A C T U A R I A L RESEARCH C L E A R I N G HOUSE 1 9 9 0 VOL. 2
INTEREST, AMORTIZATIONAND SIMPLICITY by Thomas M. Zavist, A.S.A.
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Interest m A m o r t i z a t i o n
and S i m p l i c i t y
C o n s i d e r s i m p l e i n t e r e s t for a moment. S u p p o s e y o u h a v e an e f f e c t i v e i n t e r e s t rate of 12% per year, and you w a n t to b r i n g a q u a n t i t y f o r w a r d w i t h i n t e r e s t for t h r e e months. Obviously, you m u l t i p l y by 1.03. S i m p l e i n t e r e s t seems so easy, but w h a t if you h a v e to go b a c k w a r d w i t h i n t e r e s t for t h r e e m o n t h s i n s t e a d ? Do you m u l t i p l y b y 0.97 or d i v i d e by 1.037 S u p p o s e you h a v e a m i d - y e a r adjustment. Do you m u l t i p l y by 1.06, m a k e the a d j u s t m e n t , and then m u l t i p l y b y 1.06 a g a i n to get to the end of the y e a r ? You w i l l end up w i t h 0.36% too m u c h i n t e r e s t for the year. P e r h a p s you d i v i d e by 1.06 and m u l t i p l y b y 1.12 a f t e r the a d j u s t m e n t , in o r d e r to end up w i t h e x a c t l y 12% a n n u a l l y . Does that m e a n t h a t the q u a n t i t y in the first e x a m p l e s h o u l d h a v e b e e n d i v i d e d by 1.09 and m u l t i p l i e d by 1.12, i n s t e a d of Just m u l t i p l i e d b y 1.037 S i m p l e i n t e r e s t is not so simple a f t e r all. In fact, s i m p l e i n t e r e s t is v e r y c o m p l i c a t e d . It is m u c h m o r e c o n f u s i n g to c o m p u t e t h a n c o m p o u n d interest, b e c a u s e w i t h c o m p o u n d i n t e r e s t , e v e r y s t e p is d e f i n e d p r e c i s e l y , and no c o n f u s i o n ever arises. Moreover, c o m p o u n d i n t e r e s t d e s c r i b e s how i n t e r e s t b e h a v e s in the real w o r l d m u c h b e t t e r than s i m p l e i n t e r e s t does. A m o r t i z a t i o n s c h e d u l e s are a n a l o g o u s to i n t e r e s t s c h e d u l e s in m a n y ways, and a l i n e a r a m o r t i z a t i o n s c h e d u l e has all the p r o b l e m s of s i m p l e i n t e r e s t and t h e n some. On the one hand, b a s e s w i t h the same a m o r t i z a t i o n p e r i o d but d i f f e r e n t s t a r t i n g d a t e s c a n n o t be combined, so d e t a i l e d records of p r i o r bases m u s t be m a i n t a i n e d from year to year. On the other hand, bases w i t h d i f f e r i n g signs can p r o d u c e p e c u l i a r and u n d e s i r a b l e effects w h e n a g g r e g a t e d . Imagine that you h a v e a $50,000 gain one year and a $50,000 loss the next. S u p p o s e you n e g l e c t interest, as a c c o u n t a n t s are w o n t to do, and a m o r t i z e e a c h base o v e r ten years. A f t e r one year, $5,000 of the g a i n has b e e n realized, but the u n r e a l i z e d p o r t i o n of the gain, c o u p l e d w i t h the loss, p r o d u c e s an u n r e a l i z e d net loss of $5,000. B e c a u s e the annual a m o u n t s of a m o r t i z a t i o n of the two b a s e s cancel one another, the u n r e a l i z e d net loss of $5,000 r e m a i n s on the books u n d i s t u r b e d for nine m o r e years, then v a n i s h e s in the c o u r s e of the s i n g l e final year. W h y should an a c c o u n t a n t w i t h an u n r e a l i z e d net loss of $5,000 sit a r o u n d t w i d d l i n g his thumbs for n i n e full y e a r s and s u d d e n l y r e a l i z e all of it in the e l e v e n t h y e a r ? To do so is n o t h i n g less t h a n silly. The s t a r t i n g date of an u n r e a l i z e d gain or loss b e i n g a m o r t i z e d is of no m o r e r e l e v a n c e than the y e a r p r i n t e d on a d o l l a r bill that is e a r n i n g i n t e r e s t in a bank. Each d o l l a r of u n r e a l i z e d g a i n or loss s h o u l d be t r e a t e d equal. The o n l y w a y to a g g r e g a t e s e v e r a l g a i n s and losses, and always a m o r t i z e t h e m smoothly, is to a m o r t i z e a f i x e d p r o p o r t i o n of the u n r e a l i z e d b a l a n c e e v e r y period, w h i c h means u s i n g an e x p o n e n t i a l a m o r t i z a t i o n s c h e d u l e i n s t e a d of a linear a m o r t i z a t i o n schedule.
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Recall that the force of mortality is the negative of the derivative (with respect to time) of the logarithm of an expected population. By analogy, let the force of amortization .(t) at time t be equal to the negative of the derivative (with respect to time) of the logarithm of the balance b(t) of a base. The outstanding balance b(t) of a base amortized linearly from time t = 0 to time t = n is given by
An_tJ
b(t) b(O),
so the force of amortization
is given by log b(t)
d
.(t) = -
dt log d [ dt
d log dt a =
~ n-t
J
+ log b(0) n-t
v -
- log A
nT]
1
n-tl
An_tl
,n_tl
With simple interest, the force of interest varies periodically, but with linear amortization, the force of amortization diverges to infinity at time t = n, inasmuch as the denominator in the last expression vanishes at time t = n. Not only does the force of amortization diverge to infinity, but the unweighted mean force of amortization between time t = 0 and time t = n is infinite as well and is given by
in
0
~(t) dt
I n - -d dt
0
log n-t J Y&
dt
n
log
=
log
n
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Fortunately,
outstanding
In in
.(t)
the m e a n force of a m o r t i z a t i o n b a l a n c e is f i n i t e and is given by
weighted
by
the
b(t) dt
0
b(t) dt
inn_t in
v --b(0)
dt
0
n-t[
b(0) dt
0
0
s
in In
v
n-t dt
0
a
dt
n-']
0
n-t[
~ 1 - exp(-¢n) ]
1 - exp(-~n) D~n~ n ~n~ n
2
~
n
I - (I - ~n) ] 2
n
1 -
1 - £n +
2
2
By contrast, if the force of a m o r t i z a t i o n b(t) = exp(-.t) b(0). = is constant, then
Thus, a c o n s t a n t force of a m o r t i z a t i o n produces an e x p o n e n t i a l a m o r t i z a t i o n schedule. Since the e f f e c t i v e rate of i n t e r e s t i is given by l+i = exp(~), d e f i n e the e f f e c t i v e rate of a m o r t i z a t i o n to be m such that l-m = exp(-~). Therefore, the o u t s t a n d i n g b a l a n c e is g i v e n by t b(t) = (l-m) b(0)
: (l-m)
b(t-1),
and
and the p a y m e n t m a d e at the end of each year to pay i n t e r e s t a m o r t i z e the p r i n c i p a l is given by (m+i) b(t-l).
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By h o l d i n g m fixed, a c h a n g e of i n t e r e s t rate c h a n g e s o n l y the y e a r l y p a y m e n t but not the s c h e d u l e of u n a m o r t i z e d b a l a n c e s , t h e r e b y m a k i n g a c h a n g e in interest r a t e e a s y to i m p l e m e n t . Furthermore, the e f f e c t i v e rate of a m o r t i z a t i o n is a p p r o x i m a t e d by
2
n U s i n g m ~ 2/n = 0.2 to a p p r o x i m a t e the e f f e c t i v e r a t e of a m o r t i z a t i o n in the $50,000 example above, e x a m i n e the u n r e a l i z e d net loss u n d e r the t r a d i t i o n a l linear a m o r t i z a t i o n s c h e d u l e as c o m p a r e d to an e x p o n e n t i a l a m o r t i z a t i o n schedule, as follows: U n r e a l i z e d Net Loss Linear Amortization U n r e a l i z e d Net L o s s Exponential Amortization
Year
o
1 2 3
$ (50,000)
5,000 5,000
$ (50,000) I0,000 8,000
6,400 5,120 4,096
3.277 2.621 2.097 1.678 1,342 1,074
5,000
4 5
6 7 8 9 10 11 12 13 14 15
5,000
5,000
5,000
5,000 5,000 5,000 5,000 0 0 0 0 0
859
687 550 440
B e s i d e s p r o d u c i n g a s m o o t h e r a m o r t i z a t i o n schedule, the e x p o n e n t i a l a m o r t i z a t i o n s c h e d u l e a l l o w s all the bases, w h i c h h a v e the s a m e fprce of a m o r t i z a t i o n , to be c o m b i n e d and m u l t i p l i e d each y e a r by the simple f a c t o r of (l-m). C o n s e q u e n t l y , like b a s e s o c c u r r i n g at d i f f e r e n t d a t e s can be c o m b i n e d into a single base, and the r e c o r d k e e p i n g b e c o m e s simpler. B e w a r e that the e x p o n e n t i a l l y fully. B e c a u s e like bases can bases is limited. In o t h e r b a s e s do not p r o l i f e r a t e like a m o r t i z e d bases are n e v e r a m o r t i z e d be combined, h o w e v e r , the n u m b e r of words, the e x p o n e n t i a l l y a m o r t i z e d rabbits, d e s p i t e b e i n g immortal.
E x p o n e n t i a l a m o r t i z a t i o n is not v e r y u s e f u l for m o r t g a g e s , but it has p r a c t i c a l a p p l i c a t i o n s for the p e n s i o n actuary, e s p e c i a l l y w i t h r e g a r d to the a c t u a r i a l v a l u e of assets, f u n d i n g s t a n d a r d a c c o u n t , and FASB e x p e n s e and disclosure. It m a y a p p l y to the i n s u r a n c e b u s i n e s s , but I am not f a m i l i a r w i t h h o w i n s u r a n c e a c t u a r i e s use amortization. Of course, there are legal o b s t a c l e s to o v e r c o m e .
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Compound interest is an improvement upon simple interest that makes the calculation of interest simpler. Likewise, exponential amortization is an improvement upon linear amortization that makes the calculation of amortization simpler. Most important, however, an exponential amortization schedule avoids the peculiar effects produced by the interaction of several bases amortized linearly.
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