Growth rate

McGraw-Hill Science & Technology Dictionary:

growth rate

(′grōth ′rāt)

(microbiology) Increase in the number of bacteria in a population per unit time.

Search unanswered questions...

Browse: Unanswered questions | Most-recent questions | Reference library

Growth rate

Top

Amount of change in some financial characteristic of a company.

1. percentage change in earnings per share, dividends per share, revenue, market price of stock, or total assets compared to a base year amount.
For example, growth in earnings per share equals:
Growth Rate

2. percentage change in an item such as net income considering the time value of money. Here, a future value of $1 table must be used.
Assume net income in 2009 is $300,000 and in 2013 is $500,000. The future value of $1 table factor is 1.667 ($500,000/$300,000).
The future value of $1 table indicates that the intersection of n = four years and a factor of 1.667 is about 14%, as evidenced here:
Growth Rate

3. change in retained earnings divided by beginning stockholders' equity.

4. net income less dividends divided by common stockholders' equity.
A high ratio reflects a company's ability to generate internal funds, and thus it does not have to rely on external sources.

Previous:	Group Financial Statement, Group Depreciation, Gross-Profit Ratio
Next:	Growth Stocks, Guaranteed Bond, HUB

McGraw-Hill Dictionary of Architecture & Construction:

growth rate

Top

Rate of wood growth expressed as the number of annual rings per inch measured from pith to bark; sometimes used to rate soft-woods for strength.

Investopedia Financial Dictionary:

Growth Rates

Top

The amount of increase that a specific variable has gained within a specific period and context. For investors, this typically represents the compounded annualized rate of growth of a company's revenues, earnings, dividends and even macro concepts - such as the economy as a whole.

Expected forward-looking or trailing growth rates are two common kinds of growth rates used for analysis.

Investopedia Says:
Different types of industries have different benchmarks for rates of growth. For instance, companies that are on the cutting edge of technology would be more likely to have higher annual rates of growth compared to a mature industry, like retail sales.

The use of historical growth rates is one of the simplest methods of estimating future growth. However, historically high growth rates don't always mean a high rate of growth looking into the future, because industrial and economic conditions change constantly.

For example, the auto industry has higher rates of revenue growth during good economic times. However, in times of recession, consumers would be more inclined to be frugal and not spend disposable income on a new car.

Related Links:
The CAGR is a good and valuable tool to evaluate investment options, but it does not tell the whole story. Compound Annual Growth Rate: What You Should Know
If you're looking to get a job as an analyst, you'll need to know how to work it. Style Matters In Financial Modeling
Learn how this simple calculation can help you determine a stock's earnings potential. PEG Ratio Nails Down Value Stocks

Wikipedia on Answers.com:

Growth rate (group theory)

Top

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (March 2011)

In group theory, the growth rate of a group with respect to a symmetric generating set describes the size of balls in the group. Every element in the group can be written as a product of generators, and the growth rate counts the number of elements that can be written as a product of length n.

Contents

1 Definition
2 Polynomial and exponential growth
3 Examples
4 See also
5 References
6 Further reading

Definition

Suppose G is a finitely generated group; and T is a finite symmetric set of generators (symmetric means that if $x \in T$ then $x^{-1} \in T$ ). Any element $x \in G$ can be expressed as a word in the T-alphabet

$x = a_1 \cdot a_2 \cdots a_k \mbox{ where } a_i\in T.$

Let us consider the subset of all elements of G which can be presented by such a word of length ≤ n

$B_n(G,T) = \{x\in G | x = a_1 \cdot a_2 \cdots a_k \mbox{ where } a_i\in T \mbox{ and } k\le n\}.$

This set is just the closed ball of radius n in the word metric d on G with respect to the generating set T:

$B_n(G,T) = \{x\in G | d(x, e)\le n\}.$

More geometrically, $B n (G, T)$ is the set of vertices in the Cayley graph with respect to T which are within distance n of the identity.

Given two nondecreasing positive functions a and b one can say that they are equivalent ( $a\sim b$ ) if there is a constant C such that

$a(n/ C) \leq b(n) \leq a(Cn),\,$

for example $p n \sim q n$ if $p, q > 1$ .

Then the growth rate of the group G can be defined as the corresponding equivalence class of the function

$\#(n)=|B_n(G,T)|,$

where $| B n (G, T) |$ denotes the number of elements in the set $B n (G, T)$ . Although the function $\#(n)$ depends on the set of generators T its rate of growth does not (see below) and therefore the rate of growth gives an invariant of a group.

The word metric d and therefore sets $B n (G, T)$ depend on the generating set T. However, any two such metrics are bilipschitz equivalent in the following sense: for finite symmetric generating sets E, F, there is a positive constant C such that

${1\over C} \ d_F(x,y) \leq d_{E}(x,y) \leq C \ d_F(x,y).$

As an immediate corollary of this inequality we get that the growth rate does not depend on the choice of generating set.

Polynomial and exponential growth

$\#(n)\le C(n^k+1)$

for some $C,k<\infty$ we say that G has a polynomial growth rate. The infimum $k 0$ of such k's is called the order of polynomial growth. According to Gromov's theorem, a group of polynomial growth is virtually nilpotent, i.e. it has a nilpotent subgroup of finite index. In particular, the order of polynomial growth $k 0$ has to be a natural number and in fact $\#(n)\sim n^{k_0}$ .

If $\#(n)\ge a^n$ for some $a > 1$ we say that G has an exponential growth rate. Every finitely generated G has at most exponential growth, i.e. for some $b > 1$ we have $\#(n)\le b^n$ .

If $\#(n)$ grows more slowly than any exponential function, G has a subexponential growth rate. Any such group is amenable.

Examples

A free group with a finite rank k > 1 has an exponential growth rate.

A finite group has constant growth – polynomial growth of order 0 – and includes fundamental groups of manifolds whose universal cover is compact.

If M is a closed negatively curved Riemannian manifold then its fundamental group $π 1 (M)$ has exponential growth rate. Milnor proved this using the fact that the word metric on $π 1 (M)$ is quasi-isometric to the universal cover of M.

Z^d has a polynomial growth rate of order d.

The discrete Heisenberg group H₃ has a polynomial growth rate of order 4. This fact is a special case of the general theorem of Bass and Guivarch that is discussed in the article on Gromov's theorem.

The lamplighter group has an exponential growth.

The existence of groups with intermediate growth, i.e. subexponential but not polynomial was open for many years. It was asked by Milnor in 1968 and was finally answered in the positive by Grigorchuk in 1984. There are still open questions in this area and a complete picture of which orders of growth are possible and which are not is missing.

The triangle groups include 3 finite groups (the spherical ones, corresponding to sphere), 3 groups of quadratic growth (the Euclidean ones, corresponding to Euclidean plane), and infinitely many groups of exponential growth (the hyperbolic ones, corresponding to the hyperbolic plane).

References

J. Milnor, A note on curvature and fundamental group, J. Differential Geometry 2 (1968), 1–7.
R. I. Grigorchuk, Degrees of growth of finitely generated groups and the theory of invariant means., Izv. Akad. Nauk SSSR Ser. Mat. 48:5 (1984), 939–985 (Russian).

		McGraw-Hill Science & Technology Dictionary. McGraw-Hill Dictionary of Scientific and Technical Terms. Copyright © 2003, 1994, 1989, 1984, 1978, 1976, 1974 by McGraw-Hill Companies, Inc. All rights reserved. Read more
		Barron's Accounting Dictionary. Dictionary of Accounting Terms. Copyright © 2010 by Barron's Educational Series, Inc. All rights reserved. Read more
		McGraw-Hill Dictionary of Architecture & Construction. McGraw-Hill Dictionary of Architecture and Construction. Copyright © 2003 by McGraw-Hill Companies, Inc. All rights reserved. Read more
		Investopedia Financial Dictionary. Copyright ©2010, Investopedia.com - Owned and Operated by Investopedia US, A Division of ValueClick, Inc. All rights reserved. Read more
		Wikipedia on Answers.com. This article is licensed under the Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article Growth rate (group theory). Read more

	Compound Annual Growth Rate (CAGR) (finance term)
	trimethylene glycol
	Economic Growth Rate (business term)

	What does the rate of growth do during exponential growth? Read answer...
	What is the growth rate for fashion modeling? Read answer...
	What is the growth rate of sunflowers? Read answer...

	How do you determine the continuous growth rate given the growth rate?
	What is similar about negative growth rate and zero growth rate?
	Can a company\'s growth rate be different from its sustainable growth rate?

Growth rate

growth rate

Growth rate

growth rate

Growth Rates

Growth rate (group theory)

Definition

Polynomial and exponential growth

Examples

See also

References

Further reading