# Duration

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```					Duration  While you receive the whole present value (price) of a zero at the date of its maturity, with a coupon bond you receive part of the present value before maturity.  The duration of a bond measures the average lenght of time you have to wait until receiving its whole present value.  Assume a flat term structure. The price of a bond paying x1 , x 2 , , x n at t  1,2 n is: [l] P 

x1 x2 xn   1  R 1  R 2 1  R n





 Macaulay (NBER, 1938) defined the bond’s duration as:

x 1  R [2] D  1  1 P

1

x 1  R  2 2 P

2

x 1  R n  n P

n

i.e. as the weighted average of the waiting times until cash payments are made. The weights are given by the fractions of the total present value to be received at t  1,2 n . For a zero: D=n; for a coupon bond D<n; for a perpetuity it can be shown that: D  1  R / R .  The duration of a bond is important because it represents a measure of the sensitivity of the bond’s price to interest rate changes. By taking the derivative of [1] with respect to R, we obtain:

P 2 3 4  n 1   x1 1  R  2 x 2 1  R  3x 3 1  R  x n 1  R R
P 1 1 2 3 n  1x1 1  R  2x 2 1  R  3x 3 1  R  x n 1  R R 1 R




approximately:

and, with [2], we get: [3]

P 1   D P or R 1 R P D  R   DR
P 1 R

which shows that an increase of 1 basis point in R (say from 3.00% to 3.01%) will cause a drop in the bond price of D basis points (0.10% if D=10).
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 For values of R which are not very small, instead of D, it is better to use the modified duration: D/1+R.  It is possible to show that:

D  0 and, for bonds with fixed coupons c, R D D  0 . Moreover:  0 when P  1 while, for P  1, D has that: c n
a max.

 A variable coupon rate bond is equivalent to a 1 period zero. Hence its duration is equal to 1 period (1year, 6 months or the shorter period until the first coupon’s maturation date. If the bond pays a constant spread s in addition to the variable coupon rate, its duration is:

D s 

1  c1 v1  s tvt  1  c1 v1  s vt 
2 2 n

n

It appears that D(s)=1 if s=0. Moreover: dD(s)/ds>0. Convexity  The first-order derivative of eq. [3] gives a good approximation for very small changes in R. The relation between P and R in fact is not linear but convex: log P

R
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 The tangent to the curve, whose slope is measured by the modified duration:

 log P D  , lies below the curve. Therefore, when R 1 R R
the rises and

changes, the modified duration underestimates overestimates the falls in the bond’s price.

 A better approximation can be obtained by the second-order Taylor’s series expansion:

P 12P P  R   R 2 R 2 R 2
Hence, by defining the convexity of a bond as:

C
we have:

 P1  2 R P
2

i 1

 xi

n

ii  1

1  R
P

i 2



i 1

 xi ii  11  R P

n

i  2

P D 1 2  R  C R  P 1 R 2

 Consider 2 bonds having the same duration. When R changes, the price of the bond with higher convexity increases more or decreases less than the price of the bond with lower convexity. Hence, it is better to have in portfolio the more convex bond. But the conclusion is correct for parallel shifts of the term structure, not in general.  It is possible to show that: C 

1

1  R

2

D  D

2

  2 where  2 is



a measure of the dispersion of the maturing dates of the cash-flows around their average given by D.

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Immunization  When R changes, the effects on capital and income risks have opposite signs. If the absolute values of the 2 effects are equal, the interest rate risk is nullified: it has got a complete immunization.  Assume a flat term structure and consider a bond paying x1 , x2 ,, xn . Its present value is given by [1]. If all the bond’s cash flows are invested at R, after m periods (m<n) the accumulation (future value) of the investment will be: [4]

M m  R  1  R P R
m

Example

M 2  R  x1 1  R  x 2  x 3 1  R  1  R x1 1  R
2

 1  R P R
2





 3 2

 x n 1  R
n

n2

1

 x 2 1  R

2

 x n 1  R





 The investment in the bond is made at t=0. Suppose that at t=t1, with 0  t1  m , R has a change but remains unchanged thereafter. Then, if R goes up, 1  R the future value [5]
m

P R i.e. the capital effect decreases. If they exactly offset one another,
M m  R

i.e. the income (reinvestment) effect increases while will be independent of the interest rate changes:

M m  R m1 m P R   m1  R  P R  1  R  0 R R

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Hence, taking account of [3]: P R / R  1  R easily seen that:

1

D R P R , it is

M m  R 0 R

if

m = D(R)

which is the conclusion of the Fisher and Weil “classical” immunization: an m period investment is immunized if it is made in a bond with a duration equal to m.  If R changes again after t=t1, it is necessary to adjust the (portfolio) duration so as to keep it always equal to the time remaining until t=n.  The assumption of a flat term structure is not necessary. The condition m=D holds for any term structure provided that it can shift only in a parallel way (only additive shifts assumption).See Appendix 2. Immunization of a portfolio of A and L’s: non parallel shifts  Classical immunization(Fisher-Weil and Redington):

A 1 1 2  D A R  C A  R A 1 R 2 L 1 1 2  DL R  C L  R L 1 R 2
A  L and DA  DL are not sufficient for A  L . It is also necessary C A  CL .  With parallel shifts: C A  C L  A  L independently of the sign of R .

17

 Example. Initial situation: flat term structure with
ASSETS * 1 year Zero coupon (41.1351 Mln, Price= 95.2381) * 9 years Zero coupon (60.7753 Mln, Price = 64.4609 Total A Mln 39.1763 Mln 78.3526 Total L

R  5% .
Mln 78.3526

LIABILITIES * 5 years Zero coupon Mln 39.1763 (100 Mln, Price = 78.3526)

Mln 78.3526

 At

t    0, the term structure curve has an upward parallel shift: R  55% . .
ASSETS LIABILITIES * 5 years Zero coupon Mln 38.9906 (100 Mln, Price = 76.5134) Mln 76.5134

* 1 year Zero coupon (41.1351 Mln, Price= 94.7867) *9 years Zero coupon (60.7753 Mln, Price = 61.7629 Total Mln 37.5366

A  A

Mln 76.5272 Total

L  L

Mln 76.5134

 Note that

C A  CL produces a gain both in the case of R  0 and in that of R  0 when the shifts are parallel. If only parallel shifts
were possible, there would be a violation of the non-arbitrage principle. In fact one could construct a portfolio with 2 zeros(1year and 9 years and finance it with a zero of 5 years. The duration of assets and liabilities is the same but C A  CL since the assets’ cash flows are more dispersed than the liabilities’ ones.

AL = 13,800 euro

18

 If the shift increases the slope of the term structure curve:

R1  5.25%,R5  550%,R9  5.75% , the results are: .
ASSETS LIABILITIES * 5 years Zero coupon Mln 39.0832 (100 Mln, Price = 76.5134)

* 1 year Zero coupon (41.1351 Mln, Price= 95.0119) * 9 years Zero coupon (60.7753 Mln, Price = 60.4612 Total Mln 36.7455

Mln 76.5134

A  A

Mln 75.8287 Total

L  L

Mln 76.5134

AL =  684,700 euro
 For

similar: AL =  761,100 euro.

R1  4.25%,R5  4.50%,R9  4.75% ,

a

slope

increasing

downward

shift:

the results are

 The cause of the big losses of the example, when the shifts are slope increasing, is the large value of the convexity gap CG.  Minimum risk immunization: To minimize the risk of losses, choose a portfolio with DA = DL and:
 CG  MinCG CG  0 .

19

Appendix 1:continuous capitalization Present value of 1 euro to be received at t: Future value of 1 euro at t:

v ( t )  1  Rt  1 t m( t )   1  Rt  v(t )

t

Let:

 ( t )  ln m( t )
and:

then:
t    ( u ) du 0

m( t )  e

t   ( u ) du 0

v(t )  e

.

 The present value (price) of a bond paying:

x1 , x2 ,xn

at:

t  1,2, n is given by:W ( t ) 
D( x) 
i 1 n

i 1

 xi v (i ) and its duration is:

n

 ixi v (i )
where
i 1

n

 xi v (i )

v( t )

represents the market term

structure of interest rates (not necessarily flat). Appendix 2: Fisher and Weil immunization theorem (any term structure; additive shifts)  Let ( t ) be the continuous time TSIR, L  0 an amount to be paid

at

T  0 and x  0 flows to be cashed at t  1,2, n. Let the
Wt ( x ) 
it 



present value of assets and liabilities be the same at time t:

 xi v (i )

n

 Lv (T )  Wt ( L)

Assume that at time t  t   1 (infinitesimal) there occurs an additive shift in ( t ) . Then the post-shift values:

Wt  ( x )  Wt  ( L)





if and only if:
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D( X )  T .

Wt ( x ) Proof: Let: Qt ( x , L)  1 Wt ( L) and let:  (t  , t )   (t )  Y  For any t up to t it is:
Qt ( x , L) 
 xi e
i    ( u ) du t T    ( u ) du t

for

t  t

1 n  i1 xi e L

T   ( u ) du i

Le

The post-shift value of Q will be a function of Y (and
 T    ( u )  Y  du i

t  ) like:
T   ( u ) du i

1 n 1 n Qt ( t , x , L , Y )  i1 xi e  i1xi e eY ( T  i ) L L with Q  1 for Y  0. Now, calculate the 1st and 2nd derivatives of Q
with respect to Y, to get:

1 n Q  (Y )  i1( T  i ) xi e L

T   ( u ) du i

eY ( T  i )

1 n Q  (Y )  i1( T  i ) 2 xi e eY ( T  i )  0 L Since Q( t )  1 for Y  0 and it is a convex function  Q (Y )  0, then for any Y  0 it is that Qt  0 if it has a min for Y  0 i.e. if and only if: Q (Y )  0 . This implies that:

T   ( u ) du i

21

i 1

 (T

n

 i ) xi e Le
n

i    ( u ) du t

T    ( u ) du t
n



i 1

 (T

n

 i ) xi v (i )

Lv (T )

0

T  xi v (i )
i.e.
i 1

 i 1 Lv ( T ) Lv ( T )

 ixi v (i )
and taking account of the budget

constraint:

22

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