The Effect of Shortfall as a Risk Measure for Portfolios with Hedge Funds

Abstract:  Current research suggests that the large downside risk in hedge fund returns disqualifies the variance as an appropriate risk measure. For example, one can easily construct portfolios with nonlinear pay-offs that have both a high Sharpe ratio and a high downside risk. This paper examines the consequences of shortfall-based risk measures in the context of portfolio optimization. In contrast to popular belief, we show that negative skewness for optimal mean-shortfall portfolios can be much greater than for mean-variance portfolios. Using empirical hedge fund return data we show that the optimal mean-shortfall portfolio substantially reduces the probability of small shortfalls at the expense of an increased extreme crash probability. We explain this by proving analytically under what conditions short-put payoffs are optimal for a mean-shortfall investor. Finally, we show that quadratic shortfall or semivariance is less prone to these problems. This suggests that the precise choice of the downside risk measure is highly relevant for optimal portfolio construction under loss averse preferences.     

INTEREST RATE RISK,MARKET VALUE,AND HEDGING FINANCIAL PORTFOLIOS

This paper derives theoretical hedge ratios for the financial portfolio that preserve its present value in the presence of interest rate risk. From a practical point of view and for any given portfolio, the existence of the financial futures market allows the investor to employ any of a number of different hedges, each of which approximately satisfies the theoretical condition. The theory indicates that wealth-preserving hedges depend on the interest elasticities (durations) of the spot assets and liabilities contained in the portfolio, portfolio leverage, and the interest elasticity (duration) of the financial instrument underlying the futures contract that is employed in constructing the hedge. Also, hedges designed to maintain net interest margin or net cash flow do not minimize exposure to interest rate risk.     

Risk minimization in stochastic volatility models: model risk and empirical performance

In this paper the performance of locally risk-minimizing delta hedge strategies for European options in stochastic volatility models is studied from an experimental as well as from an empirical perspective. These hedge strategies are derived for a large class of diffusion-type stochastic volatility models, and they are as easy to implement as usual delta hedges. Our simulation results on model risk show that these risk-minimizing hedges are robust with respect to uncertainty and misconceptions about the underlying data generating process. The empirical study, which includes the US sub-prime crisis period, documents that in equity markets risk-minimizing delta hedges consistently outperform usual delta hedges by approximately halving the standard deviation of the profit-and-loss ratio.     

Reducing reporting risk: Designating foreign currency forward contracts as cash flow hedges

We present an example that compares the effects on earnings of designating a foreign currency forward contract as either a cash-flow or fair-value hedge of a foreign currency denominated receivable. Entities engaging in exchange transactions not denominated in their functional currency frequently enter into foreign currency forward contracts in order to mitigate their foreign exchange rate risk exposure. The aggregate effect on earnings of the transaction gain or loss on the foreign currency receivable and the gain or loss on the forward contract is known on the date the forward contract is initiated. The effect on each period’s earnings during the term of a forward contract designated as a cash-flow hedge is also known on the date the contract is initiated; whereas the effect on each periods’ earnings from a fair-value hedge cannot be determined until the respective balance sheet dates. Therefore, designating forward contracts as cash-flow hedges may suppress volatility in reported earnings compared to designating forward contracts as fair-value hedges. In addition, the reporting risk (the amount of uncertainty surrounding the pending measure of an item to be reported in the financial statements) is lower when a forward contract is designated as a cash-flow hedge relative to designating it as a fair-value hedge. This suggests foreign currency forward contracts designated as cash-flow hedges are more consistent with the purpose of hedge accounting: to mitigate the effects on earnings of applying different measurement criteria for the hedge and the hedged item.     

Do commodities make effective hedges for equity investors?

The purpose of this paper is to evaluate whether commodities are effective hedges for equity holders. We employ three different methodologies to calculate time varying hedge ratios. First, we examine time-varying hedge ratios and how much portfolio risk can be reduced relative to a long position in the S&P 500. We calculate hedge ratios from realized variances and covariances; second, we estimate a recursive multivariate GARCH (BEKK) model and calculate the hedge ratios from the estimated covariances; and thirdly, we calculate the hedge ratios by estimating recursive OLS regressions. The results of our paper are very clear. First, commodities are not effective hedges for the S&P 500. Equity market investors and asset managers looking for a way to manage and reduce portfolio risk will be well advised to search for alternative hedges for the S&P 500 than commodities. Second, our results do not support the claim that commodities were a good hedge for the equity market during the financial crisis.     

Litigation risk, strategic disclosure and the underpricing of initial public offerings

Using word content analysis on the time-series of IPO prospectuses, we show that issuers tradeoff underpricing and strategic disclosure as potential hedges against litigation risk. This tradeoff explains a significant fraction of the variation in prospectus revision patterns, IPO underpricing, the partial adjustment phenomenon, and litigation outcomes. We find that strong disclosure is an effective hedge against all types of lawsuits. Underpricing, however, is an effective hedge only against Section 11 lawsuits, those lawsuits which are most damaging to the underwriter. Underwriters who fail to adequately hedge litigation risk experience economically large penalties, including loss of market share.     

A generalized approach to optimal hedging with option contracts

In this paper, we develop a theoretical model in which a firm hedges a spot position using options in the presence of both quantity (production) and basis risks. Our optimal hedge ratio is fairly general, in that the dependence structure is modeled through a copula function representing the quantiles of the hedged position, and hence any quantile risk measure can be employed. We study the sensitivity of the exercise price which minimizes the risk of the hedged portfolio to the relevant parameters, and we find that the subjective risk aversion of the firm does not play any role. The only trade-off is between the effectiveness and cost of the hedging strategy.     

全球流动性过剩中的对冲基金:国际新动向及其对中国的启示

全球流动性过剩导致全球金融资产激增,推动对冲基金业规模迅速膨胀。与此同时,对冲基金业也出现了新的变化:投资者结构出现了机构化趋势,并受到监管当局日趋严厉的监管。这也使得对冲基金的收益和风险水平有了新的变化,呈现稳中趋降的态势。全球经济失衡和流动性过剩也推动对冲基金在全球寻找套利和投机机会,对全球的金融稳定产生一定的影响。中国需要审慎地推进资本项目开放和加强对冲基金的监管。     

Arbitrage-free interval and dynamic hedging in an illiquid market

This paper derives two pricing PDEs for a general European option under liquidity risk. We provide two modified hedges: one hedge replicates a short option and the other replicates a long option inclusive of liquidity costs under continuous rebalancing. We identify an arbitrage-free interval by calculating the costs of the two hedges. Unlike in a setting with infinite overall transaction costs, the overall liquidity cost in our model is proved to be finite even under continuous rebalancing. Numerical results on option pricing and the moments of hedge errors of Black–Scholes and our modified hedges are also presented.     

Using Corporate Inflation Protected Securities to Hedge Interest Rate Risk

In the first half of 2008, rising inflation became a concern, but by the fall the focus was on deflation. Such shifts in the outlook for inflation represent a significant risk for some companies, particularly those whose revenues and profits are negatively affected by increases in inflation and rates. For such companies, the use of long‐term fixed‐rate debt will provide at least a partial hedge against increased rates. Less widely appreciated is that even companies whose profits move up and down with inflation face considerable risk from fluctuations in interest rates. Conventional wisdom holds that floating‐rate debt hedges this risk. But this article argues that floating‐rate debt still leaves a company exposed to increases in real interest rates. Inflation‐sensitive companies such as utilities can use corporate inflation‐protected securities (CIPS) to hedge their real interest rate risk as well as inflation risk. In addition to its hedging benefits, CIPS also have the potential to reduce borrowing costs by satisfying growing investor demand for high‐quality securities that provide inflation protection (including demand sources like the recent restoration of French savings accounts to inflation).     

Hedging with Two Futures Contracts: Simplicity Pays

We propose to use two futures contracts in hedging an agricultural commodity commitment to solve either the standard delta hedge or the roll‐over issue. Most current literature on dual‐hedge strategies is based on a structured model to reduce roll‐over risk and is somehow difficult to apply for agricultural futures contracts. Instead, we propose to apply a regression based model and a naive rules of thumb for dual‐hedges which are applicable for agricultural commodities. The naive dual strategy stems from the fact that in a large sample of agricultural commodities, De Ville, Dhaene and Sercu (2008) find that GARCH‐based hedges do not perform as well as OLS‐based ones and that we can avoid estimation error with such a simple rule. Our semi‐naive hedge ratios are driven from two conditions: omitting exposure to spot price and minimising the variance of the unexpected basis effects on the portfolio values. We find that, generally, (i) rebalancing helps; (ii) the two‐contract hedging rules do better than the one‐contract counterparts, even for standard delta hedges without rolling‐over; (iii) simplicity pays: the naive rules are the best one–for corn and wheat within the two‐contract group, the semi‐naive rule systematically beats the others and GARCH performs worse than OLS for either one‐contract or two‐contract hedges and for soybeans the traditional naive rule performs nearly as well as OLS. These conclusions are based on the tests on unconditional variance ( Diebold and Mariano, 1995 ) and those on conditional risk ( Giacomini and White, 2006 ).     

Performance analysis of a collateralized fund obligation (CFO) equity tranche

This article examines the performance of the junior tranche of a collateralized fund obligation (CFO), i.e. the residual claim (equity) on a securitized portfolio of hedge funds. We use a polynomial goal programming model to create optimal portfolios of hedge funds, conditional to investor preferences and diversification constraints (maximum allocation per strategy). For each portfolio, we build CFO structures that have different levels of leverage, and analyze both the stand-alone performance as well as potential diversification benefits (low systematic risk exposures) of investing in the equity tranche of these structures. We find that the unconstrained mean-variance portfolio yields a high performance, but greater exposure to systematic risk. We observe the exact opposite picture in the case of unconstrained optimization, where a skewness bias is added, thus proving the existence of a trade-off between stand-alone performance and low exposure to systematic risk factors. We provide evidence that leveraged exposure to these hedge fund portfolios through the structuring of CFOs creates value for the equity tranche investor, even during the recent financial crisis.     

Property and Inflation: The Hedging Characteristics of U.K. Commercial Property, 1967–1994

This paper examines the statistical similarities between U.K. commercial property capital and rental values and the price level. Our aim is to determine whether commercial property is an inflation hedge and, if so, what type of inflation it hedges against. To answer these questions, we use both a multivariate unobserved components model and structural vector autoregressions. We find that commercial property is an inflation hedge but only a weak one. More specifically, we find that property offers some form of partial hedge against changes in the underlying inflation rate but not to either temporary or permanent changes to the price level. We also find that capital values offer a stronger hedge than rental values and that industrial and retail property account for most of this hedging capacity. We find no evidence that property responds differently to high or low inflation but we do find capital and rental values respond more to unexpected inflation than anticipated price changes.     

Interest Rate Risk Rewards in Stock Returns of Financial Corporations: Evidence from Germany

The interest rate sensitivity of stock returns of financial and non-financial corporations is a well-known phenomenon. However, only little is known about the part of total stock returns that is attributable to the compensation an investor receives for being exposed to interest rate risk when investing in equity securities. We pursue here a benchmark portfolio approach, constructing benchmark portfolios having the same interest rate risk exposure as a particular stock. By studying the time series of returns of these asset-specific benchmarks, we find: i) Regardless of the industry considered, the interest rate risk benchmarks of German corporations have mostly earned a significantly positive reward. ii) Returns of interest rate risk benchmarks of financial institutions exceeded significantly those of non-financial corporations. iii) An investor willing to bear nothing but the average interest rate risk of German financial institutions would have earned a mean return of about or even exceeding 70% of the corresponding total stock returns. iv) Returns of the interest rate risk benchmarks of the German insurance sector were significantly higher than those of German banks, which seems to contradict conventional market wisdom that insurances hedge interest rate risks.     

BASIS VOLATILITY: IMPLICATIONS FOR HEDGING

Most hedges placed in futures markets must be lifted before contract expiration, which necessitates incurring “basis risk.” The focus of this paper is on quantifying such risk as a function of the timing of a hedge, its duration, distance from contract expiration, hedge life, and other market-observable variables. The development of basis-risk profiles provides a hedger with estimates of hedging risks that reasonably can be expected before the actual placement of hedges, thus serving as a useful input in the hedging decision.     

Locked Up by a Lockup: Valuing Liquidity as a Real Option

Hedge funds often impose lockups and notice periods to limit the ability of investors to withdraw capital. We model the investor's decision to withdraw capital as a real option and treat lockups and notice periods as exercise restrictions. Our methodology incorporates time-varying probabilities of hedge fund failure and optimal early exercise. We estimate a two-year lockup with a three-month notice period costs approximately 1% of the initial investment for an investor with constant relative risk aversion utility and risk aversion of three. The cost of illiquidity can easily exceed 10% if the hedge fund manager can arbitrarily suspend withdrawals.     

Currency hedging for single-currency equity portfolios: Does cross-asset risk matter?

Foreign investors who are fully invested in a single-currency domestic equity portfolio are exposed to domestic equity risk, but also to currency risk. The standard approach to hedging the currency risk optimally is to estimate a single optimal hedge ratio, but this approach hedges only exchange rate risk, not cross-asset risk. We provide an alternative approach that estimates two optimal hedge ratios to adjust the currency exposures—one associated with the domestic currency and one associated with the foreign currency—and hedges both exchange rate risk and cross-asset risk. This alternative approach can significantly reduce risk.     

Ex Ante Hedging Effectiveness of UK Stock Index Futures Contracts: Evidence for the FTSE 100 and FTSE Mid 250 Contracts

Ex ante hedging effectiveness of the FTSE 100 and FTSE Mid 250 index futures contracts is examined for a range of portfolios, consisting of stock market indexes and professionally managed portfolios (investment trust companies). Previous studies which focused on ex post hedging performance using spot portfolios that mirror market indexes are shown to overstate the risk reduction potential of index futures. Although ex ante hedge ratios are found to be characterised by intertemporal instability, ex ante hedging performance of direct hedges and cross hedges approaches that of the ex post benchmark when hedge ratios are estimated using a sufficient window size.     

Estimation of multiple period expected shortfall and median shortfall for risk management

With the regulatory requirements for risk management, Value at Risk (VaR) has become an essential tool in determining capital reserves to protect the risk induced by adverse market movements. The fact that VaR is not coherent has motivated the industry to explore alternative risk measures such as expected shortfall. The first objective of this paper is to propose statistical methods for estimating multiple-period expected shortfall under GARCH models. In addition to the expected shortfall, we investigate a new tool called median shortfall to measure risk. The second objective of this paper is to develop backtesting methods for assessing the performance of expected shortfall and median shortfall estimators from statistical and financial perspectives. By applying our expected shortfall estimators and other existing approaches to seven international markets, we demonstrate the superiority of our methods with respect to statistical and practical evaluations. Our expected shortfall estimators likely provide an unbiased reference for setting the minimum capital required for safeguarding against expected loss.