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0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
Bond Price Elasticity
Business 4179
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0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
Bond Price Elasticity
The sensitivity of bond prices (BP) to changes in the
required rate of return (I) is commonly measured by bond
price elasticity (BPe), which is estimated as
iinchangepercentBPinchangepercentBPe
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0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
Example of Elasticity
If the required rate of return changes from 10 percent to 8
percent, the bond price of a zero coupon bond will rise
from $386 to $463. Thus the bond price elasticity is
997.%20
%9.19
%10
%10%8386$
386$463$
e
BP
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0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
997.%20
%9.19
%10
%10%8
386$
386$463$
e
BP
Example of Elasticity
This implies that for each 1 percent change in interest
rates, bond prices change by 0.997 percent in the oppositedirection.
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Bond Price Elasticity and Bond Price
Theorums
The following table demonstrates how bond price elasticity
measures the effects of a given change in interest rates on
bonds with different coupon rates. Zero coupon or stripped bonds have the longest durations
because there are no intermediate cash flows, hence they
exhibit the greatest elasticity.
The higher the coupon rate, the lower the elasticity allother things being equal.
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0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
Sensitivity of 10-year bonds with different
coupon rates to interest rate changes
Effects of a Decline in the Required Rate of Return
(1)Bonds withA CouponRate of:
(2)Initial PriceOf Bonds
(When i=10%)
(3)Price of
Bonds wheni=8%
(4)=[(3)-(2)]/(2)Percentage
Changein Bond Price
(5)Percentage
Changein i
(6)=(4)/(5)Bond PriceElasticity
(BPe)
0% $386 $463 +19.9% -20% -.9975 693 799 +15.3 -20% -.765
10 1,000 1,134 +13.4 -20% -.670
15 1,307 1,470 +12.5 -20% -.624
Effects of an Increase in the Required Rate of Return:
(1)Bonds with
a CouponRate of:
(2)Initial Price
Of Bonds(When i=10%)
(3)Price of
Bonds wheni=12%
(4)=[(3)-(2)]/(2)Percentage
Change in BondPrice
(5)Percentage
Changein i
(6)=(4)/(5)Bond Price
Elasticity(BP
e)
0% $386 $322 -16.6% +20% -.830
5 693 605 -12.7 +20% -.635
10 1,000 887 -11.3 +20% -.565
15 1,307 1,170 -10.5 +20% -.525
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0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
Bond Price Sensitivity and Term to
Maturity
The following chart explores the impact of the term to
maturity on bond price sensitivity
clearly, the longer the term to maturity, the greater the
bond price elasticity.
When interest rates rise, the bond price will rise by a
greater percentage, than the fall in bond price in response
to an equal but opposite increase in interest rates.
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0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
Sensitivity of 10-year bonds with different
coupon rates to interest rate changes
Effects of a Decline in the Required Rate of Return on a 10% Coupon Rate Bond
(1)Bonds witha Term to
Maturity of:
(2)Initial PriceOf Bonds
(When i=10%)
(3)Price of
Bonds wheni=8%
(4)=[(3)-(2)]/(2)Percentage
Changein Bond Price
(5)Percentage
Changein i
(6)=(4)/(5)Bond PriceElasticity
(BPe)
1 $1,000 $1,019.40 +1.9% -20% -.095
5 1,000 1,079.87 +8.0 -20% -.4
10 1,000 1,134.21 +13.4 -20% -.67
30 1,000 1,225.20 +22.5 -20% -1.126
Effects of an Increase in the Required Rate of Return on a 10% Coupon Rate Bond
(1)Bonds witha Term to
Maturity of:
(2)Initial PriceOf Bonds
(When i=10%)
(3)Price of
Bonds when
i=12%
(4)=[(3)-(2)]/(2)Percentage
Change in Bond
Price
(5)Percentage
Change
in i
(6)=(4)/(5)Bond PriceElasticity
(BPe
)1 $1,000 $982.19 -1.8% +20% -.09
5 1,000 927.88 -7.2 +20% -.36
10 1,000 887.02 -11.3 +20% -.565
30 1,000 838.92 -16.0 +20% -.80
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0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
Bond Prices and Term to Maturity
R e q u ire d re t 1 0 .0 0 % 1 0 .0 0 % 1 0 .0 0 % 1 0 .0 0 % 1 0 .0 0 %
C o u p o n R a t 10 .00 % 8 .00% 12 .00 % 6 .00% 14 .00 %
Term to M at B o n d P r ice B o n d P rice B o n d P rice B o n d P r ic e B o n d P r ice
1 $ 1 ,0 0 0 .0 0 $ 9 8 1 .8 2 $ 1 ,0 1 8 .1 8 $ 9 6 3 .6 4 $ 1 ,0 3 6 .3 6
2 $ 1 ,0 0 0 .0 0 $ 9 6 5 .2 9 $ 1 ,0 3 4 .7 1 $ 9 3 0 .5 8 $ 1 ,0 6 9 .4 2
3 $ 1 ,0 0 0 .0 0 $ 9 5 0 .2 6 $ 1 ,0 4 9 .7 4 $ 9 0 0 .5 3 $ 1 ,0 9 9 .4 7
4 $ 1 ,0 0 0 .0 0 $ 9 3 6 .6 0 $ 1 ,0 6 3 .4 0 $ 8 7 3 .2 1 $ 1 ,1 2 6 .7 9
5 $ 1 ,0 0 0 .0 0 $ 9 2 4 .1 8 $ 1 ,0 7 5 .8 2 $ 8 4 8 .3 7 $ 1 ,1 5 1 .6 3
6 $ 1 ,0 0 0 .0 0 $ 9 1 2 .8 9 $ 1 ,0 8 7 .1 1 $ 8 2 5 .7 9 $ 1 ,1 7 4 .2 1
7 $ 1 ,0 0 0 .0 0 $ 9 0 2 .6 3 $ 1 ,0 9 7 .3 7 $ 8 0 5 .2 6 $ 1 ,1 9 4 .7 4
8 $ 1 ,0 0 0 .0 0 $ 8 9 3 .3 0 $ 1 ,1 0 6 .7 0 $ 7 8 6 .6 0 $ 1 ,2 1 3 .4 0
9 $ 1 ,0 0 0 .0 0 $ 8 8 4 .8 2 $ 1 ,1 1 5 .1 8 $ 7 6 9 .6 4 $ 1 ,2 3 0 .3 6
1 0 $ 1 ,0 0 0 .0 0 $ 8 7 7 .1 1 $ 1 ,1 2 2 .8 9 $ 7 5 4 .2 2 $ 1 ,2 4 5 .7 8
1 1 $ 1 ,0 0 0 .0 0 $ 8 7 0 .1 0 $ 1 ,1 2 9 .9 0 $ 7 4 0 .2 0 $ 1 ,2 5 9 .8 0
1 2 $ 1 ,0 0 0 .0 0 $ 8 6 3 .7 3 $ 1 ,1 3 6 .2 7 $ 7 2 7 .4 5 $ 1 ,2 7 2 .5 5
1 3 $ 1 ,0 0 0 .0 0 $ 8 5 7 .9 3 $ 1 ,1 4 2 .0 7 $ 7 1 5 .8 7 $ 1 ,2 8 4 .1 3
1 4 $ 1 ,0 0 0 .0 0 $ 8 5 2 .6 7 $ 1 ,1 4 7 .3 3 $ 7 0 5 .3 3 $ 1 ,2 9 4 .6 7
1 5 $ 1 ,0 0 0 .0 0 $ 8 4 7 .8 8 $ 1 ,1 5 2 .1 2 $ 6 9 5 .7 6 $ 1 ,3 0 4 .2 4
1 6 $ 1 ,0 0 0 .0 0 $ 8 4 3 .5 3 $ 1 ,1 5 6 .4 7 $ 6 8 7 .0 5 $ 1 ,3 1 2 .9 5
1 7 $ 1 ,0 0 0 .0 0 $ 8 3 9 .5 7 $ 1 ,1 6 0 .4 3 $ 6 7 9 .1 4 $ 1 ,3 2 0 .8 6
1 8 $ 1 ,0 0 0 .0 0 $ 8 3 5 .9 7 $ 1 ,1 6 4 .0 3 $ 6 7 1 .9 4 $ 1 ,3 2 8 .0 6
1 9 $ 1 ,0 0 0 .0 0 $ 8 3 2 .7 0 $ 1 ,1 6 7 .3 0 $ 6 6 5 .4 0 $ 1 ,3 3 4 .6 0
2 0 $ 1 ,0 0 0 .0 0 $ 8 2 9 .7 3 $ 1 ,1 7 0 .2 7 $ 6 5 9 .4 6 $ 1 ,3 4 0 .5 4
2 1 $ 1 ,0 0 0 .0 0 $ 8 2 7 .0 3 $ 1 ,1 7 2 .9 7 $ 6 5 4 .0 5 $ 1 ,3 4 5 .9 5
2 2 $ 1 ,0 0 0 .0 0 $ 8 2 4 .5 7 $ 1 ,1 7 5 .4 3 $ 6 4 9 .1 4 $ 1 ,3 5 0 .8 6
2 3 $ 1 ,0 0 0 .0 0 $ 8 2 2 .3 4 $ 1 ,1 7 7 .6 6 $ 6 4 4 .6 7 $ 1 ,3 5 5 .3 3
2 4 $ 1 ,0 0 0 .0 0 $ 8 2 0 .3 1 $ 1 ,1 7 9 .6 9 $ 6 4 0 .6 1 $ 1 ,3 5 9 .3 9
2 5 $ 1 ,0 0 0 .0 0 $ 8 1 8 .4 6 $ 1 ,1 8 1 .5 4 $ 6 3 6 .9 2 $ 1 ,3 6 3 .0 8
2 6 $ 1 ,0 0 0 .0 0 $ 8 1 6 .7 8 $ 1 ,1 8 3 .2 2 $ 6 3 3 .5 6 $ 1 ,3 6 6 .4 4
2 7 $ 1 ,0 0 0 .0 0 $ 8 1 5 .2 6 $ 1 ,1 8 4 .7 4 $ 6 3 0 .5 1 $ 1 ,3 6 9 .4 9
2 8 $ 1 ,0 0 0 .0 0 $ 8 1 3 .8 7 $ 1 ,1 8 6 .1 3 $ 6 2 7 .7 4 $ 1 ,3 7 2 .2 6
2 9 $ 1 ,0 0 0 .0 0 $ 8 1 2 .6 1 $ 1 ,1 8 7 .3 9 $ 6 2 5 .2 2 $ 1 ,3 7 4 .7 8
3 0 $ 1 ,0 0 0 .0 0 $ 8 1 1 .4 6 $ 1 ,1 8 8 .5 4 $ 6 2 2 .9 2 $ 1 ,3 7 7 .0 8
T e r m t o M a t u r i ty a n d B o n d P r i c e
0
20 0
40 0
60 0
80 0
1 0 0 0
1 2 0 0
1 4 0 0
1 6 0 0
1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 15 16 1 7 1 8 1 9 2 0 21 22 2 3 2 4 2 5 2 6 27 28
Y e a r s L e f t t o M a t u r i t y
BondP
rice(
$)
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Duration
An alternative measure of bond price sensitivity is the
bonds duration.
Duration measures the life of the bond on a present value
basis.
Duration can also be thought of as the average time to
receipt of the bonds cashflows.
The longer the bonds duration, the greater is its sensitivity
to interest rate changes.
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0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
Duration and Coupon Rates
A bonds duration is affected by the size of the coupon rate
offered by the bond.
The duration of a zero coupon bond is equal to the bonds
term to maturity. Therefore, the longest durations are
found in stripped bonds or zero coupon bonds. These are
bonds with the greatest interest rate elasticity.
The higher the coupon rate, the shorter the bonds duration.
Hence the greater the coupon rate, the shorter the duration,and the lower the interest rate elasticity of the bond price.
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0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
Duration
The numerator of the duration formula represents the present value of
future payments, weighted by the time interval until the payments
occur. The longer the intervals until payments are made, the larger will
be the numerator, and the larger will be the duration. The denominatorrepresents the discounted future cash flows resulting from the bond,
which is the bonds present value.
maturitytoyieldsbondthei
providedarepaymentsthewhichattimethet
bondthebygeneratedpaymentprincipalorcoupontheCwhere
i
C
i
tC
DUR
t
n
tt
t
n
tt
t
'
:
)1(
)1(
)(
1
1
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0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
Duration Example
As an example, the duration of a bond with $1,000 par value and a 7
percent coupon rate, three years remaining to maturity, and a 9 percent
yield to maturity is:
years
DUR
80.2)09.1(
1070$
)09.1(
70$
)09.1(
70$
)09.1(
)3(1070$
)09.1(
)2(70$
)09.1(
70$
321
321
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0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
Duration Example ...
As an example, the duration of a zero-coupon bond with $1,000 par
value and three years remaining to maturity, and a 9 percent yield to
maturity is:
years
DUR
0.3
)09.1(
1000$
)09.1(
)3(1000$
3
3
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0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
23
Duration is a handy tool because it can encapsule interest rate exposure in a
single number.
rather than focus on the formula...think of the duration calculation as a
process... semi-annual duration calculations simply call for halving the annual
coupon payments and discounting every 6 months.
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0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
24
Duration Rules-of-Thumb duration of zero-coupon bond (strip bond) = the term left until
maturity.
duration of a consol bond (ie. a perpetual bond) = 1 + (1/R)
where: R = required yield to maturity duration of an FRN (floating rate note) = 1/2 year
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0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
25
Other Duration Rules-of-ThumbDuration and Maturity
duration increases with maturity of a fixed-income asset, but at a
decreasing rate.
Duration and Yield
duration decreases as yield increases.
Duration and Coupon Interest
the higher the coupon or promised interest payment on the security, the
lower its duration.
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0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
26
Economic Meaning of Duration duration is a direct measure of the interest rate sensitivity or elasticity
of an asset or liability. (ie. what impact will a change in YTM have on
the price of the particular fixed-income security?)
interest rate sensitivity is equal to:
dP = - D [ dR/(1+R)]
P
Where: P = Price of bond
C = Coupon (annual)
R = YTM
N = Number of periods
F = Face value of bond
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0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
27
Interest Rate Elasticity the percent change in the bonds price
caused by a given change in interest rates
(change in YTM)
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0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
28
Economic Meaning of Duration interest rate sensitivity is equal to:
dP = - D [ dR/(1+R)]
P
dP/P = change in bond price
[ dR/(1+R)] = change in interest rate
Obviously, the relationship is an inverse function of Duration (D)
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0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
29
Example of Calculation of Interest
Rate Sensitivity given:
n = 6 years (Eurobond ... annual coupon payments)
8 percent coupon
8 YTM
if yields are expected to rise by 10%, what impact will that have on the price of
the bond?
the first step is to calculate the duration of the bond.
If there were no coupon payments the duration would be = 6.
since there are coupon payments the duration must be less than 6 years.
D = 4.993 years the second step is to calculate the % change in price for the bond.
= -(4.993)(.1/1.08) = - 0.4623 = - 46.23%
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0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
30
Immunization fully protecting or hedging an FIs equity holders against
interest rate risk.
elimination of interest rate risk by matching the duration of
both assets and liabilities. (not their average lives or final
maturities).
when immunized:
the gains or losses on reinvestment income that result from an
interest rate change are exactly offset by losses or gains from thebond proceeds on sale of the bond.
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0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
459K. Hartviksen
Example of Bond Price
The Canada 10.25 1 Feb 04 is quoted at 123.95 yielding 5.27%. Thismeans that for a $1,000 par value bond, these bonds are trading apremium price of $1,239.50
The figure represents bond prices as of June 17, 1998.
This bond has 5 years and 8 months (approximately) until maturity =5+(8/12) = 5.7 years
Bond Price = $102.50(PVIFAn=5.7 ,r=5.27%) + $1,000 / (1.0527)5.7
= $102.50(PVIFAr=5.27%%, n= 5.7) + $746.21= $102.50(4.8156653) + $746.21
= $493.61 + $743.42 = $1,237.03
Can you explain why the quoted price might differ from your answer?
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0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
Example of a Duration Calculation
Example
Assume a 10% coupon bond with three years left to maturity and a required return of 8%.
Coupon Rate = 10.00%
Required Return = 8.00%
Time Cashflow PVIF Present Value Weight
Time
Weighted
CFs
0
0.5 50 0.96225 $48.11 4.55% 0.022767679
1 50 0.925926 $46.30 4.38% 0.043816419
1.5 50 0.890973 $44.55 4.22% 0.063243554
2 50 0.857339 $42.87 4.06% 0.0811415172.5 50 0.824975 $41.25 3.90% 0.097598077
3 1050 0.793832 $833.52 78.89% 2.36662759
Bond Price = $1,056.60 100.00% 2.675194837 =Duration
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459K. Hartviksen
Sensitivity Analysis of Bonds
B o n d
C o u p o n
R a t e
T i m e to
m a t u r i ty
Y i e l d to
M a tu ri ty P V IF A P V IF B o n d P ric e
B o n d X 7 .0 % 1 3 6 .0 % 8 .8 5 2 6 8 3 0 .4 6 8 8 3 9 $ 1 ,0 8 8 .5 3
7 .0 % 1 2 6 .0 % 8 .3 8 3 8 4 4 0 .4 9 6 9 6 9 $ 1 ,0 8 3 .8 4
7 .0 % 1 0 6 .0 % 7 .3 6 0 0 8 7 0 .5 5 8 3 9 5 $ 1 ,0 7 3 .6 0
7 .0 % 5 6 .0 % 4 .2 1 2 3 6 4 0 .7 4 7 2 5 8 $ 1 ,0 4 2 .1 2
7 .0 % 1 6 .0 % 0 .9 4 3 3 9 6 0 .9 4 3 3 9 6 $ 1 ,0 0 9 .4 3
B o n d Y 5 .0 % 1 3 8 .0 % 7 .9 0 3 7 7 6 0 .3 6 7 6 9 8 $ 7 6 2 .8 9
5 .0 % 1 2 8 .0 % 7 .5 3 6 0 7 8 0 .3 9 7 1 1 4 $ 7 7 3 .9 2
5 .0 % 1 0 8 .0 % 6 .7 1 0 0 8 1 0 .4 6 3 1 9 3 $ 7 9 8 .7 0
5 .0 % 5 8 .0 % 3 .9 9 2 7 1 0 .6 8 0 5 8 3 $ 8 8 0 .2 2
5 .0 % 1 8 .0 % 0 .9 2 5 9 2 6 0 .9 2 5 9 2 6 $ 9 7 2 .2 2
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0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
459K. Hartviksen
Prices over time
Time To M Bond X Bond Y
13 $1,088.53 $762.89
12 $1,083.84 $773.92
10 $1,073.60 $798.70
5 $1,042.12 $880.221 $1,009.43 $972.22
0 1000 1000
Bond Prices over Time
0
500
1000
1500
13 12 10 5 1 0
Years Left Until Maturity
Bond
Pric
e
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0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
Duration of a Portfolio
Bond portfolio mangers commonly attempt to immunize
their portfolio, or insulate their portfolio from the effects of
interest rate movements.
For example, a life insurance company knows that they need $100million 30 years from now cover actuarially-determined claims
against a group of life insurance policies just no sold to a group of 30
year olds.
The insurance company has invested the premiums into 30-year
government bonds. Therefore there is no default risk to worry about.The company expects that if the realized rate of return on this bond
portfolio equals the yield-to-maturity of the bond portfolio, there
wont be a problem growing that portfolio to $100 million. The
problem is, that the coupon interest payments must be reinvested and
there is a chance that rates will fall over the life of the portfolio.
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0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
Duration of a Portfolio ...
The life insurance company example illustrates a keep risk
in fixed-income portfolio management - interest rate risk.
The portfolio manager, before-the-fact calculates the bond
portfolios yield-to-maturity. This is an ex ante
calculation. As such, a nave assumption assumption is
made that the coupon interest received each year is
reinvested at the yield-to-maturity for the remaining years
until the bond matures. Over time, however, interest rates will vary and
reinvestment opportunities will vary from that which was
forecast.
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0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
Duration of a Portfolio ...
The insurance company will want to IMMUNIZE their
portfolio from this reinvestment risk.
The simplest way to do this is to convert the entire bond
portfolio to zero-coupon/stripped bonds. Then the ex ante
yield-to-maturity will equal ex post(realized) rate of
return. (ie. the ex ante YTM is locked in since there are
no intermediate cashflows the require reinvestment).
If the bond portfolio manager matches the duration of thebond portfolio with the expected time when they will
require the $100 m, then interest rate risk will be
eliminated.
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