Bond risk and portfolios pdf

30
Bond Risk and Por-olios

description

This slide set is a work in progress and is embedded in my Principles of Finance course that I teach to computer scientists and engineers. http://financefortechies.weebly.com/

Transcript of Bond risk and portfolios pdf

Page 1: Bond risk and portfolios pdf

Bond  Risk  and  Por-olios    

Page 2: Bond risk and portfolios pdf

Interest  Rate  Risk    

y Δy

ΔP

P ΔP = dP

dyΔy+ 1

2!d2Pdy2

(Δy)2 + 13!d3Pdy3

(Δy)3 +...

ΔPP=1P⋅ΔPdy

⋅Δy&+&...

dydP

P1D* ⋅−≡

yDPP

           yDPP

*

*

Δ⋅⋅−=Δ

Δ⋅−=Δ

‘Modified’  Dura6on  

Slope=dPdy

****

curvature= d2P

dy2

Page 3: Bond risk and portfolios pdf

Modified  Dura6on    

           yDPP * Δ⋅−=Δ

If  a  bond’s  modified  dura6on,  D*,  is  equal  to  4,  and  the  yield  decreases  by  1%        (Δy  =  -­‐1%)  then  the  first  order  es6mate  for  the  change  in  bond  price  is  a  4%  increase.      

           %4%14PP

=−⋅−=Δ

Page 4: Bond risk and portfolios pdf

Modified  Dura6on  Calcula6on  

∑∑=

=

⎟⎠⎞

⎜⎝⎛ +⋅=

⎟⎠⎞

⎜⎝⎛ +

=M

1i

i

i

M

1ii

i

my1CF        

my1

CFP

! dPdy

=& CFi ⋅im⋅ 1+ y

m

"

#$

%

&'

&i&1

i=1

M

!!!

∑=

⎟⎠⎞

⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ +

−=

M

1ii

i

my1

CFmi

my1

1dydP

...)dy(dyPd

P1

21dy

dydP

P1

PdP 2

2

2

+⋅⋅+⋅=

and as Δy è dy

Page 5: Bond risk and portfolios pdf

Modified  Dura6on  Calcula6on  

∑=

⎟⎠⎞

⎜⎝⎛ +⋅

⎟⎠⎞

⎜⎝⎛ +

−=⋅

M

1ii

i

my1P

CFmi

my1

1dydP

P1 Macaulay  

Dura6on  

ii

i

my1P

CFw

⎟⎠⎞

⎜⎝⎛ +⋅

∑=

⎟⎠⎞

⎜⎝⎛ +

−=⋅

M

1iiw

mi

my1

1dydP

P1

i

M

1i

wmiD ⋅≡∑

=

⎟⎠⎞

⎜⎝⎛ +

−=⋅

my1

DdydP

P1

Page 6: Bond risk and portfolios pdf

Modified  Dura6on  Calcula6on  

⎟⎠⎞

⎜⎝⎛ +

=−≡

my1

DdydP

P1D*

⎟⎠⎞

⎜⎝⎛ +

my1

DD*

$ $ $ $

$  

 1                      2                      3                      4                        5        

Macaulay  dura6on,  D=  3  years  

Time  to  maturity,  T  =  5  years    

y

my1

DyDPP * Δ⋅

⎟⎠⎞

⎜⎝⎛ +

−=Δ⋅−=Δ

The  Macaulay  dura6on  is  the  weighted  average  6me  to  maturity  of  all  of  the  bonds  cash  flows    

Page 7: Bond risk and portfolios pdf

Bond  Dura6on  

Page 8: Bond risk and portfolios pdf

Dura6on  Calcula6on  

9954.5)11.1(

6549.6)y1(

DD* =+

=+

=

y1]y)[(1cy)(cNy)(1

yy1D N +−+⋅

−⋅++−

+=

years    6549.6%111])%11[(110%)%11(10%01)%11(1

%11%111D 10 =

+−+⋅

−⋅++−

+=

Closed  Form  For  Annual  Coupons    

42.56$            01.09954.511.941$            

yDPP *

−=

⋅⋅−=

Δ⋅⋅−=Δ

)00.887$    is    %11y    at    price    exact(

68.884$      42.56$11.941$      

PPP

=

=

−=

Δ−=

it CF DF DCF w w·∙i/m

1 1.0 100.00$             0.9009 90.09$               0.0957 0.09572 2.0 100.00$             0.8116 81.16$               0.0862 0.17253 3.0 100.00$             0.7312 73.12$               0.0777 0.23314 4.0 100.00$             0.6587 65.87$               0.0700 0.28005 5.0 100.00$             0.5935 59.35$               0.0631 0.31536 6.0 100.00$             0.5346 53.46$               0.0568 0.34097 7.0 100.00$             0.4817 48.17$               0.0512 0.35838 8.0 100.00$             0.4339 43.39$               0.0461 0.36899 9.0 100.00$             0.3909 39.09$               0.0415 0.373810 10.0 1,100.00$       0.3522 387.40$         0.4116 4.1165

P 941.11$         D   6.6549D* 5.9954

DurationPrice   $1000  bond  10%  annual  coupon,  10  yrs  to  maturity,  11%  yield      

Closed  form:    Annual  coupons    

Page 9: Bond risk and portfolios pdf

Dura6on  

$ $ $ $ $ $ $ $ $

$

1                2              3                4                5              6                7                8                9              10  Macaulay  dura6on  =  6.6549  years  

Time  to  maturity  =  10  years    

When interest rates increase, bond price decreases, but coupons can get reinvested at the higher interest rate The Macaulay duration is the time at which the two effects offset each other (first order accuracy)

The Macaulay duration is also an indicator of interest rate risk, but the modified duration is better

Page 10: Bond risk and portfolios pdf

Dura6on  Example    

$1000  bond  with  coupon  rate  of  10%  paid  annually.    The  yield  was  11%  un6l  just  a_er  the  t=0  coupon  was  paid,  then  the  yield  jumped  to  11.1%.    The  dura6on  at  y=11%  was  6.6549  years.    Now  with  a  change  of  yield,  the  change  of  total  value  is  neutral  at  about  6.65  years.    

$-­‐

$1

$2

$3

$4

$5

$6

$7

$8

$9

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

Years

Increase  in  value  of  reinvested  coupons  relative  to  case  of  11%  yield  

Decrease  in  price  of  bond  relative  to  case  of  11%  yield  

Page 11: Bond risk and portfolios pdf

Bond  Poraolios    

¨  Poraolio  Dura6on        ¤ Exact  if  all  bonds  have  the  same  yield  to  maturity    

¨  Bond  Funds    ¨  Ac6ve  v.  Passive  Poraolio  Management  ¨  Poraolio  Immuniza6on  ¨  Cash  Flow  Matching  /  Bond  Ladder    

DP= wi⋅&Dii=1

NB

Page 12: Bond risk and portfolios pdf

Dura6on  Calcula6on:  Semi  Annual  Coupon  

years    not  ,periods    in    duration

my1

my1

mc

my

mcM

my1

mymy1

DM

+⎥⎦

⎤⎢⎣

⎡−⎟

⎠⎞

⎜⎝⎛ +⋅

⎟⎠⎞

⎜⎝⎛ −⋅+⎟

⎠⎞

⎜⎝⎛ +

−+

=

 periods    annual    semi    7418.7          

212%1

212%1

27%

212%

27%9

212%1

212%

212%1

D9

=

+⎥⎦

⎤⎢⎣

⎡−⎟

⎠⎞

⎜⎝⎛ +⋅

⎟⎠⎞

⎜⎝⎛ −⋅+⎟

⎠⎞

⎜⎝⎛ +

−+

=

6518.3

2%121

8709.3D

years    8709.327414.7D

* =

⎟⎠⎞

⎜⎝⎛ +

=

==

i t CF DF DCF w w·∙i/m1 0.5 35.00$                   0.9434 33.02$               0.0398 0.01992 1.0 35.00$                   0.8900 31.15$               0.0375 0.03753 1.5 35.00$                   0.8396 29.39$               0.0354 0.05314 2.0 35.00$                   0.7921 27.72$               0.0334 0.06685 2.5 35.00$                   0.7473 26.15$               0.0315 0.07886 3.0 35.00$                   0.7050 24.67$               0.0297 0.08927 3.5 35.00$                   0.6651 23.28$               0.0280 0.09828 4.0 35.00$                   0.6274 21.96$               0.0265 0.10589 4.5 1,035.00$       0.5919 612.61$         0.7381 3.3216

P 829.958$     D   3.8709D* 3.6518

Price   Duration

$1000  bond,  7%  coupon  yield  w/  semi  annual  coupon,  and  12%  yield  

Closed  form:    Periodic  coupons    

Page 13: Bond risk and portfolios pdf

Dura6on  Determinants    

Higher  coupon  yield  decreases  dura6on  and  interest  rate  risk  

6.06.57.07.58.08.59.09.510.0

0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11%

D

c

Constant:    F=$1000,  T=10  yrs,  y=7%,  m=1  

Page 14: Bond risk and portfolios pdf

Dura6on  Determinants  

Higher  6me  to  maturity  increases  dura6on  and  interest  rate  risk  

Constant:    F=$1000,  c=5%,  y=6%,  m=1  

012345678910

-­‐ 1   2   3   4   5   6   7   8   9   10   11  

D

T

Page 15: Bond risk and portfolios pdf

Dura6on  Determinants  

Higher  yield  to  maturity  decreases  dura6on  and  interest  rate  risk  

Constant:    F=$1000,  T=10  yrs,  c=5%,  m=1  

7.57.67.77.87.98.08.18.28.38.48.5

0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11%

D

y

Page 16: Bond risk and portfolios pdf

Convexity    

...)y(dyPd

21yD

PP 2

2

2* +Δ+Δ⋅−=

Δ

2

2

dyPd

P1C ≡

...)y(CP21yDPP 2* +Δ⋅⋅⋅+Δ⋅⋅−=Δ

     my1

miCF

dydP  

M

1i

1i

i∑=

−−

⎟⎠⎞

⎜⎝⎛ +⋅⋅−=

∑=

−−

⎟⎠⎞

⎜⎝⎛ +⋅

+⋅⋅=

M

1i

2i

2i2

2

my1

m)1i(iCF

dyPd

∑=

⎟⎠⎞

⎜⎝⎛ +

⋅+⋅

⎟⎠⎞

⎜⎝⎛ +

=M

1ii

i222

2

my1

CFm

)1i(i

my1

1dyPd

Convexity    

     my1P

CFm

)1i(i

my1

1dyPd

P1 M

1ii

i222

2

∑=

⎟⎠⎞

⎜⎝⎛ +⋅

⋅+⋅

⎟⎠⎞

⎜⎝⎛ +

=⋅

i

M

1i222

2

w  m

)1i(i

my1

1dyPd

P1C ∑

=

⋅+⋅

⎟⎠⎞

⎜⎝⎛ +

=⋅=

Page 17: Bond risk and portfolios pdf

Convexity  Example    

Convexity

it CF DF DCF w w·∙i/m w·∙i·∙(i+1)/m2

1 1.0 100.00$             0.9009 90.09$               0.0957 0.0957 0.19152 2.0 100.00$             0.8116 81.16$               0.0862 0.1725 0.51743 3.0 100.00$             0.7312 73.12$               0.0777 0.2331 0.93234 4.0 100.00$             0.6587 65.87$               0.0700 0.2800 1.39995 5.0 100.00$             0.5935 59.35$               0.0631 0.3153 1.89186 6.0 100.00$             0.5346 53.46$               0.0568 0.3409 2.38607 7.0 100.00$             0.4817 48.17$               0.0512 0.3583 2.86618 8.0 100.00$             0.4339 43.39$               0.0461 0.3689 3.31989 9.0 100.00$             0.3909 39.09$               0.0415 0.3738 3.738510 10.0 1,100.00$       0.3522 387.40$         0.4116 4.1165 45.2810

P 941.11$         D   6.6549 62.5243D* 5.9954 50.7461 C

DurationPrice  

03.54$39.2$42.56$            

0001.07461.5011.9415.01.9954.511.941$            

)y(CP21yDPP 2*

−=+−=

⋅⋅⋅+⋅⋅−=

Δ⋅⋅⋅+Δ⋅⋅−=Δ

07.887$03.54$11.941$      PPP

=−=

Δ+=

$1000  bond  10%  annual  coupon,  10  yrs  to  maturity,  11%  yield      

Exact  price:  $887.00  

Page 18: Bond risk and portfolios pdf

Convexity:  Semi  Annual  Coupon  

Convexityi t CF DF DCF w w·∙i/m w·∙i·∙(i+1)/m2

1 0.5 35.00$                   0.9434 33.02$               0.0398 0.0199 0.01992 1.0 35.00$                   0.8900 31.15$               0.0375 0.0375 0.05633 1.5 35.00$                   0.8396 29.39$               0.0354 0.0531 0.10624 2.0 35.00$                   0.7921 27.72$               0.0334 0.0668 0.16705 2.5 35.00$                   0.7473 26.15$               0.0315 0.0788 0.23636 3.0 35.00$                   0.7050 24.67$               0.0297 0.0892 0.31227 3.5 35.00$                   0.6651 23.28$               0.0280 0.0982 0.39268 4.0 35.00$                   0.6274 21.96$               0.0265 0.1058 0.47639 4.5 1,035.00$       0.5919 612.61$         0.7381 3.3216 16.6079

P 829.958$     D   3.8709 18.3747D* 3.6518 16.3534 C

Price   Duration

63.29$68.0$31.30$            

0001.03534.1696.829$5.01.6518.396.829$            

)y(CP21yDPP 2*

−=+−=

⋅⋅⋅+⋅⋅−=

Δ⋅⋅⋅+Δ⋅⋅−=Δ

)32.800$    is    price    exact(

33.800$63.29$96.829$      PPP

=−=

Δ+=

$1000  bond,  7%  coupon  yield  w/  semi  annual  coupons,  and  12%  yield  

Page 19: Bond risk and portfolios pdf

$750.00

$775.00

$800.00

$825.00

$850.00

$875.00

$900.00

$925.00

$950.00

$975.00

$1,000.00

$1,025.00

$1,050.00

$1,075.00

$1,100.00

$1,125.00

$1,150.00

-­‐3.0% -­‐2.5% -­‐2.0% -­‐1.5% -­‐1.0% -­‐0.5% 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0%

Exact  price  (purple)  Second  order  es6mate  (red)  First  order  es6mate  (blue)  

$1000  bond  10%  annual  coupon,  10  yrs  to  maturity,  11%  yield,  P=$941.11  

Page 20: Bond risk and portfolios pdf

Convexity    

¨  Note  from  previous  slide  ¤ Convexity  benefits  the  price  increase  when  yields  fall    ¤ Convexity  minimizes  the  price  decrease  when  yields  rise    

¤ Thus  convexity  is  a  posi6ve  characteris6c    

 

Page 21: Bond risk and portfolios pdf

Alternate  Calcula6on  of  D  and  C  

inom

iM

1inom

*

my

1P

CFmi

my

1

1D

⎟⎠

⎞⎜⎝

⎛ +⋅

⎟⎠

⎞⎜⎝

⎛ +≡ ∑

=

miti =

( ) iteff

iM

1ii

nom

*

y1PCF

t

my

1

1D+⋅

⎟⎠

⎞⎜⎝

⎛ +≡ ∑

=

1m

y1y

mNOM

EFF −⎟⎠

⎞⎜⎝

⎛ +=

     m

y1P

CFm

)1i(i

my

1

1CM

1ii

NOM

i22

NOM∑=

⎟⎠

⎞⎜⎝

⎛ +⋅

⋅+⋅

⎟⎠

⎞⎜⎝

⎛ +

=

itmi= tt

m1i

i Δ+=+

( )

     

y1PCF

)tt(t

my

1

1CM

1it

eff

iii2

nomi∑

= +⋅⋅Δ+⋅

⎟⎠

⎞⎜⎝

⎛ +

=

( ) ( )      

y1PCF

)tt(ty11C

M

1it

eff

iii

effi∑

= +⋅⋅Δ+⋅

+=

For  semi-­‐annual  coupons  m=2  

Page 22: Bond risk and portfolios pdf

Alternate  Calcula6on  of  D  and  C  

Convexityi t CF DF DCF w w·∙t w·∙t·∙(t+Δt)1 0.5 35.00$                 0.9434 33.02$             0.0398 0.0199 0.01992 1.0 35.00$                 0.8900 31.15$             0.0375 0.0375 0.05633 1.5 35.00$                 0.8396 29.39$             0.0354 0.0531 0.10624 2.0 35.00$                 0.7921 27.72$             0.0334 0.0668 0.16705 2.5 35.00$                 0.7473 26.15$             0.0315 0.0788 0.23636 3.0 35.00$                 0.7050 24.67$             0.0297 0.0892 0.31227 3.5 35.00$                 0.6651 23.28$             0.0280 0.0982 0.39268 4.0 35.00$                 0.6274 21.96$             0.0265 0.1058 0.47639 4.5 1,035.00$       0.5919 612.61$         0.7381 3.3216 16.6079

P 829.96$         D   3.8709 18.3747D* 3.6518 16.3534 C

Price   Duration F=$1000  

c=7%  with  semi-­‐annual  coupons,  m=2  

C=$35  

yNOM=12%  

Δt=0.5  yrs    

T=4.5  yrs      

%36.1212%121y

2

EFF =−⎟⎠⎞

⎜⎝⎛ +=

Page 23: Bond risk and portfolios pdf

Ac6ve  v.  Passive  Poraolio  Management    

¨  Passive  ¤  Match  a  bond  poraolio  index  

such  as  those  maintained  by  Lehman  Brothers  n  hhp://www.lehman.com/fi/

indices/factsheets.htm#  n  Example  Fund    

   

¨  Ac6ve    ¤  Interest  rate  forecas6ng  or  

an6cipa6on  ¤  Credit  spread  forecas6ng  or  

an6cipa6on    

Page 24: Bond risk and portfolios pdf

Poraolio  Immuniza6on    

∑=

⋅=M

1iiiP DwD

Bond  A Bond  Bc 4.5%y 4% 4%N 1 3F 1,000$         1,000$              C -­‐$                 45.00$              P 961.54$     1,013.88$    D 1.0000 2.8736

21.556,924$%)41(000,000,1$20 =

+=π

%3729.53w

%6271.46w

0.28736.2)w1(0000.1w

0.28736.2w0000.1w

0.2DwDw

1ww

B

A

AA

BA

BBAA

BA

=

=

=⋅−+⋅

=⋅+⋅

=⋅+⋅

=+

A  bond  fund  manager  must  have  a  poraolio  valued  at  $1M  a_er  two  years  and  can  buy  any  number  of      • Bond  A:    $1000  par  value  zero  coupon  bond  with  1  year  to  maturity    • Bond  B:    $1000  par  value  4.5%  annual  coupon  bond  with  3  years  to  maturity        The  yield-­‐to-­‐maturity,  y,  is  4%  for  both  bonds.    

Page 25: Bond risk and portfolios pdf

Poraolio  Immuniza6on  

81.093,431$21.556,924$%6271.460A =⋅=π

41.462,493$21.556,924$%3729.530B =⋅=π

48788.013,1$41.462,493$n

44854.961$81.093,431$n

B

A

==

== type  A  bonds        type  B  bonds        

Page 26: Bond risk and portfolios pdf

Poraolio    Immuniza6on  

Yr  1                            Yr  2                              Yr  3

$961.54

$1000.00

$1045.00

$45.00

$1013.88

y    

5%  4%  3%    y

Bond  A  cash  flows  

Bond  B  cash  flows  

Ini6al  yield  curve  flat  at  4%,  either  shi_s  to  a  flat  5%  or    flat  3%

574,470$%)51(1000$448              )y1(Fn AA2A

=+⋅⋅=

+⋅⋅=π

$529,290  $484,391$21,902$22,997              %)51/(1045$48745$487%)51(45$487              

)y1/()CF(n  Cn    )y1(Cn BBBBBBB0B

=++=

+⋅+⋅++⋅⋅=

++⋅+⋅++⋅⋅=π

Π2 = ΠA2 + ΠB2 = $470,754 + $529,290 = $1,000,045

PA2 PB2 P2

3.00% 461,788$         538,258$         1,000,046$        3.25% 462,909$         537,117$         1,000,026$        3.50% 464,029$         535,982$         1,000,011$        3.75% 465,150$         534,853$         1,000,003$        4.00% 466,271$         533,729$         1,000,000$        4.25% 467,392$         532,611$         1,000,003$        4.50% 468,513$         531,499$         1,000,011$        4.75% 469,634$         530,392$         1,000,025$        5.00% 470,754$         529,290$         1,000,045$        

Page 27: Bond risk and portfolios pdf

Cash  Flow  Matching  

Year  Reqd  Payout

1 15,000$        2 17,000$        3 19,000$        4 20,000$        5 23,000$        6 27,000$        

Bond T c y

A 6 10% 9%B 6 7% 8%C 5 8% 8%D 5 6% 7%E 4 7% 8%F 4 5% 6%G 3 4% 5%H 3 3% 4%I 2 3% 3%J 1 0% 3%

A  bond  poraolio  manager  has  required  minimum  payouts  for  the  next  six  years.    She  may  choose  any  number  of  the  10  bonds  labeled  A  –  J.    To  maximize  return  she  should  create  the  poraolio  at  the  lowest  cost  possible.    Each  bond  has  annual  coupons.  

Page 28: Bond risk and portfolios pdf

Cash  Flow  Matching  

where    

Pi  ,  Π      price  of  each  bond  i  and  the  bond  poraolio    

ni      number  of  bond  i  in  poraolio    

M      number  of  available  bonds  from  which  to  choose  (10  in  this  example)    

     

subject  to  these  constraints:    

 

     for  each  year  j  from  1  to  6  in  this  example    

ni≥  0    no  shor6ng  of  bond  

ni  is  an  integer  no  frac6onal  bonds  allowed    

reqdjj CFCF ≥

i

M

1ii0 n  P    minimize ∑

=

⋅=π

Page 29: Bond risk and portfolios pdf

Cash  Flow  Matching  

Bond A B C D E F G H I J1 100$                           70$                       80$                               60$                       70$                       50$                       40$                       30$                       30$                               1,000$            2 100$                           70$                       80$                               60$                       70$                       50$                       40$                       30$                       1,030$                    3 100$                           70$                       80$                               60$                       70$                       50$                       1,040$             1,030$            4 100$                           70$                       80$                               60$                       1,070$             1,050$            5 100$                           70$                       1,080$                     1,060$            6 1,100$                     1,070$            

Payout  of  each  type  of  bond  over  six  years  

Solu6on  from  linear  programming  and  integer  programming    in  Excel    

Year   Portfolio  Payout

Reqd  Payout

1                   15,000$     15,000$  2                   17,000$     17,000$  3                   19,670$     19,000$  4                   20,110$     20,000$  5                   24,060$     23,000$  6                   27,500$     27,000$  

Portfolio  Cost   93,899$  

Flt  and  Int  Solutions  A B C D E F G H I J Cost  

24.55 0.00 19.02 0.00 14.98 0.00 13.44 0.00 11.10 9.10 92,165$      25.00 0.00 18.00 2.00 15.00 0.00 14.00 0.00 11.00 9.00 93,899$      

Page 30: Bond risk and portfolios pdf

Essen6al  Points