Portfolio models for fixed income securities
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Transcript of Portfolio models for fixed income securities
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Portfolio models for fixed income securities
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Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Agenda
Portfolio dedication Modell for porteføljeimmunisering Modeller for faktorimmunisering
StatsobligasjonerSelskapsobligasjoner
Oppsummering
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Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Risk management for fixed income securities Risk associated with changes in interest rates
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Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Risk management for fixed income securities, cont. Price of a bond which makes predetermined risk free payments
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Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Risk management for fixed income securities, cont. Yield to maturity
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Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Risk management for fixed income securities, cont. Risk measure: sensitivity of price with respect to term stucture
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Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Risk management for fixed income securities, cont. Approximation of price/yield curve
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Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Risk management for fixed income securities, cont. Different concepts of duration
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Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Risk management for fixed income securities, cont. Different concepts of duration, cont.
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Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Risk management for fixed income securities, cont. Convexity: quadratic approximation
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Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Risk management for fixed income securities, cont. Factor analysis of the term structure
Small parallel shifts are not what happen in real markets
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Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Risk management for fixed income securities, cont. Factor models, cont.
Estimate factors Keep only a few first ones
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Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Faktorimmunisering
T
j jt tt 1
jt
Tar eksplisitt høyde for endringer i rentekurvens form.
Lineærfaktormodell: Har en T*T korrelasjonsmatrise over rentenesterminsstruktur.
Prinsipalkomponenter: f r , j=1,2,...k
der (factor l
toading) måler følsomheten til rentesatsen r
for endringer i den j-te faktoren.
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Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Faktorimmunisering (2)
t
k
t jt j tj 1
Tr t
i ti tt 1
Endringer i terminstrukturen som en lineær- kombinasjon av de k uavhengige faktorene:
r f + (3)
Endring i obligasjonspris med endring i terminstrukturen:
P F te r
Setter
t
T kr t
i ti jt jt 1 j 1
inn (3):
P (F te f )
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Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Faktorimmunisering (3)
t
Ti r t
ti jtt 1j
For små endringer i terminsstrukturen er da følsomheten i obligasjonsprisen:
PF t e
f
Faktormodifisert varighet for obligasjon i m.h.t.faktor j er definert som relativ prissensitivitet:
t
Ti r t
ij i ti jtt 1j i
P 1d / P F t e (4)
f P
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Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Statsobligasjoner
n
i i Li 1
n
i ik ki=1 L
j jj=1 j=1j j
Krever at
P x P
holder når faktorene endrer verdi.
Deriverer m.h.p. faktorene:
( P x )P
df df (5)f f
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Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Statsobligasjoner (2)j
n
i ii=1 L
j j
ij i i
Faktorendringene df er pr. def. uavhengige.
Likning(5) vil gjelde for hver enkelt faktor.Får da:
( P x )P
f f
Bruker likning (4) og får førsteordens betingelser for faktorimmunisering:
d P x
n
Lj Li 1
Lj
d P , for all j=1,2,...,k
der d er den faktormodifiserte varigheten av forpliktelsen.
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Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Statsobligasjoner (3)
n
i i Li 1
n
ij i i Lj Li 1
Faktorimmuniseringsmodell:Max F(x)
s.t. P x P ,
d P x d P , for all j=1,2,...,k
x 0.
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Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Selskapsobligasjoner
Avkastningene fra obligasjoner i ulike kreditt-klasser er korrelerte.
Ser først bort fra dette:La C være antall kredittklasser og c angi den c-tekredittvurderingen. Indeksert likning for obligasjonspri
ct
Tr tc c
i tit 1
s:
P F e
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Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Selskapsobligasjoner (2)
kc c ct jt j t
j 1
j
Lineærfaktormodell for endringer i avkastning for en obligasjon med kredittvurdering c:
y df
der vi antar at k uavhengige faktorer forklarer avkastnings-
endringene for hver klasse og
ct
t
t
er den korresponderende
følsomheten til renta r for endringer i den j-te faktoren. er feilestimatet.
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Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Selskapsobligasjoner (3)
ct
c Ti r tc c
ti jtct 1j
c Tic c c
ij i ti jc ct 1j i
Obligasjonsprisenes følsomhet for endringeri faktorene:
PF t e
f
Faktormodifisert varighet m.h.t. den j-te faktoren er relativ prissensitivitet:
P 1d / P F t
f P
ctr tc
te
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Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Selskapsobligasjoner (4)
ci
c ci i
For en obligasjonsportefølje med en beholdning xav obligasjon i med kredittvurdering c, blir den nød-vendige betingelsen for immunisering:
P xC n
Lc 1 i 1
P
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Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Selskapsobligasjoner (5)
cj
nc c c cij i i jL L
i 1
cij
Deriverer m.h.p. faktorene og krever at følsomheten
m.h.t. hver faktor f er lik.
Førsteordens betingelse:
d P x d P
der d er faktormodifisert varighet for obligasjon i
tilhørende kredit
tklasse c m.h.t. faktor j .
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Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Selskapsobligasjoner (6)
C nc ci i L
c 1 i 1
1jL Lc c c
ij i i
Faktorimmuniseringsmodell for selskapsobligasjoner(antar at c=1 for forpliktelsene):
Max F(x)
s.t. P x P ,
d P , for all j=1,2,...,k, c=1 d P x
0, for all j=1,2,...k, c=2,
n
i 1 ...,C x 0
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Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Selskapsobligasjoner (7)
Tar høyde for at avkastningene til selskaps-obligasjoner i ulike kredittklasser er korrelerte.
Faktoranalyse på hele korrelasjonsmatrisa over avkastningene til obligasjoner i ulike kredittklasser.
k fa ktorer har innvirkning på alle obligasjonenesavkastning.
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Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Selskapsobligasjoner (8)cij
Lj
Lar d angi faktormodifisert varighet for obligasjon i
tilhørende kredittklasse c m.h.t. faktor j og d angi
faktormodifisert varighet for forpliktelsene.
Faktorimmuniseringsmodell for selskapsoblig
C nc ci i L
c 1 i 1
nc c cij i i Lj L,
i 1
asjoner med korrelerte kredittklasser:Max F(x)
s.t. P x P ,
d P x d P for alle j=1,2,...,k, c=1,2,...C
x 0.
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Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Oppsummering/konklusjon
”Matching” av kontantstrømmer Modell for porteføljeimmunisering Modeller for faktorimmunisering for stats-
og selskapsobligasjoner Nyttige for problemer med (tilnærmet)
kjente kontantstrømmer