Exact Region of Stability For An Investment Plan With Three Parameters
Exact Region of Stability For An Investment Plan With Three Parameters
Exact Region of Stability For An Investment Plan With Three Parameters
Abstract
Necessary and sufficient conditions for the asymptotic stability of a class of
difference equations with three parameters are obtained. These conditions are
expressed in terms of subsets of the parameter space.
1 Introduction
A principal u0 is invested for k interest periods, where k is a positive integer, at an
effective rate of interest r per period. If we denote the accumulated principal at the
end of n interest periods (n = 1, 2, ..., k) by un and consider the growth in the n-th
interest period, we obtain the simple difference equation
un+1 − un = run , n ∈ N = {0, 1, 2, ..., k, ...}. (1)
Since the above equation has the simple solution
un = u0 (1 + r)n , n ∈ N,
much can be said about its quantitative as well as the qualitative behavior. In con-
trast, if the original principal is divided into different parts and invested in financial
instruments that may regenerate and/or take different time periods to yield interests,
the corresponding difference equations may have solutions which are too complicated
to analyze. In such cases, alternate means are necessary in order to gain insight into
the nature of the investment policies.
In this paper, we will interpret un as the wealth of a person at the end of the
time period n. Here the wealth is measured in real monetary units and may take on
negative value if his liabilities are greater than his assets. To demonstrate the type of
mathematical means mentioned above, we suppose he engages in investments so that
his increase in wealth is governed by an equation of the form
un+1 − un = aun−2 + bun−1 + cun , n ∈ N = {0, 1, 2, ...}, (2)
∗ Mathematics Subject Classifications: 91B28, 39A11
† Department of Mathematics, Tsing Hua University, Hsinchu, Taiwan 30043, R. O. China.
‡ Department of Mathematics, Fujian Normal University, Fuzhou, Fujian 350007, P. R. China
194
S. S. Cheng and Y. Z. Lin 195
2 Bounding Regions
For the sake of convenience, we will denote the maximum of the moduli of the roots of
f (λ|x, y, z) by ρ(x, y, z). It is well known that ρ(x, y, z) is a continuous function with
respect to (x, y, z). Therefore, the region of stability Ω is open and its boundary is
contained in the set of points (x, y, z) such that f (λ|x, y, z) has a normal root. For this
reason, let us first consider the case when f (1|x, y, z) = 0 and f (−1|x, y, z) = 0 as well
as f (e±iθ |x, y, z) = 0 where θ ∈ (0, π). The first case leads to
f (1|x, y, z) = −(x + y + z) = 0, (3)
while the second leads to
f (−1|x, y, z) = y − z − x − 2 = 0, (4)
196 Region of Stability
sin 3θ − (z + 1) sin 2θ
y= ,
sin θ
equation (5) can then be rewritten as
(z + 1) sin θ − sin 2θ
x= = (z + 1) − 2 cos θ. (7)
sin θ
Since θ ∈ (0, π), thus z − 1 < x < z + 3 and
The surface in R3 defined by (3) separates R3 into two parts: x + y + z > 0 and
x + y + z < 0. We assert that Ω is contained in the latter subset. To see this, note that
limλ∈R,λ→∞ f (λ|x, y, z) = +∞ and limλ∈R,λ→−∞ f (λ|x, y, z) = −∞. If a + b + c ≥ 0,
then f (1|a, b, c) ≤ 0. Thus there exists a real root λ∗ ≥ 1 such that f (λ∗ |a, b, c) = 0.
This is contrary
to the definition of Ω. Similarly,
we can show that Ω is contained in
the region (x, y, z) ∈ R3 | − x + y − z − 2 < 0 .
The following is now clear.
LEMMA 1. The region of stability Ω is contained in the set
Γ = (x, y, z) ∈ R3 | x + y + z < 0, −x + y − z − 2 < 0 ,
and the set of points (x, y, z) ∈ R3 such that f (λ|x, y, z) has a normal root is contained
in
(x, y, z) ∈ R3 | x + y + z = 0
(x, y, z) ∈ R3 | − x + y − z − 2 = 0
or
(x, y, z) ∈ R3 | y = x2 − (z + 1)x − 1, z − 1 < x < z + 3 .
3 Region of Stability
In order to visualize the three dimensional region of stability Ω, we will consider its
level sets at each given z = c. To this end, we will denote such a level set by Ωc , that
is,
Ωc = (x, y) ∈ R2 | (x, y, c) ∈ Ω .
S. S. Cheng and Y. Z. Lin 197
We will also denote the level set of Γ at the point z = c by Γc , that is,
Γc = (x, y) ∈ R2 | x + y + c < 0, −x + y − c − 2 < 0 .
We will denote its graph by P, and the level set of P at z = c by Pc . Note further that
Pc is part of a parabola in the x, y-plane.
THEOREM 1. The region of stability Ω is contained in (x, y, z) ∈ R3 | z < 2 .
Proof. We first show that if c ≥ 2, Pc is outside the region Γc . Indeed, note that
when c ≥ 2, Γc is just the set of points (x, y) ∈ R2 which satisfies y < −x − c and
y < x + c + 2, that is, y < x + c + 2 for x ≤ −c − 1 and y < −x − c for x ≥ −c − 1.
Therefore, for x ∈ (c − 1, c + 3), the function that describes the bounding line segment
of this set is given by
Since
p(c − 1) − h(c − 1) = 0,
p (c − 1) − h (c − 1) = c − 2,
and
p (c − 1) − h (c − 1) = 2,
thus if c − 2 ≥ 0, then Pc will be strictly above the line segment defined by h(x) for
c − 1 < x < c + 3. In view of Lemma 1, we have shown that f (λ|a, b, c) does not have
any normal roots for any (a, b) ∈ Γc .
On the other hand, if we pick x = 0 and y = −(c + 1)2 /4 where c ≥ 2, then
(x, y) ∈ Γc . Furthermore, since
2
c+1
f λ|0, −(c − 1)2 /4, c = λ λ − =0
2
has simple root λ = 0 and double root λ = (c + 1)/2, thus ρ(0, −(c − 1)2 /4, c) > 1.
We now assert that if c ≥ 2, ρ(x, y, c) > 1 for any (x, y) ∈ Γc . Indeed, it this is not
the case, then there is some (x0 , y0 ) ∈ Γc such that ρ(x0 , y0 , c) < 1. If we now connect
the points (0, −2(c + 1)2 /4) and (x0 , y0 ) by a continuous curve completely contained
inside Γc (which can be done in view of the special form of Γc ), then by continuity,
there would be some point (x∗ , y ∗ ) on this curve such that ρ(x∗ , y ∗ , c) = 1. But then
f (λ|x∗ , y ∗ , c) has a normal root. This is contrary to what we have shown above. The
proof is complete.
198 Region of Stability
The above result and the next can be proved in a very simple manner. Indeed,
since |z + 1| is equal to the absolute value of the sum of the three roots of f (λ|x, y, z),
thus |z + 1| < 3 is a necessary condition for all roots of f (λ|x, y, z) to be subnormal.
However, we remark that the last part of the above proof makes use of the pathwise
connectedness property of the region in concern and continuity arguments. Similar
ideas can also be used again several times in the following discussions. In particular,
we may show the following result. Since the proof is similar to that above, it will only
be sketched.
THEOREM 2. The region of stability Ω is contained in (x, y, z) ∈ R3 | z > −4 .
Sketch of Proof. We first show that if c ≤ −4, Pc is outside the region Γc . This is
shown by noting that the corresponding Γc is just the set of points (x, y) ∈ R2 which
satisfies y < −x − c and y < x + c + 2 so that function the describes the bounding
boundary for x ∈ (c − 1, c + 3) is given by
Comparing the function p(x) and h(x), we see that if c ≤ −4, then the cross section
Pc will be strictly above the line segment defined by h(x) for c − 1 < x < c + 3. In
other words, f (λ|a, b, c) does not have any normal roots for any (a, b) ∈ Γc . On the
other hand, we can pick (u, v) ∈ Γc such that f (λ|u, v, c) has a real supernormal root
λ∗ . Finally, as in the proof of Theorem 1, we may show that ρ(x, y, c) ≥ 1 for any
(x, y) ∈ Γc by continuity arguments. The proof is complete.
After we have shown that Ω is bounded between the planes z = −4 and z = 2,
we may now consider three different cases; (i) 0 ≤ c < 2, (ii) −2 < c < 0, and (iii)
−4 < c ≤ −2.
THEOREM 3. For 0 ≤ c < 2, the corresponding level set Ωc is equal to
Ac = (x, y) ∈ R2 | c − 1 < x < 1, x2 − (c + 1)x − 1 < y < −x − c . (9)
Proof. For each c ∈ [0, 2), Γc is given by the set of points (x, y) that satisfy
y < x + c + 2 for x ≤ −c − 1 and y < −x − c for x ≥ −c − 1. It is not difficult to verify
that the parabola defined by
p̃(x) = x2 − (c + 1)x − 1, x ∈ R,
h(x) = −x − c, x ≥ −c − 1 (10)
at (x, y) = ((c − 1), 1 − 2c) and (x, y) = (1, −(c + 1)), but does not intersect the straight
line segment q(x) = x + c + 2 over x ≤ −c − 1. Thus, the part of Pc for x ∈ (c − 1, 1)
lies inside Γc and separates Γc into two disjoint open and pathwise connected regions
one of which is given by the set Ac defined by (9). For the sake of convenience, let us
denote the other region by Bc . We will show that there is a point (u2 , v2 ) in Bc such
that f (λ|u2 , v2 , c) has a supernormal root, and a point (u1 , v1 ) in Ac such that all roots
S. S. Cheng and Y. Z. Lin 199
of f (λ|u1 , v1 , c) are subormal. To see this, let (u2 , v2 ) = (0, −5). Then it is easily seen
that (u2 , v2 ) is in Bc . Furthermore,
f (λ|u2 , v2 , c) = λ λ2 − (c + 1)λ + 5 ,
q s r
so that its roots are 0, 12 c + 1 ± i 20 − (c + 1)2 and therefore the corresponding
√
ρ(u2 , v2 , c) = 5 > 1. Next, if c ∈ [0, 1), let (u1 , v1 ) = (0, −(c + 1)/2) and if c ∈ [1, 2),
we let
# 3 $
c+1 1 2
(u1 , v1 ) = , − (c + 1) .
3 3
It is easy to see that (0, −(c + 1)/2) ∈ Ac if c ∈ [0, 1). Furthermore, the corresponding
2 c+1
f (λ|u1 , v1 , c) = λ λ − (c + 1)λ +
2
√
has roots 0 and 12 c + 1 ± i 1 − c2 . Therefore the corresponding ρ(u1 , v1 , c) = (c +
1)/2 < 1.
To see that (u1 , v1 ) ∈ Ac for c ∈ [1, 2), first note that c < 2 implies (c + 1)/3 < 1
and (c + 1)3 /33 < 1. Furthermore, since the function
3
x+1
g(x) = − (x − 1)
3
is strictly decreasing on [1, 2) and g(2) = 0, we see that (c − 1) < (c + 1)3 /33 for
1 ≤ c < 2. We have thus shown that c − 1 < u1 < 1. Next, consider
3
x+1 (x + 1)2
w(x) = −x − − − , 1 ≤ x < 2.
3 3
Since w(2) = 0 and
(x − 2)2
w (x) = − < 0, 1 ≤ x < 2,
9
we have w(x) > 0 for x ∈ [1, 2) so that
3
1 c+1
− (c + 1)2 < −c − , 1 ≤ c < 2.
3 3
Next, consider
+ 6 3 ,
1 x+1 x+1
q(x) = − (x + 1)2 − − (x + 1) − 1 , 1 ≤ x < 2.
3 3 3
4 2
2 10 x+1 x+1
q (x) = − − +4 ,
3 3 3 3
Proof. The proof is similar to that of Theorem 3 and will thus be sketched. For
each c ∈ (−4, −2], Γc is given by the set of points (x, y) that satisfy y < x + c + 2 for
x ≤ −c − 1 and y < −x − c for x ≥ −c − 1. It is not difficult to verify that the parabola
defined by
p̃(x) = x2 − (c + 1)x − 1, x ∈ R,
q(x) = x + c + 2, x ≤ −c − 1 (12)
at (x, y) = (−1, c+1) and (x, y) = (c+3, 2c+5), but does not intersect the straight line
segment h(x) = x + c + 2 over x ≥ −c − 1. Thus the part of Pc for x ∈ (−1, c + 3) lies
inside Γc and separates Γc into two disjoint open and pathwise connected regions one of
which is given by the set Dc defined by (11). Next, we may show that (0, −5) belongs
to the complement of Dc relative to Γc and the corresponding characteristic polynomial
f (λ|0, −5, c) has supernomal roots. We may also show that for c ∈ (−3, −2], the point
(0, (c − 1)/4) belongs to Dc , and for c ∈ (−4, −3], the point (c + 1)3 /33 , −(c + 1)2 /3
S. S. Cheng and Y. Z. Lin 201
4 Special Cases
It is interesting to consider two special cases of (2):
un = αun−2 + βun−3 , n ∈ N, (14)
and
un = αun−1 + βun−3 , n ∈ N. (15)
Their characteristic polynomials are z 3 − αz − β and z 3 − αz 2 − β respectively. Thus
(14) is asymptotically stable if, and only if, (α, β) lies in the plane region defined by
α + β − 1 < 0, α − β − 1 < 0, α < β 2 − 1,
while (15) is asymptotically stable if, and only if, (α, β) lies in the plane region defined
by
−3 < α < 3, α + β − 1 < 0, β − α − 1 < 0, 0 < β 2 − αβ − 1.
References
[1] G. Gandolfo, Economic Dynamics: Methods and Models, North-Holland, 1980.
[2] E. J. Barbeau, Polynomials, Springer-Verlag, 1989.