Continuous Time Portfolio Selection under Conditional Capital at Risk

Abstract

Portfolio optimization with respect to different risk measures is of interest to both practitioners and academics. For there to be a well-defined optimal portfolio, it is important that the risk measure be coherent and quasiconvex with respect to the proportion invested in risky assets. In this paper we investigate one such measure—conditional capital at risk—and find the optimal strategies under this measure, in the Black-Scholes continuous time setting, with time dependent coefficients.

1. Introduction

The choice of risk measure has a significant effect on portfolio investment decisions. Downside risk measures—that focus attention on the downside tail of the distribution of portfolio returns—have received considerable attention in the financial world. Value at risk (VaR) is probably the most famous among these measures, having featured heavily in various regulatory frameworks. It can be defined for a random variable 𝑋 and a confidence level 𝛼 by VaR(𝑋)=𝐸[𝑋]−𝑞_𝛼, where 𝑞_𝛼 is the 𝛼-quantile of 𝑋 (see e.g., [1, Equation (1.2)]) (Another common definition is that the VaR of a loss distribution 𝐿 is the smallest number 𝑥_𝛼 such that 𝑃[𝐿>𝑥_𝛼]=𝛼. This is equivalent to the definition given here if we define the loss of the portfolio 𝑋 to be given by 𝐿=𝐸[𝑋]−𝑋, and identify 𝑥_𝛼=VaR(𝑋).). A closely-related downside risk measure is capital at risk (CaR), defined in [2] (see also [3, 4]) as the difference between the riskless investment and the quantile 𝑞_𝛼.

Quantile-based risk measures such as VaR and CaR suffer from several shortcomings. First, while they measure the best of the worst outcomes at the 100(1−𝛼)% confidence level, they do not answer the question of how severe the loss can be. Also, one of the most important concerns is that these measures are not in general subadditive; that is, when used to measure risk, they do not always satisfy the notion that the diversification should not create more risk [5]. Finally, as illustrated in [6], VaR can exhibit multiple local extrema.

These issues were addressed in the widely cited article by Artzner at al. [7], where the authors define coherent risk measures by four conditions that such measures should satisfy. The article motivated a number of authors [5, 8–13] to propose and investigate different types of coherent risk measures, all of which are tail mean-based risk measures.

One such measure, that does not suffer from the critical shortcomings of VaR and CaR, is conditional capital at risk (CCaR). This is defined (in [14]) as the difference between the riskless investment and the conditional expected wealth, under the condition that the wealth is smaller than the corresponding quantile, for a given risk level. As such, this measure provides an indication of the likely severity of the loss in the event that the loss exceeds a given quantile. In this paper we prove that CCaR is strongly quasiconvex as a function of the portfolio, which is an essential property for optimization. We investigate conditional capital at risk in a multiasset Black-Scholes setting, in continuous time, and with time-dependent coefficients. We generalize and extend the optimization approach of Emmer at al. (see [2, 14]) to the continuous-time setting.

The outline of this paper is as follows. In Section 2, we give the notation and define the portfolio process and CCaR. Section 3 provides the proof that CCaR is a coherent risk measure and that it satisfies the property of strong quasiconvexity. In Section 4, we derive an analytical solution for the minimal CCaR problem, up to a scalar constant which has to be evaluated numerically. Section 5 is devoted to the derivation of an analytical strategy for the maximal expected wealth, subject to constrained CCaR. Section 6 provides some numerical examples, and Section 7 concludes this paper.

2. Preliminaries

We introduce the following notation. The 𝑚-dimensional column vector with each component equal to 1 is denoted by 𝑒, the Euclidean norm of a matrix or vector by ‖⋅‖, and the space of ℝ^𝑛-valued, square-integrable functions defined on [0,𝑡] by 𝐿²([0,𝑡], ℝ^𝑛), or just 𝐿². The natural inner product of this space is denoted by ⟨⋅,⋅⟩_𝑡, and the corresponding norm by ‖⋅‖_𝑡.

We work under the following assumptions.

Assumption. (i) The securities are perfectly divisible.
(ii) Negative positions in securities are possible.
(iii) Rebalancing of the holdings does not lead to transaction costs.

Assumption. (i) 𝑚+1 assets are traded continuously over a finite horizon [0,𝑇].
(ii) 𝑚 of these assets are stocks that follow the generalized Black-Scholes dynamics: 𝑑𝑆_𝑖(𝑡)=𝑆_𝑖(𝑡)(𝑏_𝑖(𝑡)𝑑𝑡+^𝑚_𝑗=1𝜎_𝑖𝑗(𝑡)𝑑𝑊^𝑗(t)),𝑡∈[0,𝑇],𝑆_𝑖(0)>0,𝑖=1,…,𝑚, (2.1) where 𝑊^𝑗(𝑡), 𝑗=1,…,𝑚, are independent standard Brownian motions.
(iii) One of the assets is a bond, whose price 𝑆₀(𝑡),𝑡≥0, evolves according to the differential equation: 𝑑𝑆₀(𝑡)=𝑟(𝑡)𝑆₀(𝑡)𝑑𝑡,𝑡∈[0,𝑇],𝑆₀(0)=𝑆₀>0, (2.2) where 𝑟(𝑡)(>0) is the interest rate of the bond. Throughout this work, we assume that borrowing in the bond is unconstrained.
(iv) The volatility matrix 𝜎(𝑡) (with ijth element 𝜎_𝑖𝑗(𝑡)), its inverse 𝜎⁻¹(𝑡), the drift vector 𝑏(𝑡)∶=(𝑏₁(𝑡),…,𝑏_𝑚(𝑡))^′, and the interest rate 𝑟(𝑡) are deterministic, Borel measurable, bounded functions over [0,𝑇], so that they belong to the appropriate 𝐿² spaces.
(v) 𝜎(𝑡) satisfies the nondegeneracy condition: 𝑥^′𝜎(𝑡)𝜎(𝑡)^′𝑥≥𝛿𝑥^′𝑥,∀𝑡∈[0,𝑇],∀𝑥∈ℝ^𝑚, (2.3) where 𝛿>0 is a given constant.

We note that, under the above assumptions, the market is complete.

At any time 𝑡, 𝑁_𝑖(𝑡) shares are held in the asset 𝑆_𝑖(𝑡), leading to the wealth 𝑋^𝜋(𝑡)=∑𝑁_𝑖(𝑡)𝑆_𝑖(𝑡). The 𝑚+1-dimensional vector-valued function 𝑁(𝑡)=(𝑁₀(𝑡),…,𝑁_𝑚(𝑡))^′ is called the trading strategy. We denote the fraction of the wealth 𝑋^𝜋(𝑡) invested into the risky asset 𝑆_𝑖(𝑡) by 𝜋_𝑖𝑁(𝑡)=^𝑖(𝑡)𝑆_𝑖(𝑡)𝑋^𝜋(𝑡),𝑖=1,…,𝑚, (2.4) and call 𝜋(𝑡)∶=(𝜋₁(𝑡),…,𝜋_𝑚(𝑡))^′∈ℝ^𝑚 the portfolio. The fraction held in the bond is 𝜋₀(𝑡)=1−𝜋^′(𝑡)𝑒. Under the assumption that the trading strategy is self-financing, the wealth process follows the dynamics𝑑𝑋^𝜋(𝑡)=𝑋^𝜋(𝑡)((𝑟(𝑡)+𝐵(𝑡)^′𝜋(𝑡))𝑑𝑡+𝜋^′(𝑡)𝜎(𝑡)𝑑𝑊(𝑡)),𝑋(0)=𝑋₀, (2.5) where 𝑋₀ is the initial wealth, and the risk premium vector 𝐵(𝑡) is defined by𝐵(𝑡)∶=𝑏(𝑡)−𝑟(𝑡)𝑒,𝑡∈[0,𝑇]. (2.6)

Proceeding as in [3], to ensure a minimal tractability of the optimization problems which we solve through the following sections, we restrict our attention in this work to the class 𝒬 of portfolios 𝜋(⋅) which are Borel measurable, deterministic, and bounded over [0,𝑇]. Such portfolios are called admissible. Note that for an admissible portfolio 𝜋(⋅), (2.5) is guaranteed to have a strong solution 𝑋^𝜋(⋅) (see [15, Theorem 5.2.9]). Note that, by allowing for time-dependent coefficients, this generalises the class of portfolios considered in [14].

Under condition (2.3) the market price of risk is uniquely defined by𝜃(𝑡)=𝜎(𝑡)⁻¹𝐵(𝑡). (2.7) It will be shown throughout this work that the magnitude of the 𝐿²-norm of the market price of risk is the determining criterion for optimal investment strategies, which turn out to be the weighted averages of the bond and Merton's portfolio defined by 𝜋_𝑀(𝑡)∶=(𝜎(𝑡)𝜎(𝑡))⁻¹𝐵(𝑡). (2.8) This result is an illustration of the mutual fund theorem in complete markets.

We now recall the definitions of the quantiles and quantile-based risk measures for 𝑋^𝜋(𝑡).

Definition 2.3. Suppose that 𝐹_𝑡(𝑥) is the cumulative distribution function of the wealth 𝑋^𝜋(𝑡), at time 𝑡∈[0,𝑇]. For a risk level 𝛼∈(0,0.5), the 𝛼-quantile of 𝑋^𝜋(𝑡) is defined as 𝑞_𝛼(𝑋₀,𝜋,𝑡)∶=inf{𝑥∈ℝ∣𝐹(𝑥)≥𝛼}. (2.9) The tail mean or expected shortfall of the wealth process 𝑋^𝜋(𝑡), which we denote by TM_𝛼(𝑋^𝜋(𝑡)), is the expected value of 𝑋^𝜋(𝑡) conditional on 𝑋^𝜋(𝑡)≤𝑞_𝛼(𝑋₀,𝜋,𝑡), that is, TM_𝛼(𝑋^𝜋(𝑡))∶=𝐸[𝑋^𝜋(𝑡)∣𝑋^𝜋(𝑡)≤𝑞_𝛼(𝑋₀,𝜋,𝑡)]. (2.10) The conditional capital at risk, which we denote by CCaR(𝑋₀,𝜋,𝑡), is defined to be the difference between the riskless investment and the tail mean, that is, CCaR(𝑋₀,𝜋,𝑡)∶=𝑋₀𝑅(𝑡)−TM_𝛼(𝑋^𝜋(𝑡)), (2.11) where 𝑅(𝑡) is defined as 𝑅(𝑡)=exp(^𝑡₀𝑟(𝑠)𝑑𝑠). (2.12)

Remark 2.4. Note that, with CCaR defined in this way, we get the following.(i)An investment in a riskless asset corresponds to zero CCaR.(ii)An increase in the tail mean corresponds to a decrease in CCaR. Thus, negative CCaR corresponds to the case when the tail mean is above the riskless return, that is, a desired result, while positive CCaR corresponds to the case when the tail mean is below the riskless return which we try to reverse by minimizing CCaR.

It was shown in [3] that the 𝛼-quantile of the wealth process 𝑋^𝜋(𝑡) can be written as𝑞_𝛼(𝑋₀,𝜋,𝑡)=𝑋₀𝑅(𝑡)exp(⟨𝐵,𝜋⟩_𝑡−12‖𝜎^′𝜋‖²_𝑡−|𝑧_𝛼|‖𝜎^′𝜋‖_𝑡), (2.13) where, for a given risk level 𝛼, 𝑧_𝛼 denotes the corresponding 𝛼-quantile of the standard normal distribution (note that this is negative when 𝛼<0.5). In the following, we let 𝜑 and Φ denote the density and cumulative distribution functions of the standard normal random variable. We have the following proposition.

Proposition 2.5. The tail mean of the wealth process 𝑋^𝜋(𝑡) solving (2.5) can be expressed as TM_𝛼(𝑋^𝜋(𝑡))=𝑋₀1𝑅(𝑡)𝛼exp(⟨𝐵,𝜋⟩_𝑡)Φ(−|𝑧_𝛼|−‖𝜎^′𝜋‖_𝑡), (2.14) and the expected value is given by 𝐸[𝑋^𝜋(𝑡)]=𝑋₀𝑅(𝑡)exp(⟨𝐵,𝜋⟩_𝑡). (2.15)

The proof is given in the appendix. From Proposition 2.5 we get the following corollary.

Corollary 2.6. The conditional capital at risk of the solution to (2.5) can be written as CCaR(𝑋₀,𝜋,𝑡)=𝑋₀1𝑅(𝑡)(1−𝛼exp(⟨𝐵,𝜋⟩_𝑡)Φ(−|𝑧_𝛼|−‖𝜎^′𝜋‖_𝑡)). (2.16)

Throughout the paper we will use the following function:𝑔_𝛼(𝜋,𝑡)∶=⟨𝐵,𝜋⟩_𝑡+ln(Φ(−|𝑧_𝛼|−‖𝜎^′𝜋‖_𝑡))−ln𝛼, (2.17) which transforms the expression for conditional capital at risk intoCCaR(𝑋₀,𝜋,𝑡)=𝑋₀𝑅(𝑡)(1−exp(𝑔_𝛼(𝜋,𝑡))). (2.18)

We now turn our attention to investigating the properties of CCaR.

3. Coherency and Quasiconvexity of CCaR

The notion of coherent risk measures has been widely discussed in the recent literature (see [5, 7–13] e. g.). Below we recall the definition of coherency, confirm that CCaR is a coherent risk measure, and prove that it is quasiconvex.

Definition 3.1. Let 𝑉 be the set of real-valued random processes on a complete probability space (Ω,ℱ,𝑃), with its natural filtration {ℱ_𝑡}_{𝑡∈[0,𝑇]}, satisfying 𝐸[𝑆(𝑡)]<∞for all 𝑆(𝑡)∈𝑉, and for 𝑡∈[0,𝑇]. Then𝜌∶𝑉→ℝis a coherent risk measure if it satisfies the following properties.(1)Subadditivity: 𝜌 is subadditive if, for all random processes 𝑆(𝑡),𝑌(𝑡)∈𝑉, 𝜌(𝑆(𝑡)+𝑌(𝑡))≤𝜌(𝑆(𝑡))+𝜌(𝑌(𝑡))for𝑡∈[0,𝑇]. (3.1)(2)Positive Homogeneity: 𝜌 is positive homogeneous if, for all 𝑆(𝑡)∈𝑉 and constant 𝑐>0, 𝜌(𝑐𝑆(𝑡))=𝑐𝜌(𝑆(𝑡))for𝑡∈[0,𝑇]. (3.2)(3)Monotonicity: 𝜌 is monotone if, for all 𝑆(𝑡),𝑌(𝑡)∈𝑉, such that 𝑆(𝑡)≥𝑌(𝑡) almost everywhere, and 𝑆₀=𝑌₀, 𝜌(𝑆(𝑡))≤𝜌(𝑌(𝑡))for𝑡∈[0,𝑇]. (3.3)(4)Translation Invariance: 𝜌 is translation invariant if, for all 𝑆(𝑡)∈𝑉 and 𝑐∈ℝ, 𝜌(𝑆(𝑡)+𝑐)=𝜌(𝑆(𝑡))for𝑡∈[0,𝑇]. (3.4)

To prove that CCaR(𝑋₀,𝜋,𝑡) of a wealth process 𝑋^𝜋(𝑡)∈𝑉 is a coherent risk measure we refer to [8, Definition 2.6], where the expected shortfall of a random process 𝑋(𝑡), with the corresponding 𝛼-quantile 𝑞_𝛼, is defined as ES_𝛼(𝑋(𝑡))=−𝛼⁻¹(𝐸[𝑋(𝑡)𝐼_{𝑋(𝑡)≤𝑞𝑥}]+𝑞_𝑥(𝛼−𝑃(𝑋(𝑡)≤𝑞_𝑥))) (3.5) and is shown to be coherent [8, Proposition 3.1]. We should note that the definition of coherency used in [8] involves translation invariance in the sense that 𝜌(𝑋(𝑡)+𝑐)=𝜌(𝑋(𝑡))−𝑐, for all 𝑋(𝑡)∈𝑉 and 𝑐∈ℝ.

In order to relate the result in [8] to the coherency of CCaR in the sense used here, we first note that, if 𝑋(𝑡) is a random process with a continuous probability distribution, we have ES_𝛼(𝑋(𝑡))=−TM_𝛼(𝑋(𝑡)), so that (with a slight abuse of notation) CCaR(𝑋(𝑡))=𝑋₀𝑅(𝑡)+ES_𝛼(𝑋(𝑡)). If we consider the shifting of the portfolio value by an amount 𝑐, we have CCaR(𝑋(𝑡)+𝑐)=𝑋₀𝑅(𝑡)+𝑐+ES_𝛼(𝑋(𝑡)+𝑐)=𝑋₀𝑅(𝑡)+𝑐+ES_𝛼(𝑋(𝑡))−𝑐=𝑋₀𝑅(𝑡)+ES_𝛼(𝑋(𝑡))=CCaR(𝑋(𝑡)), (3.6) and so we have the following result.

Corollary 3.2. Conditional capital at risk CCaR(𝑋₀,𝜋,𝑡) of a wealth process 𝑋^𝜋(𝑡)∈𝑉, at time 𝑡∈[0,𝑇], for a risk level 𝛼∈(0,0.5), is a coherent risk measure.

Remark 3.3. (i) While properties 2 and 3 are quite natural, subadditivity has the obvious, yet important consequence that diversification does not create more risk.
(ii) Properties 1 and 2 together are very important as they guarantee the convexity of the risk measure which is essential for optimization.
(iii) In our definition of a coherent risk measure we say that a risk measure is translation invariant, if 𝜌(𝑋(𝑡)+𝑐)=𝜌(𝑋(𝑡)), for all 𝑋(𝑡)∈𝑉, and 𝑐∈ℝ. While in article [8], It is said that arisk measure is translation invariant, if 𝜌(𝑋(𝑡)+𝑐)=𝜌(𝑋(𝑡))−𝑐, for all 𝑋(𝑡)∈𝑉 and 𝑐∈ℝ It is our belief that the definition of the translation invariance, as given in this paper, corresponds more with the intuition behind the notion of translation invariance than the definition given in [8].

Remark 3.4. In the above remarks we investigated the impact of diversification to the portfolio risk, as measured by CCaR. We saw that the axiom of subadditivity requires that the risk of the sum of two risky processes, say two stock price processes, be less than or equal to the sum of the individual risks associated with these stock price processes. This means that diversification does not create more risk, as measured by CCaR, and that, as long as we diversify, we expect risk reduction.
However, if the portfolio consists of all market assets, the diversification is completed. Then the risk can be further reduced by optimization, that is, by rebalancing the positions across these assets. We then look at the risk measure CCaR as a function of the portfolio 𝜋, and prove that it is strongly quasiconvex in 𝜋 which further implies the uniqueness of the corresponding optimization problems' solutions.

We now turn to the notion of strong quasiconvexity, which we note has not been discussed in the context of portfolio optimization in any of the quoted references except in [3, 4]. Its usefulness lies in its role in establishing the existence of unique solutions to portfolio optimization problems.

We first recall (see [16, Definition 3.5.8]) that a function 𝑓∶𝑈⊂ℝ^𝑚→ℝ is said to be strongly quasiconvex if 𝑓(𝜆𝜋+(1−𝜆)𝜉)<𝑚𝑎𝑥{𝑓(𝜋),𝑓(𝜉)}∀𝜋,𝜉∈𝑈,𝜋≠𝜉,𝜆∈(0,1). (3.7) In the following theorem we prove that CCaR has this important property, when viewed as a function of the portfolio weights.

Theorem 3.5. For all distinct 𝜋(⋅),𝜉(⋅)∈𝒬 (where the set {𝑡∈[0,𝑇]∣𝜋(𝑡)≠𝜉(𝑡)} has a positive Lebesgue measure), and for all 𝜆∈(0,1)CCaR(𝑋₀,𝜆𝜋+(1−𝜆)𝜉,𝑇)<max{CCaR(𝑋₀,𝜋,𝑇),CCaR(𝑋₀,𝜉,𝑇)}. (3.8)

Proof of Theorem 3.5. We suppose, without loss of generality, that CCaR(𝑋₀,𝜋,𝑇)>CCaR(𝑋₀,𝜉,𝑇), (3.9) in which case, from (2.18), we have 𝑔_𝛼(𝜋,𝑇)<𝑔_𝛼(𝜉,𝑇). (3.10) Let 𝜆∈(0,1). We claim that CCaR(𝑋₀,𝜆𝜋+(1−𝜆)𝜉,𝑇)<CCaR(𝑋₀,𝜋,𝑇), (3.11) which is equivalent to the statement 𝑔_𝛼(𝜆𝜋+(1−𝜆)𝜉,𝑇)>𝑔_𝛼(𝜋,𝑇). (3.12) For ease of notation, we define the function 𝛾(𝑥)∶=ln(Φ(−|𝑧_𝛼|−𝑥)). (3.13) Since 𝑔_𝛼(𝜋,𝑇)=⟨𝐵,𝜋⟩_𝑇−ln𝛼+𝛾(‖𝜎^′𝜋‖_𝑇), (3.10) is equivalent to ⟨𝐵,𝜉−𝜋⟩_𝑇+𝛾(‖𝜎^′𝜉‖_𝑇)−𝛾(‖𝜎^′𝜋‖_𝑇)>0, (3.14) while (3.12) becomes (1−𝜆)⟨𝐵,𝜉−𝜋⟩_𝑇+𝛾(‖𝜎^′(𝜆𝜋+(1−𝜆)𝜉)‖_𝑇)−𝛾(‖𝜎^′𝜋‖_𝑇)>0. (3.15) Clearly, from (3.14), the left-hand side of (3.15) is greater than 𝛾(‖𝜎^′(𝜆𝜋+(1−𝜆)𝜉)‖_𝑇)−𝜆𝛾(‖𝜎^′𝜋‖_𝑇)−(1−𝜆)𝛾(‖𝜎^′𝜉‖_𝑇), (3.16) and the theorem will be proved if we can establish that this is nonnegative. In order to do this, we make use of the following lemma, the proof of which is given in the appendix.Lemma 3.6. The function 𝑘(𝑥)=lnΦ(−𝑥) is decreasing and strictly concave for all 𝑥>0.Since 𝛾(𝑥)=𝑘(|𝑧_𝛼|+|𝑥|), one can make use of Lemma 3.6 to establish the following: 𝛾(‖𝜎^′(𝜆𝜋+(1−𝜆)𝜉)‖_𝑇)≥𝛾(𝜆‖𝜎^′𝜋‖_𝑇+(1−𝜆)‖𝜎^′𝜉‖_𝑇)≥𝜆𝛾(‖𝜎^′𝜋‖_𝑇)+(1−𝜆)𝛾(‖𝜎^′𝜉‖_𝑇). (3.17) The proof is complete.

This theorem has an immediate, important consequence. Namely, from [16, Theorem 3.5.9], if a function 𝑓∶𝑈⊂ℝ^𝑚→ℝ is strongly quasiconvex, then its local minimum is its unique global minimum. Therefore, the following corollary is true.

Corollary 3.7. If CCaR(𝑋₀,𝜋,𝑇) has a local minimum at 𝜋^∗(⋅)∈𝑄, then 𝜋^∗(⋅) is its unique global minimum.

This guarantees that minimization of CCaR(𝑋₀,𝜋,𝑇) with respect to 𝜋 gives the true, global minimum. To illustrate strong quasiconvexity, we give the plot of CCaR for the following example (where the parameters were chosen to represent rather extreme conditions, for the purposes of illustration).

Example 3.8. We consider a market consisting of one stock following the SDE𝑑𝑆(𝑡)=𝑆(𝑡)(0.15𝑑𝑡+0.2𝑑𝑊(𝑡)),𝑡∈[0,16],𝑆(0)=10, (3.18) and the bond with the constant interest rate 𝑟=0.05. The graph of CCaR(𝜋) is given in Figure 1.

Figure 1 

The graph of $\text{CCaR}$ as a function of the fraction of wealth $\pi$ invested in the stock, for $\pi \in [-\mathrm{5,5}]\text{,}$$T=16\text{,}$$S(0)=10\text{,}$$r=0.05\text{,}$$\sigma =0.2\text{,}$$b=0.15$. Observe that, in this case (with a long-time horizon and constantly high-performing stocks), the global minimum ${\pi }_{\varepsilon }$ lies in the interval $[\mathrm{0,1}]$ and satisfies $\text{CCaR}({\pi }_{\varepsilon })<0$.

4. Minimal Conditional Capital at Risk

The first portfolio selection problem we consider is to minimize risk as measured by CCaR, that is, to determine its minimal value over all 𝜋∈𝒬. The problem can be stated asmin_𝜋∈𝒬CCaR(𝑋₀,𝜋,𝑇). (4.1) Using (2.18), problem (4.1) transforms intomin_𝜋∈𝒬𝑋₀𝑅(𝑇)(1−exp(𝑔_𝛼(𝜋,𝑇))). (4.2) Since 𝑅(𝑇) is a constant, problem (4.2) is equivalent tomax_𝜋∈𝒬𝑔_𝛼(𝜋,𝑇). (4.3)

We now introduce the fundamental dimension reduction procedure, used throughout this work. Following [2, 14], we project the optimization problems considered in this paper onto the family of surfaces 𝒬_𝜀={𝜋(⋅)∈𝐿²∶‖𝜎^′𝜋‖²_𝑇=𝜀²}, and note that 𝒬=⋃_𝜀≥0𝒬_𝜀.

We denote by 𝑔^𝜀_𝛼 the restriction of 𝑔_𝛼 to 𝒬_𝜀, so that𝑔^𝜀_𝛼(𝜋,𝑇)=⟨𝐵,𝜋⟩_𝑇+ln(Φ(−|𝑧_𝛼|−𝜀))−ln𝛼. (4.4) Taking into account the definition of 𝒬_𝜀, problem (4.3) can be stated asmax_𝜀≥0max_{𝜋∈𝒬𝜀}𝑔^𝜀_𝛼(𝜋,𝑇). (4.5) We deal with this problem in two stages. First, fixing 𝜀 reduces the problem tomax_{𝜋∈𝒬𝜀}𝑔^𝜀_𝛼(𝜋,𝑇). (4.6) If 𝜋_𝜀 denotes the unique maximising portfolio for this problem, then (4.5) can be solved through the one-dimensional problem:max_𝜀≥0𝑔^𝜀_𝛼(𝜋_𝜀,𝑇). (4.7) It remains to solve the subproblem (4.6). Since 𝜀 is fixed, we see from (4.4) that (4.6) is equivalent to the problemmax_{𝜋∈𝒬𝜀}⟨𝐵,𝜋⟩_𝑇, (4.8) the solution of which is given by the following proposition (the proof of which is given in [6, Proposition 2.1]).

Proposition 4.1. For a fixed 𝜀, the solution of problem (4.6) is attained by 𝜋_𝜀𝜀(𝑡)=‖𝜃‖_𝑇(𝜎(𝑡)𝜎(𝑡)^′)⁻¹𝐵(𝑡), (4.9) where 𝜃 denotes the market price of risk, defined in (2.7).

We are now ready to solve problem (4.3), and we have the following theorem.

Theorem 4.2. Let 𝜃(⋅) be the market price of risk (2.7). 𝑔_𝛼(𝜋,𝑇) attains its maximum when 𝜋(𝑡)∶=𝜋_𝜀∗(𝑡)=𝜀^∗‖𝜃‖_𝑇⁻¹(𝜎(𝑡)𝜎(𝑡)^′)⁻¹𝐵(𝑡), (4.10) where, if ‖𝜃‖_𝑇≥𝜑(|𝑧_𝛼|)/𝛼, 𝜀^∗ is defined to be the unique solution of the equation 𝜑(|𝑧_𝛼|+𝜀)Φ(−|𝑧_𝛼|−𝜀)=‖𝜃‖_𝑇, (4.11) and, if ‖𝜃‖_𝑇<𝜑(|𝑧_𝛼|)/𝛼, 𝜀^∗=0. The corresponding minimum conditional capital at risk is CCaR(𝑋₀,𝜋_𝜀∗,𝑇)=𝑋₀1𝑅(𝑇)(1−𝛼exp(𝜀^∗‖𝜃‖_𝑇)Φ(−|𝑧_𝛼|−𝜀^∗)) (4.12) (which is 0 if 𝜀^∗=0), and the expected wealth is 𝐸[𝑋^𝜋𝜀∗(𝑇)]=𝑋₀𝑅(𝑇)exp(𝜀^∗‖𝜃‖_𝑇). (4.13)

Proof of Theorem 4.2. Using the definition of 𝜃(𝑡) and substituting (4.9) into (4.5) allows us to rewrite (4.5) as max_𝜀≥0𝑔^𝜀_𝛼(𝜋_𝜀,𝑇)∶=𝜀‖𝜃‖_𝑇+ln(Φ(−|𝑧_𝛼|−𝜀))−ln𝛼. (4.14) If we define the function 𝑓(𝜀)∶=𝑔^𝜀_𝛼(𝜋_𝜀,𝑇)=𝜀‖𝜃‖_𝑇+ln(Φ(−|𝑧_𝛼|−𝜀))−ln𝛼, (4.15) we get 𝑓^′(𝜀)=‖𝜃‖_𝑇−𝜑(−|𝑧_𝛼|−𝜀)Φ(−|𝑧_𝛼|−𝜀)=‖𝜃‖_𝑇−𝜑(|𝑧_𝛼|+𝜀)Φ(−|𝑧_𝛼,𝑓|−𝜀)_′(0)=‖𝜃‖_𝑇−𝜑(|𝑧_𝛼|)Φ(−|𝑧_𝛼|)=‖𝜃‖_𝑇−𝜑(|𝑧_𝛼|)𝛼,𝑓^′′(𝜀)=𝜑(|𝑧_𝛼|+𝜀)(Φ(−|𝑧_𝛼|−𝜀))²((1−Φ(|𝑧_𝛼|+𝜀))(|𝑧_𝛼|+𝜀)−𝜑(|𝑧_𝛼|+𝜀)). (4.16)
We see that 𝑓^′′(𝜀) has the same form as 𝑘^′′(𝑥), where 𝑘(𝑥) is defined in Lemma 3.6, with |𝑧_𝛼|+𝜀, instead of 𝑥, so that 𝑓^′′(𝜀)≤0. Since 𝑓^′(𝜀) is a decreasing function of 𝜀 for𝜀>0, we have two cases.(i)If 𝑓^′(0)≥0, that is, ‖𝜃‖_𝑇≥𝜑(|𝑧_𝛼|)/𝛼, then the equation 𝑓^′(𝜀)=‖𝜃‖_𝑇−𝜑(|𝑧_𝛼|+𝜀)Φ(−|𝑧_𝛼|−𝜀)=0 (4.17) has a unique solution which we denote by 𝜀^∗. Since 𝑓(𝜀) is a concave function, it reaches its maximum at 𝜀^∗.(ii)If 𝑓^′(0)<0, that is, ‖𝜃‖_𝑇<𝜑(|𝑧_𝛼|)/𝛼, then 𝑓^′(𝜀)<0 for all 𝜀≥0, and so (4.17) has no solution. This implies that 𝑓(𝜀) is decreasing for all 𝜀≥0, so that the optimal solution to problem (4.14) is 𝜀=0.
To complete the proof of Theorem 4.2, note that (4.9), with 𝜀=𝜀^∗, is the optimal solution of problem (4.3). One can then write ⟨𝐵,𝜋_𝜀∗⟩_𝑇=𝜀^∗‖𝜃‖_𝑇‖𝜎⁻¹𝐵‖²_𝑇=𝜀^∗‖𝜃‖_𝑇,‖𝜎^′𝜋‖²_𝑡=𝜀^∗2, (4.18) leading to CCaR(𝑋₀,𝜋_𝜀∗,𝑇)=𝑋₀1𝑅(𝑇)(1−𝛼exp(𝜀^∗‖𝜃‖_𝑇)Φ(−|𝑧_𝛼|−𝜀^∗)),𝐸[𝑋^𝜋𝜀∗(𝑡)]=𝑋₀𝑅(𝑇)exp(𝜀^∗‖𝜃‖_𝑇). (4.19)

Remark 4.3. (i) Note that, if ‖𝜃‖_𝑇≥𝜑(|𝑧_𝛼|)/𝛼, we can deduce by using (A.12) that ‖𝜃‖_𝑇≥𝜑(|𝑧_𝛼|)𝛼=𝜑(|𝑧_𝛼|)1−Φ(|𝑧_𝛼|)≥|𝑧_𝛼|. (4.20) Therefore, the condition ‖𝜃‖_𝑇≥𝜑(|𝑧_𝛼|)/𝛼, which has to be satisfied for investing into stocks under conditional capital at risk, is stronger than the condition ‖𝜃‖_𝑇≥|𝑧_𝛼|. The latter condition is sufficient in order to include stocks into the optimal strategy using capital at risk as a risk measure. Otherwise stated, conditional capital at risk is a more conservative risk measure than capital at risk, which is consistent with its definition.
(ii) Increasing the time horizon 𝑇 leads to increasing the 𝐿² norm of the market price of risk, so that, in case (ii), the optimal strategy changes from a pure bond strategy to a mixed bond-stocks strategy. In case (i), increasing the 𝐿² norm ‖𝜃‖_𝑇 leads to increasing the expected wealth 𝐸[𝑋^𝜋(𝑇)] and decreasing conditional capital at risk CCaR(𝑋₀,𝜋,𝑇).
(iii) As was noted in the preliminary remarks, the optimal portfolio is a weighted average of Merton's portfolio and the bond, which is an illustration of the two-fund separation theorem.

We note that the solution provided in Theorem 4.2 is not an explicit analytical solution, but it is expressed in terms of the solution of the one-dimensional equation (4.11), whose solution 𝜀^∗ can be easily computed. Analytical upper and lower bounds for 𝜀^∗ are given in the following lemma, whose proof is given in the appendix.

Lemma 4.4. The unique solution 𝜀^∗ of (4.17) satisfies the inequality (‖𝜃‖_𝑇1(1−|𝑧_𝛼|²)−|𝑧_𝛼|)⁺≤𝜀^∗≤‖𝜃‖_𝑇−|𝑧_𝛼|, (4.21) where 𝑎⁺∶=max{𝑎,0}.

Remark 4.5. Note that, for 𝛼=0.05, that is, at the 95% confidence level, (1−1/|𝑧_𝛼|²)≈0.63, (4.21) approximates the result from [14], which states (23‖𝜃‖_𝑇−|𝑧_𝛼|)⁺≤𝜀^∗≤‖𝜃‖_𝑇−|𝑧_𝛼|for𝛼<0.15. (4.22) However, at a higher confidence level, that is, 𝛼<0.05, (4.21) gives a better approximation for 𝜀^∗, that is, a smaller interval to which 𝜀^∗ belongs.

5. Portfolio Optimization with Respect to Conditional Capital at Risk

We now turn to the problem of maximizing wealth subject to constrained CCaR, that is,max_𝜋∈𝒬𝐸[𝑋^𝜋(𝑇)]subjecttoCCaR(𝑋₀,𝜋,𝑇)≤𝐶. (5.1) Using (2.18), the above problem can be written in the form max_𝜋∈𝒬𝑋₀𝑅(𝑇)exp(⟨𝐵,𝜋⟩_𝑇)subjectto𝑋₀𝑅(𝑇)(1−exp(𝑔_𝛼(𝜋,𝑇)))≤𝐶, (5.2) which is equivalent tomax_𝜋∈𝒬⟨𝐵,𝜋⟩_𝑇subjecttoexp(𝑔_𝛼𝐶(𝜋,𝑇))≥1−𝑋₀𝑅(𝑇). (5.3) Since CCaR, from its definition, is smaller than total wealth, to avoid trivial cases we only consider 𝐶 such that 𝐶<𝑋₀𝑅(𝑇) (5.4) Under condition (5.4), problem (5.3) can be written asmax_𝜋∈𝒬⟨𝐵,𝜋⟩_𝑇subjectto𝑔_𝛼(𝜋,𝑇)≥𝑐, (5.5) where 𝐶𝑐=ln(1−𝑋₀𝑅(𝑇)). (5.6) Note that condition (5.4) guarantees that 𝑐 is well defined. Using the dimension reduction procedure of [2, 14] (see Section 4), we can write problem (5.5) as a one-parameter optimization problem:max_𝜀≥0max_{𝜋∈𝒬𝜀}⟨𝐵,𝜋⟩_𝑇subjectto𝑔_𝛼(𝜋,𝑇)≥𝑐. (5.7) The solution to (5.7) is given in the following theorem.

Theorem 5.1. Suppose that the constant risk level C satisfies the following condition: 𝑋₀1𝑅(𝑇)(1−𝛼exp(𝜀^∗‖𝜃‖_𝑇)Φ(−|𝑧_𝛼|−𝜀^∗))≤𝐶<𝑋₀𝑅(𝑇),if‖𝜃‖_𝑇≥𝜑(|𝑧_𝛼|)𝛼0≤𝐶<𝑋₀𝑅(𝑇),if‖𝜃‖_𝑇<𝜑(|𝑧_𝛼|)𝛼, (5.8) where 𝜀^∗ is defined in Theorem 4.2. Then the optimal solution to problem (5.7) is 𝜋_𝜀∗∗𝜀(𝑡)=^∗∗‖𝜃‖_𝑇(𝜎(𝑡)𝜎(𝑡)^′)⁻¹𝐵(𝑡), (5.9) where 𝑐 is defined in (5.6), and 𝜀^∗∗∈[𝜀^∗,∞) is the unique solution of the equation ℎ(𝜀)∶=𝜀‖𝜃‖_𝑇+ln(Φ(−|𝑧_𝛼|−𝜀))−ln𝛼−𝑐=0. (5.10) The corresponding expected wealth is equal to 𝐸[𝑋^{𝜋𝜀∗∗}(𝑇)]=𝑋₀𝑅(𝑇)exp(𝜀^∗∗‖𝜃‖_𝑇), (5.11) with the corresponding conditional capital at risk: CCaR(𝑋₀,𝜋_𝜀∗∗,𝑇)=𝐶. (5.12) The efficient frontier is given by the following curve, whose first component is an increasing function of 𝐶 defined implicitly through (5.6), (5.10), and (5.11): {(𝐶,𝐸[𝑋^{𝜋𝜀∗∗}(𝑇)])∣Csatisfies(5.8)}. (5.13)

Proof of Theorem 5.1. Proposition 4.1 states that the problem max_{𝜋∈𝒬𝜀}⟨𝐵,𝜋⟩_𝑇 (5.14) has the optimal solution 𝜋_𝜀𝜀(𝑡)=‖𝜃‖_𝑇(𝜎(𝑡)𝜎(𝑡)^′)⁻¹𝐵(𝑡). (5.15) Substituting (5.15) into (5.7) transforms it into the problem max_𝜀≥0𝜀‖𝜃‖_𝑇subjectto𝑔_𝛼(𝜋_𝜀,𝑇)≥𝑐. (5.16) Using (5.10), the above problem can be written as max_𝜀≥0𝜀‖𝜃‖_𝑇subjecttoℎ(𝜀)=𝜀‖𝜃‖_𝑇+ln(Φ(−|𝑧_𝛼|−𝜀))−ln𝛼−𝑐≥0. (5.17) Clearly, the problem achieves its optimal solution for maximal 𝜀 for which the constraint is satisfied. The solution to this problem is given in the following lemma, whose proof is given in the appendix.Lemma 5.2. Under condition (5.8), the equation ℎ(𝜀)=0 has a maximal solution 𝜀^∗∗∈[𝜀^∗,∞). Furthermore, ℎ^′(𝜀)<0, for 𝜀>𝜀^∗.We substitute 𝜀^∗∗ into (5.15) to get the optimal portfolio 𝜋_𝜀∗∗𝜀(𝑡)=^∗∗‖𝜃‖_𝑇(𝜎(𝑡)𝜎(𝑡)^′)⁻¹𝐵(𝑡), (5.18) which yields 𝐸[𝑋^{𝜋𝜀∗∗}(𝑇)]=𝑋₀𝑅(𝑇)exp(𝜀^∗∗‖𝜃‖_𝑇),CCaR(𝑋₀,𝜋_𝜀∗∗,𝑇)=𝐶. (5.19) Finally, to prove that the expected wealth is an increasing function of the risk constant 𝐶, we rewrite (5.10) in the form ℎ(𝜀^∗∗)=𝜀^∗∗‖𝜃‖_𝑇+ln(Φ(−|𝑧_𝛼|−𝜀^∗∗𝐶))−ln𝛼−ln(1−𝑋₀𝑅(𝑇))=0, (5.20) which defines 𝜀^∗∗ as an implicit function of 𝐶. Differentiating the above, we get 𝜕ℎ𝜕𝜀^∗∗𝑑𝜀^∗∗+𝑑𝐶𝜕ℎ𝜕𝐶=0, (5.21) so that 𝑑𝜀^∗∗𝑑𝐶=−𝜕ℎ/𝜕𝐶𝜕ℎ/𝜕𝜀^∗∗. (5.22) Since (𝜕ℎ/𝜕𝐶)=(1/𝑋₀𝑅(𝑇)−𝐶), and since 𝐶<𝑋₀𝑅(𝑇), it follows that 𝜕ℎ/𝜕𝐶>0. From Lemma 5.2 it follows that 𝑑ℎ/𝑑𝜀<0, for all 𝜀>𝜀^∗, and hence for 𝜀^∗∗, so that 𝑑𝜀^∗∗/𝑑𝐶>0. Thus, 𝜀^∗∗ is an increasing function of 𝐶. From (5.11), we see that the expected wealth is an increasing function of 𝜀^∗∗, and consequently of 𝐶. This completes the proof of the theorem.

Remark 5.3. If a risk-averse investor decides to take the minimal risk which, in the first case, means taking for the risk constant 𝐶=𝑋₀1𝑅(𝑇)(1−𝛼exp(𝜀^∗‖𝜃‖_𝑇)Φ(−|𝑧_𝛼|−𝜀^∗)), (5.23) implying 𝑐=−ln𝛼+𝜀^∗‖𝜃‖_𝑇+ln(Φ(−|𝑧_𝛼|−𝜀^∗)), (5.24) then the unique solution of the equation 𝜀‖𝜃‖_𝑇+ln(Φ(−|𝑧_𝛼|−𝜀))−ln𝛼−𝑐=0 (5.25) is 𝜀^∗∗=𝜀^∗.
In the second case, for a risk averse investor, the minimal risk constant is 𝐶=0, which implies 𝑐=0, leading to the equation ℎ(𝜀)=𝜀‖𝜃‖_𝑇+ln(Φ(−|𝑧_𝛼|−𝜀))−ln𝛼=0,withℎ(0)=0,ℎ^′(𝜀)≤0, (5.26) so that the unique solution of the equation ℎ(𝜀)=0 is 𝜀^∗∗=0.
Therefore, in both cases, choosing the minimal risk constant leads to the optimal strategies coinciding with the strategies from the previous chapter, in which minimal CCaR(𝑋₀,𝜋,𝑇) was determined.

6. Applications

To illustrate the results from previous sections, we give some numerical examples. We recall that the system of SDEs that models the stocks' prices is𝑑𝑆_𝑖(𝑡)=𝑆_𝑖(𝑡)(𝑏_𝑖(𝑡)𝑑𝑡+^𝑚_𝑗=1𝜎_𝑖𝑗(𝑡)𝑑𝑊^𝑗(𝑡)),𝑆_𝑖(0)>0,𝑖=1,…,𝑚, (6.1) and that the stocks' returns variance-covariance matrix, which we denote by Γ(𝑡), is equal to 𝜎(𝑡)𝜎(𝑡)^′. We also recall that Γ(𝑡) can be decomposed asΓ(𝑡)=𝜈(𝑡)Δ(𝑡)𝜈(𝑡), (6.2) where Δ(𝑡) is the stocks' returns correlation matrix, and 𝜈(𝑡) is a diagonal matrix with the entries equal to the stocks' returns standard deviations. Therefore, from (6.2) we getΓ(𝑡)=𝜎(𝑡)𝜎(𝑡)^′=𝜈(𝑡)Δ(𝑡)𝜈(𝑡). (6.3) Although, theoretically, we assume that the vector of independent Brownian motions 𝑊(𝑡) is {ℱ_𝑡}_{𝑡∈[0,𝑇]}-adapted, that is, known at time 𝑡∈[0,𝑇], it is a common practice that we only observe Γ(𝑡) or, equivalently, 𝜈(𝑡) and Δ(𝑡), but not 𝜎(𝑡). From (6.3) we see that this leads to a nonunique decomposition of Γ(𝑡) into the product 𝜎(𝑡)𝜎(𝑡)^′. Despite that fact, the Euclidean norm, and consequently, the 𝐿²-norm of the market price of risk, is uniquely determined by ‖𝜃(𝑡)‖²=‖𝜎(𝑡)⁻¹𝐵(𝑡)‖²=𝐵(𝑡)^′(𝜎(𝑡)𝜎(𝑡))⁻¹𝐵(𝑡), (6.4) or, in the terms of the standard deviation and the correlation matrix, as‖𝜃(𝑡)‖²=𝐵(𝑡)^′𝜈(𝑡)⁻¹Δ(𝑡)⁻¹𝜈(𝑡)⁻¹𝐵(𝑡). (6.5)

To keep the exposition simple we consider the cases where the interest rate 𝑟(𝑡), as well as the volatility matrix, is constant, and the number of stocks is 𝑚=3. In all numerical computations and the corresponding plots we use an annual time scale for the drifts, standard deviations, and the correlation matrix Δ(𝑡) of the 3 stocks. We model time dependency in the drift 𝑏_𝑖(𝑡) by assuming that it oscillates periodically around a constant level 𝜇_𝑖. In order to capture cycles in the economy or in the dynamics of the stocks we model the drifts as𝑏_𝑖(𝑡)=𝜇_𝑖+𝛽_𝑖cos(𝜑_𝑖𝑡),𝑖=1,2,3. (6.6) We note that the above model for stocks' drifts was already used in [17], as the amplitude and frequency coefficients 𝛽_𝑖 and 𝜑_𝑖 allow a high degree of flexibility in adjusting the shape of this time dependency. We also note that, when modeling real market data, it is quite easy to deal with the above functional form and estimate these two parameters by maximum likelihood techniques, rather than detrending the data.

We now look at four special cases with the following characteristics.(i)We let 𝜑₁=𝜑₂=𝜑₃=𝜑, with 𝜑=0.75; that is, the economic cycles of all three stocks are the same. We consider 𝛽₁=0.75𝛽, 𝛽₂=0.5𝛽, 𝛽₃=0.25𝛽, with 𝛽=0.015, which corresponds to a 1.5% deviation around the constant values 𝜇_𝑖.(ii)We assume that the interest rate is 𝑟=0.05 and numerically explore the sensitivity of the optimal strategies with respect to 𝜇_𝑖 and 𝜌.(iii)We assume that the stocks' returns have constant standard deviations given as follows: 𝜈₁=20%,𝜈₂=25%,𝜈₃=30%. (6.7)

To emphasize the importance of the diversification effect and, consequently, of the market price of risk, for the optimal strategies, in Examples 6.2, 6.3, and 6.4 we keep the same drift and volatility coefficients and change the correlation matrix.

Example 6.1. We assume that 𝜇₁=0.12, 𝜇₂=0.10, and 𝜇₃=0.08, and that the correlation matrix is Δ=1.0−0.6−0.8−0.61.00.5−0.80.51. (6.8)

In Figure 2, we show (a) the stocks' drifts over a ten year period, with daily granularity, and (b) the optimal strategy, under minimal CCaR.

(a)

(b)

Figure 2 

Plot of the Stock Drifts (a) and portfolio weights (b) for Example 6.1.

In this example, we see the expected result, that is, stock 1, which has the largest constant part in the drift, and the smallest volatility is present in the optimal portfolio in the highest percentage. However, to assume that just drifts and volatilities determine the optimal strategy would be misleading, as the following examples show.

Example 6.2. We assume that 𝜇₁=0.08, 𝜇₂=0.10, 𝜇₃=0.12, and that the correlation matrix is the same as in Example 6.1.

The stocks' drifts and portfolio weights are given in Figure 3.

(a)

(b)

Figure 3 

Plot of the Stock Drifts (a) and portfolio weights (b) for Example 6.2.

We see that, despite the fact that stock 1 has the smallest constant part of the drift, it is still present in the optimal portfolio in the highest percentage, due to its high negative correlations with both stocks 2 and 3.

Example 6.3. In this example we assume that 𝜇₁=0.08, 𝜇₂=0.10, and 𝜇₃=0.12; that is, the stocks have the same constant parts of the drifts as in Example 6.2 but a different correlation matrix given by Δ=1.0−0.30.5−0.31.0−0.90.5−0.91. (6.9)

The optimal strategy, under minimal CCaR, is given in Figure 4.

Figure 4 

Portfolio Weights under Minimal CCaR for Example 6.3.

In this example, stocks 2 and 3, which are highly negatively correlated, are present in the optimal portfolio in percentages above 400%, while stock 1 and the bond are being borrowed.

Example 6.4. In this example, we assume again that the stocks have the same constant parts of the drifts as in Examples 6.2 and 6.3, while the correlation matrix is Δ=1.00.2−0.30.21.00.1−0.30.11. (6.10)

The criterion for investing into stocks, under the minimal CCaR, is ‖𝜃‖_𝑇≥𝜑(|𝑧_𝛼|)/𝛼. For 𝛼=0.05, |𝑧_𝛼|=1.645, so that we must have ‖𝜃‖_𝑇≥2.0620, in order to start investing into stocks. In the case of low correlation coefficients, such as in the above example, the market price of risk increases very slowly, so that it takes the time horizon of 34 years to invest into stocks with the risk level of 0.05. For 𝛼=0.01, |𝑧_𝛼|=2.33, so that we must have ‖𝜃‖_𝑇≥2.6424 to start investing into stocks. If we choose the risk level to be 0.01, we have to assume the very long time horizon, of 57 years, in order to include stocks into the optimal strategy, under minimal CCaR. The relation between ‖𝜃‖_𝑇 and 𝑇 can be found in Table 1.

Table 1 

Efficient Frontiers
Figure 5 shows the efficient frontier for Example 6.1 created using Theorem 5.1. The theorem states that the expected wealth is an increasing function of 𝐶, which is bounded above by 𝐶_max=𝑋₀𝑅(𝑇). (6.11)
In order to avoid extremely risky strategies as 𝐶→𝐶_max, and for the sake of more transparency of the graphs, we restrict 𝐶 to the interval [0,0.5𝐶_max]. We note that the efficient frontiers for the other three examples are of the same exponential form, so that we omit their graphs.
The graph given in Figure 5 illustrates the fact that the efficient frontiers are an increasing function of the risk constant 𝐶 which bounds CCaR.

Figure 5 

Efficient Frontier for Example 6.1.

Time-CCaR-Expected Wealth Surfaces
In Figure 6 we give a three-dimensional plot of the expected wealth as a function of time and the risk constant 𝐶. From Theorem 5.1 we know that the expected wealth increases when the risk constant 𝐶 increases. We recall that the expected wealth is also an increasing function of the time horizon 𝑇. The following figure illustrates these facts for Example 6.1. In order to avoid extremely risky strategies, and to get a better representation, we restrict the upper bound for CCaR to the interval [0,0.5𝐶_max].

Figure 6 

Time-CCaR-Expected Wealth Surface for Example 6.1.

7. Conclusion

In this work we investigated continuous time portfolio selection under the notion of conditional capital at risk, within the Black-Scholes asset pricing paradigm, with time dependent coefficients. We showed that conditional capital at risk is a coherent risk measure and proved that it satisfies an important property of strong quasiconvexity. Based on an idea from [14], generalized in [3], we introduced the fundamental dimension reduction procedure which transforms 𝑚-dimensional optimization problems into one-dimensional problems, within the class of admissible portfolios, which are Borel measurable, bounded, and deterministic. We further developed optimal strategies for portfolio optimization under constrained conditional capital at risk. It is important to emphasize that we considered time dependent portfolios, where the methods developed in [14] no longer work. We illustrated the two-fund separation theorem, by showing that all optimal strategies are the weighted averages of Merton's portfolio and the bond, and the weights depend on the investor's risk tolerance only. Finding optimal strategies requires solving nonlinear equations which include the cumulative distribution function of the normal random variable, so that the weights can be only found numerically. We provide several numerical examples which illustrate the importance of diversification, given by the correlation matrix. The correlation matrix significantly impacts the magnitude of the 𝐿² norm of the market price of risk, which, in turn, is the determining criterion for investment strategies.

Appendix

Proof of Proposition 2.5. Let 𝜇(𝑡)=𝑟(𝑡)+𝐵(𝑡)^′𝜋(𝑡),𝜂(𝑡)=𝜎(𝑡)^′𝜋(𝑡). (A.1) Then the differential equation of the wealth process given by (2.5) can be written as 𝑑𝑋^𝜋(𝑡)=𝑋^𝜋(𝑡)(𝜇(𝑡)𝑑𝑡+𝜂(𝑡)𝑑𝑊(𝑡)),𝑋^𝜋(0)=𝑋₀. (A.2) For convenience, set 𝑌(𝑡)=ln𝑋^𝜋(𝑡). Applying the multidimensional version of Itô's Lemma, it can be shown (see the proof of Proposition 2.1 of [3]) that 𝑌(𝑡) follows the dynamics 1𝑑𝑌(𝑡)=(𝜇(𝑡)−2‖𝜂(𝑡)‖²)𝑑𝑡+𝜂(𝑡)𝑑𝑊(𝑡),𝑌(0)=ln(𝑋₀)=𝑌₀. (A.3) The 𝛼-quantile of 𝑌(𝑡) is equal to 𝑞_𝛼(𝑌₀,𝜋,𝑡)=𝑌₀+^𝑡₀1(𝜇(𝑠)−2‖𝜂(𝑠)‖²)𝑑𝑠−|𝑧_𝛼|^𝑡₀‖𝜂(𝑠)‖²𝑑𝑠, (A.4) so that the 𝛼-quantile of 𝑋(𝑡) is equal to 𝑞_𝛼(𝑋₀,𝜋,𝑡)=exp(𝑞_𝛼(𝑌₀,𝜋,𝑡)). We will further simplify the notation by introducing 𝑞_𝑥=𝑞_𝛼(𝑋₀,𝜋,𝑡),𝑞_𝑦=𝑞_𝛼(𝑌₀,𝜋,𝑡). (A.5) From 𝑋^𝜋(𝑡)𝐼_𝑋^𝜋_≤𝑞𝑥=exp(𝑌(𝑡))𝐼_{𝑌≤𝑞𝑦}, (A.6) where 𝐼_(⋅) is the corresponding indicator function, and from Bayes' theorem, we obtain TM_𝛼(𝑋^𝜋(𝑡))=𝛼⁻¹𝐸[exp(𝑌(𝑡))𝐼_{𝑌≤𝑞𝑦}]. (A.7) Choose 𝑡∈[0,T]. We will evaluate (A.7) using the characteristics of the distribution of 𝑌(𝑡), for fixed 𝑡. From (A.3) we have that 𝑌(𝑡) is a normal random variable with the parameters 𝜇_𝑦=𝑌₀+^𝑡₀1(𝜇(𝑠)−2‖𝜂(𝑠)‖²)𝑑𝑠,𝜎_𝑦=^𝑡₀‖𝜂(𝑠)‖²𝑑𝑠. (A.8) Taking into account (A.7) and (A.8), we can write TM_𝛼(𝑋^𝜋(𝑡))=𝛼⁻¹𝐸[exp(𝑌(𝑡))𝐼_{𝑌≤𝑞𝑦}]=𝛼^−1𝑞𝑦_−∞exp(𝑦)𝜑(𝑦)𝑑𝑦. (A.9) Using the standard integration techniques, and taking into account that 𝑧_𝛼<0, we get TM_𝛼(𝑋^𝜋(𝑡))=𝛼⁻¹exp(𝜇_𝑦+𝜎²_𝑦2)Φ(−|𝑧_𝛼|−𝜎_𝑦). (A.10) If we substitute the original notation from (A.8) and (A.1), we get (2.10). This completes the proof of Proposition 2.5.

Proof of Lemma 3.6. Let 𝑥>0. The function 𝑘(𝑥)=lnΦ(−𝑥) is clearly decreasing. To prove that it is strictly concave, we differentiate twice to get 𝑘^′′(𝑥)=𝜑(𝑥)(Φ(−𝑥))²((1−Φ(𝑥))𝑥−𝜑(𝑥)). (A.11) To evaluate 𝑘^′′(𝑥), we use the following standard inequality. For 𝑥>0, 1𝑥−1𝑥³≤1−Φ(𝑥)≤1𝜑(𝑥)𝑥. (A.12) Applying the above, we get that 𝑘^′′(𝑥)≤0 which means that 𝑘(𝑥) is concave. To prove that 𝑘(𝑥) is strictly concave we need to show that 𝑥(1−Φ(𝑥))<𝜑(𝑥)for𝑥>0. (A.13) If we define the function 𝑤(𝑥)∶=𝑥(1−Φ(𝑥))−𝜑(𝑥), (A.14) it is an easy exercise to prove that 𝑤(𝑥)<0, for all 𝑥>0; that is, the function 𝑘(𝑥) is strictly concave for all 𝑥>0, which ends the proof of Lemma 3.6.

Proof of Lemma 4.4. We prove that the optimal solution 𝜀^∗ of (4.17) satisfies (4.21). From (4.17) we have that (1−Φ(|𝑧_𝛼|+𝜀^∗))𝜑(|𝑧_𝛼|+𝜀^∗)=1‖𝜃‖_𝑇. (A.15) We use again the standard inequality (A.12). From inequality (A.12), with 𝑥=|𝑧_𝛼|+𝜀^∗, we have 1|𝑧_𝛼|+𝜀^∗−1(|𝑧_𝛼|+𝜀^∗)³≤1‖𝜃‖_𝑇=(1−Φ(|𝑧_𝛼|+𝜀^∗))𝜑(|𝑧_𝛼|+𝜀^∗)≤1|𝑧_𝛼|+𝜀^∗. (A.16) The right-hand side of the inequality implies 𝜀^∗≤‖𝜃‖_𝑇−|𝑧_𝛼|. (A.17) From the left-hand side of the inequality we have ‖𝜃‖_𝑇1(1−(|𝑧_𝛼|+𝜀^∗)²)≤|𝑧_𝛼|+𝜀^∗. (A.18) Since −1(|𝑧_𝛼|+𝜀^∗)²1≥−|𝑧_𝛼|², (A.19) inequality (A.18) implies ‖𝜃‖_𝑇1(1−|𝑧_𝛼|²)−|𝑧_𝛼|≤𝜀^∗. (A.20) The fact that 𝜀^∗≥0 and (A.20) give the desired result, so that Lemma 4.4 is proved.

Proof of Lemma 5.2. In this proof we find a maximal solution of the equation ℎ(𝜀)=𝜀‖𝜃‖_𝑇+ln(Φ(−|𝑧_𝛼|−𝜀))−ln𝛼−𝑐=0. (A.21) Since ℎ^′(𝜀)=𝑓^′(𝜀), defined by (4.15), we will apply some arguments from the proof of Theorem 4.2.(i)Suppose that ‖𝜃‖_𝑇≥𝜑(|𝑧_𝛼|)/𝛼, which means that ℎ^′(0)=𝑓^′(0)≥0. Then the unique maximum of ℎ(𝜀) is achieved at 𝜀^∗. From condition (5.8) (i) we have that 𝐶𝑐=ln(1−𝑋₀)1𝑅(𝑇)≤ln(1−(1−𝛼exp(𝜀^∗‖𝜃‖_𝑇))Φ(−|𝑧_𝛼|−𝜀^∗))=−ln𝛼+𝜀^∗‖𝜃‖_𝑇+ln(Φ(−|𝑧_𝛼|−𝜀^∗)). (A.22) Hence ℎ(𝜀^∗)=𝜀^∗‖𝜃‖_𝑇+ln(Φ(−|𝑧_𝛼|−𝜀^∗))−ln𝛼−𝑐≥𝜀^∗‖𝜃‖_𝑇+ln(Φ(−|𝑧_𝛼|−𝜀^∗))−ln𝛼+ln𝛼−𝜀^∗‖𝜃‖_𝑇−ln(Φ(−|𝑧_𝛼|−𝜀^∗))=0. (A.23) Therefore, ℎ(𝜀^∗)≥0. We further distinguish the following two subcases.(a)Suppose that 𝐶<0. Then (A.21) implies that ℎ(0)=−𝑐<0. From the proof of Theorem 4.2, for 𝜀>𝜀^∗, ℎ^′(𝜀)≤0, ℎ(𝜀) is concave, and since ℎ(𝜀^∗)≥0, it follows that the equation ℎ(𝜀)=0 has at least one solution, with the bigger solution 𝜀^∗∗∈[𝜀^∗,∞).(b)For 𝐶≥0, ℎ(0)=−𝑐≥0. Using the same arguments as in case (a), we get that (A.21) has a unique solution 𝜀^∗∗∈[𝜀^∗,∞).(ii)Suppose ‖𝜃‖_𝑇<𝜑(|𝑧_𝛼|)/𝛼, that is, ℎ^′(0)<0. We recall that in this case 𝜀^∗=0. The definition of ℎ(𝜀) yields ℎ(0)=ln(Φ(−|𝑧_𝛼|))−ln𝛼−𝑐=−𝑐. (A.24)From condition (5.8) (ii), we have that 𝑐≤0, that is, ℎ(0)≥0. Using ℎ^′(𝜀)=𝑓^′(𝜀), and the proof of Theorem 4.2, we have ℎ^′′(𝜀)≤0;ℎ^′(0)=‖𝜃‖_𝑇−𝜑(|𝑧_𝛼|)𝛼<0. (A.25) Therefore, ℎ^′(𝜀)<0 for all 𝜀≥0, with ℎ(0)≥0, so that ℎ(𝜀)=0 has a unique solution 𝜀^∗∗∈[𝜀^∗,∞). This completes the proof of Lemma 5.2.

Acknowledgments

This research was partially supported by the National Science and Engineering Research Council of Canada and the Network Centre of Excellence, Mathematics of Information Technology and Complex Systems.