A General Framework for Portfolio Theory. Part I: theory and various models

Utility and risk are two often competing measurements on the investment success. We show that efficient trade-off between these two measurements for investment portfolios happens, in general, on a convex curve in the two dimensional space of utility and risk. This is a rather general pattern. The modern portfolio theory of Markowitz [H. Markowitz, Portfolio Selection, 1959] and its natural generalization, the capital market pricing model, [W. F. Sharpe, Mutual fund performance , 1966] are special cases of our general framework when the risk measure is taken to be the standard deviation and the utility function is the identity mapping. Using our general framework, we also recover the results in [R. T. Rockafellar, S. Uryasev and M. Zabarankin, Master funds in portfolio analysis with general deviation measures, 2006] that extends the capital market pricing model to allow for the use of more general deviation measures. This generalized capital asset pricing model also applies to e.g. when an approximation of the maximum drawdown is considered as a risk measure. Furthermore, the consideration of a general utility function allows to go beyond the"additive"performance measure to a"multiplicative"one of cumulative returns by using the log utility. As a result, the growth optimal portfolio theory [J. Lintner, The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets, 1965] and the leverage space portfolio theory [R. Vince, The Leverage Space Trading Model, 2009] can also be understood under our general framework. Thus, this general framework allows a unification of several important existing portfolio theories and goes much beyond.


Introduction
The Markowitz modern portfolio theory [15] pioneered the quantitative analysis of financial economics. The most important idea proposed in this theory is that one should focus on the trade-off between expected return and the risk measured by the standard deviation. Mathematically, the modern portfolio theory leads to a quadratic optimization problem with linear constraints. Using this simple mathematical structure Markowitz gave a complete characterization of the efficient frontier for trade-off the return and risk. Tobin [26] showed that the efficient portfolios as an affine function of the expected return. Markowitz portfolio theory was later generalized by Lintner [9], Mossin [17], Sharpe [22] and Treynor [25] in the capital asset pricing model (CAPM) by involving a riskless bond. In the CAPM model, both the efficient frontier and the related efficient portfolios are affine in terms of the expected return [22,26].
The nice structures of the solutions in the modern portfolio theory and the CAPM model afford many applications. For example, the CAPM model is designed to provide reasonable price for risky assets in the market place. Sharpe used the ratio of excess return to risk (called the Sharpe ratio) to provide a measurement for investment performance [23]. Also the affine structure of the efficient portfolio in terms of the expected return leads to the concept of a market portfolio as well as the two fund theorem [26] and the one fund theorem [22,26]. These results provided a theoretical foundation for passive investment strategies.
While using the expected return and standard deviation as measures for reward and risk of a portfolio brings much convenience in the mathematical analysis, many other measures are more realistic. Since Bernoulli studied the St. Petersburg paradox [2], concave utility functions have been widely accepted as a more appropriate measure of the reward. General expected utilities have been used in many cases to measure the performance of a portfolio. On the other hand, current drawdown [13], maximum drawdown and its approximations [10,12,30], deviation measure [20], conditional value at risk [19] and more abstract coherent risk measures [1] are widely used as risk measures in practices. A common thread in these risk measures is that they are convex reflecting the belief that diversification reduces risk. The goal of this paper is to extend the modern portfolio theory into a general framework under which one can analyze efficient portfolios that trade-off between a convex risk measure and a reward captured by an expected utility. We phrase our primal problem as a convex portfolio optimization problem of minimizing a convex risk measure subject to the constraint that the expected utility of the portfolio is above a certain level. Thus, convex duality plays a crucial role and the structure of the solutions to both the primal and dual problems often have significant financial implications. We show that, in the space of risk measure and expected utility, efficient trade-off happens on an increasing concave curve. We also show that the efficient portfolios continuously depend on the level of the expected utility.
The Markowitz modern portfolio theory and the capital asset pricing model are, of course, special cases of this general theory. Markowitz determines portfolios of purely risky assets which provide an efficient trade-off between expected return and risk measured by the standard deviation (or equivalently the variance). Mathematically, this is a class of convex programming problems of minimizing the standard deviation of the portfolio parameterized by the level of the expected returns. The capital asset pricing model, in essence, extends the Markowitz modern portfolio theory by including a riskless bond in the portfolio. We observe that the space of the risk-expected return is, in fact, the space corresponding to the dual of the Markowitz portfolio problem. The shape of the famous Markowitz bullet is a manifestation of the well known fact that the optimal value function of a convex programming problem is convex with respect to the level of constraint. As mentioned above, the Markowitz portfolio problem is a quadratic optimization problem with linear constraint. This special structure of the problem dictates the affine structure of the optimal portfolio as a function of the expected return (see Theorem 4.1). This affine structure leads to the important two fund theorem that provides a theoretical foundation for the passive investment method. For the capital asset pricing model, such an affine structure appear in both the primal and dual representation of the solutions which leads to the two fund separation theorem in the portfolio space and the capital market line in the dual space of risk-return trade-off (cf. Theorem 4.5).
The flexibility in choosing different risk measures allows us to extend the analysis of the essentially quadratic risk measure pioneered by Markowitz to a wider range. For example, when the risk measure is a deviation measure [20], which happens e.g. when an approximation of the current drawdown is considered (see [14]), and the expected return is used to gauge the performance we show that the affine structure of the efficient solution in the classical capital market pricing model is preserved (cf. Theorem 5.1), recovering in particular the results in [20]. This is significant in that it shows that the passive investment strategy is justifiable in a wide range of settings.
The consideration of a general utility function, however, allows us to go beyond the "additive" performance measure in modern portfolio theory to a "multiplicative" one including cumulative returns when, for example, using the log utility. As a result the growth optimal portfolio theory [9] and the leverage space portfolio theory [28] can also be understood under our general framework. The optimal growth portfolio pursues to maximize the expected log utility which is equivalent to maximize the expected cumulative compound return. It is known that the growth optimal portfolio is usually too risky. Thus, practitioners often scale back the risky exposure from a growth optimal portfolio. In our general framework, we consider the portfolio that minimizes a risk measure given a fixed level of expected log utility. Under reasonable conditions, we show that such portfolios form a path parameterized by the level of expected log utility in the portfolio space that connects the optimal growth portfolio and the portfolio of a riskless bond (see Theorem 6.4). In general, for different risk measures we will derive different paths. These paths provide justifications for risk reducing curves proposed in the leverage space portfolio theory [28]. The dual problem projects the efficient trade-off path into a concave curve in the risk-expected log utility space parallel to the role of Markowitz bullet in the modern portfolio theory and the capital market line in the capital asset pricing model. Unlike the modern portfolio theory and the capital asset pricing model, under the no arbitrage assumption, the efficient frontier here is usually a finite increasing concave curve. The lower left endpoint of the curve corresponds to the portfolio of pure riskless bond and the upper right endpoint corresponds to the growth optimal portfolio. The increasing nature of the curve tells us that the more risk we take the more cumulative return we can expect. The concavity of the curve indicates, however, that with the increase of the risk the marginal increase of the expected cumulative return will decrease. Thus, a risk averse investor will usually not choose the optimal growth portfolio. It is also interesting to observe that considering the dual problem corresponding to the growth optimal portfolio problem will leads to a version of the fundamental theorem of asset pricing (see Theorem 6.10) that connects the existence of an equivalent martingale measure to no arbitrage.
Besides unifying the several important results laid out above, the general framework has many new applications. In this first installment of the paper, we layout the framework, derive the theoretical results of crucial importance and illustrate them with a few examples. More new applications will appear in the subsequent papers [3,14]. We arrange the paper as follows: First we discuss necessary preliminaries in the next section. Section 3 is devoted to our main result: a framework to trade-off between risk and utility of portfolios and its properties. In Section 4 we give a unified treatment of Markowitz portfolio theory, capital asset pricing model, and the Sharpe ratio. Section 5 is devoted to a discussion on the conditions under which the optimal trade-off portfolio possesses an affine structure. Section 6 discusses growth optimal portfolio theory and leverage portfolio theory. We also highlight some related important applications such as the fundamental theorem of asset pricing. We conclude in Section 7 pointing to applications worthy of further investigation.

A portfolio model
We consider a simple one period financial market model S on an economy with finite states represented by a sample space Ω = {ω 1 , ω 2 , . . . , ω N }. We use a probability space (Ω, 2 Ω , P ) to represent the states of the economy and their corresponding probability of occurring, where 2 Ω is the algebra of all subsets of Ω. The space of random variables on (Ω, 2 Ω , P ) is denoted RV (Ω, 2 Ω , P ) and it is used to represent the payoff of risky financial assets. Since the sample space Ω is finite, RV (Ω, 2 Ω , P ) is a finite dimensional vector space. We use RV + (Ω, 2 Ω , P ) to represent of the cone of nonnegative random variables in RV (Ω, 2 Ω , P ). Introducing the inner product RV (Ω, 2 Ω , P ) becomes a (finite dimensional) Hilbert space.

Definition 2.1. (Financial Market)
We say that S t = (S 0 t , S 1 t , . . . , S M t ), t = 0, 1 is a financial market in a one period economy provided that S 0 ∈ R M +1 + and S 1 ∈ (0, ∞) × RV + (Ω, 2 Ω , P ) M . Here S 0 0 = 1, S 0 1 = R > 0 represents a risk free bond with a positive return when R > 1. The rest of the components S m t , m = 1, . . . , M represent the price of the m-th risky financial asset at time t. We will use the notation S t = (S 1 t , · · · , S M t ) when we need to focus on the risky assets. We assume that S 0 is a constant vector representing the prices of the assets in this financial market at t = 0. The risk is modeled by assuming S 1 = (S 1 1 , . . . , S M 1 ) to be a nonnegative random vector on the probability space (Ω, 2 Ω , P ), that is S m 1 ∈ RV + (Ω, 2 Ω , P ), m = 1, 2, . . . , M . A portfolio is a column vector x ∈ R M +1 whose components x m represent the share of the m-th asset in the portfolio and S m t x m is the portion of capital invested in asset m at time t. Hence x 0 corresponds to the investment in the risk free bond and x = (x 1 , . . . , x M ) ⊤ is the risky part.
We often need to restrict the selection of portfolios. For example, in many applications we consider only portfolios with unit initial cost, i.e. S 0 · x = 1. Thus, the following definition.

Definition 2.2. (Admissible Portfolio)
We say that A ⊂ R M +1 is a set of admissible portfolios provided that A is a nonempty closed and convex set. We say that A is a set of admissible portfolios with unit initial price provided that A is a closed convex subset of {x ∈ R M +1 : S 0 · x = 1}.

Convex programming problems
Let X be a finite dimensional Banach space. Recall that a set C ⊂ X is convex if, for any x, y ∈ C and s ∈ [0, 1], sx + (1 − s)y ∈ C. For an extended valued function f : X → R ∪ {+∞} we define its domain by We say f is lower semicontinuous if epi(f ) is a closed set. The following proposition characterizes an epigraph of a function.
The key is to observe that, for a set F with the structure in (2.1), a function is well defined and then F = epi(f ) holds.
Q.E.D. We say a function f is convex if epi(f ) is a convex set. Alternatively, f is convex if and only if, for any x, y ∈ dom(f ) and s ∈ [0, 1], +∞). We say f is concave when −f is convex and we say f is upper semicontinuous if −f is lower semicontinuous. Define the hypograph of a function f by Then a symmetric version of Proposition 2.3 is Moreover, the function f can be defined by Since utility functions are concave and risk measures are usually convex, the analysis of a general trade-off between utility and risk naturally leads to a convex programming problem. The general form of such convex programming problems is where f , g and h satisfy the following assumption. Convex programming problems have nice properties due to the convex structure. We briefly recall the pertinent results related to convex programming. First the optimal value function v is convex. This is a well-known result that can be found in standard books on convex analysis, e.g. [4]. It is, however, crucial for our applications below and, thus, we list it as a lemma and give a brief proof below for completeness. Proposition 2.7. (Convexity of Optimal Value Function) Let f , g and h satisfy Assumption 2.6. Then the optimal value function v in the convex programming problem (2.5) is convex and lower semicontinuous.
Proof. Consider (y i , z i ) ∈ dom(v), i = 1, 2 in the domain of v and an arbitrary ε > 0. We can find x i ε feasible to the constraint of problem v(y i , z i ) such that Now for any λ ∈ [0, 1], we have It is easy to check that λx 1 . Combining with inequality (2.7) and letting ε → 0 we arrive at v(λ(y 1 , z 1 The lower semicontinuity of v is easier to verify. Q.E.D. By and large, there are two (equivalent) general approaches to help solving a convex programming problem: by using the related dual problem and by using Lagrange multipliers. The two methods are equivalent in the sense that a solution to the dual problem is exactly a Lagrange multiplier (see [5]). Using Lagrange multipliers is more accessible to practitioners outside the special area of convex analysis. We will take this approach. The Lagrange multipliers method tells us that under mild assumptions we can expect there exists a Lagrange multiplier λ = (λ y , λ z ) with λ y ≥ 0 such thatx is a solution to the convex programming problem (2.5) if and only if it is a solution to the unconstrained problem of minimizing The function L(x, λ) is called the Lagrangian. To understand why and when does a Lagrange multiplier exist, we need to recall the definition of the subdifferential. Definition 2.8. (Subdifferential) Let X be a finite dimensional Banach space and X * its dual space. The subdifferential of a lower semicontinuous convex function ϕ : Geometrically, an element of the subdifferential gives us the normal vector of a support hyperplane for the convex function at the relevant point. It turns out that Lagrange multipliers of problem (2.5) are simply the negative of elements of the subdifferential of v. We summarize and prove the sufficiency in the lemma below which we will actually use.
andx is a solution of (2.5). Then Proof. Observe that v(y, z) is a nonincreasing function with respect to the minorization ≤ in y. Using −λ ∈ ∂v(y, z), for any vector ∆y ≥ 0, we have It follows that λ y ≥ 0 verifying (i).
By the definition of the subdifferential and the fact that v(g(x), h(x)) = v(y, z), we then have It follows that the complementary slackness condition Finally, by the definition of the subdifferential we have Thus, for any x, Using the fact thatx is a solution to problem in (2.5) and the complementary slackness condition (2.10) we have (2.12) Combining (2.11) and (2.12) verifies (ii). Q.E.D.
Remark 2.10. By Theorem 2.9 Lagrange multipliers exist when (2.5) has a solutionx and ∂v(y, z) ̸ = ∅. Calculating ∂v(y, z) requires to know the value of v in a neighborhood of (y, z) and is not realistic. Fortunately, the well-known Fenchel-Rockafellar theorem (see e.g. [4]) tells us when (y, z) belongs to the relative interior of dom(v), then ∂v(y, z) ̸ = ∅. This is a very useful sufficient condition. A particularly useful special case is the Slater condition (see also [4]): when there is only an inequality constraint g(x) ≤ y, if there exists x ∈ dom(f ) such that g(x) < y implies already that ∂v(y) ̸ = ∅.

Efficient trade-off between risk and utility
We consider the financial market described in Definition 2.1 and consider a set of admissible portfolios A ⊂ R M +1 (see Definition 2.2). The payoff of each portfolio x ∈ A at time t = 1 is S 1 · x. The merit of a portfolio x is often judged by its expected utility E[u(S 1 ·x)] where u is an increasing concave utility function. The increasing property of u models the more payoff the better. The concavity reflects the fact that with the increase of payoff, its marginal utility to an investor decreases. On the other hand investors are often sensitive to the risk of a portfolio which can be gauged by a risk measure. Because diversification reduces risk, the risk measure should be a convex function.

Technical Assumptions
Some standard assumptions on the utility and risk functions are often needed in the more technical discussion below. We collect them here. (r1n) (Normalization) There is at least one portfolio of purely bonds in A. Furthermore, (r2) (Diversification Reduces Risk) The risk function r is convex.
(r2s) (Diversification Strictly Reduces Risk) The risk function r is strictly convex.

Assumption 3.3. (Conditions on Utility Function) Utility functions u : R → R ∪ {−∞}
are usually assumed to satisfy some of the following properties.
(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.
Another important condition which often appears in the financial literature is no arbitrage.
We say market S t has no arbitrage if there does not exist any arbitrage portfolio for the financial market S t .
An arbitrage is a way to make return above the risk free rate without taking any risk of losing money. If such an opportunity exists then investors will try to take advantage of it. In this process they will bid up the price of the risky assets and cause the arbitrage opportunity to disappear. For this reason, usually people assume a financial market does not contain any arbitrage.
The following is a weaker requirement than arbitrage: We say the market has no nontrivial riskless portfolio if there does not exist a riskless portfolio x with x ̸ = 0.
A trivial riskless portfolio of investing everything in the riskless asset S 0 t always exists. A nontrivial riskless portfolio, however, is not to be expected and we will often use this assumption.
It turns out that the difference between no nontrivial riskless portfolio and no arbitrage is exactly the following: The three conditions in Definitions 3.4, 3.5 and 3.6 are related as follows:

Proposition 3.7. Consider financial market S t of Definition 2.1. There is no nontrivial riskless portfolio in S t if and only if S t has no arbitrage portfolio and no nontrivial bond replicating portfolio.
Proof. The conclusion follows directly from Definitions 3.4, 3.5 and 3.6. Q.E.D.

Corollary 3.8. No nontrivial riskless portfolio implies no arbitrage portfolio.
Assuming the financial market has no arbitrage then no nontrivial riskless portfolio is equivalent to no nontrivial bond replicating portfolio and has the following characterization. (i) There is no nontrivial bond replicating portfolio.
Q.E.D. A rather useful corollary of Theorem 3.9 is that any of the conditions (i)-(iii) of that theorem ensures the covariance matrix of the risky assets to be positive definite.

Corollary 3.10. (Positive Definite Covariance Matrix) Assume the financial market S t in Definition 2.1 has no nontrivial riskless portfolio. Then the covariant matrix of the risky assets
is positive definite.
Proof. We note that under the assumption of the corollary, for any nontrivial risky portfolio x, S 1 · x cannot be a constant. Otherwise, ( S 1 − R S 0 ) · x would be a constant which contradicts S t has no nontrivial riskless portfolio. It follows that for any nontrivial risky portfolio x, Thus, Σ is positive definite. Q.E.D.

Efficient Frontier for the Risk-Utility trade-off
We note that to increase the utility one often has to take on more risk and as a result the risk increases. The converse is also true. For example, if one allocates all the capital to the riskless bond then there will be no risk but the price to pay is that one has to forgo all the opportunities to get a high payoff on risky assets so as to reduce the expected utility. Thus, the investment decision of selecting an appropriate portfolio becomes one of trading-off between the portfolio's expected return and risk. To understand such a trade-off we define, for a set of admissible portfolios A ⊂ R M +1 in Definition 2.2, the set on the two dimensional risk-expected utility space for a given risk measure r and utility u. Given a financial market S t and a portfolio x, we often measure risk by observing Proof. Since x → S 1 · x is a linear mapping, the risk measure r inherits the properties of ρ so that it satisfies properties (r1), (r2) and (r3) in Assumption 3.1. One sufficient condition forr to preserve the strict convexity of ρ is that the matrix G in (3.3) is of full rank since all portfolios have unit initial cost. It follows from Theorem 3.9 that this condition follows from no nontrivial riskless portfolio in the financial market S t . Q.E.D.
(2) When ρ is restricted to a set of admissible portfolios A with unit initial cost. In this case we can see that Similarly, we are interested in when the expected utility Below is a set of useful sufficient conditions. Then the expected utility E[u(S 1 · x)] as a function of the portfolio x is strictly concave on A.
To prove that this function is strictly concave on A, consider two distinct portfolios x 1 , x 2 ∈ A. By assumption (c), both x 1 and x 2 have unit initial cost and thus x 1 ̸ = x 2 . Assumption (a) and Proposition 3.7 implies that for the matrix G defined in (3.3), G x 1 ̸ = G x 2 . Thus, using again the fact that both x 1 and x 2 have unit initial cost, we have The strictly concavity of x → E[u(S 1 · x)] now follows from the strict concavity of the utility function u as assumed in (b).
Q.E.D. When r(x) = ρ(S 1 · x) is induced by ρ as in Corollary 3.12 we also use the notation The following assumption will be needed in concrete applications.

Proposition 3.16. Assume that A is a set of admissible portfolios as in Definition 2.2. We claim: (a) Assume that the risk measure r satisfies (r2) in Assumption 3.1 and the utility function u satisfies (u2) in Assumption 3.3. Then set G(r, u; A) is convex and
Then convexity of r in x yields and (u2) gives Thus, (3.7) By Assumption 3.15 a subsequence of x n (denoted again by x n ) converges to, say,x ∈ A. Taking limits in (3.7) we arrive at Thus, (r, µ) ∈ G(r, u; A) and hence G(r, u; A) is a closed set.
in the two dimensional risk-expected utility space. Investors prefer portfolios with lower risk if the expected utility is the same or with higher expected utility given the same level of risk.

Definition 3.17. (Efficient Portfolio)
We say that a portfolio x ∈ A is Pareto efficient provided that there does not exist any portfolio x ′ ∈ A such that either

Definition 3.18. (Efficient Frontier) We call the set of images of all efficient portfolios in the two dimensional risk-expected utility space the efficient frontier and denote it by
The next theorem characterizes efficient portfolios in the risk-expected utility space. Proof. If a portfolio x represented in the risk-expected utility space as (r, µ) is not on the (non vertical or horizontal) boundary of the G(r, u; A), then for ε small enough we have either (r − ε, µ) ∈ G(r, u; A) or (r, µ + ε) ∈ G(r, u; A). This means x can be improved.
Q.E.D. The following relationship is straightforward but very useful.

Remark 3.21. (Empty Efficient Frontier)
If (α, 0) ∈ A for all α ∈ R and the increasing utility function u has no upper bound then for any risk measure r satisfying (r1) and

Representation of Efficient Frontier
In view of Remark 3.21, in this section we will consider a set of admissible portfolios A with unit initial cost as in Definition 2.   G(r, u, A) is closed and convex according to Proposition 3.16.
Alternatively, we can also directly apply Proposition 2.7 to the second representation in (3.9) and (3.10) to derive the convexity and concavity of γ and ν, respectively.
The increasing property of γ and ν follows directly from the second representation in (3.9) and (3.10), respectively.
Q.E.D. It also follows

Efficient Portfolios
We have seen that the efficient trade-off between risk and expected utility of a portfolio can be represented as the graph of a lower semicontinuous convex function µ → γ(µ) that relates the level of expected return µ to a minimum risk. Alternatively, these points in the expected utility-risk space can also be represented as the graph of an upper semicontinuous concave function r → ν(r) that relates the level of risk r to a maximum possible utility. We now turn to analyze how the corresponding efficient portfolios behave. Ideally we would want that each point on the efficient trade-off frontier corresponds to exactly one portfolio. For this purpose we need additional assumptions on risk measures and utility functions. Then, in case there exists some x ∈ A with E[u(S 1 · x)] finite, we can define

13)
and and claim the following: (a) For µ ∈ (µ min , µ max ) there exists exactly one portfolio x(µ) on the efficient frontier G ef f (r, u, A) which corresponds to (γ(µ), µ). Moreover, the mapping µ → x(µ) is continuous on (µ min , µ max ). Furthermore, when µ max and/or µ min are/is attained by some x ∈ A the above statement holds on the interval (µ min , (b) For r ∈ (r min , r max ) there exists exactly one portfolio y(r) on the efficient frontier G ef f (r, u, A) which corresponds to (r, ν(r)). Moreover, the mapping r → y(r) is continuous on (r min , r max ). Furthermore, when r min is a minimum and/or r max are/is attained by some x ∈ A, the above statement holds on the interval [r min , r max ), (r min , r max ], or [r min , r max ]. (c) If in addition, r satisfies (r1n) in Assumption 3.1 then r min = 0, µ min = u(R) and x(µ min ) = y(r min ) = (1, 0) ⊤ (see Figure 2).

Proof. (a)
We focus on the case when condition (c1) is satisfied and will comment on the modifications needed for the similar case when (c2) is satisfied.
Consider µ ∈ (µ min , µ max ). Then we can find a portfoliox ∈ A with E[u(S 1 ·x)] ≥ µ. By (3.12) r min ≤ r(x). Thus, the set x ∈ A} is nonempty. Moreover, Assumption 3.15 ensures that A µ is compact. It follows that there exists at least one portfolio x(µ) such that Clearly, x(µ) corresponds to the point (γ(µ), µ) on the efficient frontier G ef f (r, u, A).
Finally, we show the continuity of x(µ) by contradiction. Suppose this mapping is discontinuous at µ 0 . Then, for a fixed positive number ε 0 > 0, there exists a sequence (3.15) By Assumption 3.15 we may assume without loss of generality that x(µ n ) converges to some portfolio But the uniqueness of the efficient portfolio (3.16) implies that x * = x(µ 0 ), which is a contradiction. If µ min and/or µ max is finite and attained at some x ∈ A then with the same arguments as above the unique continuous portfolio extends to the respective bound of (µ min , µ max ).
The proof for the case when condition (c2) holds is similar. The only difference is that uniqueness of the efficient portfolio now follows from the strict concavity of the mapping x → E[u(S 1 · x)] (by Lemma 3.14) and the convexity of r(x).
(c) Since A contains only portfolios of unit initial cost, (1, 0) ⊤ ∈ A when (r1n) is satisfied. Then we can directly verify the conclusion in (c). Q.E.D.

Remark 3.25. (a) When Assumption 3.15 (b) holds, then
is also finite by (3.13). A typical efficient frontier corresponding to this case is illustrated in Figure 1. (b) It is possible that µ max and/or r max to be +∞. Suppose µ max is finite and attained at an efficient portfolio x(µ max ). Under the conditions of the theorem the portfolio κ := x(µ max ) is unique and independent of the risk measure. A graphic illustration is given in Figure 3.
(c) Trade-off between utility and risk is thus implemented by portfolios x(µ) which trace out a curve in the leverage space of Vince [28]. Note that the curve x(µ) depends on the risk measure r as well as the utility function u. This provides a method for systematically selecting portfolios in the leverage space to reduce risk exposure.

Markowitz Portfolio Theory and CAPM Model
Let us now turn to applications of the general theory. We show that the results in the previous section provide a general unified framework for several familiar portfolio theories. They are Markowitz portfolio theory, CAPM model, growth optimal portfolio theory and leverage space portfolio theory. Of course, when dealing with concrete risk measures and expected utilities related to these concrete theories additional helpful structure in the solutions often emerge. Although many different expositions of these theories do already exist in the literature, for convenience of readers we include brief arguments using Lagrange multiplier methods. In this entire section we will assume that the market S t from Definition 2.1 has no nontrivial riskless portfolio.

Markowitz Portfolio Theory
Markowitz [15] portfolio theory which considers only risky assets can be understood as a special case of the framework discussed in Section 3. The risk measure is the standard deviation σ and the utility function is the identity function. So we face the problem We assume E[ S 1 ] is not proportional to S 0 , that is, for any α ∈ R, Since the variance is a monotone increasing function of the standard deviation we can minimize half of variance for convenience.
Optimization problem (4.3) is already in the form (3.9) with A = {x ∈ R M +1 : S 0 · x = 1, x 0 = 0}. We can check condition (c1) in Theorem 3.24 is satisfied. Moreover, Corollary 3.10 implies that Σ is positive definite since S t has no nontrivial riskless portfolio. Hence, the risk function r has compact level sets. Thus, Assumption 3.15 is satisfied and Theorem 3.24 is applicable. Let x(µ) be the optimal portfolio corresponding to µ. Consider the Lagrangian where λ 1 ≥ 0. Thanks for Theorem 2.9 we have In other words We must have λ 1 > 0 because otherwise x ⊤ (µ) would be unrelated to the payoff S 1 . The complementary slackness condition implies that E[ S 1 · x(µ)] = µ. Right multiplying (4.5) by x(µ) we have To determine the Lagrange multipliers, we need the numbers α = E[ Solving (4.8) and (4.9) we derive since Σ −1 is positive definite and condition (4.2) holds. Substituting (4.10) into (4.7) we see that the efficient frontier is determined by the curve usually referred to as the Markowitz bullet due to its shape. A typical Markowitz bullet is shown in Figure 4 with an asymptote Thus, relationships (4.12) and (4.13) describe the efficient frontier G ef f (σ, id, {S 0 ·x = 1, x 0 = 0}) as in Definition 3.18. Also note that (4.12) implies that µ min = β/γ and r min = 1/ √ γ.
Thus, as a corollary of Theorem 3.24, we have

They correspond to the upper boundary of the Markowitz bullet given by
) .
The optimal portfolio x(µ) can be determined by (4.6) and (4.10) as which is affine in µ.
The structure of the optimal portfolio in (4.14) implies the well known two fund theorem derived by Tobin in [26].

Capital Asset Pricing Model
The capital asset pricing model (CAPM) is a theoretical model independently proposed by Lintner [9], Mossin [17], Sharpe [22] and Treynor [25] for pricing a risky asset according to its expected payoff and market risk, often referred to as the beta. The core of the capital asset pricing model is an extension of the Markowitz portfolio theory to include a riskless bond. Thus we can apply the general framework in Section 3 with the same setting as in Section 4.1. Similar to the previous section we can consider the equivalent problem of Similar to the last section problem (4.15) is in the form (3.9) with A = {x ∈ R M +1 : S 0 · x = 1}. We can check condition (c1) in Theorem 3.24 is satisfied. Again the risk function r has compact level sets since Σ is positive definite. Thus, Assumption 3.15 is satisfied and Theorem 3.24 is applicable. The Lagrangian of this convex programming problem is (4.16) where λ 1 ≥ 0. Again we have Using S 0 1 = R and S 0 0 = 1, the first component of (4.17) implies So that (4.17) becomes (4.20) by right multiplying x(µ) in (4.19). Solving x ⊤ (µ) from (4.19) we have Right multiplying with E[ S ⊤ 1 ] and S ⊤ 0 and using the α, β and γ introduced in the previous section we derive respectively. Multiplying (4.23) by R and subtract it from (4.22) we get Combining (4.20) and (4.24) we arrive at (4.25) It only makes sense to involve risky assets when we can expect an excess return. Thus, µ ≥ R. Relation (4.25) defines a straight line on the (σ, µ)-plane since Σ is positive definite. The line given in (4.26) is called the capital market line. Also combining (4.21), (4.23) and (4.24) we have Again we see the affine structure of the solution. In particular, when µ = R and µ = (α − βR)/(β − γR) we derive, respectively, the portfolio (1, 0) ⊤ that contains only the riskless bond and the portfolio (0, (E[ S 1 ] − R S 0 )Σ −1 /(β − γR)) ⊤ that contains only risky assets. We call this portfolio the market portfolio and denote it x M . The market portfolio corresponds to the coordinates . (4.29) Since the risk σ is non negative we see that the market portfolio exists only when This condition is Note that (4.30) also implies (4.27). Again note that although the computation is done in terms of the risk function r( x) = Thus, the market portfolio has to reside on the Markowitz efficient frontier. Moreover, by (4.28) we can see that the market portfolio x M is the only portfolio on the CAPM efficient frontier that consists of purely risky assets. Thus, (4.32) so that the capital market line is tangent to the Markowitz bullet at (σ M , µ M ) as illustrated in Figure 5. The affine structure of the solutions is summarized in the following one fund theorem [22,26].

Theorem 4.5. (One Fund Theorem)
Assume that the financial market S t has no nontrivial riskless portfolio. Moreover assume that condition (4.30) holds. All the optimal portfolios in the CAPM model (4.15) are generalized convex combinations of the riskless bond and the market portfolio Optimal portfolios x(µ) are affine in µ (see (4.28)) and can be represented as points in the (σ, µ)-plane as located on the capital market line The capital market line is tangent to the boundary of the Markowitz bullet at the coordinates of the market portfolio (σ M , µ M ) and intercepts the µ axis at (0, R) (see Fig.  5).

Remark 4.6. The one fund theorem combined with the two fund theorem provides a theoretical foundation for the passive investment strategy. The two fund theorem implies that if two broad based indices are approximately on the Markowitz frontier then we can use a linear combination of these two indices to derive the market portfolio. Thus, by the one fund theorem in order to construct an efficient portfolio in the sense of the CAPM model we only need to consider a mix of the bond and the two indices.
Alternatively we can write the slope of the capital market line as This quantity is called the price of risk and we can rewrite the equation for the capital market line (4.26) as In the risk-return space this is the slope of the line representing portfolios mixing x with a riskless bond. Clearly the larger this ratio the better the portfolio serves this purpose. Sharpe [23] proposed to use this ratio, later called Sharpe ratio, to measure the performance of mutual funds.
We can also use the capital market line to price a risky asset as we initially set out to do. The pricing principle in the capital asset pricing model is that adding a fair priced risky asset to the market should not change the capital market line. For convenience we assume that the price is implied by the expected return of the asset. Thus, given a risky asset a i , we try to determine its expected return µ i . Here is the covariance of a i 1 /a i 0 and the payoff of the market portfolio.
Proof. Consider a portfolio relies on the parameter α that mixes the risky asset a i and the market portfolio: Denote the expected return and the standard deviation of p(α) by µ α and σ α , respectively. Hence we have and where σ 2 i is the variance of a i 1 /a i 0 . The parametric curve (σ α , µ α ) must lie below the capital market line because the latter consists of optimal portfolios. On the other hand it is clear that when α = 0 this curve coincides with the capital market line. Thus, the capital market line is tangent to the line of the parametric curve (σ α , µ α ) at α = 0. Since the slope of the capital market line is (µ M − R)/σ M , it follows that (4.40) Solving for µ i we derive Q.E.D.

Affine Structure of the Efficient Portfolios
The affine dependence of the efficient portfolio on the return µ observed in the CAPM still holds when the standard deviation is replaced by the more general deviation measure (see [20]. In this section we derive this affine structure using the general framework discussed in Section 3 and provide a proof different from that of [20]. We also construct a counterexample showing that the two fund theorem (Theorem 4.2) fails in this setting. Let's consider a risk measure r that satisfies (r1), (r1n), (r2) and (r3) in Assumption 3.1 and the related problem of finding efficient portfolios becomes Since for µ = R there is an obvious solution x(R) = (1, 0) corresponding to r(x(R)) = r( 0) = 0, we have r min = 0 and µ min = R. In what follows we will only consider µ > R. Moreover, we note that for r satisfying the positive homogeneous property (r3) in Assumption 3.1, y ∈ ∂ r( x) implies that In fact, for any t ∈ (−1, 1), Then there exists an efficient portfolio x 1 corresponding to (r 1 , µ 1 ) = (r(x 1 ), R + 1) on the efficient frontier for problem (5.1) such that the efficient frontier for problem (5.1) in the risk-expected return space is a straight line that passes through the points (0,R) corresponding to a portfolio of pure bond (1, 0) ⊤ and (r 1 , µ 1 ) corresponding to the portfolio x 1 , respectively. Moreover, the straight line connecting (1, 0) ⊤ and x 1 in the portfolio space, namely for µ ≥ R, represents a set of efficient portfolios that corresponds to in the risk-expected return space (see Definition 3.18 and (3.9)).
Proof. The Lagrangian of this convex programming problem (5.1) is where λ 1 ≥ 0 and λ 2 ∈ R. Condition (5.4) implies that, for any µ there exists a portfolio of the form y = (y 0 , 0, . . . , 0, ym, 0, . . . , 0)  Fixing µ 1 = R + 1 > R, denote x 1 = x(µ 1 ). Then Since r is independent of x 0 we have Substituting (5.11) into (5.10) we have because at the optimal solution x 1 the constraint is binding. Using (r3) it follows from (5.2) and (5.12) that Thus, we can write (5.13) as We can verify that S 0 · x t = 1 and On the other hand it follows from assumptions (r1) and (r3) that Thus, for any x satisfying S 0 · x = 1 and it follows from (5.15) that For any µ > R, letting t µ := µ − R, we have µ = R + t µ . Thus, by inequality (5.19) we have r( x(µ)) ≥ t µ r( x 1 ). On the other hand x(µ) is an efficient portfolio implies that r( x(µ)) ≤ r( x tµ ) = t µ r( x 1 ) yielding equality In other words γ(µ) is an affine function in µ. Also, we conclude that points (γ(µ), µ) on this efficient frontier correspond to efficient portfolios as an affine mapping of the parameter µ into the portfolio space. Also using r 1 we can write (5.20) as That is to say the efficient frontier of (5.1) in the risk-expected return space is given by the parameterized straight line (5.6). Q.E.D.
which is a similar representation of the efficient portfolios as (5.5). The portfolio x M is called a master fund in [20].  (5.25) as illustrated in Figure 6. (c) If x 1 0 = 1 then the efficient portfolios in (5.5) are related to µ in a much simpler fashion In this case there is no master fund as observed in [20]. In the language of [20], portfolio x 1 is called a basic fund. Thus, Theorem 5.1 recovers the results in Theorem 2 and Theorem 3 in [20] with a different proof and a weaker condition (condition (5.4) is weaker than (A2) on page 752 of Rockafellar et al [20]).
Since the standard deviation satisfies Assumptions (r1), (r1n), (r2) and (r3), the result above is a generalization of the relationship between the CAPM model and the Markowitz portfolio theory. We note that the standard deviation is not the only risk measure that satisfies these assumptions. For example, some forms of approximation to the expected drawdowns also satisfy these assumptions (cf. [14]).
Theorem 5.1 is a full generalization of the one fund theorem (Theorem 4.5) in the previous section. On the other hand it has been noted in footnote 10 in [20] that a similar generalization of the two fund theorem (Theorem 4.2) is not to be expected. We construct a concrete counter-example below. r( x) (5.27) Subject to Choose the payoff S 1 such that E[ S 1 · x] = x 1 so that x 1 = µ at the optimal solution. Finally, let's construct r( x) so that the optimal solution x(µ) is not affine in µ.
We do so by constructing a convex set G with 0 ∈ intG (interior of G) and then set r( x) = 1 for x ∈ ∂G (boundary of G) and extend r to be positive homogeneous. Then Obviously for µ = 1 the optimal solution is x(1) = (1, 0, 0) ⊤ with r( x(1)) = 1/10 For

Growth Optimal and Leverage Space Portfolio
Growth portfolio theory is proposed by Lintner [9] and is also related to the work of Kelly [8]. It is equivalent to maximizing the expected log utility: Subject to S 0 · x = 1.
is compact (and possibly empty in some cases).
Proof. Since u is continuous, the set in (6.3) is closed. Thus, we need only to show it is also bounded. Assume the contrary that there exists a sequence of portfolios x n with S 0 · x n = 1 (6.4) and ∥x n ∥ → ∞ satisfying Equation (6.4) implies that ∥ x n ∥ → ∞. Then without loss of generality we may assume x n /∥ x n ∥ converges to x * = (x * 0 , x * ) ⊤ where ∥ x * ∥ = 1. Condition (u3) and (6.5) for arbitrary µ ∈ R imply that, for each natural number n, Dividing (6.4) and (6.6) by ∥ x n ∥ and taking limits as n → ∞ we derive S 0 · x * = 0 (6.7) and Combining (6.7) and (6.8) we have and thus x * is a nontrivial riskless portfolio, which is a contradiction. Q.E.D. Proof. of Theorem 6. 2 We can verify that the utility function u = ln satisfies conditions (u1), (u2s), (u3) and (u4). Also {x : E[ln(S 1 · x)] ≥ ln(R), S 0 · x = 1} ̸ = ∅ because it contains (1, 0) ⊤ . Thus, Lemma 6.3 implies that problem (6.1) has at least one solution and µ max = max is finite. By Lemma 3.14, x → E[ln(S 1 · x)] is strictly concave. Thus problem (6.1) has a unique optimal portfolio. Q.E.D. The growth optimal portfolio has the nice property that it provides the fastest compounded growth of the capital. By Remark 3.25 (b) it is independent of any risk measures. In the special case that all the risky assets are representing a certain gaming outcome, κ is the Kelly allocation in [8]. However, the growth portfolio is seldomly used in investment practice for being too risky. The book [11] edited by MacLean, Thorp, and Ziemba provides an excellent collection of papers with chronological research on this subject. These observations motivated Vince [28] to introduce his leverage space portfolio to scale back from the growth optimal portfolio. Recently, [10,30] further introduce systematical methods to scale back from the growth optimal portfolio by, among other ideas, explicitly accounts for limiting a certain risk measure. The analysis in [10,30] can be phrased as solving where r is a risk measure that satisfies conditions (r1) and (r2). Alternatively, to derive the efficient frontier we can also consider (b) problem (6.11) defines ν(r) : [0, r(κ)] → R as a continuous increasing concave function, where κ is the optimal growth portfolio. Moreover, problem (6.11) has a continuous path of unique solutions y(r) that maps the interval [0, r(κ)] into a curve in the leverage portfolio space R M +1 . Finally, y(0) = (1, 0) ⊤ , y(r(κ)) = κ, ν(0) = ln(R) and ν(r(κ)) = µ κ .
Proof. Note that Assumption 3.15 (a) holds due to Lemma 6.3 and (c2) in Theorem 3.24 is also satisfied. Then (a) follows straight forward from conclusions (a) and (c) in Theorem 3.24 where µ max = µ κ and r min = 0 are finite and attained and (b) follows from conclusions (b) and (c) in Theorem 3.24 with µ min = ln(R) and r max = γ(κ).
Q.E.D. Theorem 6.4 relates the leverage portfolio space theory to the framework setup in Section 3. It becomes clear that each risk measure satisfying conditions (r1), (r1n) and (r2) generates a path in the leverage portfolio space connecting the portfolio of a pure riskless bond to the growth optimal portfolio. Theorem 6.4 also tells us that different risk measures usually correspond to different paths in the portfolio space. Many commonly used risk measures satisfy conditions (r1) and (r2). The curve x(µ) provides a pathway to reduce risk exposure along the efficient frontier in the risk-expected log utility space. As observed in [10,30], when investments have only a finite time horizon then there are additional interesting points along the path x(µ) such as the inflection point and the point that maximizes the return/risk ratio. Both of which provide further landmarks for investors.
Similar to the previous sections we can also consider the related problem of using only portfolios involving risky assets, i.e., Subject to S 0 · x = 1.

Theorem 6.5. (Existence of Solutions) Suppose that
Then problem (6.12) has a solution.
In fact, a far more common restriction to the set of admissible portfolios are limits of risk. For this example if, for instance, we restrict the risk by r 1 (x) ≤ 0.5 then we will create a shared efficient frontier of (6.1) with that of (6.11) where r is a priori restricted (see Figure 10). Remark 6.7. (Efficiency Index) Although the growth optimal portfolio is usually not implemented as an investment strategy, the maximum utility µ max corresponding to the growth optimal portfolio κ, empirically estimated using historical performance data, can be used as a measure to compare different investment strategies. This is proposed in [31] and called the efficiency index. When the only risky asset is the payoff of a game with two outcomes following a given playing strategy, the efficiency coefficient coincides with Shannon's information rate (see [8,21,31]). In this sense, the efficiency index gauges the useful information contained in the investment strategy it measures.
Also related to the growth optimal portfolio theory is the fundamental theorem of asset pricing (FTAP). FTAP characterizes the no arbitrage condition with the existence of a martingale measure, which is defined below. Subject to S 0 · x = 1.
First we observe that when a utility function u satisfies condition (u4) we can also characterize the no arbitrage condition in terms of the supremum of the expected utility. Proof. Note that We can easily verify that when a utility function u satisfies condition (u4) and there exists an arbitrage portfolio then On the other hand, by Proposition 3.7 when S t has no nontrivial portfolio equivalent to the bond and no arbitrage implies that S t has no nontrivial riskless portfolio. By (ii) The optimal value of the portfolio utility optimization problem (6.15) is finite and attained.
(iii) There is an equivalent martingale measure for the financial market S t proportional to a subgradient of −u at the optimal solution of (6.15).

Proof.
Observe that (i) equivalent to (ii) is already derived in Theorem 6.9.

Conclusion and Open Problems
Following the pioneering idea of Markowitz to trade-off the expected return and standard deviation of a portfolio, we consider a general framework to efficiently trade-off between a concave expected utility and a convex risk measure for portfolios. Under reasonable assumptions we show that (i) the efficient frontier in such a trade-off is a convex curve in the expected utility-risk space, (ii) the optimal portfolio corresponding to each level of the expected utility is unique and (iii) the optimal portfolios continuously depend on the level of the expected utility. Moreover, we provide an alternative treatment of the results in [20] showing that the one fund theorem (Theorem 4.5) holds in the trade-off between a deviation measure and the expected return (Theorem 5.1) and construct a counter-example illustrating that the two fund theorem (Theorem 4.2) fails in such a general setting. Furthermore, the efficiency curve in the leverage space is supposedly an economic way to scale back risk from the growth optimal portfolio (Theorem 6.4). This general framework unifies a group of well known portfolio theories. They are Markowitz portfolio theory, capital asset pricing model, the growth optimal portfolio theory, and the leverage portfolio theory. It also extends these portfolio theories to more general settings.
The new framework also leads to many questions of practical significance worthy further explorations. For example, quantities related to portfolio theories such as the Sharpe ratio and efficiency index can be used to measure investment performances. What other performance measurements can be derived using the general framework in Section 3? Portfolio theory can also inform us about pricing mechanisms such as those discussed in the capital asset pricing model and the fundamental theorem of asset pricing. What additional pricing tools can be derived from our general framework?
Clearly, for the purpose of applications we need to focus on certain special cases. Drawdown related risk measures coupled with the log utility attracts much attention in practice. In Part II of this series [14] several drawdown related risk measures are constructed and analyzed. We will conduct a related case study in the third part of this series [3].