# A General Framework for Portfolio Theory. Part III: Multi-Period Markets and Modular Approach

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Multi-Period Market and Trading Strategies

#### 2.1. Definitions

**Definition**

**1**(Risk-free asset)

**.**

**Definition**

**2**

**.**For $M\in \mathbb{N}$ let $S:={\left({S}_{n}\right)}_{0\le n\le N}\in {\mathcal{L}}^{2}(N;{\mathbb{R}}_{>0}^{M+1})$ with

**Definition**

**3**(Trading strategy)

**.**

- ${x}_{n}$ may depend on ${S}_{0},\dots ,{S}_{n-1}$ but not on later prices;
- ${S}_{n}^{i}$ absolute price of the ith asset at time n;
- ${x}_{n}^{i}$: number of shares invested into the ith asset from time step $n-1$ to n;
- ${S}_{n-1}^{i}{x}_{n}^{i}$: amount of money invested into the ith asset;
- ${S}_{n}^{i}{x}_{n}^{i}$: absolute value of this investment after the time step from $n-1$ to n; and
- ${S}_{n}^{\top}{x}_{n}={\sum}_{i=0}^{M}{S}_{n}^{i}{x}_{n}^{i}$: absolute value of all investments after the time step from $n-1$ to n.

**Definition**

**4**(Wealth of trading strategy)

**.**

**Definition**

**5**(Admissible trading strategy)

**.**

#### 2.2. Properties of the Multi-Period Market Model

**Definition**

**6**

**Proposition**

**1.**

**Proof.**

**Remark**

**1**(Bond)

**.**

**Definition**

**7**(Arbitrage opportunity, bond replicating, and risk-free)

**.**

- (a)
- We say X is risk-free if$${S}_{n-1}^{\top}{x}_{n}\le {\mathcal{W}}_{n-1}\left(X\right)\phantom{\rule{1.em}{0ex}}\phantom{\rule{4.pt}{0ex}}a.s.\phantom{\rule{4.pt}{0ex}}\mathit{for}\phantom{\rule{4.pt}{0ex}}\mathit{all}\phantom{\rule{4.pt}{0ex}}n=1,\dots ,N\phantom{\rule{4.pt}{0ex}}\mathit{and}\phantom{\rule{2.em}{0ex}}{\mathcal{W}}_{N}\left(X\right)\ge {\mathcal{W}}_{0}\frac{{S}_{N}^{0}}{{S}_{0}^{0}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{4.pt}{0ex}}a.s.$$We say market model S has no nontrivial risk-free trading strategy if there does not exist a risk-free trading strategy X with $\widehat{X}\not\equiv 0$ (i.e., besides the trivial ones with $\widehat{X}=0$ a.s. there are no risk-free trading strategies).
- (b)
- We say X is an arbitrage opportunity if$${S}_{n-1}^{\top}{x}_{n}\le {\mathcal{W}}_{n-1}\left(X\right)\phantom{\rule{1.em}{0ex}}\phantom{\rule{4.pt}{0ex}}a.s.\phantom{\rule{4.pt}{0ex}}\mathit{for}\phantom{\rule{4.pt}{0ex}}\mathit{all}\phantom{\rule{4.pt}{0ex}}n=1,\dots ,N,\phantom{\rule{2.em}{0ex}}{\mathcal{W}}_{N}\left(X\right)\ge {\mathcal{W}}_{0}\frac{{S}_{N}^{0}}{{S}_{0}^{0}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{4.pt}{0ex}}a.s.,$$and$$\mathrm{P}\left({\mathcal{W}}_{N}\left(X\right)>{\mathcal{W}}_{0}\frac{{S}_{N}^{0}}{{S}_{0}^{0}}\right)>0.$$We say market model S is arbitrage-free, if there does not exist any arbitrage opportunity.
- (c)
- We say X is bond replicating if$${S}_{n-1}^{\top}{x}_{n}\le {\mathcal{W}}_{n-1}\left(X\right)\phantom{\rule{1.em}{0ex}}\phantom{\rule{4.pt}{0ex}}a.s.\phantom{\rule{4.pt}{0ex}}\mathit{for}\phantom{\rule{4.pt}{0ex}}\mathit{all}\phantom{\rule{4.pt}{0ex}}n=1,\dots ,N\phantom{\rule{4.pt}{0ex}}\mathit{and}\phantom{\rule{2.em}{0ex}}{\mathcal{W}}_{N}\left(X\right)={\mathcal{W}}_{0}\frac{{S}_{N}^{0}}{{S}_{0}^{0}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{4.pt}{0ex}}a.s.$$We say market model S has no nontrivial bond replicating trading strategy, if there does not exist a bond replicating trading strategy X with $\widehat{X}\not\equiv 0$ (i.e., besides the trivial ones with $\widehat{X}=0$ a.s. there are no bond replicating trading strategies).

**Remark**

**2**(Interpretation of Definition 7)

**.**

**Theorem**

**1**(Multi-period market model with no nontrivial risk-free trading strategy)

**.**

- (a)
- S has no nontrivial risk-free trading strategy.
- (b)
- S is arbitrage-free and has no nontrivial bond replicating trading strategy.
- (c)
- For all $n=1,\dots ,N$ and all $\mathit{\eta}\in {\mathcal{L}}^{0}(\Omega ,{\mathcal{F}}_{n-1},\mathrm{P};{\mathbb{R}}^{M+1})$ with $\widehat{\mathit{\eta}}\not\equiv 0$ it is$$P\left({\left({S}_{n}-\frac{{S}_{n}^{0}}{{S}_{n-1}^{0}}{S}_{n-1}\right)}^{\top}\mathit{\eta}<0\right)>0.$$
- (d)
- S is arbitrage-free and the following holds for all trading strategies X and Y:$$\mathcal{W}\left(X\right)=\mathcal{W}\left(Y\right)\phantom{\rule{4.pt}{0ex}}a.s.\phantom{\rule{4.pt}{0ex}}\mathit{implies}\phantom{\rule{4.pt}{0ex}}\widehat{X}=\widehat{Y}\phantom{\rule{4.pt}{0ex}}a.s.$$

**Proof.**

**Remark**

**3**

**.**In the one-period case $N=1$, we can define $R:={S}_{1}^{0}/{S}_{0}^{0}$ and $x:={x}_{1}$. If we have ${S}_{0}^{\top}x={\mathcal{W}}_{0}$, then we obtain from Equation (4) that

#### 2.3. Trading Strategy Generating Function

**Definition**

**8**(Trading strategy generating function)

**.**

**Lemma**

**1.**

**Proof.**

**Example**

**1**(Buy and hold; constant number of shares).

**Example**

**2**(Constant weight/fixed fraction).

**Remark**

**4.**

## 3. Efficient Portfolios

#### 3.1. Performance Criteria

**Definition**

**9**(Absolute/relative drawdown process).

**Definition**

**10**(Multi-path expected log drawdown)

**.**

**Remark**

**5.**

#### 3.2. Optimization

**Setting**

**1.**

- (i)
- Multi-period market model$S\in {\mathcal{L}}^{2}(N;{\mathbb{R}}_{>0}^{M+1})$, $M,N\in \mathbb{N}$ (see Definition 2).
- (ii)
- Trading strategy, which is defined by a given trading strategy generating function $v:A\to {\mathcal{L}}^{0}(N-1;{\mathbb{R}}^{M+1})$ as in Definition 8 with non-empty and convex domain $A\subset {\mathbb{R}}^{M+1}$.
- (iii)
- Utility function $\mathfrak{u}:A\to \mathbb{R}\cup \{-\infty \}$, which is assumed to be proper concave.
- (iv)
- Risk function $\mathfrak{r}:A\to \mathbb{R}\cup \{\infty \}$, which is assumed to be proper convex.

**Problem**

**1.**

- (a)
- Let $\beta >0$ and $\mu \in \mathbb{R}$ be fixed. The minimum risk optimization problem is defined by$$\underset{x\in A}{min}\mathfrak{r}\left(x\right)\phantom{\rule{1.em}{0ex}}\mathit{subject}\phantom{\rule{4.pt}{0ex}}\mathit{to}\phantom{\rule{4.pt}{0ex}}\mathfrak{u}\left(x\right)\ge \mu ,\phantom{\rule{4pt}{0ex}}{S}_{0}^{\top}{v}_{1}\left(x\right)=\beta .$$
- (b)
- Let $\beta >0$ and $r\in \mathbb{R}$ be fixed. The maximum utility optimization problem is defined by$$\underset{x\in A}{max}\mathfrak{u}\left(x\right)\phantom{\rule{1.em}{0ex}}\mathit{subject}\phantom{\rule{4.pt}{0ex}}\mathit{to}\phantom{\rule{4.pt}{0ex}}\mathfrak{r}\left(x\right)\le r,\phantom{\rule{4pt}{0ex}}{S}_{0}^{\top}{v}_{1}\left(x\right)=\beta .$$

#### 3.3. Efficient Frontier

**Definition**

**11**(Risk utility space)

**.**

**Remark**

**6.**

**Remark**

**7.**

**Proposition**

**2**(Properties in risk utility space)

**.**

- (a)
- $\mathfrak{r}$ is closed proper convex if and only if ${\mathcal{B}}_{\mathfrak{r},A}\left(r\right)$ is closed for all $r\in \mathbb{R}$.$\mathfrak{u}$ is closed proper concave if and only if ${\mathcal{B}}_{\mathfrak{u},A}\left(\mu \right)$ is closed for all $\mu \in \mathbb{R}$.
- (b)
- Assume ${\mathcal{B}}_{\mathfrak{u},A}\left(\mu \right)$ and ${\mathcal{B}}_{\mathfrak{r},A}\left(r\right)$ are closed for all $\mu ,r\in \mathbb{R}$. If either ${\mathcal{B}}_{\mathfrak{u},A}\left(\mu \right)$ is compact for all $\mu \in \mathbb{R}$ or ${\mathcal{B}}_{\mathfrak{r},A}\left(r\right)$ is compact for all $r\in \mathbb{R}$, then ${\mathcal{B}}_{\mathfrak{r},\mathfrak{u},A}(r,\mu )$ is convex and compact for all $r,\mu \in \mathbb{R}$.
- (c)
- $\mathcal{G}(\mathfrak{r},\mathfrak{u};A)$ is convex and $(r,\mu )\in \mathcal{G}(\mathfrak{r},\mathfrak{u};A)$ implies that, for any $k>0$, we have $(r+k,\mu )\in \mathcal{G}(\mathfrak{r},\mathfrak{u};A)$ and $(r,\mu -k)\in \mathcal{G}(\mathfrak{r},\mathfrak{u};A)$.
- (d)
- If ${\mathcal{B}}_{\mathfrak{r},\mathfrak{u},A}(r,\mu )$ is compact for all $r,\mu \in \mathbb{R}$, then $\mathcal{G}(\mathfrak{r},\mathfrak{u};A)$ is closed.

**Proof.**

**Definition**

**12**(Efficient portfolio and efficient frontier)

**.**

**Theorem**

**2**(Properties of efficient frontier)

**.**

- (a)
- The efficient frontier ${\mathcal{G}}_{\mathit{eff}}(\mathfrak{r},\mathfrak{u};A)$ is located in the boundary of $\mathcal{G}(\mathfrak{r},\mathfrak{u};A)$ and has no vertical and no horizontal line segments.
- (b)
- If ${\mathcal{B}}_{\mathfrak{r},\mathfrak{u},A}(r,\mu )$ is compact for all $r,\mu \in \mathbb{R}$, then ${\mathcal{G}}_{\mathit{eff}}(\mathfrak{r},\mathfrak{u};A)$ is non-empty and equals to the non-vertical and non-horizontal part of the boundary of $\mathcal{G}(\mathfrak{r},\mathfrak{u};A)$, i.e.,$${\mathcal{G}}_{\mathit{eff}}(\mathfrak{r},\mathfrak{u};A)=\left\{(r,\mu )\in \partial \mathcal{G}(\mathfrak{r},\mathfrak{u};A)\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.166667em}{0ex}}(r-k,\mu ),(r,\mu +k)\notin \partial \mathcal{G}(\mathfrak{r},\mathfrak{u};A)\phantom{\rule{4pt}{0ex}}\forall k>0\right\},$$where ∂$\mathcal{G}(\mathfrak{r},\mathfrak{u};A)$ denotes the boundary of $\mathcal{G}(\mathfrak{r},\mathfrak{u};A)$ in ${\mathbb{R}}^{2}$.
- (c)
- If $B\subset A$ is convex, then ${\mathcal{G}}_{\mathit{eff}}(\mathfrak{r},\mathfrak{u};A)\cap \mathcal{G}(\mathfrak{r},\mathfrak{u};B)\subset {\mathcal{G}}_{\mathit{eff}}(\mathfrak{r},\mathfrak{u};B)$.

**Proof.**

- “⊂” follows from (a).
- Show “⊃”: Let $({r}_{0},{\mu}_{0})\in \left\{\right(r,\mu )\in \partial \mathcal{G}(\mathfrak{r},\mathfrak{u};A)\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.166667em}{0ex}}(r-k,\mu ),(r,\mu +k)\notin \partial \mathcal{G}(\mathfrak{r},\mathfrak{u};A)\phantom{\rule{4pt}{0ex}}\forall k>0\}$ be arbitrary. Then, since $\mathcal{G}(\mathfrak{r},\mathfrak{u};A)$ is closed by Proposition 2 (d), it has to be $({r}_{0},{\mu}_{0})\in \mathcal{G}(\mathfrak{r},\mathfrak{u};A)$. Hence, there must exist an ${x}_{0}\in A$ such that $\mathfrak{r}\left({x}_{0}\right)\le {r}_{0}$ and $\mathfrak{u}\left({x}_{0}\right)\ge {\mu}_{0}$. In addition, it must be $(r,\mu )\notin \mathcal{G}(\mathfrak{r},\mathfrak{u};A)$ for all $r\le {r}_{0}$ and $\mu \ge {\mu}_{0}$ with $(r,\mu )\ne ({r}_{0},{\mu}_{0})$, because $\mathcal{G}(\mathfrak{r},\mathfrak{u};A)$ is convex and unbounded from below and unbounded to the right by Proposition 2 (c). Consequently, even $\mathfrak{r}\left({x}_{0}\right)={r}_{0}$ and $\mathfrak{u}\left({x}_{0}\right)={\mu}_{0}$ must hold and ${x}_{0}$ thus is efficient, i.e., ${x}_{0}\in {\mathcal{G}}_{\mathrm{eff}}(\mathfrak{r},\mathfrak{u};A)$.
- Show ${\mathcal{G}}_{\mathrm{eff}}(\mathfrak{r},\mathfrak{u};A)\ne \varnothing $: Because of Setting 1, we have $dom\left(\mathfrak{u}\right)\cap dom\left(\mathfrak{r}\right)\ne \varnothing $, i.e., there exists ${x}_{1}\in dom\left(\mathfrak{u}\right)\cap dom\left(\mathfrak{r}\right)$ and it is $({r}_{1},{\mu}_{1}):=(\mathfrak{r}\left({x}_{1}\right),\mathfrak{u}\left({x}_{1}\right))\in \mathcal{G}(\mathfrak{r},\mathfrak{u};A)$. Since $\mathfrak{r}$ is convex and $\mathfrak{u}$ is concave, $\mathfrak{r}$ is bounded below and $\mathfrak{u}$ is bounded above on each compact set. The set ${\mathcal{B}}_{\mathfrak{r},\mathfrak{u},A}({r}_{1},{\mu}_{1})$ is compact by assumption. Hence, by definition of ${\mathcal{B}}_{\mathfrak{r},\mathfrak{u},A}({r}_{1},{\mu}_{1})$, the function $\mathfrak{r}$ on ${\mathcal{B}}_{\mathfrak{r},\mathfrak{u},A}({r}_{1},{\mu}_{1})$ is contained in say $[{r}_{*},{r}_{1}]$ and the function $\mathfrak{u}$ on ${\mathcal{B}}_{\mathfrak{r},\mathfrak{u},A}({r}_{1},{\mu}_{1})$ is contained in say $[{\mu}_{1},{\mu}^{*}]$. Therefore, the image of $(\mathfrak{r},\mathfrak{u})$ restricted on ${\mathcal{B}}_{\mathfrak{r},\mathfrak{u},A}({r}_{1},{\mu}_{1})$ is a subset of $\mathcal{G}(\mathfrak{r},\mathfrak{u};A)$ and $\varnothing \ne \mathcal{G}(\mathfrak{r},\mathfrak{u};A)\cap Q\subset [{r}_{*},{r}_{1}]\times [{\mu}_{1},{\mu}^{*}]$ for $Q=\{({r}^{\prime},{\mu}^{\prime})\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.166667em}{0ex}}{r}^{\prime}\le {r}_{1},\phantom{\rule{4pt}{0ex}}{\mu}^{\prime}\ge {\mu}_{1}\}$, see Figure 1. Clearly, there must be a point $({r}_{2},{\mu}_{2})\in \partial \mathcal{G}(\mathfrak{r},\mathfrak{u};A)\cap Q$ such that $({r}_{2}-k,{\mu}_{2})$ and $({r}_{2},\mu +k)$ do not belong to $\mathcal{G}(\mathfrak{r},\mathfrak{u};A)$ for all $k>0$. Since $\mathcal{G}(\mathfrak{r},\mathfrak{u};A)$ is closed, by Equation (33), the point $({r}_{2},{\mu}_{2})$ belongs to ${\mathcal{G}}_{\mathrm{eff}}(\mathfrak{r},\mathfrak{u};A)$, i.e., ${\mathcal{G}}_{\mathrm{eff}}(\mathfrak{r},\mathfrak{u};A)\ne \varnothing $.

**Definition**

**13.**

**Lemma**

**2**(Infima/Suprema of ${\mathcal{G}}_{\mathrm{eff}}(\mathfrak{r},\mathfrak{u};A)$)

**.**

**Proof.**

**Definition**

**14.**

**Corollary**

**1.**

- $I=[{r}_{min},{r}_{max}]$ and $J=[{\mu}_{min},{\mu}_{max}]$;
- $I=[{r}_{min},\infty )$ and $J=[{\mu}_{min},{\mu}_{max})$, where ${\mu}_{max}=\infty $ is possible;
- $I=({r}_{min},{r}_{max}]$ and $J=(-\infty ,{\mu}_{max}]$, where ${r}_{min}=-\infty $ is possible; or
- $I=({r}_{min},\infty )$ and $J=(-\infty ,{\mu}_{max})$, where ${\mu}_{max}=\infty $ and/or ${r}_{min}=-\infty $ is possible.

**Proof.**

**Proposition**

**3**(Functions related to efficient frontier)

**.**

**Proof.**

**Corollary**

**2**(Parametrization of efficient frontier as graph)

**.**

**Proof.**

#### 3.4. Efficient Portfolios

**Theorem**

**3**(Existence for Problem 1)

**.**

- (a)
- For each $\mu \in J$, there exists an efficient element ${x}_{\mu}\in A$ with $\mathfrak{u}\left({x}_{\mu}\right)=\mu $. The element ${x}_{\mu}$ also solves Equation (MinR).
- (b)
- For each $r\in I$, there exists an efficient element ${y}_{r}\in A$ with $\mathfrak{r}\left({y}_{r}\right)=r$. The element ${y}_{r}$ also solves Equation (MaxU).
- (c)

**Proof.**

**Theorem**

**4**(Uniqueness and efficient portfolio path)

**.**

- (a)
- For each $\mu \in J$, there is exactly one efficient element ${x}_{\mu}\in A$ with $\mathfrak{u}\left({x}_{\mu}\right)=\mu $, which in addition is the unique solution of Equation (MinR).Furthermore, the mapping $\tilde{\gamma}:J\to A,\phantom{\rule{4pt}{0ex}}\mu \mapsto {x}_{\mu}$ is continuous.For each $\mu \notin J$ and $\mu \ge {\mu}_{max}=supJ$, there does not exist any solution of Equation (MinR).If ${\mu}_{min}>-\infty $, then for $\mu \notin J$ and $\mu \le {\mu}_{min}$ (i.e., $\mu <{\mu}_{min}$, see Corollary 1) the solution of Equation (MinR) is not necessarily unique and can be an element in A which is not efficient.
- (b)
- For each $r\in I$, there is exactly one efficient element ${y}_{r}\in A$ with $\mathfrak{r}\left({y}_{r}\right)=r$, which in addition is the unique solution of Equation (MaxU).Furthermore, the mapping $\tilde{\nu}:I\to A,\phantom{\rule{4pt}{0ex}}r\mapsto {y}_{r}$ is continuous.For each $r\notin I$ and $r\le {r}_{min}=infI$, there does not exist any solution of Equation (MaxU).If ${r}_{max}<\infty $, then for $r\notin I$ and $r\ge {r}_{max}$ (i.e., $r>{r}_{max}$) the solution of Equation (MaxU) is not necessarily unique and can be an element in A, which is not efficient.

**Proof.**

**Remark**

**8**

## 4. Application

**Lemma**

**3**(utility function; logarithm of TWR)

**.**

**Proof.**

**Lemma**

**4**(risk function; logarithm of TWR)

**.**

**Proof.**

**Remark**

**9**

**Remark**

**10**(ρ

_{ln}and u

_{logTWR}in finite probability space)

**.**

**Remark 11**(Notes on $dom\left({\mathfrak{u}}_{\mathrm{logTWR}}\right)$ and ${A}_{\mathrm{twr}}$)

**.**

- (a)
- Clearly $dom\left({\mathfrak{u}}_{\mathrm{logTWR}}\right)\subset {A}_{\mathrm{twr}}$.
- (b)
- If $f\in {A}_{\mathrm{twr}}$, then, using Equation (20), it follows that ${\mathcal{W}}_{n}\left({v}_{\mathrm{twr}}\left(f\right)\right)={\mathcal{W}}_{0}{\prod}_{k=1}^{n}(1+{T}_{k}^{\top}f)>0$ a.s. Of course, this is trivial and directly follows from the definition of ${A}_{\mathrm{twr}}$ in Equation (42). In fact, ${A}_{\mathrm{twr}}$ is defined as the admissible set of ${v}_{\mathrm{twr}}$ (see Example 2).
- (c)
- We have$$\begin{array}{cc}dom\left({\mathfrak{u}}_{\mathrm{logTWR}}\right)\hfill & =\left\{f\in {\mathbb{R}}^{M+1}\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.166667em}{0ex}}E\left[ln\left(1+{T}_{n}^{\top}f\right)\right]>-\infty \phantom{\rule{4pt}{0ex}}\mathit{for}\phantom{\rule{4.pt}{0ex}}\mathit{all}\phantom{\rule{4.pt}{0ex}}n=1,\dots ,N\right\}\hfill \\ \hfill & =\left\{f\in {\mathbb{R}}^{M+1}\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.166667em}{0ex}}{\int}_{\Omega}ln\left(1+{T}_{n}{\left(\omega \right)}^{\top}f\right)\phantom{\rule{0.166667em}{0ex}}d\mathrm{P}\left(\omega \right)>-\infty \phantom{\rule{4pt}{0ex}}\mathit{for}\phantom{\rule{4.pt}{0ex}}\mathit{all}\phantom{\rule{4.pt}{0ex}}n=1,\dots ,N\right\}.\hfill \end{array}$$Proof: The second equality holds by definition. For the first one, the relation “⊃” is obvious. Let now $f\in dom\left({\mathfrak{u}}_{\mathrm{logTWR}}\right)\subset {A}_{\mathrm{twr}}$ be arbitrary. Since ${\mathfrak{u}}_{\mathrm{logTWR}}\left(f\right)<\infty $ by Lemma 3 we have ${\mathfrak{u}}_{\mathrm{logTWR}}\left(f\right)\in \mathbb{R}$. In addition, $E[ln(1+{T}_{n}^{\top}f)]<\infty $ holds for $n=1,\dots ,N$ (cf. Equation (45)). Hence, it must be $E[ln(1+{T}_{n}^{\top}f)]>-\infty $ for $n=1,\dots ,N$, which shows the relation “⊂” and therefore the equality.
- (d)
- Define$${A}_{\mathrm{twr}}^{*}:=\left\{f\in {\mathbb{R}}^{M+1}\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.166667em}{0ex}}\mathit{there}\phantom{\rule{4.pt}{0ex}}\mathit{exists}\phantom{\rule{4.pt}{0ex}}\epsilon >0\phantom{\rule{4.pt}{0ex}}\mathit{such}\phantom{\rule{4.pt}{0ex}}\mathit{that}\phantom{\rule{4pt}{0ex}}1+{T}_{n}^{\top}f\ge \epsilon \phantom{\rule{4.pt}{0ex}}a.s.\phantom{\rule{4.pt}{0ex}}\mathit{for}\phantom{\rule{4.pt}{0ex}}\mathit{all}\phantom{\rule{4.pt}{0ex}}n=1,\dots ,N\right\}.$$Then, we obtain ${A}_{\mathrm{twr}}^{*}\subset dom\left({\mathfrak{u}}_{\mathrm{logTWR}}\right)$.Proof: Let $f\in {A}_{\mathrm{twr}}^{*}$ be arbitrary. Then, $ln(1+{T}_{n}^{\top}f)\ge ln\left(\epsilon \right)>-\infty $ a.s. This, of course, gives $E[ln(1+{T}_{n}^{\top}f)]\ge ln\left(\epsilon \right)>-\infty $.

**Theorem**

**5**(Existence and uniqueness for u

_{logTWR}and ρ

_{ln})

**.**

- (a)
- (Growth optimal trading strategy) The problem in Equation (MaxTWR) without risk restriction, i.e.,$$\underset{f\in A}{max}{\mathfrak{u}}_{\mathrm{logTWR}}\left(f\right)\phantom{\rule{1.em}{0ex}}\mathit{subject}\phantom{\rule{4.pt}{0ex}}\mathit{to}\phantom{\rule{4.pt}{0ex}}{S}_{0}^{\top}{\left({v}_{\mathrm{twr}}\right)}_{1}\left(f\right)=\beta $$
- (b)
- (Risk minimal trading strategy) The problem in Equation (MinDD) without utility restriction, i.e.,$$\underset{f\in A}{min}{\rho}_{\mathrm{ln}}\left(f\right)\phantom{\rule{1.em}{0ex}}\mathit{subject}\phantom{\rule{4.pt}{0ex}}\mathit{to}\phantom{\rule{4.pt}{0ex}}{S}_{0}^{\top}{\left({v}_{\mathrm{twr}}\right)}_{1}\left(f\right)=\beta ,$$
- (c)
- For each $\mu \in J=[{\mu}_{min},{\mu}_{max}]\ne \varnothing $, there is exactly one efficient element ${f}_{\mu}^{*}\in A$ with ${\mathfrak{u}}_{\mathrm{logTWR}}\left({f}_{\mu}^{*}\right)=\mu $, which is also the unique solution of Equation (MinDD). The mapping $\tilde{\gamma}:J\to A,\phantom{\rule{4pt}{0ex}}\mu \mapsto {f}_{\mu}^{*}$ is continuous.
- (d)
- For each $r\in I=[{r}_{min},{r}_{max}]\ne \varnothing $, there is exactly one efficient element ${\tilde{f}}_{r}^{*}\in A$ with ${\rho}_{\mathrm{ln}}\left({\tilde{f}}_{r}^{*}\right)=r$, which is also the unique solution of Equation (MaxTWR). The mapping $\tilde{\nu}:I\to A,\phantom{\rule{4pt}{0ex}}\mu \mapsto {\tilde{f}}_{\mu}^{*}$ is continuous.

**Proof.**

**Lemma**

**5.**

**Proof.**

**Lemma**

**6.**

**Proof.**

**Corollary**

**3.**

**Proof.**

**Example 3**($dom\left({\mathfrak{u}}_{\mathrm{logTWR}}\right)$, ${A}_{\mathrm{twr}}$ and ${A}_{\mathrm{twr}}^{*}$)

**.**

- (a)
- Let ${T}_{1}\left(t\right):=exp(-1/t)-1\in (-1,0)$ for $t\in (0,1)$. Then$${\mathfrak{u}}_{\mathrm{logTWR}}\left(f\right)={h}_{1}\left(f\right)={\int}_{0}^{1}ln\left(1+{T}_{1}\left(t\right)f\right)\phantom{\rule{0.166667em}{0ex}}dt,\phantom{\rule{2.em}{0ex}}f\in dom\left({h}_{1}\right)=dom\left({\mathfrak{u}}_{\mathrm{logTWR}}\right).$$$${h}_{1}\left(1\right)={\int}_{0}^{1}ln(1+{T}_{1}\left(t\right))\phantom{\rule{0.166667em}{0ex}}dt={\int}_{0}^{1}-\frac{1}{t}\phantom{\rule{0.166667em}{0ex}}dt=-\infty ,$$
- (b)
- Let ${\tilde{T}}_{1}\left(t\right):=exp(-1/\sqrt{t})-1\in (-1,0)$ for $t\in (0,1)$. Reasoning as in (a), we again get ${\tilde{A}}_{\mathrm{twr}}^{*}=(-\infty ,1)$ and ${\tilde{A}}_{\mathrm{twr}}=(-\infty ,1]$. However, this time$${\tilde{h}}_{1}\left(1\right)={\int}_{0}^{1}ln(1+{\tilde{T}}_{1}\left(t\right))\phantom{\rule{0.166667em}{0ex}}dt={\int}_{0}^{1}-\frac{1}{\sqrt{t}}\phantom{\rule{0.166667em}{0ex}}dt=-2.$$Hence, ${f}^{*}:=1\in dom\left({\tilde{h}}_{1}\right)=dom\left({\mathfrak{u}}_{\mathrm{logTWR}}\right)=(-\infty ,1]$ and therefore ${\tilde{A}}_{\mathrm{twr}}^{*}\u2acbdom\left({\mathfrak{u}}_{\mathrm{logTWR}}\right)$. Again ${\mathfrak{u}}_{\mathrm{logTWR}}$ is closed proper concave by Corollary 3.

## 5. Conclusions and Outlook

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Illustration to show ${\mathcal{G}}_{\mathrm{eff}}(\mathfrak{r},\mathfrak{u};A)\ne \varnothing $.

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**MDPI and ACS Style**

Maier-Paape, S.; Platen, A.; Zhu, Q.J.
A General Framework for Portfolio Theory. Part III: Multi-Period Markets and Modular Approach. *Risks* **2019**, *7*, 60.
https://doi.org/10.3390/risks7020060

**AMA Style**

Maier-Paape S, Platen A, Zhu QJ.
A General Framework for Portfolio Theory. Part III: Multi-Period Markets and Modular Approach. *Risks*. 2019; 7(2):60.
https://doi.org/10.3390/risks7020060

**Chicago/Turabian Style**

Maier-Paape, Stanislaus, Andreas Platen, and Qiji Jim Zhu.
2019. "A General Framework for Portfolio Theory. Part III: Multi-Period Markets and Modular Approach" *Risks* 7, no. 2: 60.
https://doi.org/10.3390/risks7020060