Optimal Portfolios of Illiquid Assets

Merton’s problem, introduced in Merton [12], aims to determine the strategies followed by utility optimizing investors in continuous time market models. To this end, we now consider the most general portfolio strategy, or control process, that trades within the market impact model (1) over some time interval $[s,t]\subset\mathbb{R}_{+}$. In our setting, each possible strategy will be simply a $d-$ dimensional trading rate process $v=(v_{u})_{u\in[s,t]}$ that is adapted to the information filtration $\{{\cal F}_{t}\}$: We denote the set of such admissible strategies by $\Pi^{ad}[s,t]$. The subclass of deterministic strategies where each value $v_{u},u\in[s,t]$ is $\mathcal{F}_{s}$ measurable is denoted by $\Pi^{det}[s,t]$.

Given any control process $v\in\Pi^{ad}[0,s]$ for $0<t<s$, the cash net of debt owed $C_{t}:=C^{v}_{t}$ and marked-to-market equity, or assets net of debt owed, $X_{t}:=X^{v}_{t}:=C_{t}+q_{t}^{\prime}S_{t}$ are given by:

	$\displaystyle C_{t}$	$\displaystyle=$	$\displaystyle C_{0}-\int_{0}^{t}v_{u}^{\prime}\tilde{S}_{u}\ du=C_{0}-\int_{0}^{t}v_{u}^{\prime}S_{u}\ du-\int_{0}^{t}v_{u}^{\prime}\Gamma v_{u}\ du\ ,$		(2)
	$\displaystyle X_{t}$	$\displaystyle=$	$\displaystyle X_{0}+\int_{0}^{t}q_{u}\ dS_{u}-\int_{0}^{t}v_{u}^{\prime}\Gamma v_{u}\ du\ .$		(3)

where the second equation is obtained by integration by parts. Note that here and henceforth, the superscript $v$ that labels processes controlled by $v$ will be omitted.

The interpretation of (2) and (3) in terms of the firm’s balance sheet is that assets are stochastic due to fluctuations of $S$, while the debt, thought of as deposits, is assumed to be constant and sufficient to fund all trades. In other words, we focus on market illiquidity without funding illiquidity. It is consistent with the Principle of Limited Liability that a firm becomes insolvent when its equity $X_{t}$ becomes negative. In the following, an insolvent firm with negative equity $X_{T}<0$ at a time $T$, will be declared to be in default, implying that the laws of bankruptcy will be applied to the firm.

The FI can now try to solve Merton’s optimal problem of a CARA investor with constant absolute risk aversion parameter $\lambda>0$ over any period $[t,T]$. For each $t$, they may express the value function $J_{t}$ achieved in terms of a certainty equivalent value $W_{t}$,

$$J_{t}:=-e^{-\lambda W_{t}}:=\text{sup}_{v\in\Pi^{ad}[t,T]}\ \left(-\mathbb{E}[-e^{-\lambda X_{T}}|{\cal F}_{t}]\right)\ .$$

(4)

If the supremum exists, it is achieved by adopting an optimal control denoted by $v^{*}(t)=(v_{u}^{*}(t))_{u\in[t,T]}$, which will be an adapted process over $[t,T]$. The CARA investment problem in general always satisfies the dynamic programming principle (see Schied et al. [17]), which means that for any $s\leq t\leq T$, $v_{u}^{*}(t)=v_{u}^{*}(s)$ for all $u\geq t$ and

$$-e^{-\lambda W_{s}}=\text{sup}_{v\in\Pi^{ad}[s,t]}\ \left(-\mathbb{E}[e^{-\lambda W_{t}}|{\cal F}_{s}]\right)\ .$$

(5)

An investor restricted to deterministic strategies over $[t,T]$ cannot achieve a higher certainty equivalent value than equation (4). Therefore, if $\widetilde{W}_{t}$ is defined by

$$-e^{-\lambda\widetilde{W}_{t}}:=\text{sup}_{v\in\Pi^{det}[t,T]}\ \left(-\mathbb{E}[-e^{-\lambda X_{T}}|{\cal F}_{t}]\right)$$

(6)

then $\widetilde{W}_{t}\leq W_{t}$. The first result of this paper, stated next, is that (4) is always optimized by deterministic strategies and therefore $W_{t}=\widetilde{W}_{t}$ for $t\geq 0$. Moreover, it will be found in subsequent sections that the optimal control and value functions can be expressed in closed forms involving one-dimensional integrals that solve a system of ordinary differential equations of Riccati type. First, however, a note about notation: Because $v^{*}$ and $q^{*}$ turn out to be deterministic, we henceforth replace the stochastic process notation $v_{u}$ by function notation $v(u)$ and moreover suppress the dependence on the investment period $[t,T]$.

The proof of this theorem is found in the Appendix. As we shall see in Section 3, $V(\tau,q)$ is a quadratic form in $q$ with time-dependent coefficients and thus the ODE (8) for $q^{*}$ is linear and can be solved explicitly.

From equations (2) and (3) we can deduce that, if $v$ is deterministic, then for any $0\leq s\leq t\leq T$, the equity $X_{t}$ conditioned on ${\cal F}_{s}$ is normally distributed with mean and variance given by

	$\displaystyle\mathbb{E}[X_{t}|{\cal F}_{s}]$	$\displaystyle=$	$\displaystyle X_{s}+\int_{s}^{t}\Bigl{(}q^{\prime}(u)(\Lambda v(u)+\mu)-v^{\prime}(u)\Gamma v(u)\Bigr{)}\ du,$		(9)
	$\displaystyle{\mbox{Var}}[X_{t}|{\cal F}_{s}]$	$\displaystyle=$	$\displaystyle\int_{s}^{t}\ q^{\prime}(u)\Sigma\Sigma^{\prime}q(u)\ du\ .$		(10)

In particular, the fact that $X_{T}|{\cal F}_{t}$ is always normal implies that

$$\mathbb{E}[e^{-\lambda X_{T}}|{\cal F}_{t}]=e^{-\lambda\left(\mathbb{E}[X_{T}|{\cal F}_{t}]-\frac{\lambda}{2}{\mbox{Var}}[X_{T}|{\cal F}_{t}]\right)}$$

(11)

and hence from (6) and Theorem 2.1 one deduces that

$$W_{t}=\widetilde{W}_{t}=\text{sup}_{v\in\Pi^{det}[t,T]}\left(\mathbb{E}[X_{T}|{\cal F}_{t}]-\frac{\lambda}{2}{\mbox{Var}}[X_{T}|{\cal F}_{t}]\right)\ .$$

(12)

This demonstrates the well-known equality of the certainty equivalent value for CARA optimization with the value function for Markowitz’ mean-variance (M-V) optimization, as well as the coincidence of their optimal strategies, when the optimal equity processes under consideration are all normally distributed.

In practice, the firm’s default probability (DP), meaning the probability that $X_{T}<0$, may be preferable to variance as a risk measure for institutional investors, as it gives more information about bad scenarios that need to be controlled. In the Bachelier model, the normality that follows for deterministic strategies implies that over any time horizon $[t,T]$, the Mean-Variance (M-V) criterion

Since Merton’s optimal problem (4) satisfies Bellman’s Dynamic Programming Principle at all times, its optimal strategies are “time-consistent”, which means that the optimal strategies computed for any two periods $[t,T]$ and $[s,T]$ always coincide on the intersection $[s\vee t,T]$. On the other hand, it is known () that mean-variance optimization is generally time-inconsistent and optimal adapted strategies starting at one time do not usually appear optimal at a later time. Surprisingly, Theorem 2.1 combined with equation 12 implies that in the present context, both the mean-variance and mean-default probability problems (2.2) and (2.2) are in fact time consistent, provided the optimization is restricted to deterministic strategies. The following result summarizes these relationships.

In the single risky asset case, one can verify that the scalar functions $A,B,C$ and the optimal trading strategy $q^{*}(u)$ have comparatively simple formulas obtained by reducing those given in Theorem 3.2. Notice that several distinct possibilities are determined by the relation between $D=\Sigma\sqrt{\frac{\lambda}{2\Gamma}}$ and $E=\frac{\Lambda}{2\Gamma}.$

The third and fourth cases are the cases where $T^{*}<\infty$, and one finds the solutions become unbounded: $\lim_{\tau\rightarrow T^{*}}A(\tau)=\infty$. In cases 1 and 2, the solutions are bounded for all $\tau$, and $T^{*}=\infty$.

In his original paper Merton [12], Robert Merton presented the exact solution to the problem of optimal investment in a frictionless market for an asset price that follows a geometric Brownian motion. His solution technique also leads to an exact solution of our present model in the limit of zero market impact, $\Lambda=0,\Gamma=0$, which we will call the “Merton solution”. It is of some interest to consider the explicit general solution from the previous section as a perturbation of the Merton solution, and to investigate the nature of its convergence as market impact goes to zero. We suppose that $\Lambda=\epsilon\Lambda_{1},\Gamma=\epsilon\Gamma_{1}$ with small $\epsilon$ and denote by $W(t,T,x,q,\epsilon)$ the certainty equivalent value function with its dependence on $\epsilon$. For simplicity, we confine our attention to the single asset case of the previous section.

The Merton solution over the period $[t,T]$ with initial conditions $X_{t}=x,q_{t}=q$ involves an instantaneous trade that incurs no trading cost, to the optimal value $q^{M}:=\frac{\mu}{\lambda\Sigma^{2}}$. This portfolio is then held constant. One can show that this strategy achieves the certainty equivalent value function $W(t,T,x,q,0)=x+\frac{\mu^{2}(T-t)}{2\lambda\Sigma^{2}}$ which we note is independent of $q$.

Now, for small $\epsilon$, the general solution of our model is given by Case 1 of Proposition 3.3, which leads to the following perturbative expansion

$\displaystyle W(t,T,x,q,\epsilon)=W(t,T,x,q,0)+\epsilon^{1/2}Q(q)+o(\epsilon^{3/2}).$

(47)

where

$\displaystyle Q(q)$

$\displaystyle:=$

$\displaystyle\Gamma_{1}D_{1}q^{2}+\frac{\mu}{D_{1}}q+(2E-1)D_{1}.$

(48)

Here we define $D_{1}=\Sigma\sqrt{\frac{\lambda}{2\Gamma_{1}}}$ which does not depend on $\epsilon$ and we have $D=\epsilon^{-1/2}D_{1}\to\infty$ as $\epsilon\to 0.$ It is obvious that $E$ does not depend on $\epsilon$ either. Thus the value function of our problem converges to the value function of the Merton solution with rate of convergence $\epsilon^{1/2}.$

The optimal holding at the terminal time is given by

$\displaystyle q_{T}^{*}=q^{M}+(q_{t}-q^{M})\frac{\cosh K}{\cosh(D\tau-K)}+Eq^{M}\frac{1-\tanh(D\tau-K)}{D(1-\tanh^{2}K)}.$

(49)

Here $\tau:=T-t.$ It is straightforward that $\lim_{\epsilon\to 0}K=0,$ hence $\lim_{\epsilon\to 0}q_{T}^{*}=q^{M}.$

Let $\tilde{A}(\tau):=\frac{U(\tau)}{V(\tau)}=D\ {\tanh}(D\tau-K),$ the trading rate at the initial time $t$ is given by

	$\displaystyle v_{t}$		$\displaystyle=\lim_{s\to t}\dot{q}_{s}$		(52)
			$\displaystyle=q_{t}\tilde{A}(\tau)+U(\tau)\frac{\mu}{2\Gamma D^{2}}[\frac{E(D^{2}-(\tilde{A}(\tau))^{2})}{D^{2}-E^{2}}-\frac{\tilde{A}(\tau)}{U(\tau)}]$
			$\displaystyle=(q_{t}-q^{M})D\tanh(D\tau-K)+\frac{\mu E\cosh K}{2\Gamma_{1}D_{1}^{2}\cosh(D\tau-K)}.$

Note that $\lim_{\epsilon\to 0}{\tanh}(D\tau-K)=1$ and $\lim_{\epsilon\to 0}{\cosh}(D\tau-K)=\infty$, we have $\lim_{\epsilon\to 0}v_{t}=\pm\infty$ depending on if $q_{t}>q^{M}$ or $q_{t}<q^{M},$ i.e. the optimal strategy is to trade rapidly in the beginning. We then conclude that the optimal trajectory $q^{*}(u,\epsilon)$ converges to an $L$–shaped or $\Gamma$–shaped curve when the market impact tends to zero.

This result implies that when market impact is low, the firm will follow an optimal trading strategy very close to the constant holding strategy of the Merton problem. A more surprising fact is the portfolio which starts at the Merton portfolio will remain constant if permanent impact has $\Lambda_{1}=0$, and all strategies regardless of initial portfolios will move towards the Merton portfolio for sometime initially. In the following subsection, we will show similar results for the Multi-Asset case.

Figure 1 shows for each of the three firms how the expected rate of return on equity (ERR) and default probability (DP) for their CARA/MV/DP optimal strategies depend as $\lambda$ varies over the set of feasible values $[0,\infty)$. These quantities are computed by the formulas

$${\rm ERR}(\lambda)=\frac{1}{T}(\frac{E(\lambda)}{X_{0}}-1)\ ,\quad{\rm DP}(\lambda)=N\left(-\frac{E(\lambda)}{\sqrt{V(\lambda)}}\right)$$

(53)

where $N(\cdot)$ denotes the CDF of the standard normal and $E(\lambda),V(\lambda)$ are given by (26) and (27) . Such a graph is called an efficient frontier, and it summarizes the results a firm may achieve by adopting different possible risk aversion parameters.

As explained earlier, the three firms each select the optimal investment strategy given by the value $\lambda$ that leads to ${\rm DP}(\lambda)=0.01$: with the benchmark parameters given in Table 1, the three values they compute are $\lambda=[2.56\times 10^{-7},6.7\times 10^{-8},1.83\times 10^{-8}]$. While Figure 1 suggests that, ceteris paribus, larger firms have a lower efficient frontier, this ordering can be made to reverse by increasing the permanent impact parameter.

To better understand the properties of the optimal investment strategies that result from our method, we now investigate how the three hypothetical firms’ optimal trading in the single asset case compare as important model parameters are varied away from the benchmark parameters of Table 1. Figures 2 and 3 summarize the results of four experiments, and show how the firms’ optimal trading strategies over the time period $[0,1/2]$ years change as the asset rate of return, asset volatility, temporary market impact and permanent market impact are made to vary one at a time. In each figure, the red curve denotes the benchmark parametrization, while the other two curves show the result as one specific parameter is varied upwards (blue curve) and downwards (green curves).

One point needs to be reiterated: for each choice of a set of parameters excluding the risk aversion parameter $\lambda$, $\lambda$ is computed to ensure that the firm’s default probability (DP) is exactly 1%. Thus each curve in these figures corresponds to a different value of $\lambda$.

The effect on the optimal strategy of varying the asset rate of return $\mu$ and volatility $\Sigma$ is shown in Figure 2. It is not a surprise to observe that the optimal strategy will include more of the risky asset as the rate of return is raised, or as the volatility is lowered. There is a threshold value of $\Sigma$ below which the firm switches from sell strategies to buy strategies. Although not shown in the graph, one finds the reverse is the case for $\mu$. Finally, the velocity of selling strategies seems to retain a similar shape over time under these variations. Each of these observations are borne out by more extensive investigations of the dependence on these parameters.

In Figure 3a, the main effect of decreasing temporary impact $\Gamma$ is seen to be to move more quickly to the final holding level early in the period. This can be understood as a change in the optimal balance between reducing temporary impact costs and price uncertainty due to the asset volatility. To a lesser extent, one also sees in these examples that the level of the final holdings decreases slightly as $\Gamma$ increases.

The effect of permanent impact $\Lambda$ on the strategy is more subtle. From Figure 3b, a higher permanent impact parameter $\Lambda$ leads to an optimal strategy ending with a higher holding level. It also causes more curvature for the trading strategies, especially towards the closing time where all trading curves seem to have positive slope. Indeed, directly from (8), the general formula for the trading velocity, one verifies that at the close of the period $\frac{dq}{du}\mid_{u=T}=\frac{1}{2}{\Gamma^{-1}\Lambda}$. This means that as long as $\Lambda$ is positive, every trader holding long positions, whether leveraging up or down, will always end the period by buying more shares. The reason is because permanent impact gives any trader a small opportunity to push the asset prices in a favourable direction at the last moment. We call this the Ponzi property of our market impact model: the gains it implies cannot be converted to cash without bursting the small price bubble the trader has created.

The perturbative analysis of Section 3.2 provides an alternative framework for understanding the effect of permanent and temporary market impact. We investigate the middle-sized firm with $q_{0}=2\times 10^{5}$ and market impact parameters $\Gamma(\epsilon)=\epsilon\times 10^{-7},\ \Lambda(\epsilon)=\epsilon\times 4\times 10^{-8}$ for a sequence of values $\epsilon_{n}=10^{-n},n=0,1,\dots$ approaching zero. Figure 4(a) shows how the optimal strategies converge for $\epsilon\to 0$ to the constant Merton solution for $u\in(t,T)$, but show rapid transient effects for $u$ near both endpoints. The small Ponzi effect near $u=T$ can be turned off by taking $\Lambda(\epsilon)=0$, as shown in Figure 4(b).

These figures suggest that for reasonable parameter values and small market impact, our model will deliver strategies that are effectively similar to the Merton solution. The observed relationship between the optimal strategies and the Merton solution, valid for small market impact, actually remains true for intermediate levels of market impact such as our benchmark parametrizations. One observes in Figures 2 and 3 that all strategies tend to flatten as $u$ approaches $T$, albeit with a small Ponzi effect at the end of the period. It will be well worth studying the extent that the value of the holdings at which the strategy flattens is well approximated by the Merton solution. As the market impact parameters decrease, the flat portion of the curve becomes wider, and closer to the Merton solution.