Bond risk and portfolios pdf
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Transcript of Bond risk and portfolios pdf
Bond Risk and Por-olios
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
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
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
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
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
Bond Dura6on
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
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
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
$9
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10
Years
Increase in value of reinvested coupons relative to case of 11% yield
Decrease in price of bond relative to case of 11% yield
Bond Poraolios
¨ Poraolio Dura6on ¤ Exact if all bonds have the same yield to maturity
¨ Bond Funds ¨ Ac6ve v. Passive Poraolio Management ¨ Poraolio Immuniza6on ¨ Cash Flow Matching / Bond Ladder
DP= wi⋅&Dii=1
NB
∑
Dura6on Calcula6on: Semi Annual Coupon
years not ,periods in duration
my1
my1
mc
my
mcM
my1
mymy1
DM
+⎥⎦
⎤⎢⎣
⎡−⎟
⎠⎞
⎜⎝⎛ +⋅
⎟⎠⎞
⎜⎝⎛ −⋅+⎟
⎠⎞
⎜⎝⎛ +
−+
=
periods annual semi 7418.7
212%1
212%1
27%
212%
27%9
212%1
212%
212%1
D9
=
+⎥⎦
⎤⎢⎣
⎡−⎟
⎠⎞
⎜⎝⎛ +⋅
⎟⎠⎞
⎜⎝⎛ −⋅+⎟
⎠⎞
⎜⎝⎛ +
−+
=
6518.3
2%121
8709.3D
years 8709.327414.7D
* =
⎟⎠⎞
⎜⎝⎛ +
=
==
i t CF DF DCF w w·∙i/m1 0.5 35.00$ 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
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
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
Dura6on Determinants
Higher yield to maturity decreases dura6on and interest rate risk
Constant: F=$1000, T=10 yrs, c=5%, m=1
7.57.67.77.87.98.08.18.28.38.48.5
0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11%
D
y
Convexity
...)y(dyPd
21yD
PP 2
2
2* +Δ+Δ⋅−=
Δ
2
2
dyPd
P1C ≡
...)y(CP21yDPP 2* +Δ⋅⋅⋅+Δ⋅⋅−=Δ
my1
miCF
dydP
M
1i
1i
i∑=
−−
⎟⎠⎞
⎜⎝⎛ +⋅⋅−=
∑=
−−
⎟⎠⎞
⎜⎝⎛ +⋅
+⋅⋅=
M
1i
2i
2i2
2
my1
m)1i(iCF
dyPd
∑=
⎟⎠⎞
⎜⎝⎛ +
⋅+⋅
⎟⎠⎞
⎜⎝⎛ +
=M
1ii
i222
2
my1
CFm
)1i(i
my1
1dyPd
Convexity
my1P
CFm
)1i(i
my1
1dyPd
P1 M
1ii
i222
2
∑=
⎟⎠⎞
⎜⎝⎛ +⋅
⋅+⋅
⎟⎠⎞
⎜⎝⎛ +
=⋅
i
M
1i222
2
w m
)1i(i
my1
1dyPd
P1C ∑
=
⋅+⋅
⎟⎠⎞
⎜⎝⎛ +
=⋅=
Convexity Example
Convexity
it CF DF DCF w w·∙i/m w·∙i·∙(i+1)/m2
1 1.0 100.00$ 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: $887.00
Convexity: Semi Annual Coupon
Convexityi t CF DF DCF w w·∙i/m w·∙i·∙(i+1)/m2
1 0.5 35.00$ 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
$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=$941.11
Convexity
¨ Note from previous slide ¤ Convexity benefits the price increase when yields fall ¤ Convexity minimizes the price decrease when yields rise
¤ Thus convexity is a posi6ve characteris6c
Alternate Calcula6on of D and C
inom
iM
1inom
*
my
1P
CFmi
my
1
1D
⎟⎠
⎞⎜⎝
⎛ +⋅
⋅
⎟⎠
⎞⎜⎝
⎛ +≡ ∑
=
miti =
( ) iteff
iM
1ii
nom
*
y1PCF
t
my
1
1D+⋅
⋅
⎟⎠
⎞⎜⎝
⎛ +≡ ∑
=
1m
y1y
mNOM
EFF −⎟⎠
⎞⎜⎝
⎛ +=
m
y1P
CFm
)1i(i
my
1
1CM
1ii
NOM
i22
NOM∑=
⎟⎠
⎞⎜⎝
⎛ +⋅
⋅+⋅
⎟⎠
⎞⎜⎝
⎛ +
=
itmi= tt
m1i
i Δ+=+
( )
y1PCF
)tt(t
my
1
1CM
1it
eff
iii2
nomi∑
= +⋅⋅Δ+⋅
⎟⎠
⎞⎜⎝
⎛ +
=
( ) ( )
y1PCF
)tt(ty11C
M
1it
eff
iii
effi∑
= +⋅⋅Δ+⋅
+=
For semi-‐annual coupons m=2
Alternate Calcula6on of D and C
Convexityi t CF DF DCF w w·∙t w·∙t·∙(t+Δt)1 0.5 35.00$ 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=$35
yNOM=12%
Δt=0.5 yrs
T=4.5 yrs
%36.1212%121y
2
EFF =−⎟⎠⎞
⎜⎝⎛ +=
Ac6ve v. Passive Poraolio Management
¨ Passive ¤ Match a bond poraolio index
such as those maintained by Lehman Brothers n hhp://www.lehman.com/fi/
indices/factsheets.htm# n Example Fund
¨ Ac6ve ¤ Interest rate forecas6ng or
an6cipa6on ¤ Credit spread forecas6ng or
an6cipa6on
Poraolio Immuniza6on
∑=
⋅=M
1iiiP DwD
Bond A Bond Bc 4.5%y 4% 4%N 1 3F 1,000$ 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.
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
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$
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.
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
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$
Essen6al Points