, developed to value listed options, to value assets, businesses and equity stakes in businesses. These applications are often categorized loosely as real options, but they have to be used with caution. Basis for Approach A contingent claim or option is an asset which pays off only under certain contingencies - if the value of the underlying asset exceeds a pre-specified value for a call option, or is less than a pre-specified value for a put option. Much work has been done in the last few decades in developing models that value options, and these option-pricing models can be used to value any assets that have option-like features. Figure 1.2 illustrates the payoffs on call and put options as a function of the value of the underlying asset: Figure 1.2: Payoffs on Options as a Function of the Underlying Assets Value An option can be valued as a function of the following variables - the current value and the variance in value of the underlying asset, the strike price and the time to expiration of the option and the riskless interest rate. This was first established by Black and Scholes (1972) and has been extended and refined subsequently in numerous variants. 4 While the Black-Scholes option-pricing model ignored dividends and assumed that options would not be exercised early, it can be modified to allow for both. A discrete-time variant, the Binomial option-pricing model, has also been developed to price options. An asset can be valued as a call option if the payoffs on it are a function of the value of an underlying asset if that value exceeds a pre-specified level, the asset is worth the difference if not, it is worth nothing. It can be valued as a put option if it gains value as the value of the underlying asset drops below a pre - specified level, and if it is worth nothing when the underlying assets value exceeds that specified level. There are many assets that generally are not viewed as options but still share several option characteristics. A patent can be analyzed as a call option on a product, with the investment outlay needed to get the project going considered the strike price and the patent life becoming the life of the option. An undeveloped oil reserve or gold mine provides its owner with a call option to develop the reserve or mine, if oil or gold prices increase. The essence of the real options argument is that discounted cash flow models understate the value of assets with option characteristics. The understatement occurs because DCF models value assets based upon a set of expected cash flows and do not fully consider the possibility that firms can learn from real time developments and respond to that learning. For example, an oil company can observe what the oil price is each year and adjust its development of new reserves and production in existing reserves accordingly rather than be locked into a fixed production schedule. As a result, there should be an option premium added on to the discounted cash flow value of the oil reserves. It is this premium on value that makes real options so alluring and so potentially dangerous. Applicability and Limitations Using option-pricing models in valuation does have its advantages. First, there are some assets that cannot be valued with conventional valuation models because their value derives almost entirely from their option characteristics. For example, a biotechnology firm with a single promising patent for a blockbuster cancer drug wending its way through the FDA approval process cannot be easily valued using discounted cash flow or relative valuation models. It can, however, be valued as an option. The same can be said about equity in a money losing company with substantial debt most investors buying this stock are buying it for the same reasons they buy deep out-of-the-money options. Second, option-pricing models do yield more realistic estimates of value for assets where there is a significant benefit obtained from learning and flexibility. Discounted cash flow models will understate the values of natural resource companies, where the observed price of the natural resource is a key factor in decision making. Third, option-pricing models do highlight a very important aspect of risk. While risk is considered almost always in negative terms in discounted cash flow and relative valuation (with higher risk reducing value), the value of options increases as volatility increases. For some assets, at least, risk can be an ally and can be exploited to generate additional value. This is not to suggest that using real options models is an unalloyed good. Using real options arguments to justify paying premiums on discounted cash flow valuations, when the options argument does not hold, can result in overpayment. While we do not disagree with the notion that firms can learn by observing what happens over time, this learning has value only if it has some degree of exclusivity. We will argue that it is usually inappropriate to attach an option premium to value if the learning is not exclusive and competitors can adapt their behavior as well. There are also limitations in using option pricing models to value long-term options on non-traded assets. The assumptions made about constant variance and dividend yields, which are not seriously contested for short term options, are much more difficult to defend when options have long lifetimes. When the underlying asset is not traded, the inputs for the value of the underlying asset and the variance in that value cannot be extracted from financial markets and have to be estimated. Thus the final values obtained from these applications of option pricing models have much more estimation error associated with them than the values obtained in their more standard applications (to value short term traded options). The Role of Valuation Valuation is useful in a wide range of tasks. The role it plays, however, is different in different arenas. The following section lays out the relevance of valuation in portfolio management, in acquisition analysis and in corporate finance. 1. Portfolio Management The role that valuation plays in portfolio management is determined in large part by the investment philosophy of the investor. Valuation plays a minimal role in portfolio management for a passive investor, whereas it plays a larger role for an active investor. Even among active investors, the nature and the role of valuation is different for different types of active investment. Market timers use valuation much less than investors who pick stocks, and the focus is on market valuation rather than on firm-specific valuation. Among security selectors, valuation plays a central role in portfolio management for fundamental analysts, and a peripheral role for technical analysts. The following sub-section describes, in broad terms, different investment philosophies and the roles played by valuation in each one. 1. Fundamental Analysts: The underlying theme in fundamental analysis is that the true value of the firm can be related to its financial characteristics -- its growth prospects, risk profile and cashflows. Any deviation from this true value is a sign that a stock is under or overvalued. It is a long-term investment strategy, and the assumptions underlying it are that: (a) The relationship between value and the underlying financial factors can be measured. (b) The relationship is stable over time. (c) Deviations from the relationship are corrected in a reasonable time period. Fundamental analysts include both value and growth investors. The key difference between the two is in where the valuation focus lies. Reverting back to our break down of assets in figure 1.1, value investors are primarily interested in assets in place and acquiring them at less than their true value. Growth investors, on the other hand, are far more focused on valuing growth assets and buying those assets at a discount. While valuation is the central focus in fundamental analysis, some analysts use discounted cashflow models to value firms, while others use multiples and comparable firms. Since investors using this approach hold a large number of undervalued stocks in their portfolios, their hope is that, on average, these portfolios will do better than the market. 2. Activist Investors: Activist investors take positions in firms that have a reputation for poor management and then use their equity holdings to push for change in the way the company is run. Their focus is not so much on what the company is worth today but what its value would be if it were managed well. Investors like Carl Icahn, Michael Price and Kirk Kerkorian have prided themselves on their capacity to not only pinpoint badly managed firms but to also create enough pressure to get management to change its ways. How can valuation skills help in this pursuit To begin with, these investors have to ensure that there is additional value that can be generated by changing management. In other words, they have to separate how much of a firms poor stock price performance has to do with bad management and how much of it is a function of external factors the former are fixable but the latter are not. They then have to consider the effects of changing management on value this will require an understanding of how value will change as a firm changes its investment, financing and dividend policies. As a consequence, they have to not only know the businesses that the firm operates in but also have an understanding of the interplay between corporate finance decisions and value. Activist investors generally concentrate on a few businesses they understand well, and attempt to acquire undervalued firms. Often, they wield influence on the management of these firms and can change financial and investment policy. 3. Chartists: Chartists believe that prices are driven as much by investor psychology as by any underlying financial variables. The information available from trading measures -- price movements, trading volume and short sales -- gives an indication of investor psychology and future price movements. The assumptions here are that prices move in predictable patterns, that there are not enough marginal investors taking advantage of these patterns to eliminate them, and that the average investor in the market is driven more by emotion than by rational analysis. While valuation does not play much of a role in charting, there are ways in which an enterprising chartist can incorporate it into analysis. For instance, valuation can be used to determine support and resistance lines 5 on price charts. 4. Information Traders: Prices move on information about the firm. Information traders attempt to trade in advance of new information or shortly after it is revealed to financial markets. The underlying assumption is that these traders can anticipate information announcements and gauge the market reaction to them better than the average investor in the market. For an information trader, the focus is on the relationship between information and changes in value, rather than on value, per se. Thus an information trader may buy an overvalued firm if he believes that the next information announcement is going to cause the price to go up, because it contains better than expected news. If there is a relationship between how undervalued or overvalued a company is, and how its stock price reacts to new information, then valuation could play a role in investing for an information trader. 5. Market Timers: Market timers note, with some legitimacy, that the payoff to calling turns in markets is much greater than the returns from stock picking. They argue that it is easier to predict market movements than to select stocks and that these predictions can be based upon factors that are observable. While valuation of individual stocks may not be of much direct use to a market timer, market timing strategies can use valuation in one of at least two ways: (a) The overall market itself can be valued and compared to the current level. (b) Valuation models can be used to value a large number of stocks, and the results from the cross-section can be used to determine whether the market is over or under valued. For example, as the number of stocks that are overvalued, using the valuation model, increases relative to the number that are undervalued, there may be reason to believe that the market is overvalued. 6. Efficient Marketers: Efficient marketers believe that the market price at any point in time represents the best estimate of the true value of the firm, and that any attempt to exploit perceived market efficiencies will cost more than it will make in excess profits. They assume that markets aggregate information quickly and accurately, that marginal investors promptly exploit any inefficiencies and that any inefficiencies in the market are caused by friction, such as transactions costs, and cannot exploited. For efficient marketers, valuation is a useful exercise to determine why a stock sells for the price that it does. Since the underlying assumption is that the market price is the best estimate of the true value of the company, the objective becomes determining what assumptions about growth and risk are implied in this market price, rather than on finding under or over valued firms. 2. Valuation in Acquisition Analysis Valuation should play a central part of acquisition analysis. The bidding firm or individual has to decide on a fair value for the target firm before making a bid, and the target firm has to determine a reasonable value for itself before deciding to accept or reject the offer. There are special factors to consider in takeover valuation. First, there is synergy, the increase in value that many managers foresee as occurring after mergers because the combined firm is able to accomplish things that the individual firms could not. The effects of synergy on the combined value of the two firms (target plus bidding firm) have to be considered before a decision is made on the bid. Second, the value of control, which measures the effects on value of changing management and restructuring the target firm, will have to be taken into account in deciding on a fair price. This is of particular concern in hostile takeovers. As we noted earlier, there is a significant problem with bias in takeover valuations. Target firms may be over-optimistic in estimating value, especially when the takeover is hostile, and they are trying to convince their stockholders that the offer price is too low. Similarly, if the bidding firm has decided, for strategic reasons, to do an acquisition, there may be strong pressure on the analyst to come up with an estimate of value that backs up the acquisition. 3. Valuation in Corporate Finance There is a role for valuation at every stage of a firms life cycle. For small private businesses thinking about expanding, valuation plays a key role when they approach venture capital and private equity investors for more capital. The share of a firm that a venture capitalist will demand in exchange for a capital infusion will depend upon the value she estimates for the firm. As the companies get larger and decide to go public, valuations determine the prices at which they are offered to the market in the public offering. Once established, decisions on where to invest, how much to borrow and how much to return to the owners will be all decisions that are affected by valuation. If the objective in corporate finance is to maximize firm value 6. the relationship between financial decisions, corporate strategy and firm value has to be delineated. As a final note, value enhancement has become the mantra of management consultants and CEOs who want to keep stockholders happy, and doing it right requires an understanding of the levers of value. In fact, many consulting firms have come up with their own measures of value (EVA and CFROI, for instance) that they contend facilitate value enhancement. 4. Valuation for Legal and Tax Purposes Mundane though it may seem, most valuations, especially of private companies, are done for legal or tax reasons. A partnership has to be valued, whenever a new partner is taken on or an old one retires, and businesses that are jointly owned have to be valued when the owners decide to break up. Businesses have to be valued for estate tax purposes when the owner dies, and for divorce proceedings when couples break up. While the principles of valuation may not be different when valuing a business for legal proceedings, the objective often becomes providing a valuation that the court will accept rather than the right valuation. Conclusion Valuation plays a key role in many areas of finance -- in corporate finance, in mergers and acquisitions and in portfolio management. The models presented will provide a range of tools that analysts in each of these areas will find of use, but the cautionary note sounded in this introduction bears repeating. Valuation is not an objective exercise, and any preconceptions and biases that an analyst brings to the process will find their way into the value. 1 There are approximately five times as many buy recommendations issued by analysts on Wall Street as there are sell recommendations. 2 The income from cash is riskless and should be discounted back at a riskless rate. Instead, analysts use risk adjusted discount rates (costs of equity or capital) to discount the cash income, thus resulting in a discount on face value. When analysts use multiples, they often will use the average PE ratio at which peer group companies as the multiple for cash income. 3 When book value weights are used, the costs of capital tend to be much lower for many U. S. firms, since book equity is lower than market equity. This then pushes up the value for these firms. While this may make it attractive to the sellers of these firms, very few buyers would be willing to pay this price for the firm, since it would require that the debt that they use in their financing will have to be based upon the book value, often requiring tripling or quadrupling the dollar debt in the firm. 4 Black, F. and M. Scholes, 1972, The Valuation of Option Contracts and a Test of Market Efficiency . Journal of Finance, v27, 399-417. 5 On a chart, the support line usually refers to a lower bound below which prices are unlikely to move and the resistance line refers to the upper bound above which prices are unlikely to venture. While these levels are usually estimated using past prices, the range of values obtained from a valuation model can be used to determine these levels, i. e. the maximum value will become the resistance level and the minimum value will become the support line. 6 Most corporate financial theory is constructed on this premise.
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