Optimal Hedging and Valuation of Nontraded Assets

This paper examines a number of valuation problems faced byan expected-utility maximizing investor who, over a given timehorizon, is constrained to hold an asset which cannot be replicatedby dynamic trading and which therefore does not have a uniqueno-arbitrage price. We first derive the private valuation whichthe investor assigns to the nontraded asset in order to determinehis optimal investment in the traded assets. We thereby showthat, as part of this portfolio, the investor hedges the privatevaluation process of the nontraded asset, rather than its marketprice process. We also study the price at which the investorwould be willing to sell the nontraded asset if he were subsequentlyprohibited from trading in it, as well as the amount the investorwould be willing to pay to remove the trading restriction. Allthree values are shown to depend in an intuitive manner on theinvestor’s risk aversion, the residual risk of the nontradedasset unhedged by the traded assets, the difference betweenthe constrained holding and optimal unconstrained holding ofthe asset and the length of the time horizon over which theasset cannot be traded. JEL Classification: G11     

Nontraded asset valuation with portfolio constraints: a binomial approach

We provide a simple binomial framework to value American-stylederivatives subject to trading restrictions. The optimal investmentof liquid wealth is solved simultaneously with the early exercisedecision of the nontraded derivative. No-short-sales constraintson the underlying asset manifest themselves in the form of animplicit dividend yield in the risk-neutralized process forthe underlying asset. One consequence is that American calloptions may be optimally exercised prior to maturity even whenthe underlying asset pays no dividends. Applications to executivestock options (ESO) are presented: it is shown that the valueof an ESO could be substantially lower than that computed usingthe Black-Scholes model. We also analyze nontraded payoffs basedon a price that is imperfectly correlated with the price ofa traded asset.     

Performance of Thinly Traded Assets: A Case in Real Estate

Thinly traded private assets do not fit into the traditional finance paradigm of a liquid and well‐functioning market where trading is continuous and instantaneous. Since private assets cannot be bought and sold easily, they bear liquidity risk. Classical finance theories cannot properly gauge the performance of illiquid private assets because they implicitly assume such illiquidity is trivial. This paper proposes an alternative performance metric for the illiquid private asset, which explicitly captures liquidity risk in a formal analysis. Applying the new performance metric, we are able to explain the decades‐old “real estate risk premium puzzle.”     

On the Optimal Hedge of a Nontraded Cash Position

In this paper, we focus on the optimal demand for futures contracts by an investor with a logarithmic utility function who attempts to hedge a nontraded cash position. When the analysis is conducted in the “cash-commodity-price” space, we show that the value function associated with the Bernoulli investor program is not additively separable, thus suggesting that this investor hedges against shifts in the opportunity set as represented by the commodity price. By establishing the equivalence between the cash formulation of the problem and the wealth formulation, we are able to analyze the problem in the “wealth-commodity-price” space. In this space, we show the additive separability of the value function when the futures settlement price process is perfectly locally correlated with the commodity price process. The demand for futures in this instance is composed of (a) a mean-variance term and (b) a minimum-variance component that is a classic feature of models with nontraded assets. Since the first-best (nonmyopic) optimum is attained, however, the deviation from a mean-variance demand should not be interpreted as the expression of a nonmyopic behavior but rather as an attempt to restore a first-best optimum. On the other hand, when the correlation between the futures price and the underlying commodity price is imperfect, in general, the value function does not separate additively, the first-best solution cannot be attained, and the optimal futures trading strategy involves a hedging term against shifts in the opportunity set.     

Optimal commodity asset allocation with a coherent market risk modeling

This paper fills a fundamental gap in commodity price risk management and optimal portfolio selection literatures by contributing a thorough reflection on trading risk modeling with a dynamic asset allocation process and under the supposition of illiquid and adverse market settings. This paper analyzes, from a portfolio managers' perspective, the performance of liquidity adjusted risk modeling in obtaining efficient and coherent investable commodity portfolios under normal and adverse market conditions. As such, the author argues that liquidity risk associated with the uncertainty of liquidating multiple commodity assets over given holding periods is a key factor in formalizing and measuring overall trading risk and is thus an important component to model, particularly in the wake of the repercussions of the recent 2008 financial crisis. To this end, this article proposes a practical technique for the quantification of liquidity trading risk for large portfolios that consist of multiple commodity assets and whereby the holding periods are adjusted according to the specific needs of each trading portfolio. Specifically, the paper proposes a robust technique to commodity optimal portfolio selection, in a liquidity-adjusted value-at-risk (L-VaR) framework, and particularly from the perspective of large portfolios that have both long and short positions or portfolios that consist of merely pure long trading positions. Moreover, in this paper, the author develops a portfolio selection model and an optimization-algorithm which allocates commodity assets by minimizing the L-VaR subject to applying credible operational and financial constraints based on fundamental asset management considerations. The empirical optimization results indicate that this alternate L-VaR technique can be regarded as a robust portfolio management tool and can have many uses and applications in real-world asset management practices and predominantly for fund managers with large commodity portfolios.     

Can exchange traded funds be used to exploit industry and country momentum?

There is overwhelming empirical evidence on the existence of country and industry momentum effects. This line of research suggests that investors who buy country and industry portfolios with relatively high past returns and sell countries and industries with relatively low past returns will earn positive risk-adjusted returns. These studies focus on country and industry indexes that cannot be traded directly by investors. This raises the question of whether country and industry momentum effects really can be exploited by investors or whether they are illusionary in nature because they exist only on non-tradable assets. We analyze the profitability of country and industry momentum strategies using actual price data on exchange traded funds (ETFs). We find that over the sample periods during which these ETFs were traded, an investor would have been able to exploit country and industry momentum strategies with an excess return of about 5 % per annum. These returns cannot be explained by unconditional exposures to the Fama–French factors. The daily average bid-ask spreads on ETFs are substantially below the implied break-even transaction cost levels. Hence, we conclude that investors who are not willing or able to trade individual stocks may use ETFs to benefit from momentum effects in country and industry portfolios.     

Optimal holding period for assets that must be liquidated: A certainty equivalent wealth approach

Consumers often invest with a specific goal in mind and often know with some precision when the investment proceeds will be needed to achieve that goal. Because different investors have different attitudes toward risk and because different asset types exhibit different risk characteristics, there is often confusion as to the appropriate investment asset for a particular investor with a known investment horizon. It is also frequently, unclear as to whether investments should be switched to a less risky asset as time to liquidation becomes short. This paper addresses the issues of initial asset choice and the advisability of switching among assets when the investment goal date is known, employing the methodology of certainty equivalent wealth. In addition to suggesting optimal investment strategies for individuals based upon holding period and degree of risk aversion, it shows that switching investment assets produces suboptimal results.     

A model of optimal portfolio selection under liquidity risk and price impact

We study a financial model with one risk-free and one risky asset subject to liquidity risk and price impact. In this market, an investor may transfer funds between the two assets at any discrete time. Each purchase or sale policy decision affects the rice of the risky asset and incurs some fixed transaction cost. The objective is to maximize the expected utility from terminal liquidation value over a finite horizon and subject to a solvency constraint. This is formulated as an impulse control problem under state constraints and we characterize the value function as the unique constrained viscosity solution to the associated quasi-variational Hamilton–Jacobi–Bellman inequality. We would like to thank Mihail Zervos for useful discussions.     

Transaction costs, trading volume, and the liquidity premium

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. The 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.     

Derivative Pricing with Liquidity Risk: Theory and Evidence from the Credit Default Swap Market

We derive an equilibrium asset pricing model incorporating liquidity risk, derivatives, and short‐selling due to hedging of nontraded risk. We show that illiquid assets can have lower expected returns if the short‐sellers have more wealth, lower risk aversion, or shorter horizon. The pricing of liquidity risk is different for derivatives than for positive‐net‐supply assets, and depends on investors' net nontraded risk exposure. We estimate this model for the credit default swap market. We find strong evidence for an expected liquidity premium earned by the credit protection seller. The effect of liquidity risk is significant but economically small.     

Revisiting the Illiquidity Discount for Private Companies: A New (and “Skeptical”) Restricted‐Stock Study

Data from restricted‐stock studies are routinely used by business‐valuation analysts and small‐business appraisers to estimate discounts for lack of marketability, or DLOMs, which are then applied in the valuation of private companies. The rationale for the use of such DLOMs is that, even after an investor is compensated for the risk associated with holding an asset, an asset held unwillingly (due to illiquidity) must be worth less than if the asset were held by choice. But the same rationale can also be applied to the DLOM on riskless assets (such as Treasuries), and the evidence is consistent with a DLOM on such assets of only about 2.5%. This in turn suggests that any DLOM larger than 2.5% amounts to a second round of discounting for risk (where the first round occurs in a DCF or similar core valuation). Discounting with conventionally measured DLOMs is likely to be redundant because liquidity or marketability is highly correlated with company size, and size is already an important determinant of discount rates. Existing evidence suggests that, before DLOMs are applied, real‐world valuations of small businesses typically include discounts of as much as 50% for lack of size. And given that restricted‐stock studies are routinely used to support DLOMs of 20% to 40%, the valuation discounts resulting from this procedure are likely to be much too large. In contrast to industry practice, the author's study of a large sample of private placements of equity produces evidence consistent with use of a DLOM no greater than 5% or 6%.     

Price discovery and the international flow of information

We document that the opening volatility of American depositary receipts (ADRs) is lower when the trading of the underlying asset overlaps trading of the ADR on the New York Stock Exchange (NYSE). This lower volatility is consistent with the notion that price discovery on the NYSE is enhanced by concurrent trading in the underlying market. We also find that ADR volatility does not change when the underlying market closes, indicating that there is no significant change in the flow of public and private information at that time. This finding suggests that the NYSE becomes the dominant market during periods when both ADRs and their underlying assets are traded.     

Arbitrage Chains

A privately informed trader will engage in costly arbitrage, that is, trade on his knowledge that the price of an asset is different from the fundamental value if: (1) his order does not move the price immediately to reflect the information; and (2) he can hold the asset until the date when the information is reflected in the price. We study a general equilibrium model in which all agents optimize. In each period, there may be a trader with a limited horizon who has private information about a distant event. Whether he acts on his information, and whether subsequent informed traders act, is shown to depend on the possibility of a sequence or chain of future informed traders spanning the event date. An arbitrageur who receives good news will buy only if it is likely that, at the end of his trading horizon, a subsequent arbitrageur's buying will have pushed up the expected price. We show that limited trading horizons result in inefficient prices, because informed traders do not act on their information until the event date is sufficiently close. We also show that limited horizons can arise because of the cost-carry associated with holding an arbitrage portfolio over an extended period of time.     

The performance of model based option trading strategies

This paper analyzes returns to trading strategies in options markets that exploit information given by a theoretical asset pricing model. We examine trading strategies in which a positive portfolio weight is assigned to assets which market prices exceed the price of a theoretical asset pricing model. We investigate portfolio rules which mimic standard mean-variance analysis is used to construct optimal model based portfolio weights. In essence, these portfolio rules allow estimation risk, as well as price risk to be approximately hedged. An empirical exercise shows that the portfolio rules give out-of-sample Sharpe ratios exceeding unity for S&P 500 options. Portfolio returns have no discernible correlation with systematic risk factors, which is troubling for traditional risk based asset pricing explanations.     

Risk-sensitive benchmarked asset management

This paper extends the risk-sensitive asset management theory developed by Bielecki and Pliska and by Kuroda and Nagai to the case where the investor's objective is to outperform an investment benchmark. The main result is a mutual fund theorem. Every investor following the same benchmark will take positions, in proportions dependent on his/her risk sensitivity coefficient, in two funds: the log-optimal portfolio and a second fund which adjusts for the correlation between the traded assets, the benchmark and the underlying valuation factors.     

Optimal Consumption and Investment with Transaction Costs and Multiple Risky Assets

We consider the optimal intertemporal consumption and investment policy of a constant absolute risk aversion (CARA) investor who faces fixed and proportional transaction costs when trading multiple risky assets. We show that when asset returns are uncorrelated, the optimal investment policy is to keep the dollar amount invested in each risky asset between two constant levels and upon reaching either of these thresholds, to trade to the corresponding optimal targets. An extensive analysis suggests that transaction cost is an important factor in affecting trading volume and that it can significantly diminish the importance of stock return predictability as reported in the literature.     

On the Optimality of Portfolio Insurance

This paper examines the optimality of an insurance strategy in which an investor buys a risky asset and a put on that asset. The put's striking price serves as the insurance level. In complete markets, it is highly unlikely that an investor would utilize such a strategy. However, in some types of less complete markets, an investor may wish to purchase a put on the risky asset. Given only a risky asset, a put, and noncontinuous trading, an investor would purchase a put as a way of introducing a risk-free asset into the portfolio. If, in addition, there is a risk-free asset and the investor's utility function displays constant proportional risk-aversion, then the investor would buy the risk-free asset directly and not buy a put. In sum, only under the most incomplete markets would an investor find an insurance strategy optimal.     

The Price Is (Almost) Right

Most previous research tests market efficiency using average abnormal trading profits on dynamic trading strategies, and typically rejects the joint hypothesis of market efficiency and an asset pricing model. In contrast, we adopt the perspective of a buy‐and‐hold investor and examine stock price levels. For such an investor, the price level is more relevant than the short‐horizon expected return, and betas of cash flow fundamentals are more important than high‐frequency stock return betas. Our cross‐sectional tests suggest that there exist specifications in which differences in relative price levels of individual stocks can be largely explained by their fundamental betas.     

Monetary Policy and Asset Prices: A Mechanism Design Approach

We investigate the effects of monetary policy on asset prices in economies where assets are traded periodically in bilateral meetings. The trading mechanism is designed to maximize social welfare taking as given the frictions in the environment and monetary policy. We show that asset price “bubbles” emerge in a constrained‐efficient monetary equilibrium only if liquidity is abundant and the first‐best allocation is implementable. In contrast, if liquidity is scarce, assets are priced at their fundamental value in any constrained‐efficient monetary equilibrium, in which case an increase in inflation has no effect on asset prices, but it reduces output and welfare.