1987 VALUATION
ACTUARY HANDBOOK
Appendix 1
T H E O R Y BEHIND M A C A U L A Y D U R A T I O N AND I L L U S T R A T I V E E X A M P L E S
Definition o f Duration
M a c a u l a y d u r a t i o n is the w e i g h t e d a v e r a g e of the t i m e to e a c h f u t u r e c a s h
flow.
The w e i g h t s are t h e p r e s e n t v a l u e of t h e c o r r e s p o n d i n g cash flows.
In
symbols,
Duration (D) =
P
=
~ tvtCFt/P , where
t=O
~ v t C F t.
t=0
Thus, d u r a t i o n p r o v i d e s us with a s e n s e f o r t h e a v e r a g e " l e n g t h " of an a s s e t
or l i a b i l i t y . N o t e t h a t in this d e f i n i t i o n , we a s s u m e a f l a t y i e l d c u r v e .
In a l a t e r
s e c t i o n , we will e x a m i n e t h e i m p l i c a t i o n s of a n o n l e v e l y i e l d c u r v e .
Example o f Duration
C o n s i d e r the f o l l o w i n g e x a m p l e :
o
Liability:
$100, 16 p e r c e n t , 6 . 9 - y e a r c o m p o u n d b u l l e t (to
p a y $Z78.46 a t m a t u r i t y ) .
A1-1
o
Asset:
$100,
16
percent,
30-year
mortgage
(to pay
$16.19 p e r a n n u m f o r 30 years).
o
D L = 6.9 = D A.
0
If i n t e r e s t r a t e s drop t o 14 p e r c e n t , m a r k e t v a l u e s (MV) c h a n g e
as f o l l o w s :
HV L = 2 7 8 . 4 6 / ( 1 . 1 4 )
HVA =
6"9 = 1 1 2 . 7 5 .
16.19 a~r~ll4 % = 113.37.
T h c cxccss of assct ovcr l i a b i l i t y value rcprcscnts a .5% gain.
O
If i n t e r e s t r a t e s i n c r e a s e to 18 p e r c e n t :
HL L = 2 7 8 . 4 6 / ( 1 . 1 8 )
NL A = 1 6 . 1 9
o
6"9 = 8 8 . 8 7 .
a3"-~ 18% = 8 9 . 3 2 .
Again, a s s e t s n o w exceed liabilities by .5%.
Thus, we a r e now well p r o t e c t e d a g a i n s t i n t e r e s t s h i f t s .
In f a c t , we
a p p e a r t o g a i n e i t h e r way. We will c o m e b a c k t o this in t h e n e x t s e c t i o n .
It is instructive to examine this example from the maturity date viewpoint:
AI-Z
Value of A s s e t s at M a t u r i t y of L i a b i l i t i e s , if
I n t e r e s t = i% t o M a t u r i t y
From cash flow reinvestment a
16%
14%
18%
$166.1Z
$155.49
$177.4Z
11Z.37
124.51
10Z.43
$Z78.49
$Z80.00
$Z79.85
F r o m l i q u i d a t i o n of a s s e t s b
[
]Total
a. 16.19
x s-~i
b.
x
16.19
x
az'-~i
(1 + i ) 0"9"
x
(1 + i ) 0 . 9 .
In e a c h c a s e , t h e t o t a l v a l u e of t h e
r e q u i r e d p a y m e n t of $Z78.46 at m a t u r i t y .
s h o w n m o v e in d i f f e r e n t d i r e c t i o n s .
on l i q u i d a t i o n
reinvestment
of a s s e t s
(versus
One
is r e a s o n a b l y
N o t e , though, t h a t
c l o s e to t h e
the two pieces
If i n t e r e s t r a t e s fall to 14 p e r c e n t , we gain
the
16 p e r c e n t
of t h e annual c a s h flow.
o p p o s i t e holds.
assets
can thus b e g i n
In t h e
base
case) b u t lose on t h e
18 p e r c e n t
to s e e h o w this
scenario, just
works.
For
a better
understanding, consider the following theoretical development.
Duration: T h e o r e t i c a l D e v e l o p m e n t
Classical Imm,mication Theory
A t = c a s h f l o w of a s s e t s at t i m e t.
B t = c a s h f l o w of l i a b i l i t i e s at t i m e t.
Goal
following:
(I)
of
classical
immunization
theory
~
t Atvt _~ ~
t Btvt
r e g a r d l e s s of t h e i n t e r e s t r a t e .
AI-3
is
to
achieve
the
the
To s i m p l i f y t h e n o t a t i o n , let's d e f i n e :
pA = ~t
Atvt, pL = t~ Btvt
To d e t e r m i n e t h e c o n d i t i o n s t h a t n e e d to hold to e n s u r e t h a t
e q u a t i o n 1 is t r u e , we d e f i n e
(Z) f(i) = pA _ pL.
If f(i)=O, w h a t we w a n t is f ( i ' ) ~ f(i) for all i n t e r e s t r a t e s i', or in
o t h e r words we w a u t f(i) to be a m i n i m u m p o i n t of t h e function, f.
This will b e t h e c a s e if t h e f i r s t d e r i v a t i v e of f(i) equals 0 and t h e
second derivative ~
0, so t h a t our c o n d i t i o n s b e c o m e
(3)
Z tvtAt = 7. tvtBt
t
t
(4)
t
The
(first derivative
t2vtAt ~ Z
t2vtBt" (second
t
following i n d i c a t o r s
were
then
= 0).
derivative~
developed
O)
as tools for
u t i l i z i n g this t h e o r y :
DA
1 = asset duration =
L
DI=
liability duration =
7"
tvtAt/PA.
t
Zt
tvtBt/PL.
A
D Z = a s s e t s p r e a d (or c o n v e x i t y ) of p a y m e n t s =
D2= L l i a b i l i t y s p r e a d of p a y m e n t s =
A1-4
Zt
7t t2vtAt/PA"
t2vtBt/PL "
Assuming that:
nA
= p L at t h e i n i t i a l i n t e r e s t r a t e i,
A
T h e n D 1 = D L1 and D AE = D L
2 a r e n e c e s s a r y and s u f f i c i e n t to
ensure that
p A :~ p L
at any i n t e r e s t rate.
J
N o t e t h a t t h e n o t a t i o n f o r M a c a u l a y d u r a t i o n m a y be D or D 1" T h e l a t t e r
is o f t e n u s e d t o d i s t i n g u i s h t h e f i r s t m o m e n t (D 1) f r o m t h e s e c o n d m o m e n t (Dz).
In t h e e x a m p l e p r e v i o u s l y g i v e n , D AZ = 8Z.Z a n d D LZ = 47. 6.
and
A
D z >
Thus, D A1 = D L1
L
DZ, w h i c h a s s u r e s us t h a t t h e p r e s e n t v a l u e o f t h e a s s e t c a s h flo w s
will e q u a l or e x c e e d t h e p r e s e n t v a l u e o f t h e l i a b i l i t y c a s h f l o w s a t all i n t e r e s t
rates.
As we did s e e , this i n e q u a l i t y h e l d f o r v a r i o u s i n t e r e s t r a t e s .
T h e r e q u i r e m e n t i n v o l v i n g t h e D2's i m p l i e s t h a t
t h e a b s o l u t e " l e n g t h " or
s p r e a d o f t h e a s s e t s is s o m e h o w " l o n g e r " t h a n t h a t of t h e l i a b i l i t i e s .
In t h e
e x a m p l e g i v e n , we saw t h a t t hi s p e r m i t t e d l o s s e s u p o n l i q u i d a t i o n t o b e o f f s e t b y
gains upon c a s h f l ow r e i n v e s t m e n t , or v i c e v e r s a .
r e q u i r e m e n t is n e c e s s a r y .
Thus, w e c a n s e e t h a t t h e D z
In f a c t , if we h a d s w i t c h e d t h e a s s e t and l i a b i l i t y in
this e x a m p l e , t h e n e v e n t h o u g h d u r a t i o n s m a t c h e d , we would h a v e (small) losses
u p on i n t e r e s t r a t e m o v e s .
F r o m t h e c a l c u l u s , we c a n s e e t h a t f(i) w o u l d be a t a
maximum of i rather than a minimum.
AI-5
The phenomenon w h e r e one can gain if i n t e r e s t r a t e s move a n y w h e r e was
n o t e d by I. T. Vanderhoof, 1 who suggests t h a t in real life, the s t r u c t u r e of
i n t e r e s t r a t e s (that is, yield curves) p r e v e n t such all-win situations.
D u r a t i o n as Measure of Change in P r i c e per C h ~ E e in I n t e r e s t
Now consider an a l t e r n a t e d e m o n s t r a t i o n . Assume
Ct
= cash flow at t i m e t.
Po
= purchase price at time o
Z
t Ct/(l+i)t"
to
We can obtain the marginal change in price due
a change in i n t e r e s t r a t e
by taking the d e r i v a t i v e of Po with r e s p e c t to (i); t h a t is,
dPo/di = - Z t Ct/(l + i)t ÷ 1
-I
~
tvtCt
(I+i)
or
(-Id+ii ) Z tvtCt
cLP° =
Dividing both sides by Po yields
dPo
di
=
Po
tCt vt
(l+i)
- ~'. tvtC t
di
=
x
X
(]+i)
Po
Po
I. T. Vanderhoof. "Immunization with (Almost) No M~thematics."
A c t u a r y 16, no. 4 (198Z): 1, 4, 7.
A1-6
The
ol"
% c h a n g e in p r i c e = % c h a n g e in (1 + i) x (-duration).
One i m p l i c a t i o n of this r e s u l t is t h a t if the d u r a t i o n s of t h e a s s e t s
and l i a b i l i t i e s a r e i d e n t i c a l , t h e n a c h a n g e in i n t e r e s t r a t e s will p r o d u c e
comparable
c h a n g e s in m a r k e t v a l u e s .
In o t h e r
words, i m m u n i z a t i o n
is
achieved.
It is i m p o r t a n t to n o t e , though, t h a t e x a c t m a t c h i n g is e n s u r e d only
f o r i n f i n i t e s i m a l c h a n g e s in i n t e r e s t , t h a t is c h a n g e s b y di.
interest rates
In f a c t , o n c e
c h a n g e b y di, t h e n e w d u r a t i o n b a s e d on this n e w i n t e r e s t
r a t e will likely no l o n g e r m a t c h ; h o w e v e r , as w e s a w in t h e }ast s e c t i o n , if
I~A~
I~Lat t h e original i n t e r e s t r a t e ,
t h e a s s e t m a r k e t v a l u e will still
e x c e e d t h e l i a b i l i t y v a l u e w i t h f u r t h e r c h a n g e s in i n t e r e s t r a t e s .
F o r e x a m p l e , a s s u m e $100 is i n v e s t e d in a 3 0 - y e a r m o r t g a g e y i e l d i n g
16 p e r c e n t ; t h e f o l l o w i n g t a b l e c o m p a r e s t h e p e r c e n t a g e c h a n g e in m a r k e t
v a l u e p r o d u c e d b y the d u r a t i o n f o r m u l a j u s t g i v e n to t h e a c t u a l p e r c e n t a g e
c h a n g e in p r i c e u n d e r v a r i o u s i n t e r e s t r a t e s :
Duration Formula
Interest
Rate
16%
17
18
20
(i)
(z)
Duration
at 16%
%Change
in (1 + i)
f r o m 1.16
6.90
6.90
6.90
6.90
0.00%
0.86
1.7Z
3.45
A c t u a l C h ~ n ~ e in Market V a l u e
(3)
-(1) x (Z)
0.00%
-5.93
-11.87
-23.81
A1-7
(4)
(s)
(6)
Market
Value
% Change
from
16% Value
R a t i o of
(3) t o (5)
$i00.00
94.38
89.3Z
80.61
0.00%
-5.6Z
-10.68
-19.39
m
105.5%
III.I
IZZ.8
Thus, as i n t e r e s t
rates
move significantly
from
the base rate, duration
b e c o m e s less u s e f u l as a m e a s u r e of p r i c e c h a n g e .
Other Properties of Duratiaa
It
is
often
liabilities.
necessary
to
combine
duration
for
several
assets
and/or
F o r t u n a t e l y , d u r a t i o n has a s i m p l e a g g r e g a t i o n p r o p e r t y , d e v e l o p e d
here:
1
L e t A t and A t d e n o t e c a s h f l o w s for t w o s e p a r a t e a s s e t s , and D and D 1
t
denote their respective durations.
The d u r a t i o n for t h e c o m b i n e d a s s e t ,
D C, e q u a l s
,:,c=L,,%
+
+
,¢
= (~vtAt +~tvtAtl) / (p + pl)
= (DP+DIp I) / (P +Pl)
D - p
p + pl
=
+ D1
p_____~l.
p + pl
Thus, if t h e d u r a t i o n s and p r e s e n t v a l u e s of t h e s e p a r a t e a s s e t s a r e known,
it is e a s y to c o m p u t e t h e d u r a t i o n of t h e c o m b i n e d a s s e t .
relationship
is u s e f u l for d e r i v i n g
additional
properties
Furthermore,
this
of i m m u n i z a t i o n
and
duration.
For
example,
it is u s u a l l y t h e
l i a b i l i t i e s and t h e a s s o c i a t e d a s s e t s ,
assumed
in
the
classical
case
that
when w o r k i n g
that pA exceeds pL
immunization
requirements.
with statutory
R e c a l l t h a t e q u a l i t y is
We
can
t r a d i t i o n a l r e q u i r e m e n t s in one of t w o w a y s , d e p e n d i n g on our goals.
Al-8
modify
the
A s s u m e f i r s t t h a t we w a n t t h e dollar a m o u n t of " e c o n o m i c surplus" (S = p A
_ pL) to b e a t l e a s t as g r e a t in all o t h e r i n t e r e s t s c e n a r i o s .
Stated differently,
w e w a n t to i m m u n i z e t h e dollar a m o u n t of surplus.
Relative
to t h e
calculus
development
used
earlier,
we still
want
the
f u n c t i o n f(i) = p A _ p L to be at a m a x i m u m at i, so t h a t we w a n t if(i) = 0, and
f " ( i ~ 0. T h e r e f o r e ,
f'(i) =
~tvt+IAt + ~.tvt+IBt = 0
ltvtA t
°
ltvtB t
pADA= pLDL or
(I)
DA - pLDL/pA
and
Zt(t+l)vt+2Bt]~
f"(i) =[Zt(t+l)v t+2 At
~t2vtAt
+ ~tvtAt>[t2vtBt
+Z
0
tvt8 t ]
pAD~+ pADA ~pLD~ + pLDL
pADA+ PA(pLDL/pA) ~pLD~ + pLDL
pADA>pLD~, or
(z)
(all a s s u m i n g c o n d i t i o n 1 holds).
Thus, if p A ~ p L
and c o n d i t i o n s 1 and Z hold, t h e v a l u e of surplus, or f(i),
will b e at l e a s t as g r e a t at o t h e r i n t e r e s t r a t e s .
s p e c i a l c a s e of this m o r e g e n e r a l c a s e .
AI-9
N o t e t h a t p A = p L is j u s t a
Some m a y i n s t e a d wish to i m m u n i z e the surplus - to - liability ratio (S/P L)
so t h a t it will be at least as g r e a t at o t h e r i n t e r e s t rates. It turns out t h a t this is
a c h i e v e d by the original conditions, D A = D L and D ~ > D L, when p A > p L .
To see this, define pS = pA _ p L and consider t h e following:
O
We can c a r v e out r e p r e s e n t a t i v e assets whose p r e s e n t value9 pAL, equals
pL at the initial i n t e r e s t r a t e .
By r e p r e s e n t a t i v e , I
mean that
D AL = D A = D L and
DzAL =
DzA
>
DzL.
Let pS equal the present value of the
Let pS equal the p r e s e n t value of the remaining assets.
A L
P1L '~ and P1S d e n o t e the p r e s e n t values at a d i f f e r e n t i n t e r e s t r a t e .
O
Let P
O
From the duration conditions~ we know t h a t P
This implies t h a t
:
S
Pl
S
P1
p--AL --
--L
Pl
AI-IO
AL
1 >P
"
0
S _AL
Since we " c a r v e d " o u t r e p r e s e n t a t i v e a s s e t s , P1/~, 1
= pS/pAL
= pS/pL
for any i n t e r e s t r a t e , so t h a t
L >
PlS/P 1 --
pS/pL.
Shortcomings of Duration
T h e r e a r e a f e w p r a c t i c a l and t h e o r e t i c a l d i f f i c u l t i e s in a c t u a l l y using
duration.
One p r o b l e m
is t h a t
we c a n be m a t c h e d (D A = D L) t o d a y , b u t
m i s m a t c h e d t o m o r r o w , e v e n w i t h o u t a n y c h a n g e s in i n t e r e s t r a t e s .
Consider the
following example:
i
Liability:
$100, 16 p e r c e n t , 6.9 - y e a r c o m p o u n d b u l l e t (to p a y
$Z78.46 at maturity).
o
Asset:
$100, 16 p e r c e n t , 30 - y e a r m o r t g a g e (to p a y $16.19 p e r
annum).
30
O
A t issue, D L = 6.9 = D A ( =
[
tvt/~
t=l
DL
Z = 47.6.
o
A
D Z = 8Z.Z.
One y e a r l a t e r ,
D L = 5.9.
A1-11
30
t=l
vt).
DA
d e p e n d s to s o m e e x t e n t on how t h e f i r s t p a y m e n t
was i n v e s t e d . If i n v e s t e d in a c a s h e q u i v a l e n t
instrument,
oA=
iz9 i [zgl
+
[txv
/
1,]Ev=
t=l
6.04.
t=l
If t h e f i r s t p a y m e n t is r e i n v e s t e d in a Z g - y e a r m o r t g a g e ,
DA
Z9
Z9
t=l
t=l
% t x v / ~ v t -- 6.8s.
N o t e e s p e c i a l l y t h a t D A m a y c h a n g e only s l i g h t l y , w h e r e as D L c h a n g e d
rather significantly.
W h a t this i m p l i e s is t h a t if d u r a t i o n s a r e m a t c h e d t o d a y ,
a n d i n t e r e s t r a t e s c h a n g e t o d a y , we should be in g o o d shape.
H o w e v e r , if one
y e a r passes and t h e n i n t e r e s t r a t e s c h a n g e , we a r e no l o n g e r p r o t e c t e d .
The
s o l u t i o n to this p r o b l e m is a c t i v e p o r t f o l i o m a n a g e m e n t .
A n o t h e r i n t e r e s t i n g p r o b l e m is t h a t the v a l u e s of D 1 and D Z u s u a l l y v a r y
with interest rates.
Thus, if i n t e r e s t r a t e s s h i f t , d u r a t i o n s m a y no l o n g e r m a t c h ,
w h i c h m a y or m a y n o t be a p r o b l e m .
A
L
A
d e t e r m i n e d D 1 = D1 a n d D _ >
If t h e s h i f t o c c u r s i m m e d i a t e l y a f t e r we
L
D Z a t t h e o r i g i n a l i, t h e n we a r e still p r o t e c t e d
against further interest rate changes that may occur immediately.
Example:
O
At time 0:
AI-IZ
Liability:
$100, 16 p e r c e n t , 6.9 - y e a r c o m p o u n d b u l l e t (to p a y
$Z78.46 at m a t u r i t y ) .
Asset:
$100, 16 p e r c e n t , 30 - y e a r m o r t g a g e (to p a y $16.19 per
annum).
D L = 6.9 = DA; D L = 47.6; D A = 8Z.Z.
Z
Z
O
A t t i m e 0, a s s u m e i n t e r e s t r a t e s s h i f t to 18 p e r c e n t ; e v a l u a t e a t t i m e = 0:
p L = 278.46 x v 6.9 = 88.87.
18%
p A = 16.19 x a 3 - ~ ! 8 %
= 89.3Z.
D L = 6.9 y e a r s .
30
t
D A= t~=ltv18%/PIA8% = 6.3.
N o t e t h a t d u r a t i o n s a n d m a r k e t v a l u e s no l o n g e r m a t c h .
This is n o t a
p r o b l e m , s i n c e a t 16 p e r c e n t we h a v e s e e n t h a t all n e c e s s a r y c o n d i t i o n s for
immunization are satisfied.
H o w e v e r , it c a n l e a d to a m i s i n t e r p r e t a t i o n of
results.
A1-13
If, at a given interest rate, our three conditions are satisfied, we are
protected. If not, it is not always clear where we stand.
Consider the following
example:
Liability:
$100, 16 p e r c e n t , 8 - y e a r c o m p o u n d b u l l e t (to p a y
$3Z7.84 at m a t u r i t y ) .
o
Asset:
$100, 16 p e r c e n t , 30 - y e a r m o r t g a g e (to p a y $16.19 p e r
annum).
o
O
At 16%
pA
o
D A = 6.9 # D L = 8.0.
o
D AZ = 8Z.Z D Lz = 64.0.
=
pL
$I00.00.
O
Since D L ~ D A, one m i g h t guess t h a t i n t e r e s t r a t e i n c r e a s e s will
c a u s e a gain, w h e r e a s i n t e r e s t r a t e d e c l i n e s should c a u s e losses.
is almost, but not quite, true:
i
8%
10
1Z
14
16
18
Z0
ZZ
pA
pL
18Z.Z6
15Z.6Z
130.41
113.37
100.00
89.3Z
177.1Z
15Z.94
13Z.41
114.93
I00.00
87.ZZ
80.61
73.40
76.Z5
5.7
66.80
9.9
AI-14
%
+Z.8%
0.Z
-1.5
-1.4
0.0
Z.4
This
o
A
L
It turns out that at IZ.75 percent, D 1 = D1 = 8.0, D AZ = 107.4 a n d
D~ = I00,
b u t p A = IZ3.51 and p L = IZS.5Z.
point of m a x i m u m
percent
loss.
I n t e r e s t i n g l y , t h e n , i = IZ.75 is t h e
If we c o u l d i n c r e a s e
our a s s e t s b y 1.6
(1Z5.SZ/1Z3.51), all c o n d i t i o n s w o u l d b e s a t i s f i e d (at 1Z.75
p e r c e n t ) and p A w o u l d t h e n e q u a l or e x c e e d p L at all o t h e r i n t e r e s t
rates.
T h e c o n c l u s i o n o f all this is t h a t i f we c a n find a n y i n t e r e s t r a t e such t h a t
i
pA
~
pL,
D A1
=
DIL, a n d
A
Dz
#
>
t h e n we a r e p r o t e c t e d
against immediate
c h a n g e s in i n t e r e s t r a t e s .
If not, w e
n e e d to r e s t r u c t u r e t h e a s s e t s a n d / o r l i a b i l i t i e s to s a t i s f y t h e s e t h r e e c o n d i t i o n s .
Call and Withdrawal Options
The
theoretical
justification
independent of the interest scenario.
for duration
assumes
that
cash flows are
If, i n s t e a d , t h e a s s e t s call when i n t e r e s t
AI-I5
r a t e s decline, or policyholders withdraw when i n t e r e s t r a t e s rise, then losses
could r e s u l t , r e g a r d l e s s of the duration tests.
This is a major s h o r t c o m i n g of
simple duration satistics.
Term Structure o f k t e r e s t R a t e s
The t e r m s t r u c t u r e of i n t e r e s t r a t e s is the s t r u c t u r e of yields on debt
i n s t r u m e n t s t h a t d i f f e r only in the t i m e r e m a i n i n g to t h e i r m a t u r i t y dates.
One purpose of this t h e o r y is to explain the relationship b e t w e e n the yield
curve and investors' e x p e c t a t i o n s of f u t u r e yields. Z N o t a t i o n used is as follows:
R
A c t u a l yield per period at t i m e t on an i n v e s t m e n t m a t u r i n g n
n,t
periods from t i m e t,
rl, t
=
E x p e c t e d yield per period at t i m e t on an i n v e s t m e n t maturing
in 1 period from t i m e t.
Thus, the Rn, t's are the yields one would observe in a s t a n d a r d yield curve
at t i m e t.
The r l , t ' s are 1-year yields t h a t investors a n t i c i p a t e will apply to
f u t u r e periods.
Z
This subject is c o v e r e d in C h a p t e r 14 of S e c u r i t y Market by Garbade.
Al-16
The
Term
interrelated.
Structure
Theory
holds
that
these
two
yields
are
closely
In f a c t ,
1
Rn, t =
(R1, t + r l , t + Z +
n
" " " + r l , t + n - 1 ) (See f o o t n o t e 3.)
N u m e r i c a l e x a m p l e s of t h e r's v e r s u s R's follow:
Case A
Expected Future Yields
Term Yields at
k
1
Z
3
4
5
rl,t + k
t, R k , t
16.0%
15.5
15.0
14.5
14.0
16.00%
15.75
15.50
15.Z5
15.00
Case B
Expected Future Yields Term Yields at
rl,t + k
t, Rk, t
1Z.0%
1Z.5
13.0
13.5
14.0
1Z.O0%
1Z.Z5
1Z.50
1Z.75
13.00
We can n o w m o d i f y t h e C l a s s i c a l I m m u n i z a t i o n T h e o r y to a c c o m m o d a t e this nonlevel yield curve.
A s s u m e t h e following:
At
=
c a s h f l o w of a s s e t s at t i m e t.
Bt
=
c a s h f l o w of l i a b i l i t i e s at t i m e t.
Both R
. and r-
a r e d e f i n e d as c o n t i n u o u s i n t e r e s t r a t e s .
Thus,
e R z ' t =n'etRl't : e t r l ' t + 1, so t h a t RZ,~ = I(R,~+,,~ r l , t + 1)/n.
The t h e o r y also
p r o v i d e s for t h e e x i s t e n c e of risk p r e m i u m s in r e a l i t y t h a t s e r v e to c r e a t e
e q u i l i b r i u m in t h e m a r k e t b e t w e e n s u p p l y and d e m a n d f o r issues of d i f f e r e n t
maturities.
A1-17
We want to achieve the following equation, regardless of the interest rate or
rates:
A 1
AZ
B1
+
(I + r l , O)
BZ
+...~
+
(i + rl,o)(l + r l , I)
(I + r l , O)
+ ....
(i + rl,o)(l + r l , I)
It turns out that if:
(1)
oo
~.
t=l
TT (l+r I i)
i =0
(2)
co
Z
t=l
co
= ~
t=l
tA t
t-1
'
tB t
TF ( l + r l i)
i =0
t2A t
t-1
i=07[ ( l + r l , i)
>
oo
Z
t=l
(or DA : D~)
t-1
,
'
t2Bt
A 2)
L
(or D2>D
t-1
i=O~l"( l = r l , i )
then if interest rates change, such that
d(l + r l , t)
d(l + rl, t)
=
d(l + r l , O)
for a l l t ,
(i + rl, O)
then pA will equal or e x c e e d the pL at the new interest r a t e or rates.
This
additional r e q u i r e m e n t implies that there cannot be drastic changes in the slope
of the yield curve. The following example will illustrate.
Al-18
Example:
O
Expectations
(rl,t):
rl, 0 =
1Z p e r c e n t i n c r e m e n t e d
y e a r to 16.5 p e r c e n t
O
Liability:
b y P~ p e r c e n t p e r
at t = 9.
$100, 1Z.9Z p e r c e n t , 4.43 - y e a r c o m p o u n d b u l l e t (to p a y
$171.31 at m a t u r i t y ) .
O
Assets:
$I00,
13.30 p e r c e n t ,
I0 - y e a r
mortgage
(to p a y $18.65
per annum).
;
o
D L1 = 4.43 = D A1 •
D LZ = Z0.
O
D ZA=
Z7 •
If (1 + r l , t) d e c r e a s e s b y 5% ( r l , 0
rl,1
O
pL
=
$IZ5.75.
pA
=
$ I Z 7 . 1 8 { I . 1 % gain).
= (1.1Z)
x 0.95
-
1 = 6.4%
= 6.9% , . . • , rl, 9 = 10.7%), then
If (1 + r l , t) i n c r e a s e s b y 5% ( r l , 0 = (1.1Z) x 1.05 - 1 = 1 7 . 6 % ,
rl,1
eL
=
$80.7z.
pA
=
$81.31 (0.7% gain).
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= 1 8 . 1 % , • • • , r l , 9 = ZZ.3%)~ t h e n
o
If the slope steepens to rl,0 = 1 2 % (rl,1 = 13% , . . . , rl,9 = Xl%),
then
O
O
pL
=
$96.87.
pA
=
$95.4Z (1.5% loss).
If the slope becomes flat at 1 6 % (rl,t
pL
=
$88.76.
pA
=
$90.14 (1.6% gain).
If the slope declines (rl,0 = IZ%, rl,1
= 16%
= 11.5%
for
all t), then
, • • • , rl, 9 = 7.5%),
then
pL
=
$107.36.
pA
=
$111.50 (3.9% gain).
This example
demonstrates
that if the rl,t's change by a constant
percentage, the p A will exceed the p L
Note, though, that in some cases, even
where the rl,t's changed by other than a constant percentage, we still gained.
Only in the case where the slope b e c a m e more than proportionately steeper did
we lose.
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I believe
that
r i g o r o u s l y p r o v e d it.
a s s e t s is g r e a t e r
this
may be a general
truth
although
we h a v e n o t y e t
The r e q u i r e m e n t of t h e Dz's i m p l i e s t h a t t h e s p r e a d of t h e
t h a n t h a t of t h e l i a b i l i t i e s .
Thus, a s t e e p e r y i e l d will likely
r e d u c e t h e m a r k e t v a l u e of a s s e t s m o r e t h a n it will r e d u c e t h e m a r k e t v a l u e of
liabilities.
A d e c l i n e in t h e s l o p e will h a v e t h e o p p o s i t e e f f e c t .
In any case, this represents another shortcoming in the concept of duration.
Even if we meet the conditions of the Dl'S and Dz's, we may still lose, should the
yield curve change shape dramatically.
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