Mean-variance theory is ideally suited to contexts where return distributions are defined by these moments and such a context is provided by multivariate normal return distributions. Under such a hypothesis across two periods the absence of autocorrelation renders returns between periods to be independent. The presence of autocorrelation in squared returns is then inconsistent with this implied independence. Much of the evidence, along with the considerations of correlation in squared returns, points towards non-Gaussian models for returns. What objective functions are then best suited to the task of designing portfolios? Much depends on the purpose of the portfolio design.

With a strict one- or two-period view of the situation, one can imagine the case of an investor placing monies in the market at the start of the period involved, liquidating the position at the end of the period and consuming the resulting accumulated wealth. Axioms of rational behaviour under uncertainty then suggest the use of the expected utility of final wealth as the appropriate decision criterion. However, in many practical situations such a formulation misconstrues the reality of the investment activity. The periods involved are fairly short with durations of a few weeks or months at the end of which no consumption of accumulated wealth is being contemplated. Instead, the portfolio is to be liquidated and reinvested into a new portfolio designed in the light of new circumstances and information. Hence, the focus is just on the market value of the portfolio at the future date, marked by the end of the period, and not on its utility, expected or otherwise. Potential market participants could then be primarily interested in maximizing the market value of the portfolio. Now, classically, the market value of a portfolio is the sum of the value of its components and is, as a consequence, independent of how it is constructed. This linearity or additivity follows from the law of one price and the absence of arbitrage opportunities.

Conic portfolio theory is based on the pricing operators of two-price economies where the law of one price is abandoned. Such economies separate bid and ask prices. Markets are viewed as offering to buy random cash flows at prices that render the resulting cash flows less its price to be market acceptable. Acceptable cash flows include all nonnegative cash flows but more generally form a convex cone of acceptable random variables. Every such cone may equivalently be represented as those cash flows that have a positive expectation under a set of test probability measures. As a consequence, the best bid price becomes the smallest expectation delivered by the test probabilities. The bid price being an infimum of expectations is then a concave function and the bid price for a package may well exceed the sum of the bid prices for the components. In conic portfolio theory portfolios are designed to maximize this bid price seen as a conservative market valuation.

Markets are also viewed as willing to sell random cash flows if the resulting price less the cash flow is market acceptable. Positivity of expectations under test probabilities renders the best ask price to be the supremum of expectations across test probabilities. The ask price is then a convex function on the space of random cash flows and, by construction, the ask exceeds the bid. It is also the case that the ask price is the negative of the bid for the negative cash flow.

Since constants come out of the infima of expectations, Madan (2016) shows that one may write the bid price as the expected value under a base probability less the ask price for the negative of the centered or demeaned cash flow. One may then also view the bid price as measuring reward by the expectation under the base probability less a risk measure given by the ask price for the negative of the centered cash flow. Since the centered negative cash flow has a zero mean by construction, and as the base probability is one of the test probabilities, the ask price is always positive and can be minimized subject to attaining a particular expectation. This naturally leads us to a mean ask price frontier and examples of such constructions may be found in Madan (2016). Here we consider the mean ask price frontier for the two-period portfolio problem.

Two additional assumptions termed comonotone additivity and law invariance simplify the evaluation of the bid price of a random cash flow. We suppose the random variables being considered are defined on a probability space \((\Omega,\mathcal {F},P)\) for a base probability *P*. In order to explain comonotone additivity, we first note that two random variables *X,Y* are said to be comonotone if, for example, one is a monotone increasing function of the other. More generally they move together in the same direction across the set of events, or have no negative comovements or a Kendall’s tau of unity. In general, the bid price of *X*+*Y* is larger than the sum of the bid prices for each, reflecting some possible advantages of diversification. Comonotone additivity asserts that for comonotone risks we have strict additivity with the bid for the sum equalling the sum of the bids in this case. Put another way, there are no diversification benefits for comonotone risks.

The second assumption of law invariance asserts that the bid price be computable from information on just the probability law of the random cash flow. How the random variable correlates with other random variables is not relevant. This is a strong assumption from the perspective of the concerns of particular agents who may well be interested in whether the cash flow being valued provides hedging benefits for other risks they are already carrying. However, the valuation attained in an abstract market, like that induced by the Walrasian auctioneer who is merely concerned with trying to clear as much risk as possible, may make correlation issues less relevant.

Under these two assumptions Kusuoka (2001) showed that the bid price *b*(*X*) of a random variable *X* with distribution function *F*
_{
X
}(*x*) is given by the expectation under concave distortion. More specifically, there exists a concave distribution function *Ψ*(*u*) defined on the unit interval such that

$$b(X)=\int_{-\infty}^{\infty}xd\Psi (F_{X}(x)). $$

The set of test probabilities under which *X*−*b*(*X*) is market acceptable or has a nonnegative expectation are shown in Madan et al. (2015) to be given by all probabilities *Q* such that for all \(A\in \mathcal {F}\)

$$Q(A)\leq \Psi (P(A)). $$

Cherny and Madan (2009) observed that expectation under concave distortion is also an expectation under the quantile based change of measure *Ψ*
^{′}(*F*
_{
X
}(*x*)). Further requiring that *Ψ*
^{′}(*u*) tends to infinity and zero as *u* tends to zero or unity, respectively, to reflect both risk aversion and an absence of gain enticement, they introduced the distortion termed *minmaxvar* and defined, for a stress level parameter *γ*, by

$$\Psi^{\gamma }(u)=1-\left(1-u^{\frac{1}{1+\gamma}}\right)^{1+\gamma}. $$

We shall use this distortion to illustrate bid price evaluations in this paper.

It is shown below that we may also write

$$\begin{array}{@{}rcl@{}} b(X) &=&E[X]-a(\widetilde{X}) \\ \widetilde{X} &=&E[X]-X \end{array} $$

and the functional *a*(*X*) is the ask price functional defined as

$$a(X)=\int_{-\infty}^{\infty}xd\Psi\left(1-F_{X}(x)\right). $$

We may now construct the mean ask price frontier and maximize the bid price on this frontier as the maximum for the mean less the ask price on the frontier.

For our two-period return in the absence of a risk-free asset we note that

$$R_{0,2}^{p}=\left(1+a_{0}^{\prime}R_{1}\right)\left(1+a_{1}^{\prime}R_{2}\right), $$

where *R*
_{1},*R*
_{2} are as in Eqs. (1) and (2). Furthermore, we have the constraints

$$\begin{array}{@{}rcl@{}} a_{0}^{\prime}\mathbf{1} &=&1\\ a_{1}^{\prime}\mathbf{1} &=&1. \end{array} $$

We may write

$$R_{0,2}^{p}=E\left[R_{0,2}^{p}\right]+\widetilde{R}_{0,2}^{p} $$

where \(\widetilde {R}_{0,2}^{p}\) is the centered two-period return.

For the bid price we then have

$$\begin{array}{@{}rcl@{}} b\left(R_{0,2}^{p}\right) &=&E\left[R_{0,2}^{p}\right]+b\left(\widetilde{R} _{0,2}^{p}\right) \\ &=&E\left[R_{0,2}^{p}\right]-\left(-b\left(-\left(-\widetilde{R}_{0,2}^{p}\right) \right) \right)\\ &=&E\left[R_{0,2}^{p}\right]-a\left(-\widetilde{R}_{0,2}^{p}\right). \end{array} $$

We may write

$$\begin{array}{@{}rcl@{}} R_{0,2}^{p} &=&\left(1+a_{0}^{\prime}\left(R_{1}-\mu_{1}+\mu_{1}\right)\right)\left(1+a_{1}^{\prime}\left(R_{2}-\mu_{2}+\mu_{2}\right)\right)\\ &=&\left(1+a_{0}^{\prime}\mu_{1}+a_{0}^{\prime}\widetilde{R}_{1}\right)\left(1+a_{1}^{\prime}\mu_{2}+a_{1}^{\prime}\widetilde{R}_{2}\right)\\ &=&1+a_{0}^{\prime}\mu_{1}+a_{1}^{\prime}\mu_{2}+a_{0}^{\prime}\mu_{1}a_{1}^{\prime}\mu_{2}+\left(1+a_{0}^{\prime}\mu_{1}\right)a_{1}^{\prime}\widetilde{R}_{2}+\left(1+a_{1}^{\prime}\mu_{2}\right)a_{0}^{\prime}\widetilde{R} _{1}+a_{0}^{\prime}\widetilde{R}_{1}a_{1}^{\prime}\widetilde{R}_{2}. \end{array} $$

We see that

$$\widetilde{R}_{0,2}^{p}=\left(1+a_{0}^{\prime}\mu_{1}\right)a_{1}^{\prime}\widetilde{R}_{2}+\left(1+a_{1}^{\prime}\mu_{2}\right)a_{0}^{\prime}\widetilde{R} _{1}+a_{0}^{\prime}\widetilde{R}_{1}a_{1}^{\prime}\widetilde{R}_{2} $$

and the mean ask price frontier requires the minimization of

$$a\left(-\widetilde{R}_{0,2}^{p}\right) $$

subject to

$$\begin{array}{@{}rcl@{}} a_{0}^{\prime}\mathbf{1} &=&1 \\ a_{1}^{\prime}\mathbf{1} &=&1 \\ a_{0}^{\prime}\mu_{1}+a_{1}^{\prime}\mu_{2}+a_{0}^{\prime}\mu_{1}a_{1}^{\prime}\mu_{2} &=&m, \end{array} $$

where *m* is the target two-period mean return. The bid price maximizing portfolio is the one on this frontier that maximizes the value of \(m-a\left (-\widetilde {R}_{0,2}^{p}\right) \).

The bid price could be maximized directly for the optimal portfolio. Alternatively one may also construct the mean ask price frontier analogous to a mean-variance frontier with the advantage that the optimal portfolio is located on the frontier where its slope is unity, as risk and reward are both measured in dollars and the bid price is precisely the reward less the risk of the ask price and there is no trade-off coefficient between the two. The situation with mean-variance is both artificial and arbitrary as reward is measured in dollars and risk in squared dollars and the use of a linear trade-off between them quite inappropriate and unsatisfactory.

The solution of this maximization problem requires the specification of the joint law across many assets, say *n*, for the vector of returns simultaneously across two consecutive periods with the resulting distributional problem being one in dimension 2*n*, the dimension of the joint vector (*a*
_{0},*a*
_{1}). This is quite a tall order and, with a view to gaining some tractability on this problem, we build our way up to this problem by first reporting on the simpler one-period subproblem. The two-period mean ask price frontier is taken up in the next section. Here we compare the one-period problem in our context with the classical mean-variance frontier that must be revised to accomodate time change conditional drifts differentiated from unconditional drifts.

This subproblem in our context is richer than the classical mean-variance problem by providing access to skewness via the drift of the time changed Brownian motion along with kurtosis via the volatility of the time change. The mean ask price frontier that we eventually employ takes account of all these dimensions of the problem. Even if we fix the kurtosis and consider just the minimization of variance, there are now two drifts to be addressed in the portfolio design for even a single period. They are unconditional mean and the mean conditional on the time change that we call a random drift. As a consequence, we observe that the classical mean-variance frontier for a single-period is now three-dimensional.

For *n* assets over a single period let the drifts of the Brownian motions to be time changed be given by a vector *θ*. The centered returns are then modeled by

$$\widetilde{R}_{1}=\theta (T-1)+\sqrt{T}Z, $$

where *Z* is multivariate Gaussian with mean zero and covariance matrix *Σ*. The time change represents a measure of economic time and is uniform across assets. Given the law of the time change, to be specified later, we may access the return distribution of any portfolio with centered return

$$\begin{array}{@{}rcl@{}} \widetilde{R}_{p} &=&a^{\prime}\widetilde{R}_{1}\\ &=&d(T-1)+\sqrt{Tv}z\\ d &=&a^{\prime}\theta\\ v &=&a^{\prime}\Sigma a \end{array} $$

and *z* is a standard normal variate. In addition, if the assets have mean returns, *μ*, the portfolio return may be accessed with the knowledge of three numbers *m*=*a*
^{′}
*μ*, the random drift *d* and the variance *v* along with the law of *T*. We recognize that these values are related via the equations for *m,d*,*v* in terms of the portfolio weights *a*.

To further describe the classical minimum variance investment opportunity set for the first period, we consider the problem

$$\begin{array}{@{}rcl@{}} v^{\ast} &=&\min_{a}a^{\prime}\Sigma a \\ s.t.~a^{\prime}\mu &=&m \\ a^{\prime }\theta &=&d \\ a^{\prime }\mathbf{1} &=&1. \end{array} $$

(5)

The solution to this problem (5) may be described in terms of three distinguished portfolios

$$\begin{array}{@{}rcl@{}} \eta &=&\frac{\Sigma^{-1}\mu }{\mathbf{1}^{\prime }\Sigma^{-1}\mu} \notag \\ \delta &=&\frac{\Sigma^{-1}\theta }{\mathbf{1}^{\prime }\Sigma^{-1}\theta } \\ \zeta &=&\frac{\Sigma^{-1}\mathbf{1}}{\mathbf{1}^{\prime }\Sigma^{-1} \mathbf{1}}; \notag \end{array} $$

(6)

their mean returns *ρ*
_{
η
},*ρ*
_{
δ
},*ρ*
_{
ζ
}; their random drifts *y*
_{
η
},*y*
_{
δ
},*y*
_{
ζ
}; and their variances and covariances \(\sigma _{\eta }^{2},\sigma _{\delta }^{2},\sigma _{\zeta }^{2},\sigma _{\eta \delta },\sigma _{\eta \zeta }\) and *σ*
_{
δ
ζ
}.

The solution here may be contrasted with classical mean-variance theory as presented, for example, in Skiadas (2009) Chapter 2 where only the first and the third portfolios are involved in describing the one-dimensional frontier. In the current context, three portfolios are distinguished and the frontier is two-dimensional.

###
**Proposition 1**

The solution to problem (5) is given by

$$v^{\ast}=\widetilde{\lambda}^{2}\sigma_{\eta}^{2}+\widetilde{\kappa}^{2}\sigma_{\delta }^{2}+\widetilde{\pi}^{2}\sigma_{\zeta}^{2}+2\widetilde{\lambda}\widetilde{\kappa}\sigma_{\eta \delta}+2\widetilde{\lambda}\widetilde{\pi}\sigma_{\eta \zeta}+2\widetilde{\kappa}\widetilde{\pi}\sigma_{\delta \zeta}, $$

where

$$\begin{array}{@{}rcl@{}} \widetilde{\lambda}(\rho_{\eta}-\rho_{\zeta})+\widetilde{\kappa}\left(\rho_{\delta}-\rho_{\zeta}\right)+\rho_{\zeta} &=&m\\ \widetilde{\lambda}\left(y_{\eta }-y_{\zeta}\right)+\widetilde{\kappa}\left(y_{\delta }-y_{\zeta}\right)+y_{\zeta} &=&d \\ 1-\widetilde{\lambda }-\widetilde{\kappa} &=&\widetilde{\pi}. \end{array} $$

We suppose the asset space is rich enough to permit the availability of variances above the minimum variance given *m,d* with all levels of *m,d* being attainable. The investment opportunity set for the first period then consists of triples *m,d*,*v* with

$$v\geq v^{\ast }(m,d). $$

This is a three-dimensional mean-variance frontier as the optimal variance now depends on the choice of both a deterministic drift *m* and a random drift *d*.

By way of an example we take the inputs

$$\begin{array}{*{20}l} \begin{array}{lll} \rho_{\eta } && 0.06 \\ \rho_{\delta } && 0.09 \\ \rho_{\zeta } && 0.03 \\ y_{\eta } && -0.1 \\ y_{\delta } && -0.12 \\ y_{\zeta } && -0.05 \\ \sigma_{\eta } && 0.15 \\ \sigma_{\delta } && 0.20 \\ \sigma_{\zeta } && 0.05 \\ \sigma_{\eta \delta } && 0.0210 \\ \sigma_{\eta \zeta } && 0.0015 \\ \sigma_{\delta \zeta } && 0.0001. \end{array} \end{array} $$

Figure 3 presents a graph of such a three-dimensional volatility efficiency frontier while Fig. 4 presents an associated contour plot.

This opportunity set is fully defined by specifying the mean returns, random drifts, variances, and covariances of the three distinguished portfolios. The investment opportunity sets may be allowed to be different in the two periods by taking different settings for the mean returns, random drifts, variances, and covariances of the distinguished portfolios in the two periods.

In the presence of risk-free assets for both periods with interest rates of *r*
_{1},*r*
_{2}, respectively, the volatility cost frontier is defined in terms of two distinguished portfolios

$$\begin{array}{@{}rcl@{}} \xi &=&\frac{\Sigma^{-1}(\mu -r\mathbf{1})}{\mathbf{1}^{\prime }\Sigma^{-1}(\mu -r\mathbf{1})} \\ \delta &=&\frac{\Sigma^{-1}\theta}{\mathbf{1}^{\prime }\Sigma^{-1}\theta}. \end{array} $$

For the frontier one needs the excess returns *x*
_{
ξ
}=*ξ*
^{′}(*μ*−*r*
**1**),*x*
_{
δ
}=*δ*
^{′}(*μ*−*r*
**1**); the random drift coefficients *y*
_{
ξ
}=*ξ*
^{′}
*θ*,*y*
_{
δ
}=*δ*
^{′}
*θ*; the variances \(\sigma _{\xi }^{2},\sigma _{\delta }^{2};\) and the covariance *σ*
_{
ξ
δ
}.

###
**Proposition 2**

In the presence of a risk-free asset, the minimum variance *v*
^{∗} for a deterministic drift of *m* and a random drift of *d* is given by

$$v^{\ast}=\widetilde{\lambda}^{2}\sigma_{\xi}^{2}+\widetilde{\kappa}^{2}\sigma_{\delta}^{2}+2\widetilde{\lambda}\widetilde{\kappa} \sigma_{\xi\delta} $$

where

$$\left[\begin{array}{cc} x_{\xi} & x_{\delta} \\ y_{\xi} & y_{\delta } \end{array}\right] \left[\begin{array}{c} \widetilde{\lambda} \\ \widetilde{\kappa } \end{array}\right] =\left[\begin{array}{c} m \\ d \end{array}\right]. $$

The ask price minimization problem may be implemented by specifying the joint law for the two time changes in the two periods. In this regard, we first observe that stochastic volatility models as formulated by GARCH models are not able to deliver the level of correlation in squared volatilities observed in data. These correlations can be higher than 0.75.

In a typical GARCH(1,1) specification, the squared returns are given by

$$\begin{array}{@{}rcl@{}} Y_{1} &=&{\sigma_{1}^{2}}{Z_{1}^{2}}\\ Y_{2} &=&{\sigma_{2}^{2}}{Z_{2}^{2}}\\ {\sigma_{2}^{2}} &=&\omega +\beta {\sigma_{1}^{2}}+\alpha Y_{1}. \end{array} $$

We may compute the correlation between *Y*
_{1},*Y*
_{2} for *Z*
_{1},*Z*
_{2} being independent standard normal variates.

###
**Proposition 3**

The correlation between *Y*
_{1},*Y*
_{2} is bounded above by

$$\frac{1}{\sqrt{\frac{\beta^{2}}{\alpha^{2}}+4+2\beta /\alpha }}. $$

For typical values of *β* near unity and *α* near zero or (1−*β*) this correlation is expected to be small.

We recognise from Proposition 3 that for a given first-period volatility the only source of correlation is the randomness in first-period squared returns that typically receives a small weight in estimated models. To give the first-period volatility some volatility of its own we simulate the first-period volatility from a log-normal distribution with its own volatility and then compute squared return correlations. Figure 5 presents a graph of squared return correlations as a function of the volatility of the first-period return volatility.

We observe that typical levels of empirically observed squared return correlations are associated with very high levels of first-period log-normal volatility. With deterministic first-period volatility there is no chance for correlations in squared returns to reach empirically observed levels.

These considerations have motivated our use of correlated time changes as the source for correlation in squared returns. Volatility in such a construction is then a random variable going forward and not a number. This introduces the possibility of substantially correlating squared returns.

Figure 6 presents results on simulated squared return correlations for different levels of *ν*,*α*. The returns are generated as

$$\begin{array}{@{}rcl@{}} X_{1} &=&-0.2(T_{1}-1)+0.25\sqrt{T_{1}}Z_{1}\\ X_{2} &=&-0.3(T_{2}-1)+0.32\sqrt{T_{2}}Z_{2}\\ T_{1} &=&G(\alpha)+G_{1}(1-\alpha)\\ T_{2} &=&G(\alpha)+G_{2}(1-\alpha). \end{array} $$

*G*(*t*),*G*
_{1}(*t*),*G*
_{2}(*t*) are independent gamma processes with mean rates unity and variance rates *ν*,1,1.*Z*
_{1} and *Z*
_{2} are standard normal variates independent of each other and *G,G*
_{1},*G*
_{2}. Though we deal with processes we are only interested in the associated random variables.

With such a formulation for the correlated time changes in the two periods one may solve for the mean ask price frontier by minimizing the ask price for a given level of the two-period mean return. In the process we determine the deterministic drifts *m*
_{1},*m*
_{2}; the random drifts *d*
_{1},*d*
_{2}; and the portfolio variances *v*
_{1},*v*
_{2}. The conservative value maximizing or bid price maximizing portfolios may then be located on this frontier. The next section implements this program.