# Capital growth

Capital growth (also called expansion capital and growth equity) refers to appreciation of an asset value over time. Capital growth is the difference between current value and price at which the asset was bought. The current value can be known for sure or predicted based on market value.

Each investment on financial markets is associated with risk. The risk can occur by positive or negative situations. In case of positive situations, value of the asset increases, while in the opposite - it decreases. The capital growth is related to increase of the asset value.

Investors seek for assets that are cheap (e.g. P/E index is low), because they have greater potential of capital growth. However, if the bear market happens, even the cheap assets value can decrease. But usually less than expensive ones.

## Origins of the model

Recognition of investment has a lot seemingly independent sources, for example "Williams [1936],Kelly [1956], Latane [1959], and Breiman [1960,1961]" don't know about theirs researches. However, in Samuelson's translation and researches we can see thet in 1738 Bernoulli came across this issue unexpectedly. Samuelson seems to be the first who finds what he wanted to find - connect geometric mean criterion with utility theory. Hakansson was the first who noticed that occures the growth optimal strategies incompatibility in consumption investment circumstances in the assumption that "optimal strategy's consumption are logarithmic". Eventually the model which analyzes a compromise between security and capital growth has been introduced by MacLean and Ziemba in 1986 (N.H. Hakansson, N.H. Ziemba 1995, p. 66).

## Capital growth model

"Capital growth model describes the concentration of capital at a point in time. In companies that continue their operations, capital grows according to the Equation$C_{t1,s,p,M}=C_{t0}\cdot{e}^{(p-s+M)\cdot{\vartriangle}t}$

where:

Ct0—the beginning concentration of capital [expressed in monetary terms] in the time moment t0,

Ct1,s,p,M—the ending concentration of capital [ex- pressed in monetary terms] in the time moment t1, which has been subdued to natural dispersion “s” (risk), risk premium “p” and a management variable “M” through the time period Δt,

S—dispersion variable (risk) [expressed as 1/year],

P—risk premium, p = E (s) [expressed as 1/year],

M—management variable [expressed as 1/year],

Δt—time period between time moments: t0 and t1 [ex- pressed in years]." (B. Kurek 2012, p.365).

## Capital growth theory

Capital growth theory could be helpful while analizing dynamic investment situations. The Kelly criterion actually the growth-optimal investment strategy which is the most favorable strategy in reinvestment fortuity certainly brings to increase capital in long term. The most attractive feature is that it asymptotically reduces alleged time needed to get concrete level of capital and what's more there is no danger of collapse.

In many environments the Kelly criterion is probably the best. It concerns environments containing coruption and non-capital income (N.H. Hakansson, W.T. Ziemba 1995, p.82).

## The Kelly Capital Growth Strategy

"The Kelly capital growth criterion, which maximizes the expected log of final wealth, provides the strategy that maximizes long run wealth growth asymptotically for repeated investments over time." There is one defect which leads to increase of variability in the short term. This defect is the occurrence of a substantially lack of risk aversion. The Kelly criterion is used by many investors, hedge funds and sports betting due to offering many positive investment sitiations (W.T. Ziemba 2016, p. 49).

## References

Author: Karolina Korbut