The Optimality of Multi-stage Venture Capital Financing: An Option-Theoretic Approach

For venture capital firms, facing undiversifiable risks, multi-staged financing is an optimal contract which offers significant risk reduction at a cost of only slightly lower potential return. The optimality does not depend on the presence of moral hazard and agency problems. Our theoretical model of multi-stage financing, largely based on Asian option pricing theory, allows us to compute the risk reduction ratio due to multi-staging. The return on a staged financing plan is equivalent to an exchange of a straight equity stake for that acquired through stochastic averaging over time. We compare standard deviation ratios for staged vs. up-front financings as well as across asset classes. We find that risk mitigation due to multi-staging is significant in and of itself and enough to markedly improve venture capital’s risk-reward ratios relative to alternatives

Optimal Liquidation of Venture Capital Stakes

We model the optimal liquidation behavior of a venture capital or non-diversified asset management firm faced with a sale of concentrated security holdings. As the firm?s stake is large, its sales can lead to permanent and temporary price depressions. At the optimum, the institution chooses the liquidation interval to balance the exposure to the market return variance against the impact of its own sales on the realized return. We obtain closed-form solutions for power impact functions uncorrelated with returns. We also consider market impact correlated with the return process, i.e. a case where liquidity evaporates during severe price dislocations

A Simple Utility Approach to Private Equity Sales

The paper examines the liquidity risk of a private equity firm that decides to dispose of a large holding in its portfolio. As the sale takes time, it requires a careful balancing act of the exposure to the fluctuations in the market value of the investment against the large sale-induced price depression. A mean-standard deviation utility framework is an appealing decision tool for optimizing protracted asset dispositions. The firm maximizes the expected profit from the sale strategy net of the price concession minus a penalty function for exposure to the price risk, with the penalty weight related to a loss confidence interval

The modeling of liquidity in the value-at-risk framework

This dissertation is an exposition of a new method of modeling liquidity in the Value-at-Risk (VaR) framework. Liquidity, in general, refers to the deviation of transaction price from the intrinsic value of a security. Prior research has focused on the size of that deviation defined as the bid-ask spread set by the dealer to compensate him for adverse selection and inventory costs. Instead, the model proposed here looks at the time it takes to liquidate a position in the market and its relationship to the VaR of the agent\u27s portfolio. This has not been done before. ^ The dissertation is divided as follows. Chapter 1 contains a critique of the VaR modeling literature from an implementation perspective of a financial institution. It provides a motivation for a liquidity adjustment model which is independent of the probability assumption for the underlying risk factors and the composition of the agent\u27s portfolio, and, specifically, whether it contains any non-linear assets (e.g. convex bonds and options). Numerical examples show that the inclusion of derivatives in the portfolio renders many specialized approximate methods inadequate. Chapter 2 develops a theoretical framework for the inclusion of the time dimension of liquidity into the VaR framework, irrespective of whether the sources of liquidity are exogenous (beyond the institution\u27s control) or endogenous (related to the size of the institution\u27s portfolio relative to the market depth). It proposes a statistical aggregation procedure which relies on conditional multivariate sampling to account for the case where liquidity is constrained by a slow speed of trading in a given market. Chapter 3 investigates the optimal behavior of an agent faced with selling his holdings, which may be large relative to the size of the entire market. He solves for the liquidation horizon that represents the best tradeoff between the volatility of the stochastic process for the equilibrium return and a depressed price due to the liquidation program\u27s market impact. The impact specifications are either general power or linear functions correlated with the underlying returns. They allow for the empirically observed phenomenon of liquidity shocks appearing during rapid market price compressions.