Short Positions in the First Principal Component Portfolio

Insurance companies and pension plans typically hold well-diversified equity portfolios. These institutions are also often restricted from taking short positions. The diversification requirement operates on the portfolio level, while the short sale constraint is at the individual security level. We examine an investment strategy that exposes a tension between these two requirements. This strategy uses the first principal component to construct the portfolio and by design meets the first requirement. Empirical portfolios based on the first principal component do an excellent job of capturing market exposure and minimizing diversifiable risk. However, in practice such portfolios sometimes contain a few short positions. So this strategy does not always meet the second requirement. We examine which features of stock returns give rise to short positions when a portfolio is based on the first principal component, and we are able to identify the characteristics of the stocks that are responsible for the short positions. These stocks tend to have negative correlations with the majority of other stocks. In contrast such stocks would typically be held long in a Markowitz portfolio. We discuss and explain this puzzle.     

PORTFOLIO MANAGEMENT WITH CONSTRAINTS

The traditional portfolio selection problem concerns an agent whose objective is to maximize the expected utility of terminal wealth over some horizon. This basic problem can be modified by adding constraints. In this paper we investigate the portfolio selection problem for an investor who desires to outperform some benchmark index with a certain confidence level. The benchmark is chosen to reflect some particular investment objective and it can be either deterministic or stochastic. The optimal strategy for this class of problems can lead to nonconvex constraints raising issues of existence and uniqueness. We solve this optimal portfolio selection problem and investigate the procedure for both deterministic and stochastic benchmarks.     

The impact of variance estimation in option valuation models

This paper examines some implications of using an estimate of the variance in option valuation models. This procedure produces biased option values. It is shown that the magnitude of this bias is not large. The dispersion induced in the option price is more significant particularly for parameter values of practical interest. The nature and extent of this dispersion is examined by numerical examples. The paper suggests how a Bayesian approach could be used to cope with the estimation error.     

Designing a Countercyclical Insurance Program for Systemic Risk

This article proposes a framework for measuring and managing systemic risk. Current solvency regulations have been criticized for their focus on individual firms rather than the system as a whole. We show how an insurance program can be designed to deal with systemic risk through a risk charge on participating institutions. The risk charge is based on the generalized co‐conditional tail expectation, a conditional risk measure adapted from conditional value‐at‐risk. Current regulations have been criticized on the grounds that their capital requirements are procyclical. They require extra capital in periods of extreme stress thus exacerbating a crisis. We show how to construct a countercyclical risk charge and illustrate the approach using a numerical example.     

Pricing of New Securities in an Incomplete Market: the Catch 22 of No-Arbitrage Pricing

There are two distinctly different approaches to the valuation of a new security in an incomplete market. The first approach takes the prices of the existing securities as fixed and uses no-arbitrage arguments to derive the set of equivalent martingale measures that are consistent with the initial prices of the traded securities. The price of the new security is then obtained by appealing to certain criteria or on the basis of some preference assumption. The second method prices the new security within a general equilibrium framework. This paper clarifies the distinction between the two approaches and provides a simple proof that the introduction of the new security will typically change the prices of all the existing securities. We are left with the paradox that a genuinely new derivative security is not redundant, but the dominant pricing paradigm in derivative security pricing is the no-arbitrage approach, which requires the redundancy of the security. Given the widespread practice of using the no-arbitrage approach to price (or bound the price of) a new security, we also comment on some justifications for this approach.     

Dynamic Fund Protection

Abstract

Dynamic fund protection provides an investor with a floor level of protection during the investment period. This feature generalizes the concept of a put option, which provides only a floor value at a particular time. The dynamic protection feature ensures that the fund value is upgraded if it ever falls below a certain threshold level. Gerber and Pafumi (2000) have recently derived a closed-form expression for the price of this protection when the basic portfolio follows geometric Brownian motion. In this paper we examine the pricing of this feature under the constant elasticity of variance process. Two approaches are used to obtain numerical results. First, we show how to extend the basic Monte Carlo approach to handle the particular features of dynamic protection. When a discrete-time simulation approach is used to value a derivative that is subject to continuous monitoring, there is a bias. We show how to remove this bias. Second, a partial differential equation approach is used to price dynamic protection. We demonstrate that the price of the dynamic protection is sensitive to the investment assumptions. We also discuss a discrete time modification of the dynamic protection feature that is suitable for practical implementation. The paper deals just with pricing and does not consider the important question of reserving for these contracts.     

Discretely adjusted option hedges

This paper analyses the distribution of returns on a hedged portfolio, consisting of a European call option and its associated stock, when the portfolio is rebalanced at discrete time intervals. Under the assumptions of the Black-Scholes model this distribution is particularly skew. In tests of the average return on a hedged portfolio this skewness leads to biased t-statistics. The paper explores the nature and extent of this bias and suggests procedures for overcoming it. Other aspects of discrete hedging are also discussed.     

Optimal Portfolio Selection with Transaction Costs

Abstract

This paper examines the lifetime portfolio-selection problem in the presence of transaction costs. Using a discrete time approach, we develop analytical expressions for the investor's indirect utility function and also for the boundaries of the no-transactions region. The economy consists of a single risky asset and a riskless asset. Transactions in the risky asset incur proportional transaction costs. The investor has a power utility function and is assumed to maximize expected utility of end-of-period wealth. We illustrate the solution procedure in the case in which the returns on the risky asset follow a multiplicative binomial process. Our paper both complements and extends the recent work by Gennotte and Jung (1994), which used numerical approximations to tackle this problem.     

Optimal design of equity-linked products with a probabilistic constraint

There is a rich variety of tailored investment products available to the retail investor. These products combine upside participation in bull markets with downside protection in bear markets. Examples include the equity-linked products sold by insurance companies and the structured products marketed by banks. This paper examines a particular contract design for products of this nature. The paper finds the optimal design from the investor's viewpoint. It is assumed that the investor wishes to maximize expected utility of the terminal wealth subject to certain constraints. These constraints include a guaranteed rate of return as well as the opportunity to outperform a benchmark portfolio with a given probability. We derive the explicit form of the optimal design assuming both constraints apply and we illustrate the nature of the solution using some specific examples.