Chapter 3web.mit.edu/astomper/www/univie/pof/Chapter_3.pdf · 2010-01-12 · 01&$’2’314%1$5)6...

!"#$%&"'()$"*

+,-./+

01&$'2'314%1$5)601&$'2'314%1$5)6

TexPoint fonts used in EMF.Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAAAAAAAAAAAA

Chapter 3: Fixed-Income Securities

Chapter 4: Common Stocks

Chapter 5: Forwards and Futures

Chapter 6: Options

!"#$%&"'()$"*

+,-./+ 01&$'2''314%1$5)6

7"'819"'4"1&6":'$81$;_ 2%*56"**':"#5*5)6*')<$"6'&":%#"'$)'914%1$5)6')<'1**"$*=>?*_ @A)'"4"B"6$*'1&"'5BC)&$16$'56'914%56D'1'>?;'$5B"'16:'&5*E_ 314%"')<'>?*'5*':"$"&B56":'56'<5616#514'B1&E"$*

?&)B'$8"'B1&E"$F'A"'#16'4"1&6_ G)A'$)'914%"'$5B"'HHH'$5B"'914%"')<'B)6"I_ G)A'$)'914%"'&5*E'HHH'&5*E'C&"B5%B

J6'C1&$5#%41&F_ 0&5#"*'56'$8"'K)6:'B1&E"$'D59"*'$8"'$5B"'914%"')<'B)6"I_ 0&5#"*'56'$8"'*$)#E'B1&E"$'D59"*'$8"'&5*E'C&"B5%B

J6'$85*'C1&$')<'$8"'#)%&*"F'A"'*$%:I'$8"'914%1$5)6')<'K)6:*F'*$)#E*F<)&A1&:*F'<%$%&"*'16:')C$5)6*-

L

J6$&):%#$5)6J6$&):%#$5)6

!"#$%&"'()$"*

+,-./+

+,-./+'?5616#"'@8")&I'J+,-./+'?5616#"'@8")&I'J

M4"NM4"N'O$)BC"&'O$)BC"&PJ@'O4)16'O#8))4')<'P161D"B"6$PJ@'O4)16'O#8))4')<'P161D"B"6$J6*$5$%$"'<)&'M:916#":'O$%:5"*F'35"661J6*$5$%$"'<)&'M:916#":'O$%:5"*F'35"661

Lecture Lecture 33: Fixed-Income Securities: Fixed-Income Securities

!"#$%&"'()$"*

+,-./+ !"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

_ ?5N":H56#)B"'*"#%&5$5"*_ R9"&95"A')<'<5N":H56#)B"'B1&E"$*_ @"&B'*$&%#$%&"')<'56$"&"*$'&1$"*_ S5*#)%6$'K)6:*'16:'#)%C)6'K)6:*_ ?)&A1&:'56$"&"*$'&1$"*_ J6$"&"*$'&1$"'&5*E_ J6<41$5)6'&5*E_ S"<1%4$'&5*E

T"1:56D*;_ 2&"14"IF'PI"&*'16:'M44"6F'>81C$"&*'.F'L._ 2):5"F'U16"'16:'P1&E%*F'>81C$"&*'+.F'+,F'+V_ O14)B)6'2&)$8"&*F'WX6:"&*$16:56D'S%&1$5)6'16:'3)41$545$IW

.

U"I'#)6#"C$*U"I'#)6#"C$*

!"#$%&"'()$"*

+,-./+

?5N":H56#)B"'*"#%&5$5"*'1&"'<5616#514'#415B*'A5$8'C&)B5*":'#1*8'<4)A*)<'<5N":'1B)%6$'C15:'1$'<5N":':1$"*-

>41**5<5#1$5)6')<'?5N":HJ6#)B"'O"#%&5$5"*;+- @&"1*%&I'O"#%&5$5"*;

– X-O-'@&"1*%&I'*"#%&5$5"*'YK544*F'6)$"*F'K)6:*Z– 2%6:*F'[\2*F'X-U-'\54$*']

L- ?":"&14'MD"6#I'O"#%&5$5"*;– O"#%&5$5"*'5**%":'KI'<":"&14'1D"6#5"*'Y?G!2F'?(PM']Z

Q- >)&C)&1$"'*"#%&5$5"*;– >)BB"&#514'C1C"&'Y>0Z– P":5%BH$"&B'6)$"*'YP@(*Z– >)&C)&1$"'K)6:*']

.- P%65#5C14'*"#%&5$5"*'YP%65"*Z,- P)&$D1D"HK1#E":'*"#%&5$5"*'YP2OZV- M**"$'K1#E":'*"#%&5$5"*'YM2OZF']

,

?5N":H56#)B"'*"#%&5$5"*?5N":H56#)B"'*"#%&5$5"* !"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

!"#$%&"'()$"* ^L//_'[516D'716D

+,-./+

X-O-'K)6:'B1&E"$':"K$'L//`'YaK5445)6*Z

O)%&#"*;''X-O-'S"C1&$B"6$')<'@&"1*%&IF'?":"&14'T"*"&9"'OI*$"BF'?":"&14'MD"6#5"*F'''''''''''''''''@8)B*)6'?5616#514F'24))BK"&DF'OJ?PM

R9"&95"A')<'?J'B1&E"$*R9"&95"A')<'?J'B1&E"$* !"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

!"#$%&"'()$"* ^L//_'[516D'716D

+,-./+

R%$*$16:56D'X-O-'K)6:'B1&E"$'':"K$'Ya'K5445)6*Z

°° P%65#5C14 @&"1*%&IP)&$D1D"T"41$":

>)&C)&1$"S"K$

?":"&14'MD"6#IO"#%&5$5"*

P)6"IP1&E"$*

M**"$H21#E": @)$14

+bb_ +F./L-` QF,.L-_ LFb,,-L LF`/_-, +FQ//-V +Fb``-_ `Q+-, +.FV+b-+

+bbb +F.,`-+ QF,Lb-, QFQQ.-L QF/.V-, +FVL/-/ LFQQ_-_ b//-_ +VFLLV-b

L/// +F._/-, QFL+/-/ QF,V,-_ QFQ,_-. +F_,.-V LFVVL-V +F/`+-_ +`FL/Q-`

L//+ +FV/Q-V QF+bV-V .F+L`-V QF_QV-. LF+.b-V LF,_`-L +FL_+-+ +_F`_L-+

L//L +F`VQ-/ QF.Vb-L .FV_V-. .F/bb-, LFLbL-_ LF,.,-` +F,.Q-Q L/FQbb-b

L//Q +Fb//-` QF_LL-+ ,FLQ_-V .F.,_-. LFVQV-` LF,+b-b +FVbQ-` LLFL`/-+

L//. LF/Q/-b .FL,`-L ,F.,,-_ .F`_,-+ LF`.,-+ LFb/.-L +F_L`-_ L.F//V-+

L//, LFLLV-/ .F,+`-Q ,Fb+,-V .FbV/-/ LFV+Q-_ QF.QQ-` +Fb,,-L L,FVL+-V

L//V LF./Q-L .FV_b-_ VF.bL-. ,FQV,-/ LFVV/-+ .F//_-_ LF+Q/-. L`F`.b-`

L//` LFVL+-/ .F_,,-b `FL+/-Q ,F_L,-. LFb.V-Q .F+./-L LF.`L-. Q/F/`+-,L//_c+ LFV,`-/ .Fbb,-_ `FQb`-/ ,Fb/,-V LFb_.-L .F+L,-b LF._/-Q Q/F,.,-_

O)%&#"*;'X-O-'S"C1&$B"6$')<'@&"1*%&IF'?":"&14'T"*"&9"'OI*$"BF'?":"&14'MD"6#5"*F'@8)B*)6'?5616#514F'24))BK"&DF'OJ?PM

R9"&95"A')<'?J'B1&E"$*R9"&95"A')<'?J'B1&E"$* !"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

!"#$%&"'()$"* ^L//_'[516D'716D

+,-./+

X-O-'K)6:'B1&E"$''5**%16#"'L//`'YaK5445)6*Z

O)%&#"*;''X-O-'S"C1&$B"6$')<'@&"1*%&IF'?":"&14'T"*"&9"'OI*$"BF'?":"&14'MD"6#5"*F'''''''''''''''''@8)B*)6'?5616#514F'24))BK"&DF'OJ?PM

R9"&95"A')<'?J'B1&E"$*R9"&95"A')<'?J'B1&E"$* !"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

!"#$%&"'()$"* ^L//_'[516D'716D

+,-./+

X-O-'K)6:'B1&E"$'5**%16#"'Ya'K5445)6*Z

°° P%65#5C14 @&"1*%&IP)&$D1D"T"41$":

>)&C)&1$"S"K$

?":"&14'MD"6#IO"#%&5$5"*

M**"$H21#E": @)$14

+bb_ L_V-_ .Q_-. +F+.Q-b/ V+/-` ,bV-. L_V-V QFQVL-`/+bbb LL`-, QV.-V +F/L,-./ VLb-L ,._ L_`-+ QF/_+-_/L/// L//-_ Q+L-. V_.-. ,_`-, ..V-V QQ` LF,V_-`/L//+ L_`-` Q_/-` +FV`+-Q/ ``V-+ b.+ Q_Q-Q .F../-+/L//L Q,`-, ,`+-V LFL.b-L/ VQV-` +F/.+-,/ .Vb-L ,FQL,-`/L//Q Q_L-` `.,-L QF/`+-+/ ``,-_ +FLV`-,/ V//-L VF_.L-,/

L//. Q,b-_ _,Q-Q +F``b-// `_/-` __+-_Y.Z _Vb-_ .FV.L-V/L//, ./_-L `.V-L +FbVV-`/ `,L-_ VVb +F+`L-+/ ,F`+,-//L//V Q_V-, `__-, +Fb_`-_/ +F/,_-b/ `.`-Q +FL,Q-+/ VFLLL-+/L//` .Lb-+ `,L-Q LF/,/-+/ +F+L_-Q/ b.+-_ b/+-L VFL/L-_/L//_c+ _,-+ L/Q-_ Q_V-` L+Q-+ .Lb-, ,_-+ +FQ`V-Q/O)%&#"*;'X-O-'S"C1&$B"6$')<'@&"1*%&IF'?":"&14'T"*"&9"'OI*$"BF'?":"&14'MD"6#5"*F'@8)B*)6'?5616#514F'''''''''''''''''24))BK"&DFOJ?PM-

R9"&95"A')<'?J'B1&E"$*R9"&95"A')<'?J'B1&E"$* !"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

!"#$%&"'()$"*

+,-./+ !"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

+/

R&D165d1$5)6')<'?J'B1&E"$*R&D165d1$5)6')<'?J'B1&E"$*

Issuers:

1. Governments2. Corporations3. Commercial Banks4. States5. Municipalities6. SPVs7. Foreign Institutions

Intermediaries:

1. Primary Dealers2. Other Dealers3. Investment Banks4. Credit-rating Agencies5. Credit Enhancers6. Liquidity Enhancers

Investors:

1. Governments2. Pension Funds3. Insurance Companies4. Commercial Banks5. Mutual Funds6. Hedge Funds7. Foreign Institutions8. Individuals

!"#$%&"'()$"*

+,-./+ !"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

>1*8'<4)A;

+- P1$%&5$I

L- 0&56#5C14

Q- >)%C)6

eN1BC4"-'M'QHI"1&'K)6:'A5$8'C&56#5C14')<'a+F///'16:'166%14'#)%C)6C1IB"6$')<',f'81*'$8"'<)44)A56D'#1*8'<4)A;

++

>1*8'<4)A')<'?J'*"#%&5$5"*>1*8'<4)A')<'?J'*"#%&5$5"*

-t = 0 1 2 3 time

6 6

6

50 50 50+ 1;000

!"#$%&"'()$"*

+,-./+ !"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

314%1$5)6;

+- @5B"'914%"

– J6$"&"*$'&1$"*

L- T5*E*;

– J6<41$5)6

– >&":5$

– @5B56D'Y#1441K545$IZ

– !5g%5:5$I

– >%&&"6#I']

+L

314%1$5)6')<'?J'*"#%&5$5"*314%1$5)6')<'?J'*"#%&5$5"*

!"#$%&"'()$"*

+,-./+ !"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

R%&')Kh"#$59"'8"&"'5*'$)'914%"'&5*E4"**'#1*8'<4)A*

_ \59"6'$8"'&5#8'*"$')<'<5N":H56#)B"'*"#%&5$5"*'$&1:":'56'$8"'B1&E"$F'$8"5&C&5#"*'C&)95:"'$8"'56<)&B1$5)6'6"":":'$)'914%"'&5*E4"**'#1*8'<4)A*'1$'816:

J6'$8"'B1&E"$F'$85*'56<)&B1$5)6')6'$8"'$5B"'914%"')<'B)6"I'5*'D59"6'56B16I':5<<"&"6$'<)&B*;

+- OC)$'56$"&"*$'&1$"*

L- 0&5#"*')<':5*#)%6$'K)6:*'Y"-D-F'd"&)H#)%C)6'K)6:*Z

Q- 0&5#"*')<'#)%C)6'K)6:*

.- ?)&A1&:'56$"&"*$'&1$"*

+Q

@"&B'*$&%#$%&"')<'56$"&"*$'&1$"*@"&B'*$&%#$%&"')<'56$"&"*$'&1$"*

!"#$%&"'()$"*

+,-./+ !"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

OC)$'56$"&"*$'&1$"'r$''5*'$8"'#%&&"6$'Y166%145d":Z'56$"&"*$'&1$"'<)&'B1$%&5$I:1$"'$

_ r$'5*'<)&'C1IB"6$*')64I')6':1$"'$

_ r$'5*':5<<"&"6$'<)&'"1#8'':5<<"&"6$':1$"'$

eN1BC4"-''OC)$'56$"&"*$'&1$"*')6'L//,-/_-/+;

@8"'*"$')<'*C)$'56$"&"*$'&1$"*'<)&':5<<"&"6$'B1$%&5$5"*

D59"*' $8"' $"&B' *$&%#$%&"' )<' 56$"&"*$' &1$"*F'A85#8' &"<"&*' $)' $8"' &"41$5)6K"$A""6'*C)$'&1$"*'16:'$8"5&'B1$%&5$5"*

+.

@"&B'*$&%#$%&"')<'56$"&"*$'&1$"*@"&B'*$&%#$%&"')<'56$"&"*$'&1$"*

Maturity (year) 1/4 1/2 1 2 5 10 20 25 25.5 (longest)I nterest R ate (%) 3.29 3.61 3.87 3.97 4.06 4.41 4.65 4.57 4.61fr1, r2, . . . , rt, . . .g

{r1,r2 ,...,rt ,....}

!"#$%&"'()$"*

+,-./+ !"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

O"C$',F'L//_'Y24))BK"&DZ

8$$C;==)6456"-A*h-#)B=B:#=C1D"=B1&E"$*:1$1-8$B4iB):j$)C619k/k///L

@"&B'*$&%#$%&"')<'56$"&"*$'&1$"*@"&B'*$&%#$%&"')<'56$"&"*$'&1$"*

!"#$%&"'()$"*

+,-./+ !"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

8$$C;==<5N":56#)B"-<5:"45$I-#)B=<5=?JG5*$)&5#14l5"4:

@"&B'*$&%#$%&"')<'56$"&"*$'&1$"*@"&B'*$&%#$%&"')<'56$"&"*$'&1$"*

G5*$)&I')<'X-O-'$"&B'*$&%#$%&"')<'56$"&"*$'&1$"*

!"#$%&"'()$"*

+,-./+ !"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

M':5*#)%6$'K)6:'Yd"&)'#)%C)6'K)6:Z'A5$8'B1$%&5$I':1$"'$''5*'1'K)6:A85#8'C1I*'a+')64I'1$'$-

eN1BC4"-''O@TJ0Om'1&"'$&1:":'1$'$8"'<)44)A56D'C&5#"*;

?)&'$8"',HI"1&'O@TJ0OF'A"'819"

m'O"C1&1$"'@&1:56D')<'T"D5*$"&":'J6$"&"*$'16:'0&56#5C14'O"#%&5$5"*

+`

S5*#)%6$'K)6:*S5*#)%6$'K)6:*

Maturity (year) 1/4 1/2 1 2 5 10 30Price 0.991 0.983 0.967 0.927 0.797 0.605 0.187

0.797 =1

(1 + r5)5 ) r5 =1

(0.797)1=5 ¡ 1 = 4.64%Bt =

1

(1+rt)t or rt =1

B1=tt

¡ 1

0.797 = 1(1+ r5 )

5 ! r5 = 4.64%

!"#$%&"'()$"*

+,-./+ !"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

!"$'B$':"6)$"'$8"'#%&&"6$'C&5#"'Y$5B"'/Z')<'1':5*#)%6$'K)6:'B1$%&56D'1$'$-@8"6

0&5#"*')<':5*#)%6$'K)6:*'C&)95:"'56<)&B1$5)6'1K)%$'*C)$'56$"&"*$'&1$"*'16:95*"'9"&*1-

+_

S5*#)%6$'K)6:*S5*#)%6$'K)6:*

Bt =1

(1 + rt)t or rt =1

B1=tt

¡ 1B1 =1

(1 + r1) $ r1

B2 =1

(1 + r2)2 $ r2

B3 =1

(1 + r3)3 $ r3

..

.

BT =1

(1 + rT )T $ rT

Bt =1

(1+ rt )t ! rt =

1

Bt1t"1

!"#$%&"'()$"*

+,-./+ !"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

M'#)%C)6'K)6:'C1I*'1'*$&"1B')<'&"D%41&'#)%C)6'C1IB"6$*'16:'1C&56#5C14'1$'B1$%&5$I-

M'#)%C)6'K)6:'5*'1'C)&$<)45)')<':5*#)%6$'K)6:*-

eN1BC4"-'M'QHI"1&'K)6:')<'a+F///'C1&'16:',f'166%14'#)%C)6-

+b

>)%C)6'K)6:*>)%C)6'K)6:*

-0 1 2 3 time

6 66

50 50 50 + 1, 000

!"#$%&"'()$"*

+,-./+ !"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

L/

>)%C)6'K)6:*>)%C)6'K)6:*

-0 1 2 3 time

6 66

50 50 50+ 1; 000

=

-0 1 2 3 time

650(50 1-year ST R I P S)

+

-0 1 2 3 time

650(50 2-year ST R I P S)

+

-0 1 2 3 time

61050(1050 3-year ST R I P S)

!"#$%&"'()$"*

+,-./+ !"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

O%CC)*"'$81$'$8"':5*#)%6$'K)6:'C&5#"*'1&"'1*'<)44)A*

781$'*8)%4:'$8"'C&5#"')<'$8"'#)%C)6'K)6:'K"i

781$'5<'6)$i

@8"'C&5#"')<'1'#)%C)6'K)6:'5*'D59"6'KI

L+

>)%C)6'K)6:*>)%C)6'K)6:*

t 1 2 3 4 5Bt 0.952 0.898 0.863 0.807 0.757

Price = (50)(0.952) + (50)(0.898) + (1050)(0.863)

= 998.65B =

TX

t= 1(Ct £ Bt) + (P £ BT )

=C11+r1

+ ¢¢¢+CT ¡ 1

(1+rT ¡ 1)T ¡ 1 +CT +P(1+rT )T

B = 50 0.952 + 50 0.898 +1050 0.863 = 998.65

B = CFtBt =t=1

T

! CFt(1+ rt )

tt=1

T

!

!"#$%&"'()$"*

+,-./+ !"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

l5"4:H$)HB1$%&5$I')<'1'K)6:F':"6)$":'KI'IF'5*'D59"6'KI

\59"6'5$*'B1$%&5$IF'$8"'C&56#5C4"'16:'$8"'#)%C)6'&1$"F'$8"&"'5*'1')6"'$))6"'B1CC56D'K"$A""6'$8"'C&5#"')<'1'K)6:'16:'5$*'l@P-

eN1BC4"-''>%&&"6$'+H'16:'LHI"1&'*C)$'56$"&"*$'&1$"*'1&"',f'16:'VfF&"*C"#$59"4I-''@8"'C&5#"')<'1'LHI"1&'@&"1*%&I'#)%C)6'K)6:'A5$8'C1&914%"')<'a+//'16:'1'#)%C)6'&1$"')<'Vf'5*

J$*'l@P'5*',-b`/Vf;

LL

l5"4:H$)HB1$%&5$I'Yl@PZl5"4:H$)HB1$%&5$I'Yl@PZ

B =TX

t= 1

Ct(1+y)t +

P

(1+y)T

B =6

1 + 0.05+

106

(1 + 0.06)2 = 100.0539100.0539 =6

1 + 0.059706+

106

(1 + 0.059706)2

B =CFt(1+ y)tt=1

T

!

B =61.05

+1061.062

= 100.0539

100.0539 = 61+ y

+106(1+ y)2

!"#$%&"'()$"*

+,-./+ !"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

O)'<1&F'A"'819"'<)#%*":')6'*C)$'56$"&"*$'&1$"*;'&1$"*'<)&'1'$&16*1#$5)6K"$A""6'$):1IF'/F'16:'1'<%$%&"':1$"F'$-

()AF'A"'*$%:I'<)&A1&:'56$"&"*$'&1$"*;'&1$"*'<)&'1'$&16*1#$5)6'K"$A""6$A)'<%$%&"':1$"*F'<)&'56*$16#"F't+'16:'tL

?)&'1'<)&A1&:'$&16*1#$5)6'$)'K)&&)A'B)6"I'56'$8"'<%$%&";_ @"&B*')<'$8"'$&16*1#$5)6'1&"'1D&"":')6'$):1IF'$'j'/_ !)16'5*'&"#"59":')6'1'<%$%&"':1$"'t+_ T"C1IB"6$')<'$8"'4)16')##%&*')6':1$"'tL

()$";_ ?%$%&"' *C)$' &1$"*' #16' K"' :5<<"&"6$' <&)B' #%&&"6$' #)&&"*C)6:56D<)&A1&:'&1$"*

LQ

?)&A1&:'56$"&"*$'&1$"*?)&A1&:'56$"&"*$'&1$"*

!"#$%&"'()$"*

+,-./+ !"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

eN1BC4"-'M*'$8"'>?R')<'1'X-O-'B%4$561$5)614F'I)%'"NC"#$'$)&"C1$&51$"'a+/'P'<&)B'1'<)&"5D6'*%K*5:51&I'56'+'I"1&F'A85#8'A544'K"%*":'$)'C1I':595:"6:*'+'I"1&'41$"&-'()$'E6)A56D'$8"'56$"&"*$'&1$"*56'+

'''''I"1&F'I)%'A)%4:'45E"'$)'4)#E'56$)'1'4"6:56D'&1$"')6"'I"1&'<&)B'6)A<)&'1'C"&5):')<')6"'I"1&-'781$'*8)%4:'I)%':)i

@8"'#%&&"6$'56$"&"*$'&1$"*'1&"

L.

?)&A1&:'56$"&"*$'&1$"*?)&A1&:'56$"&"*$'&1$"*

time to maturity t (years) 1 2spot interest rate rt 0.05 0.07

!"#$%&"'()$"*

+,-./+ !"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

O$&1$"DI;

_ 2)&&)A'ab-,L.P'6)A'<)&')6"'I"1&'1$',f

_ J69"*$'$8"'C&)#"":*'ab-,L.P'<)&'$A)'I"1&*'1$'`f

R%$#)B"'Y56'B5445)6':)441&*Z;

@8"'4)#E":H56'+HI"1&'4"6:56D'&1$"'+'I"1&'<&)B'6)A'5*'b-/.f-

L,

?)&A1&:'56$"&"*$'&1$"*?)&A1&:'56$"&"*$'&1$"*

Year 0 1 21-yr borrowing 9.524 ¡ 10.000 02-yr lending ¡ 9.524 0 10.904Repatriation 0 10.000 0Net 0 0 10.904

!"#$%&"'()$"*

+,-./+ !"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

@8"'<)&A1&:'56$"&"*$'&1$"'K"$A""6'$5B"'$H+'16:'$'5*

)&

LV

?)&A1&:'56$"&"*$'&1$"*?)&A1&:'56$"&"*$'&1$"*

(1 + rt)t = (1 + rt¡ 1)t¡ 1(1 + ft)Spot and forward rates

-

6

year1 2 3 4 5

-r1 = f1-r1 -f2

-r2 -f3-r3 -f4

-r4 -f5

(1+ rt )t = (1+ rt!1)

t!1(1+ ft )

ft =(1+ rt )

t

(1+ rt!1)t!1 !1

!"#$%&"'()$"*

+,-./+ !"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

eN1BC4"-'O%CC)*"'$81$':5*#)%6$'K)6:'C&5#"*'1&"'1*'<)44)A*;

M'#%*$)B"&'A16$*'1'<)&A1&:'#)6$&1#$'$)'K)&&)A'aL/P'$8&""'I"1&*<&)B'6)A'<)&')6"'I"1&-'>16'I)%'Y1'K16EZ'g%)$"'1'&1$"i

L`

?)&A1&:'56$"&"*$'&1$"*?)&A1&:'56$"&"*$'&1$"*

t 1 2 3 4Bt 0.9524 0.8900 0.8278 0.7629rt 0.05 0.06 0.065 0.07

f4 = 8.51%

f4 = 8.51%

!"#$%&"'()$"*

+,-./+ !"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

781$'*8)%4:'I)%':)'$):1I'$)'4)#EH56'$8"*"'#1*8'<4)A*i

+- 2%I'L/F///F///')<'Q'I"1&':5*#)%6$'K)6:*F'#)*$56D

L- ?5616#"'$85*'KI'*"4456D'.'I"1&':5*#)%6$'K)6:*')<'1B)%6$

>1*8'<4)A*'<&)B'$85*'*$&1$"DI'Y56'B5445)6':)441&*Z;

@8"'56$"&"*$'<)&'$85*'<%$%&"'569"*$B"6$'5*'D59"6'KI;

L_

?)&A1&:'56$"&"*$'&1$"*?)&A1&:'56$"&"*$'&1$"*

(20, 000, 000)(0.8278) = $16, 556, 00016, 556, 000/0.7629 = $21, 701, 403

Year 0 1{2 3 4Purchase of 3-year bonds -16.556 0 20.000 0Sale of 4-year bonds 16.556 0 0 -21.701Total 0 0 20.000 -21.701

21, 701, 403

20, 000, 000 ¡ 1 = 8.51%

21,701,40320,000,000

!1 = 8.51%

$20m0.8278 = $16.556m

$16.556m / 0.7629 = $21.701m

!"#$%&"'()$"*

+,-./+ !"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

781$':"$"&B56"*'$8"'$"&B'*$&%#$%&"')<'56$"&"*$'&1$"*i

+- eNC"#$":''<%$%&"'*C)$'&1$"*

L- T5*E')<'4)6D'K)6:*

P):"4*')<'56$"&"*$'&1$"*;

_ eNC"#$1$5)6*'GIC)$8"*5*

_ !5g%5:5$I'0&"<"&"6#"

_ SI61B5#'P):"4*'Y31*5#"EF'>)NHJ6D"&*)44'16:'T)**F']Z

Lb

GIC)$8"*5*')6'56$"&"*$'&1$"*GIC)$8"*5*')6'56$"&"*$'&1$"*

!"#$%&"'()$"*

+,-./+ !"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

eNC"#$1$5)6*'GIC)$8"*5*;''?)&A1&:'&1$"*'C&":5#$'<%$%&"'*C)$'&1$"*

JBC45#1$5)6*;_ @8"'*4)C"')<'$8"'$"&B'*$&%#$%&"'&"<4"#$*'$8"'B1&E"$n*'"NC"#$1$5)6*')<'<%$%&"*8)&$H$"&B'56$"&"*$'&1$"*

!5g%5:5$I'0&"<"&"6#"'GIC)$8"*5*;''J69"*$)&*'&"D1&:'4)6D'K)6:*'1*&5*E5"&'$816'*8)&$'K)6:*

JBC45#1$5)6*;_ !)6D'K)6:*')6'19"&1D"'&"#"59"'85D8"&'&"$%&6*'$816'*8)&$'K)6:*_ ?)&A1&:'&1$"')6'19"&1D"'oo)9"&HC&":5#$nn'<%$%&"'*8)&$H$"&B'&1$"*-_ @"&B'*$&%#$%&"'&"<4"#$*1Z '"NC"#$1$5)6*')<'<%$%&"'56$"&"*$'&1$"*F'16:KZ '&5*E'C&"B5%B':"B16:":'KI'569"*$)&*'56'4)6D'K)6:*

Q/

GIC)$8"*5*')6'56$"&"*$'&1$"*GIC)$8"*5*')6'56$"&"*$'&1$"*

ft = E [r1(t)]ft = E [r1(t)] + LiquidityPremium

ft = E[r1(t)]

ft = E[r1(t)]+ LP

!"#$%&"'()$"*

+,-./+ !"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

Q+

!5g%5:5$I'C&"<"&"6#"!5g%5:5$I'C&"<"&"6#"


Consider a situation where the short rate is 5%.

Expectations hypothesis:Suppose that the expected short rate for the following year is 6%.What is the price of a 2-year zero?

(1+ r2 )2 = (1+ r1)(1+ E[r2 ]) = 1.05 1.06B2 = $1000 / (1+ r2 )2 = $898.47

Suppose that an investor wants to invest for 1 year. Strategy A: invest into the 1-year zero.Strategy B: invest into the 2-year zero, and sell it after 1 year.

!"#$%&"'()$"*

+,-./+ !"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

QL

!5g%5:5$I'C&"<"&"6#"!5g%5:5$I'C&"<"&"6#"


What is the expected return of strategy B?

If the future short rate is 6%, the future bond price is

$1000/1.06=$943.4.

In this case, the return is ($943.4-$898.47)/$898.47=5%.

So, is the investor indifferent between strategies A and B?

What if the investor requires a discount to hold the 2-year bond?Suppose that the bond trades at $890.

B2 = $1000 / (1+ r2 )2 = $890

r2 = 6%

f2 =1.062

1.05= 7% = E[r1]+1%

!"#$%&"'()$"*

+,-./+ !"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

QQ

!5g%5:5$I'C&"<"&"6#"!5g%5:5$I'C&"<"&"6#"


The liquidity preference hypothesis is based on the idea thatinvestors require a risk-premium to invest in long-term bonds.

Why?

Consider an investor who wants to invest for a 2 year period.

Such an investor bears roll-over risk if she invests in 1 year bonds.To hold 1 year bonds, the investor will require a risk premium...

! ft < E[r1(t)]

Forward rates contain information about 1. expected future short rates and2. risk premia.

!"#$%&"'()$"*

+,-./+ !"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

Q.

!5g%5:5$I'C&"<"&"6#"!5g%5:5$I'C&"<"&"6#"


Yieldconstant liquidity premium

constant forward rate fyield curve:(1+yt)=(1+r1)(1+f)…(1+f)

constant expected short rate r1

!"#$%&"'()$"*

+,-./+ !"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

Q,

!5g%5:5$I'C&"<"&"6#"!5g%5:5$I'C&"<"&"6#"

Yield

!"#$%&"'()$"*

+,-./+ !"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

QV

O%BB1&IO%BB1&I

Bond yields

Prices in the bond market contain information aboutthe time-value of money:

Spot interest ratesForward interest rates

The term structure of interest rates

Next time: managing bond portfolios.

!"#$%&"'()$"*

+,-./+

M*'56$"&"*$'&1$"*'#816D"'Y*$)#81*$5#144IZ')9"&'$5B"F'K)6:'C&5#"*'14*)#816D"-'@8"'914%"')<'1'K)6:'5*'*%Kh"#$'$)'56$"&"*$'&1$"'&5*E-

Q`

J6$"&"*$'&1$"'&5*EJ6$"&"*$'&1$"'&5*E

-

yield (%)

6

Price (in log)

5.0

100 r

`̀`̀`̀`̀`̀

```````````````

!"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

!"#$%&"'()$"*

+,-./+

Q_

J6$"&"*$'&1$"'&5*EJ6$"&"*$'&1$"'&5*E !"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

!"#$%&"'()$"*

+,-./+

S%&1$5)6'16:'P):5<5":'S%&1$5)6'YS%&1$5)6'16:'P):5<5":'S%&1$5)6'Y1**%B"'1'<41$'$"&B'*$&%#$%&"'1$'r$'= yZ

P1#1%41I':%&1$5)6'5*'$8"'A"5D8$":'19"&1D"'$"&B'$)'B1$%&5$I

M' K)6:n*' 56$"&"*$' &1$"' &5*E' #16' K"' B"1*%&":' KI' 5$*' &"41$59"' C&5#"#816D"'A5$8'&"*C"#$'$)'1'#816D"'56'I5"4:;

@85*'5*'#144":'1'K)6:n*'B):5<5":':%&1$5)6')&'9)41$545$I-

Qb

P"1*%&"*')<'56$"&"*$'&1$"'&5*EP"1*%&"*')<'56$"&"*$'&1$"'&5*E !"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

D =PV[CFt ]

Bt = 1

Bt=1

T

! CFt(1+ y)t

tt=1

T

!

MD = !1B"B"y

=D1+ y

!"#$%&"'()$"*

+,-./+

eN1BC4"-' >)6*5:"&' 1' .HI"1&' @H6)$"' A5$8' <1#"' 914%"' a+//' 16:' `f#)%C)6F'*"4456D'1$'a+/Q-,/F'I5"4:56D'Vf-

?)&'@H6)$"*F'#)%C)6*1&"'C15:'*"B5H166%144I-@5B"'#)%6$'56'%65$*')<V'B)6$8*-

S%&1$5)6'Y56'+=L'I"1&'%65$*Z;

P):5<5":':%&1$5)6'Y9)41$545$IZ;

J<'$8"'*"B5H166%14'I5"4:'B)9"*'%C'KI'/-+fF'$8"'K)6:'C&5#"':"#&"1*"*&)%D84I'KI'/-VbLf-

./

P"1*%&"*')<'56$"&"*$'&1$"'&5*EP"1*%&"*')<'56$"&"*$'&1$"'&5*E

t CF PV(CF ) t¢PV(CF )1 3.5 3.40 3.402 3.5 3.30 6.603 3.5 3.20 9.604 3.5 3.11 12.445 3.5 3.02 15.106 3.5 2.93 17.597 3.5 2.85 19.928 103.5 81.70 653.63

103.50 738.28

D = (738.28)/103.50 = 7.13MD = D/(1 + y) = 7.13/1.03 = 6.92

!"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

D = 738.28 /103.50 = 7.13MD = 7.13 /1.03 = 6.92

!"#$%&"'()$"*

+,-./+

.+

S%&1$5)6S%&1$5)6

t CF PV(CF ) t¢PV(CF )1 3.5 3.40 3.402 3.5 3.30 6.603 3.5 3.20 9.604 3.5 3.11 12.445 3.5 3.02 15.106 3.5 2.93 17.597 3.5 2.85 19.928 103.5 81.70 653.63

103.50 738.28

D = (738.28)/103.50 = 7.13MD = D/(1 + y) = 7.13/1.03 = 6.92

!"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

Duration

How does duration depend on bond maturity?What happens if the maturity increases?

!"#$%&"'()$"*

+,-./+

.L

S%&1$5)6S%&1$5)6

t CF PV(CF ) t¢PV(CF )1 3.5 3.40 3.402 3.5 3.30 6.603 3.5 3.20 9.604 3.5 3.11 12.445 3.5 3.02 15.106 3.5 2.93 17.597 3.5 2.85 19.928 103.5 81.70 653.63

103.50 738.28

D = (738.28)/103.50 = 7.13MD = D/(1 + y) = 7.13/1.03 = 6.92

!"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

Duration

Par and premium bonds: duration increases.Discount bonds: duration can decrease!

How does the duration depend on the bond yield?What happens if the yield increases?

!"#$%&"'()$"*

+,-./+

.Q

S%&1$5)6S%&1$5)6

t CF PV(CF ) t¢PV(CF )1 3.5 3.40 3.402 3.5 3.30 6.603 3.5 3.20 9.604 3.5 3.11 12.445 3.5 3.02 15.106 3.5 2.93 17.597 3.5 2.85 19.928 103.5 81.70 653.63

103.50 738.28

D = (738.28)/103.50 = 7.13MD = D/(1 + y) = 7.13/1.03 = 6.92

!"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

Duration

Negative relation between the duration and the bond yield

!"#$%&"'()$"*

+,-./+

eN1BC4"-' Y>)6$56%":Z' .HI"1&' @H6)$"' A5$8' `f' #)%C)6' 16:' Vf' <41$I5"4:'#%&9"-

_ S%&1$5)6'5*'Sj`-+Q_ 3)41$545$I'5*'PS'j'V-bL

M*'$8"'I5"4:'#816D"*F'$8"'K)6:'C&5#"'14*)'#816D"*;

_ ?)&'*B144'I5"4:'#816D"*F'C&5#56D'KI'PS'5*'1##%&1$"_ ?)&'41&D"'I5"4:'#816D"*F'C&5#56D'KI'PS'5*'561##%&1$"

..

>)69"N5$I>)69"N5$I

Y ield Price Using MD Di®erence0.040 96.63 96.35 0.290.035 100.00 99.93 0.070.031 102.79 102.79 0.000.030 103.50 - -0.029 104.23 104.23 0.000.025 107.17 107.09 0.080.020 110.98 110.67 0.32

!"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

0.1%

0.692%

!"#$%&"'()$"*

+,-./+

2)6:'C&5#"'5*'6)$'1'456"1&'<%6#$5)6')<'$8"'I5"4:-'?)&'41&D"'I5"4:#816D"*F'$8"'"<<"#$')<'#%&91$%&"'Y5-"-F'6)6456"1&5$IZ'K"#)B"*5BC)&$16$-

>)69"N5$IF'>pF'B"1*%&"*'$8"'#%&91$%&"')<'$8"'K)6:'C&5#"'YC"&'aZ'1*'1<%6#$5)6')<'$8"'I5"4:;

.,

>)69"N5$I>)69"N5$I

-

yield (%)

6

Price (in log)

3.0

103.5 r

`̀`̀`̀`̀`̀

``````````````H̀HHHHH

H H H H HH

Duration gives the slope

? Convexity gives curvature

?

(¢ B ) = ¢ B¢ y (¢ y) + 1

2¢ 2B¢ y2 (¢ y)2 + ¢¢¢

¡ MD £ (¢ y)+ CX £ (¢ y)2

CX =1

2

1

B

!2B!y2

!"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

!B = "MD!y + CX(!y)2

CX =121B!2B!y2

!"#$%&"'()$"*

+,-./+

P)*$'K)6:*' D59"' 6)B5614' C1I)<<*-' ' J6' $8"' C&"*"6#"' )<' 56<41$5)6' &5*EF&"14'C1I)<<*'1&"'&5*EI'"9"6'A8"6'6)B5614'C1I)<<*'1&"'*1<"-

eN1BC4"-'O%CC)*"'$81$' 56<41$5)6'6"N$'I"1&' 5*'%6#"&$156'"N'16$"F'A5$8"g%144I'C)**5K4"'&1$"')<'+/fF'_f'16:'Vf-''@8"'&"14'56$"&"*$'&1$"'5*Lf-

@8"'+HI"1&'6)B5614'56$"&"*$'&1$"'A544'K"'Y&)%D84IZ'+/f-

>)6*5:"&'$8"'&"$%&6'<&)B'569"*$56D'56'1'+HI"1&'@&"1*%&I'*"#%&5$I;

.V

J6<41$5)6'&5*EJ6<41$5)6'&5*E

Year 0 value In° ation rate (%) Year 1 nom. payo® Year 1 real payo®1000 0.10 1100 10001000 0.08 1100 10191000 0.06 1100 1038

!"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

!"#$%&"'()$"*

+,-./+

?5N":H56#)B"' *"#%&5$5"*' 819"' C&)B5*":' C1I)<<*' )<' <5N":' 1B)%6$' 1$<5N":' $5B"*-' eN#4%:56D' D)9"&6B"6$' K)6:*F' )$8"&' <5N":H56#)B"*"#%&5$5"*F'*%#8'1*'#)&C)&1$"'K)6:*F'#1&&I'$8"'&5*E')<'<15456D'$)'C1I)<<'1*'C&)B5*":-

S"<1%4$' &5*E' Y#&":5$' &5*EZ' &"<"&*' $)' $8"' &5*E' $81$' 1' :"K$' 5**%"&' <154*' $)B1E"'$8"'C&)B5*":'C1IB"6$*'Y56$"&"*$')&'C&56#5C14Z-

2)6:'&1$56D*'KI'&1$56D'1D"6#5"*'Y"-D-F'P)):In*'16:'Oqr0Z'C&)95:"'56:5#1$5)6*)<'$8"'45E"458)):')<':"<1%4$'KI'"1#8'5**%"&-

_ J69"*$B"6$'D&1:";'M11'HH'211'KI'P)):In*')&'MMM'HH'222'KI'Or0_ OC"#%41$59"'Yh%6EZ;'21'16:'K"4)A'KI'P)):In*')&'22'16:'K"4)A'KI'Or0

.`

S"<1%4$'&5*ES"<1%4$'&5*E

Description Moody's S&PG ilt-edge Aaa AAAVery high grade Aa AAUpper medium grade A ALower medium grade Baa BBBLow grade Ba BB

!"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

!"#$%&"'()$"*

+,-./+

eN1BC4"-'O%CC)*"'144'K)6:*'819"'C1&'914%"'a+F///'16:_ +/HI"1&'@&"1*%&I'*$&5C'5*'*"4456D'1$'a.VQ-+bF'I5"4:56D'_f_ +/HI"1&'d"&)'5**%":'KI'pls'J6#-'5*'*"4456D'1$'aQL+-b`_ eNC"#$":'C1I)<<'<&)B'plsn*'+/HI"1&'d"&)'5*'a`VL-LL@8"'pls'K)6:;

_ 0&)B5*":'l@P;'$8"'I5"4:'5<':"<1%4$':)"*'6)$')##%&_ eNC"#$":'l@P;'$8"'C&)K1K545$IHA"5D8$":'19"&1D"')<'144'C)**5K4"'I5"4:*_ S"<1%4$'C&"B5%B;'$8"':5<<"&"6#"'K"$A""6'C&)B5*":'I5"4:'16:'"NC"#$":'I5"4:_ 2)6:'&5*E'C&"B5%B;'$8"':5<<"&"6#"'K"$A""6'$8"'"NC"#$":'I5"4:')6'1'&5*EI'K)6:'16:$8"'I5"4:')6'1'&5*EH<&""'K)6:')<'*5B541&'B1$%&5$I'16:'#)%C)6'&1$"

._

S"<1%4$'&5*ES"<1%4$'&5*E

Promised YTM =≥1000:00

321:97´1=10

¡ 1= 12%

Expected YTM =≥762:22321:97

´1=10¡ 1= 9%

Default Premium = Promised YTM ¡ Expected YTM= 12% ¡ 9% = 3%

Risk Premium = Expected YTM ¡ Default-free YTM= 9% ¡ 8% = 1%

!"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

Promised YTM = (1000 / 321.97)110 !1 = 12%

Expected YTM = (762.22 / 321.97)110 !1 = 9%

Default premium = 12% - 9% = 3%Risk premium = 9% - 8% = 1%

!"#$%&"'()$"*

+,-./+

Y ield-to-maturity for a risky bond

8% - Default-free YTM9% - Expected YTM

12% - Promised YTM

Default-freerate

R iskpremium

DefaultpremiumY ield

spread

.b

S"<1%4$'&5*ES"<1%4$'&5*E !"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

!"#$%&"'()$"*

+,-./+ !"#$%&"'Q;'?5N":'56#)B"'*"#%&5$5"*

_ ?5N":H56#)B"'*"#%&5$5"*_ R9"&95"A')<'<5N":H56#)B"'B1&E"$*_ @"&B'*$&%#$%&"')<'56$"&"*$'&1$"*_ S5*#)%6$'K)6:*'16:'#)%C)6'K)6:*_ ?)&A1&:'56$"&"*$'&1$"*_ J6$"&"*$'&1$"'&5*E_ J6<41$5)6'&5*E_ S"<1%4$'&5*E

,/

U"I'#)6#"C$*U"I'#)6#"C$*

Chapter 3web.mit.edu/astomper/www/univie/pof/Chapter_3.pdf · 2010-01-12 · 01&$’2’314%1$5)6...

Documents

Transcript of Chapter 3web.mit.edu/astomper/www/univie/pof/Chapter_3.pdf · 2010-01-12 · 01&$’2’314%1$5)6...