## Final Report Summary - PORTFOLIO (Optimal Portfolios with Random Environments, Frictions and Incentives)

This project focuses on a class of portfolio choice problems arising in Mathematical Finance. These problems share a common relevance for applications, and lead to new mathematical questions in the area of Stochastic Processes. The project investigates dynamic portfolio choice with: (i) Random Environments; (ii) Trading Frictions; (iii) Incentive Fees. These problems involve complex models of financial markets, where state variables affect investment opportunities through several channels. The following are the main results of the project:

1) Random Environments:

a) Portfolios and Risk Premia for the Long Run (with Scott Robertson)

Annals of Applied Probability, 22 (2012) no. 1 p. 239-284.

This paper develops a method to derive optimal portfolios and risk premia explicitly in a general diffusion model for an investor with power utility and a long horizon. The market has several risky assets and is potentially incomplete. Investment opportunities are driven by, and partially correlated with, state variables which follow an autonomous diffusion. The framework nests models of stochastic interest rates, return predictability, stochastic volatility and correlation risk. In models with several assets and a single state variable, long-run portfolios and risk premia admit explicit formulas up the solution of an ordinary differential equation which characterizes the principal eigenvalue of an elliptic operator. Multiple state variables lead to a quasilinear partial differential equation which is solvable for many models of interest. The paper derives the long-run optimal portfolio and the long-run optimal pricing measures depending on relative risk aversion, as well as their finite-horizon performance.

b) Static Fund Separation of Long Term Investments (with Scott Robertson)

Mathematical Finance, forthcoming.

This paper proves a class of static fund separation theorems, valid for investors with a long horizon and constant relative risk aversion, and with stochastic investment opportunities. An optimal portfolio decomposes as a constant mix of a few preference-free funds, which are common to all investors. The weight in each fund is a constant that may depend on an investor’s risk aversion, but not on the state variable, which changes over time. Vice versa, the composition of each fund may depend on the state, but not on the risk aversion, since a fund appears in the portfolios of different investors. We prove these results for two classes of models with a single state variable, and several assets with constant correlations with the state. In the linear class, the state is an Ornstein–Uhlenbeck process, risk premia are affine in the state, while volatilities and the interest rate are constant. In the square root class, the state follows a square root diffusion, expected returns and the interest rate are affine in the state, while volatilities are linear in the square root of the state.

c) Abstract, Classic, and Explicit Turnpikes (with Costantinos Kardaras, Scott Robertson, Hao Xing)

Finance and Stochastics, 18 (2014) no. 1 p. 75-114.

Portfolio turnpikes state that as the investment horizon increases, optimal portfolios for generic utilities converge to those of isoelastic utilities. This paper proves three kinds of turnpikes. In a general semimartingale setting, the abstract turnpike states that optimal final payoffs and portfolios converge under their myopic probabilities. In diffusion models with several assets and a single state variable, the classic turnpike demonstrates that optimal portfolios converge under the physical probability. In the same setting, the explicit turnpike identifies the limit of finite-horizon optimal portfolios as a long-run myopic portfolio defined in terms of the solution of an ergodic HJB equation.

d) Consumption in Incomplete Markets (with Gu Wang)

Working paper

An agent maximizes isoelastic utility from consumption with infinite horizon in an incomplete market, in which state variables are driven by diffusions. We first provide a general verification theorem, which links the solution of the Hamilton-Jacobi-Bellman equation to the optimal investment and consumption policies. To tackle the intractability of such problems, we propose approximate policies, which admit an upper bound in closed-form for their utility loss. These policies are optimal for a fictitious complete market, in which the safe rate and the state variable follow different dynamics, but excess returns remain the same. The approximate policies have closed form solutions in common models, and become optimal if the market is complete or utility is logarithmic.

2) Trading Frictions

a) The Fundamental Theorem of Asset Pricing under Transaction Costs (with Emmanuel Lepinette, Miklos Rasonyi)

Finance and Stochastics, 16 (2012) no. 4 p. 741-777.

This paper proves the fundamental theorem of asset pricing with transaction costs, when bid and ask prices follow locally bounded càdlàg (right-continuous, left-limited) processes. The robust no free lunch with vanishing risk condition (RNFLVR) for simple strategies is equivalent to the existence of a strictly consistent price system (SCPS). This result relies on a new notion of admissibility, which reflects future liquidation opportunities. The RNFLVR condition implies that admissible strategies are predictable processes of finite variation. The Appendix develops an extension of the familiar Stieltjes integral for càdlàg integrands and finite-variation integrators, which is central to modelling transaction costs with discontinuous prices.

b) Transaction Costs, Trading Volume, and the Liquidity Premium (with Stefan Gerhold, Johannes Muhle-Karbe, Walter Schachermayer)

Finance and Stochastics, 18 (2014) no. 1 p. 1-37.

In a market with one safe and one risky asset, an investor with a long horizon, constant investment opportunities, and constant relative risk aversion trades with small proportional transaction costs. We derive explicit formulas for the optimal investment policy, its implied welfare, liquidity premium, and trading volume. At the first order, the liquidity premium equals the spread, times share turnover, times a universal constant. Results are robust to consumption and finite-horizons. We exploit the equivalence of the transaction cost market to another frictionless market, with a shadow risky asset, in which investment opportunities are stochastic. The shadow price is also found explicitly.

c) Long Horizons, High Risk-Aversion, and Endogenous Spreads (with Johannes Muhle-Karbe)

Mathematical Finance, forthcoming.

For an investor with constant absolute risk aversion and a long horizon, who trades in a market with constant investment opportunities and small proportional transaction costs, we obtain explicitly the optimal investment policy, its implied welfare, liquidity premium, and trading volume. We identify these quantities as the limits of their isoelastic counterparts for high levels of risk aversion. The results are robust with respect to finite horizons, and extend to multiple uncorrelated risky assets. In this setting, we study a Stackelberg equilibrium, led by a risk-neutral, monopolistic market maker who sets the spread as to maximize profits. The resulting endogenous spread depends on investment opportunities only, and is of the order of a few percentage points for realistic parameter values.

d) Portfolio Choice with Transaction Costs: a User's Guide (with Johannes Muhle-Karbe)

Paris-Princeton Lectures in Mathematical Finance 2013

Recent progress in portfolio choice has made a wide class of problems involving transaction costs tractable. We review the basic approach to these problems, and outline some directions for future research.

e) Dynamic Trading Volume (with Marko Weber)

We derive the process followed by trading volume, in a market with finite depth and constant investment opportunities, where a representative investor, with a long horizon and constant relative risk aversion, trades a safe and a risky asset. Trading volume approximately follows a Gaussian, mean-reverting diffusion, and increases with depth, volatility, and risk aversion. The model generates an endogenous ban on leverage and short-selling.

3) Incentive Fees

a) Performance Maximization of Actively Managed Funds (with Gur Huberman, Zhenyu Wang)

Journal of Financial Economics, 101 (2011) no. 3 p. 574-595.

A growing literature suggests that, even in the absence of any ability to predict returns, holding options on the benchmarks or trading frequently can generate positive alpha. The ratio of alpha to its tracking error appraises a fund's performance. This paper derives the performance-maximizing strategy, which turns out to be a variant of a buy-write strategy, and the least upper bound on such performance enhancement. If common equity indices are used as benchmarks, the potential alpha generated from trading frequently can be substantial in magnitude, but it carries considerable risk. The statistical significance in estimated alpha is low, and the probability of a negative alpha is high. The performance enhancement from holding options can be significant – both economically and statistically – if the options' implied volatilities are higher than the volatilities of the benchmark returns. The performance-maximizing strategy derived in this paper is different from the strategies that switch portfolio exposure to the benchmarks. The exposure-switching strategies are not promising unless the switching is based on superior information.

b) The Incentives of Hedge Fund Fees and High-Water Marks (with Jan Obloj)

Mathematical Finance, forthcoming.

Hedge funds are associated with markedly asymmetric compensation: managers receive in performance fees a large fraction of their funds' profits, paid when a fund exceeds its high-water mark. We study the consequences of such incentive contracts, solving the portfolio choice problem from the viewpoints of managers and investors. We find that managers with constant relative risk aversion and constant investment opportunities, maximizing utility of fees at long horizons, choose constant Merton portfolios, with an effective risk aversion shrunk towards one in proportion to performance fees. In particular, the risk shifting implications are ambiguous and depend on the manager's own risk aversion. In a competitive equilibrium with a representative investor and heterogeneous funds, we find that funds with equal investment opportunities but different performance fees and managers' risk aversions coexist with the resulting leverage being an increasing function of fees. This theoretical prediction is verified empirically.

c) Robust Portfolios and Weak Incentives in Long Run Investments (with Johannes Muhle-Karbe, Hao Xing)

Working Paper.

When the planning horizon is long, and the safe asset grows indefinitely, iso-elastic portfolios are nearly optimal for investors who are close to iso-elastic for high wealth, and not too risk averse for low wealth. We prove this result in a general arbitrage-free, frictionless, semi-martingale model. As a consequence, optimal portfolios are robust to the perturbations in preferences induced by common option compensation schemes, and such incentives are weaker when their horizon is longer. Robust option incentives are possible, but require several arbitrarily large exercise prices, and are not always convex.

d) Hedge and Mutual Funds' Fees and the Separation of Private Investments (with Gu Wang)

A fund manager invests both the fund's assets and own private wealth in separate but potentially correlated risky assets, aiming to maximize expected utility from private wealth in the long run. If relative risk aversion and investment opportunities are constant, we find that the fund's portfolio depends only on the fund's investment opportunities, and the private portfolio only on private opportunities. This conclusion is valid both for a hedge fund manager, who is paid performance fees with a high-water mark provision, and for a mutual fund manager, who is paid management fees proportional to the fund's assets. The manager invests earned fees in the safe asset, allocating remaining private wealth in a constant-proportion portfolio, while the fund is managed as another constant-proportion portfolio. The optimal welfare is the maximum between the optimal welfare of each investment opportunity, with no diversification gain. In particular, the manager does not use private investments to hedge future income from fees.

1) Random Environments:

a) Portfolios and Risk Premia for the Long Run (with Scott Robertson)

Annals of Applied Probability, 22 (2012) no. 1 p. 239-284.

This paper develops a method to derive optimal portfolios and risk premia explicitly in a general diffusion model for an investor with power utility and a long horizon. The market has several risky assets and is potentially incomplete. Investment opportunities are driven by, and partially correlated with, state variables which follow an autonomous diffusion. The framework nests models of stochastic interest rates, return predictability, stochastic volatility and correlation risk. In models with several assets and a single state variable, long-run portfolios and risk premia admit explicit formulas up the solution of an ordinary differential equation which characterizes the principal eigenvalue of an elliptic operator. Multiple state variables lead to a quasilinear partial differential equation which is solvable for many models of interest. The paper derives the long-run optimal portfolio and the long-run optimal pricing measures depending on relative risk aversion, as well as their finite-horizon performance.

b) Static Fund Separation of Long Term Investments (with Scott Robertson)

Mathematical Finance, forthcoming.

This paper proves a class of static fund separation theorems, valid for investors with a long horizon and constant relative risk aversion, and with stochastic investment opportunities. An optimal portfolio decomposes as a constant mix of a few preference-free funds, which are common to all investors. The weight in each fund is a constant that may depend on an investor’s risk aversion, but not on the state variable, which changes over time. Vice versa, the composition of each fund may depend on the state, but not on the risk aversion, since a fund appears in the portfolios of different investors. We prove these results for two classes of models with a single state variable, and several assets with constant correlations with the state. In the linear class, the state is an Ornstein–Uhlenbeck process, risk premia are affine in the state, while volatilities and the interest rate are constant. In the square root class, the state follows a square root diffusion, expected returns and the interest rate are affine in the state, while volatilities are linear in the square root of the state.

c) Abstract, Classic, and Explicit Turnpikes (with Costantinos Kardaras, Scott Robertson, Hao Xing)

Finance and Stochastics, 18 (2014) no. 1 p. 75-114.

Portfolio turnpikes state that as the investment horizon increases, optimal portfolios for generic utilities converge to those of isoelastic utilities. This paper proves three kinds of turnpikes. In a general semimartingale setting, the abstract turnpike states that optimal final payoffs and portfolios converge under their myopic probabilities. In diffusion models with several assets and a single state variable, the classic turnpike demonstrates that optimal portfolios converge under the physical probability. In the same setting, the explicit turnpike identifies the limit of finite-horizon optimal portfolios as a long-run myopic portfolio defined in terms of the solution of an ergodic HJB equation.

d) Consumption in Incomplete Markets (with Gu Wang)

Working paper

An agent maximizes isoelastic utility from consumption with infinite horizon in an incomplete market, in which state variables are driven by diffusions. We first provide a general verification theorem, which links the solution of the Hamilton-Jacobi-Bellman equation to the optimal investment and consumption policies. To tackle the intractability of such problems, we propose approximate policies, which admit an upper bound in closed-form for their utility loss. These policies are optimal for a fictitious complete market, in which the safe rate and the state variable follow different dynamics, but excess returns remain the same. The approximate policies have closed form solutions in common models, and become optimal if the market is complete or utility is logarithmic.

2) Trading Frictions

a) The Fundamental Theorem of Asset Pricing under Transaction Costs (with Emmanuel Lepinette, Miklos Rasonyi)

Finance and Stochastics, 16 (2012) no. 4 p. 741-777.

This paper proves the fundamental theorem of asset pricing with transaction costs, when bid and ask prices follow locally bounded càdlàg (right-continuous, left-limited) processes. The robust no free lunch with vanishing risk condition (RNFLVR) for simple strategies is equivalent to the existence of a strictly consistent price system (SCPS). This result relies on a new notion of admissibility, which reflects future liquidation opportunities. The RNFLVR condition implies that admissible strategies are predictable processes of finite variation. The Appendix develops an extension of the familiar Stieltjes integral for càdlàg integrands and finite-variation integrators, which is central to modelling transaction costs with discontinuous prices.

b) Transaction Costs, Trading Volume, and the Liquidity Premium (with Stefan Gerhold, Johannes Muhle-Karbe, Walter Schachermayer)

Finance and Stochastics, 18 (2014) no. 1 p. 1-37.

In a market with one safe and one risky asset, an investor with a long horizon, constant investment opportunities, and constant relative risk aversion trades with small proportional transaction costs. We derive explicit formulas for the optimal investment policy, its implied welfare, liquidity premium, and trading volume. At the first order, the liquidity premium equals the spread, times share turnover, times a universal constant. Results are robust to consumption and finite-horizons. We exploit the equivalence of the transaction cost market to another frictionless market, with a shadow risky asset, in which investment opportunities are stochastic. The shadow price is also found explicitly.

c) Long Horizons, High Risk-Aversion, and Endogenous Spreads (with Johannes Muhle-Karbe)

Mathematical Finance, forthcoming.

For an investor with constant absolute risk aversion and a long horizon, who trades in a market with constant investment opportunities and small proportional transaction costs, we obtain explicitly the optimal investment policy, its implied welfare, liquidity premium, and trading volume. We identify these quantities as the limits of their isoelastic counterparts for high levels of risk aversion. The results are robust with respect to finite horizons, and extend to multiple uncorrelated risky assets. In this setting, we study a Stackelberg equilibrium, led by a risk-neutral, monopolistic market maker who sets the spread as to maximize profits. The resulting endogenous spread depends on investment opportunities only, and is of the order of a few percentage points for realistic parameter values.

d) Portfolio Choice with Transaction Costs: a User's Guide (with Johannes Muhle-Karbe)

Paris-Princeton Lectures in Mathematical Finance 2013

Recent progress in portfolio choice has made a wide class of problems involving transaction costs tractable. We review the basic approach to these problems, and outline some directions for future research.

e) Dynamic Trading Volume (with Marko Weber)

We derive the process followed by trading volume, in a market with finite depth and constant investment opportunities, where a representative investor, with a long horizon and constant relative risk aversion, trades a safe and a risky asset. Trading volume approximately follows a Gaussian, mean-reverting diffusion, and increases with depth, volatility, and risk aversion. The model generates an endogenous ban on leverage and short-selling.

3) Incentive Fees

a) Performance Maximization of Actively Managed Funds (with Gur Huberman, Zhenyu Wang)

Journal of Financial Economics, 101 (2011) no. 3 p. 574-595.

A growing literature suggests that, even in the absence of any ability to predict returns, holding options on the benchmarks or trading frequently can generate positive alpha. The ratio of alpha to its tracking error appraises a fund's performance. This paper derives the performance-maximizing strategy, which turns out to be a variant of a buy-write strategy, and the least upper bound on such performance enhancement. If common equity indices are used as benchmarks, the potential alpha generated from trading frequently can be substantial in magnitude, but it carries considerable risk. The statistical significance in estimated alpha is low, and the probability of a negative alpha is high. The performance enhancement from holding options can be significant – both economically and statistically – if the options' implied volatilities are higher than the volatilities of the benchmark returns. The performance-maximizing strategy derived in this paper is different from the strategies that switch portfolio exposure to the benchmarks. The exposure-switching strategies are not promising unless the switching is based on superior information.

b) The Incentives of Hedge Fund Fees and High-Water Marks (with Jan Obloj)

Mathematical Finance, forthcoming.

Hedge funds are associated with markedly asymmetric compensation: managers receive in performance fees a large fraction of their funds' profits, paid when a fund exceeds its high-water mark. We study the consequences of such incentive contracts, solving the portfolio choice problem from the viewpoints of managers and investors. We find that managers with constant relative risk aversion and constant investment opportunities, maximizing utility of fees at long horizons, choose constant Merton portfolios, with an effective risk aversion shrunk towards one in proportion to performance fees. In particular, the risk shifting implications are ambiguous and depend on the manager's own risk aversion. In a competitive equilibrium with a representative investor and heterogeneous funds, we find that funds with equal investment opportunities but different performance fees and managers' risk aversions coexist with the resulting leverage being an increasing function of fees. This theoretical prediction is verified empirically.

c) Robust Portfolios and Weak Incentives in Long Run Investments (with Johannes Muhle-Karbe, Hao Xing)

Working Paper.

When the planning horizon is long, and the safe asset grows indefinitely, iso-elastic portfolios are nearly optimal for investors who are close to iso-elastic for high wealth, and not too risk averse for low wealth. We prove this result in a general arbitrage-free, frictionless, semi-martingale model. As a consequence, optimal portfolios are robust to the perturbations in preferences induced by common option compensation schemes, and such incentives are weaker when their horizon is longer. Robust option incentives are possible, but require several arbitrarily large exercise prices, and are not always convex.

d) Hedge and Mutual Funds' Fees and the Separation of Private Investments (with Gu Wang)

A fund manager invests both the fund's assets and own private wealth in separate but potentially correlated risky assets, aiming to maximize expected utility from private wealth in the long run. If relative risk aversion and investment opportunities are constant, we find that the fund's portfolio depends only on the fund's investment opportunities, and the private portfolio only on private opportunities. This conclusion is valid both for a hedge fund manager, who is paid performance fees with a high-water mark provision, and for a mutual fund manager, who is paid management fees proportional to the fund's assets. The manager invests earned fees in the safe asset, allocating remaining private wealth in a constant-proportion portfolio, while the fund is managed as another constant-proportion portfolio. The optimal welfare is the maximum between the optimal welfare of each investment opportunity, with no diversification gain. In particular, the manager does not use private investments to hedge future income from fees.