• Illiquid assets included the portfolios of many investors
  • long-term institutional investment (pension funds, endowments, sovereign wealth funds etc.)
  • personal finance (a house, real estate, private company shares, art, jewelry, etc.)
• How to incorporate their properties into the allocation decision?
• How illiquidity affects consumption over the long term?

Liquidity is hard to define, but “you know it when you see it.” (O’Hara (1995), Market Microstructure Theory)

Ang, Papanikolaou and Westerfield: Portfolio Choice with Illiquid Assets (2013), NBER Working Paper

• Liquidity as the outcome of a random process: Search friction
  • A trade in the illiquid asset can occur only with uncertainty about the timing
  • The investor is able to consume only out of liquid wealth while allowed to transfer between the liquid and illiquid with uncertain timing
• Wealth implications when liquidity is stochastic

The Market

• Risk-free Money Market Account $$\begin{equation} dB_t / B_t = r_t dt \end{equation}$$
• Risky Assets

$$\begin{aligned} \frac{\bm{dS}_t}{\bm{S}_t} & = \bm{\mu} dt + \bm{\sigma}\bm{dZ}_t \\ & = (r \mathbb{1} + \bm{\sigma}\bm{\lambda}) dt + \bm{\sigma}\bm{dZ}_t \end{aligned}$$

where $\bm{\lambda} = \bm{\sigma} ^{-1}(\bm{\mu} - r \mathbb{1})$ is Price of Risk

• Wealth Dynamics

The Market is Complete ...

• A complete market is one, in which any future uncertain pay-off is attainable.

  • A portfolio strategy can be constructed which will produce any uncertain pay-off in the future given initial budget.

  • The same number of assets as the number of risk sources (B.M.s)

$$\frac{dS_{it}}{S_{it}} = (r + \sum_{j=1}^{n}\sigma_{ij}\lambda_{jt}) dt + \sum_{j=1}^{n}\sigma_{ij}dZ_{jt} ,\quad \quad i=1,...,n$$

... and arbitrage free...

• An arbitrage opportunity is a self-financing trading strategy which starting with zero investment cost generates
  • non-negative payoff with probability 1 and
  • positive payoff with positive probability.

• The Martingale Solution Method:
  • choose directly the optimally invested wealth in complete and arb. free market
  • prices are uniquely determined and any future payoff can be replicated by a specific trading strategy.

$$\begin{equation}\label{eq:ProblemOptStatic} \begin{aligned} & \sup_{(c_s)}{E\left[\int_{0}^{\infty}{\mathrm{e}^{-\beta s} u(C_s)}ds\right]} \\ & \text{s.t. }E \int_{0}^{\infty}{M_sC_sds} = W_0 M_0 \end{aligned} \end{equation}$$

Optimal Solution

The Market

• Riskless asset $dB_t=rB_tdt$

• Liquid risky asset $dS_{1t}/S_{1t} = \mu + \sigma dZ_{1t}$

• Risky illiquid asset $dS_{2t}/S_{2t} = \nu + \psi \rho dZ_{1t} + \psi \sqrt{1-\rho^2} dZ_{2t}$

• The investor cannot be certain that a market for asset $S_2$ will exist over time.

• The asset will be tradable with probability $p$ and will be illiquid with probability $1-p$.

Wealth

• Consume out of liquid wealth only.

• Illiquid wealth can be consumed only after converting it to liquid by transferring the amount $dI$ if there is a trading possibility (determined by probability $p$).

• Wealth evolves according to

Value

• Becomes a function of Wealth and $\xi$ only

$$V(kw_t,kx_t) = \sup_{\theta, dI, c} E_t \left[\int\limits_{t}^{\infty} {\mathrm{e}^{-\beta (s-t)} \frac{(kc_sw_s)}{1-\gamma}^{1-\gamma}}ds\right] = k^{1-\gamma}V(w_t,x_t)$$

$$V(w_t,x_t) =(W_t)^{1-\gamma}V\left(\frac{w_t}{x_t + w_t},\frac{x_t}{x_t + w_t}\right) = W_t^{1-\gamma}V\left((1-\xi),\xi\right)$$

Optimal illiquid investment $\xi$

$\xi^* = arg\max_\xi H(\xi)$

Optimal Consumption

• The Euler Equation: $$u'(cw) = V_w$$

Optimal liquid investment $\theta$

• From the HJB PDE equation by Ang et al. (2013)

The Bellman Equation Discretized & Decomposed

$\begin{aligned} V(W_t,\xi_t) = & \max_{(\theta_t,dI_t,c_t \in \mathcal{R})}\{u(c_t (1-\xi_t) W)\Delta t \\ & + \mathrm{e}^{-\beta \Delta t} E_{W,\xi}[ V(W_{t+\Delta t}, \xi_{t+\Delta t})]\} \\ W_t^{(1-\gamma)} H(\xi_t) = & \max_{(\theta_t,dI_t,c_t \in \mathcal{R})}\{W_t^{(1-\gamma)} u(c_t (1-\xi_t) )\Delta t \\ & + \mathrm{e}^{-\beta \Delta t} E_{W,\xi}[ W_{t+\Delta t}^{(1-\gamma)} H(\xi_{t+\Delta t})]\} \end{aligned}$

• State Transition Dynamics

• Trading probability $p$ can be calibrated through a the Poisson distribution

Smooth Approximation Algorithm

Initialization: Choose functional form $\hat{V}(W;a)$ and chose approximation grid for the wealth variable $\omega={\omega_1,...,\omega_n}$. Make initial guess $\hat{V}(W;a^0)$ and choose stopping criterion $\epsilon$.

• Step 1: : Compute $v_j=(\mathcal{T}\hat{V}(W,a^i)(\omega_j))$ for each $\omega_j \in \omega$, $j = 1,...,n$

• Step 2: : Compute $a^{i+1} \in R^n$ such that $\hat{V}(W;a^{i+1})$ approximates $(v_j,\omega_j)$ . One possible approach to the fitting step is to use least squares:

  $$\min_{a\in\mathcal{R}^n} \sum\limits_{j=1}^{n}(\hat{V}(\omega_j;a)-v_j)^2$$

• Step 3: If $\|V^{i+1}-V^i\|< \epsilon$ stop, otherwise go to Step 1.

Smooth Approximation applied on the Decomposed Illiquid Bellman:

• Our goal is to find the parameters $a_1, ..., a_8$

$$\hat{H}(\xi;\bm{a})= a_0+a_1\xi+a_2\xi^2+....+a_8\xi^8$$

• Value Function Iteration

$\begin{aligned} H(\xi_t) = & \max_{(\theta_t,dI_t,c_t \in \mathcal{R})}\{ u(c_t (1-\xi_t) )\Delta t \\ & + \mathrm{e}^{-\beta \Delta t} (p H^* E[ R_{t+\Delta t}^{(1-\gamma)}] +(1-p)E[ R_{t+\Delta t}^{(1-\gamma)} H(\xi_{t+\Delta t})]\} \\ & \equiv(\mathcal{T}H(\xi_{t+ \Delta t})) = h_j \end{aligned}$

• Clear adverse effect from illiquidity on the optimal illiquid asset holdings

State Dependency

• At high illiquid asset endowment levels the investor will tend to reduce
  • the allocation the liquid asset
  • consumption

1 Year Average Waiting Time

10 Year Average Waiting Time

• An infinitely lived agent is able to rebalance often enough to avoid being stuck with too much illiquid wealth

• Still creates enough situations where liquid risky holdings need to be cut down

• Fluctuations in illiquid wealth increase risk aversion

• Positive dependency between the degree of the liquidity friction, the utility cost and the liquidity premium.

Sensitivity to the Trading Interval

• Diversification benefits from low correlation between the liquid and illiquid assets are muted relative to the model with no frictions

• Utility cost is much stronger for assets with high negative correlation

Asset Class Data

• JP Morgan Capital Market Assumptions 2016: 10 Year Forecast

• Liquidity based on Ang et al (2013)

consumption

allocation