An evolutionary finance model with a risk-free asset

Abstract

The purpose of this work is to develop an evolutionary finance model with a risk-free asset playing the role of a numeraire. The model describes a market where one risk-free and several “short-lived” risky assets (securities) are traded in discrete time. The risky securities live one period, yield random payoffs at the end of it, and then are re-born at the beginning of the next period. The main goal of the study is to identify investment strategies that make it possible for an investor to “survive” in the market selection process. It is shown that a strategy of this kind exists, is in a sense asymptotically unique and can be described by a simple explicit formula amenable for quantitative investment analysis.

Appendix

To prove Theorems 1 and 2 we derive a random dynamical system describing the dynamics of the market shares $r_{t}^{i}$ of the investors $i=1,\ldots ,N$.

Theorem 3

The following equations hold:

$$\begin{aligned} r_{t+1}^{i}=\sum _{k=0}^{K}R_{t+1,k}\frac{\lambda _{t,k}^{i}r_{t}^{i}}{\langle \lambda _{t,k},r_{t}\rangle },\quad t=0,1,2,\ldots ,\ i=1,2,\ldots ,N. \end{aligned}$$

(24)

Proof

Put

$$\begin{aligned} {\bar{V}}_{t,0}:={\bar{w}}_{t}\ \left[ =\sum _{i=1}^{N}x_{t,0}^{i}=\sum _{i=1}^{N} \lambda _{t,0}^{i}w_{t}^{i}\right] ,\ {\bar{V}}_{t,k}=V_{t,k},\quad k=1,2,\ldots ,K. \end{aligned}$$

(25)

By using the notation $A_{t+1,0}=1+\beta _{t+1}$, we write

$$\begin{aligned} W_{t+1}= & {} \sum _{i=1}^{N}w_{t+1}^{i}=\sum _{i=1}^{N}\langle A_{t+1},x_{t} ^{i}\rangle =\sum _{i=1}^{N}\sum _{k=0}^{K}A_{t+1,k}x_{t,k}^{i} \nonumber \\= & {} \sum _{k=0}^{K}\left( A_{t+1,k}\sum _{i=1}^{N}x_{t,k}^{i}\right) =(1+\beta _{t+1}){\bar{w}}_{t}+\sum _{k=1}^{K}A_{t+1,k}V_{t,k}=\sum _{k=0} ^{K}A_{t+1,k}{\bar{V}}_{t,k}.\nonumber \\ \end{aligned}$$

(26)

Define $\lambda _{t,k}:=(\lambda _{t,k}^{1},\ldots ,\lambda _{t,k}^{N})$ and $w_{t}:=(w_{t}^{1},\ldots ,w_{t}^{N})$. From (7) and (10) we get

$$\begin{aligned} p_{t,k}V_{t,k}=\langle \lambda _{t,k},w_{t}\rangle ,\quad k=1,\ldots ,K, \,\,t\ge 0, \end{aligned}$$

which implies

$$\begin{aligned} x_{t,k}^{i}=\frac{\lambda _{t,k}^{i}w_{t}^{i}}{p_{t,k}}=\frac{\lambda _{t,k} ^{i}w_{t}^{i}V_{t,k}}{\langle \lambda _{t,k},w_{t}\rangle }=\frac{\lambda _{t,k}^{i}w_{t}^{i}{\bar{V}}_{t,k}}{\langle \lambda _{t,k},w_{t}\rangle },\quad k=1,\ldots ,K,\,\,t\ge 0 \end{aligned}$$

(27)

(see 11). Further, we find

$$\begin{aligned} p_{t,0}{\bar{V}}_{t,0}={\bar{V}}_{t,0}={\bar{w}}_{t}=\langle \lambda _{t,0} ,w_{t}\rangle , \end{aligned}$$

(28)

where the first equality holds because $p_{t,0}=1$, the second follows from (25), and the third is a consequence of (1) and (12). By using (12) and (12), we obtain

$$\begin{aligned} x_{t,0}^{i}=\frac{\lambda _{t,0}^{i}w_{t}^{i}}{p_{t,0}}=\frac{\lambda _{t,0} ^{i}w_{t}^{i}{\bar{V}}_{t,0}}{\langle \lambda _{t,0},w_{t}\rangle }. \end{aligned}$$

(29)

From (29) and (27) we get

$$\begin{aligned} w_{t+1}^{i}= & {} \sum _{k=0}^{K}A_{t+1,k}x_{t,k}^{i}=A_{t+1,0}{\bar{V}}_{t,0} \frac{\lambda _{t,0}^{i}w_{t}^{i}}{\langle \lambda _{t,0},w_{t}\rangle } +\sum _{k=1}^{K}A_{t+1,k}{\bar{V}}_{t,k}\frac{\lambda _{t,k}^{i}w_{t}^{i}}{\langle \lambda _{t,k},w_{t}\rangle } \\= & {} \sum _{k=0}^{K}A_{t+1,k}{\bar{V}}_{t,k}\frac{\lambda _{t,k}^{i}w_{t}^{i}}{\langle \lambda _{t,k},w_{t}\rangle }. \end{aligned}$$

Dividing the first and the last terms in this chain of equalities by $W_{t+1}$ and using (26), we get

$$\begin{aligned} r_{t+1}^{i}=\sum _{k=0}^{K}\frac{A_{t+1,k}{\bar{V}}_{t,k}}{\sum _{l=0} ^{K}A_{t+1,l}{\bar{V}}_{t,l}}\frac{\lambda _{t,k}^{i}w_{t}^{i}}{\langle \lambda _{t,k},w_{t}\rangle }. \end{aligned}$$

(30)

Observe that

$$\begin{aligned} \frac{\lambda _{t,k}^{i}w_{t}^{i}}{\langle \lambda _{t,k},w_{t}\rangle } =\frac{\lambda _{t,k}^{i}w_{t}^{i}/W_{t}}{\langle \lambda _{t,k},w_{t} /W_{t}\rangle }=\frac{\lambda _{t,k}^{i}r_{t}^{i}}{\langle \lambda _{t,k} ,r_{t}\rangle }. \end{aligned}$$

(31)

Further, we have

$$\begin{aligned}&\frac{A_{t+1,k}{\bar{V}}_{t,k}}{\sum _{l=0}^{K}A_{t+1,l}{\bar{V}}_{t,l}} =\frac{a_{t+1,k}\,{\bar{w}}_{t}{\bar{V}}_{t,k}}{\left( 1+\beta _{t+1}\right) {\bar{w}}_{t}+\sum _{l=1}^{K}a_{t+1,l}\,{\bar{w}}_{t}{\bar{V}}_{t,l}} \nonumber \\&\quad =\frac{a_{t+1,k}\,V_{t,k}}{\sum _{l=0}^{K}a_{t+1,l}\,V_{t,l}}=R_{t+1,k} \end{aligned}$$

(32)

($k=1,\ldots ,K$) and

$$\begin{aligned}&\frac{A_{t+1,0}{\bar{V}}_{t,0}}{\sum _{l=0}^{K}A_{t+1,l}{\bar{V}}_{t,l}} =\frac{\left( 1+\beta _{t+1}\right) \,{\bar{w}}_{t}}{\left( 1+\beta _{t+1}\right) {\bar{w}}_{t}+\sum _{l=1}^{K}a_{t+1,l}\,{\bar{w}}_{t}V_{t,l}} \nonumber \\&\quad =\frac{a_{t+1,0}\,V_{t,0}}{\sum _{l=0}^{K}a_{t+1,l}\,V_{t,l}}=R_{t+1,0} \end{aligned}$$

(33)

[see (18) and (19)]. By substituting (31)–(33) into (30), we arrive at (24).

The proof is complete. $\square$

We have shown that the dynamics of the market shares $r_{t}^{i}$ of the investors $i=1,2,\ldots ,N$, in the model at hand is governed by the random dynamical system (24). A detailed analysis of this system was carried out in Amir et al. (2013). Theorems 1 and 2 are direct consequences of Theorems 4 and 5 proved in Amir et al. (2013), which we formulate below.

Let $R_{t,k}(s^{t})\ge 0$ ($t=0,1,2,\ldots$; $k=0,\ldots ,K$) be measurable real-valued functions satisfying (20) and (22). Consider sequences of measurable vector functions $\Lambda =(\lambda _{t}(s^{t} ))_{t=0}^{\infty }$, where $\lambda _{t}(s^{t})=(\lambda _{t,0}(s^{t} ),\ldots ,\lambda _{t,K}(s^{t}))\in \Delta ^{K+1}$. Denote by ${\mathcal {L}}$ the set of N-tuples $(\Lambda ^{1},\ldots ,\Lambda ^{N})$ of such sequences for which the random dynamical system (24) generates well-defined vectors $r_{t}^{i}=(r_{t,0}^{i},\ldots ,r_{t,K}^{i})\in \Delta ^{K+1}$, $i=1,\ldots ,N$, $t=0,1,\ldots$, i.e., the validity of the inequality

$$\begin{aligned} \sum _{i=1}^{N}\lambda _{t,k}^{i}r_{t}^{i}>0,\quad k=0,1,\ldots ,K, \end{aligned}$$

is guaranteed for each $t\ge 0$. Define $\Lambda ^{*}=(\lambda _{t}^{*}(s^{t}))_{t=0}^{\infty }$ by (21).

Theorem 4

For each $(\Lambda ^{1},\ldots ,\Lambda ^{N} )\in {\mathcal {L}}$ with $\Lambda ^{1}=\Lambda ^{*}$, we have $\inf _{t\ge 0}r_{t}^{1}>0$ (a.s.).

Theorem 5

Let $\Lambda =(\lambda _{t}(s^{t}))_{t=0}^{\infty }$ be a sequence of measurable vector functions with values in $\Delta ^{K+1}$. If $\inf _{t\ge 0}r_{t}^{1}>0$ for any $(\Lambda ^{1},\ldots ,\Lambda ^{N})\in {\mathcal {L}}$ with $\Lambda ^{1}=\Lambda$, then (23) holds.

For proofs of Theorems 4 and 5 see Amir et al. (2013, Section 4).