# A General Framework for Portfolio Theory—Part I: Theory and Various Models

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. A Portfolio Model

**Definition**

**1.**

**Remark**

**1.**

**Definition**

**2.**

#### 2.2. Convex Programming

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Remark**

**2.**

**Assumption**

**1.**

**Proposition**

**3.**

**Definition**

**3.**

**Theorem**

**1.**

- (i)
- ${\lambda}_{y}\ge 0$,
- (ii)
- the Lagrangian $L(x,\lambda )$ defined in (6) attains a global minimum at $\overline{x}$, and
- (iii)
- λ satisfies the complementary slackness condition$$\begin{array}{c}\hfill \langle \lambda ,(g\left(\overline{x}\right)-y,h\left(\overline{x}\right)-z)\rangle =\langle {\lambda}_{y},g\left(\overline{x}\right)-y\rangle =0,\end{array}$$

**Proof.**

**Remark**

**3.**

## 3. Efficient Trade-Off between Risk and Utility

#### 3.1. Technical Assumptions

**Assumption**

**2.**

- (r1)
- (Riskless Asset Contributes No risk) The risk measure $\mathfrak{r}\left(x\right)=\widehat{\mathfrak{r}}\left(\widehat{x}\right)$ is a function of only the risky part of the portfolio, where ${x}^{\top}=({x}_{0},{\widehat{x}}^{\top})$.
- (r1n)
- (Normalization) There is at least one portfolio of purely bonds in A. Furthermore, $\mathfrak{r}\left(x\right)=0$ if and only if x contains only riskless bonds, i.e., ${x}^{\top}=({x}_{0},{\widehat{0}}^{\top})$ for some ${x}_{0}\in \mathbb{R}$.
- (r2)
- (Diversification Reduces Risk) The risk function $\mathfrak{r}$ is convex.
- (r2s)
- (Diversification Strictly Reduces Risk) The risk function $\widehat{\mathfrak{r}}$ is strictly convex.
- (r3)
- (Positive homogeneous) For $t>0$, $\widehat{\mathfrak{r}}\left(t\widehat{x}\right)=t\widehat{\mathfrak{r}}\left(\widehat{x}\right)$.
- (r3s)
- (Diversification Strictly Reduces Risk on Level Sets) The risk function $\widehat{\mathfrak{r}}$ satisfies (r3) and, for all $\widehat{x}\ne \widehat{y}$ with $\widehat{\mathfrak{r}}\left(\widehat{x}\right)=\widehat{\mathfrak{r}}\left(\widehat{y}\right)=1$ and $\alpha \in (0,1)$,$$\widehat{\mathfrak{r}}(\alpha \widehat{x}+(1-\alpha )\widehat{y})<\alpha \widehat{\mathfrak{r}}\left(\widehat{x}\right)+(1-\alpha )\widehat{\mathfrak{r}}\left(\widehat{y}\right)=1.$$

**Lemma**

**1.**

- (a)
- $\mathfrak{r}$ satisfies (r2), and
- (b)
- $f\left(x\right)=\widehat{f}\left(\widehat{x}\right)={\left[\widehat{\mathfrak{r}}\left(\widehat{x}\right)\right]}^{2}$ satisfies (r1), (r1n) and (r2s).

**Proof.**

**Remark**

**4.**

**Assumption**

**3.**

- (u1)
- (Profit Seeking) The utility function u is an increasing function.
- (u2)
- (Diminishing Marginal Utility) The utility function u is concave.
- (u2s)
- (Strict Diminishing Marginal Utility) The utility function u is strictly concave.
- (u3)
- (Bankrupcy Forbidden) For $t<0$, $u\left(t\right)=-\infty $.
- (u4)
- (Unlimited Growth) For $t\to +\infty $, we have $u\left(t\right)\to +\infty $.

**Definition**

**4.**

- (a)
- (No Nontrivial Riskless Portfolio) We say a portfolio x is riskless if$$\langle {S}_{1}-R{S}_{0},x\rangle \ge 0.$$
- (b)
- (No Arbitrage) We say x is an arbitrage if it is riskless and there exists some $\omega \in \mathsf{\Omega}$ such that$$\langle {S}_{1}\left(\omega \right)-R{S}_{0},x\rangle \ne 0.$$
- (c)
- (Nontrivial Bond Replicating Portfolio) We say that ${x}^{\top}=({x}_{0},{\widehat{x}}^{\top})$ is a nontrivial bond replicating portfolio if $\widehat{x}\ne \widehat{0}$ and$$\langle {S}_{1}-R{S}_{0},x\rangle =0.$$

**Proposition**

**4.**

**Proof.**

**Theorem**

**2.**

- (i)
- There is no nontrivial bond replicating portfolio.
- (ii)
- For every nontrivial portfolio x with $\widehat{x}\ne \widehat{0}$, there exists some $\omega \in \mathsf{\Omega}$ such that$$\begin{array}{c}\hfill \langle {S}_{1}\left(\omega \right)-R{S}_{0},x\rangle <0.\end{array}$$
- (ii*)
- For every risky portfolio $\widehat{x}\ne \widehat{0}$, there exists some $\omega \in \mathsf{\Omega}$ such that$$\begin{array}{c}\hfill \langle {\widehat{S}}_{1}\left(\omega \right)-R{\widehat{S}}_{0},\widehat{x}\rangle <0.\end{array}$$
- (iii)
- The matrix$$\begin{array}{c}\hfill G:=\left[\begin{array}{cccc}{S}_{1}^{1}\left({\omega}_{1}\right)-R{S}_{0}^{1}& {S}_{1}^{2}\left({\omega}_{1}\right)-R{S}_{0}^{2}& \dots & {S}_{1}^{M}\left({\omega}_{1}\right)-R{S}_{0}^{M}\\ {S}_{1}^{1}\left({\omega}_{2}\right)-R{S}_{0}^{1}& {S}_{1}^{2}\left({\omega}_{2}\right)-R{S}_{0}^{2}& \dots & {S}_{1}^{M}\left({\omega}_{2}\right)-R{S}_{0}^{M}\\ \vdots & \vdots & \vdots & \vdots \\ {S}_{1}^{1}\left({\omega}_{N}\right)-R{S}_{0}^{1}& {S}_{1}^{2}\left({\omega}_{N}\right)-R{S}_{0}^{2}& \dots & {S}_{1}^{M}\left({\omega}_{N}\right)-R{S}_{0}^{M}\end{array}\right]\in {\mathbb{R}}^{N\times M}\end{array}$$

**Proof.**

**Corollary**

**1.**

**Proof.**

#### 3.2. Efficient Frontier for the Risk-Utility Trade-Off

**Proposition**

**5.**

- (1)
- When ρ is invariant under adding constants, i.e., $\rho (X)=\rho (X+c)$, for any $X\in RV(\mathsf{\Omega},{2}^{\mathsf{\Omega}},P)$ and $c\in \mathbb{R}$. A useful example is when ρ is the standard deviation.
- (2)
- When ρ is restricted to a set of admissible portfolios A with unit initial cost. In this case, we can see that$$\begin{array}{c}\hfill \widehat{\mathfrak{r}}(\widehat{x}):=\rho (R+{({\widehat{S}}_{1}-R{\widehat{S}}_{0})}^{\top}\widehat{x})=\rho ({S}_{1}^{\top}x).\end{array}$$

**Proposition**

**6.**

- (a)
- the financial market ${S}_{t}$ has no nontrivial riskless portfolio,
- (b)
- the utility function u satisfies condition (u2s) in Assumption 3, and
- (c)
- A is a set of admissible portfolios with unit initial cost as in Definition 2.

**Assumption**

**A4.**

**Proposition**

**7.**

**Proof.**

**Definition**

**5.**

**Theorem**

**3.**

**Proof.**

**Remark**

**5.**

#### 3.3. Representation of Efficient Frontier

**Proposition**

**8.**

**Proof.**

**Theorem**

**4.**

**Proof.**

#### 3.4. Efficient Portfolios

**Theorem**

**5.**

- (c0)
- there exists some $\overline{x}\in A$ with $\overline{\mu}:=\mathbb{E}\left[u({S}_{1}^{\top}\overline{x})\right]$ and $\overline{r}:=\mathfrak{r}(\overline{x})$ finite.

- (c1)
- The risk measure $\mathfrak{r}$ satisfies conditions (r1) and (r2s) in Assumption 2 and the utility function satisfies conditions (u1) and (u2) in Assumption 3.
- (c2)
- The risk measure $\mathfrak{r}$ satisfies conditions (r1) and (r2) in Assumption 2 and the utility function satisfies conditions (u1) and (u2s) in Assumption 3.
- (c3)
- The risk measure $\mathfrak{r}$ satisfies conditions (r1), (r1n) and (r3s) in Assumption 2 and the utility function satisfies conditions (u1) and (u2) in Assumption 3.

**Proof.**

**Remark**

**6.**

**Proposition**

**9.**

**Proof.**

**Corollary**

**2.**

- (a)
- ${r}_{min}\in I$ if and only if ${\mu}_{min}\in J$, and ${r}_{max}\in I$ if and only if ${\mu}_{max}\in J$.
- (b)
- If ${r}_{min}\in I$ then ${\mu}_{min}=\nu ({r}_{min})$ and $\gamma ({\mu}_{min})={r}_{min}$.
- (c)
- If ${\mu}_{max}\in J$ then ${r}_{max}=\gamma ({\mu}_{max})$ and $\nu ({r}_{max})={\mu}_{max}$.
- (d)
- (i)
- If ${r}_{min}\in I$ and ${\mu}_{max}\in J$ then $I=[{r}_{min},{r}_{max}]$ and $J=[{\mu}_{min},{\mu}_{max}]$.
- (ii)
- If ${r}_{min}\notin I$ and ${\mu}_{max}\in J$ then $I=({r}_{min},{r}_{max}]$ and $J=(-\infty ,{\mu}_{max}]$.
- (iii)
- If ${r}_{min}\in I$ and ${\mu}_{max}\notin J$ then $I=[{r}_{min},\infty )$ and $J=[{\mu}_{min},{\mu}_{max})$.
- (iv)
- If ${r}_{min}\notin I$ and ${\mu}_{max}\notin J$ then $I=({r}_{min},\infty )$ and $J=(-\infty ,{\mu}_{max})$.

**Proof.**

**Remark**

**7.**

**Example**

**1.**

**Example**

**2.**

## 4. Markowitz Portfolio Theory and CAPM Model

#### 4.1. Markowitz Portfolio Theory

**Theorem**

**6.**

**Theorem**

**7.**

#### 4.2. Capital Asset Pricing Model

**Theorem**

**8.**

**Remark**

**8.**

**Theorem**

**9.**

## 5. Affine Efficient Frontier for Positive Homogeneous Risk Measure

**Theorem**

**10.**

**Proof.**

**Corollary**

**3.**

**Proof.**

**Remark**

**9.**

**Theorem**

**11.**

**Proof.**

**Remark**

**10.**

**Example**

**3.**

## 6. Growth Optimal and Leverage Space Portfolio

**Remark**

**11.**

**Theorem**

**12.**

**Lemma**

**2.**

**Proof.**

**Proof**

**of**

**Theorem**

**12.**

**Theorem**

**13.**

**Proof.**

**Remark**

**12.**

**Theorem**

**14.**

**Proof.**

**Example**

**4.**

**Remark**

**13.**

## 7. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Efficient frontier when ${r}_{min}>0$ and ${\mu}_{max}$ is finite and attained as maximum.

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**MDPI and ACS Style**

Maier-Paape, S.; Zhu, Q.J.
A General Framework for Portfolio Theory—Part I: Theory and Various Models. *Risks* **2018**, *6*, 53.
https://doi.org/10.3390/risks6020053

**AMA Style**

Maier-Paape S, Zhu QJ.
A General Framework for Portfolio Theory—Part I: Theory and Various Models. *Risks*. 2018; 6(2):53.
https://doi.org/10.3390/risks6020053

**Chicago/Turabian Style**

Maier-Paape, Stanislaus, and Qiji Jim Zhu.
2018. "A General Framework for Portfolio Theory—Part I: Theory and Various Models" *Risks* 6, no. 2: 53.
https://doi.org/10.3390/risks6020053