Downside Risk

Economists have long recognized that investors care differentlyabout downside losses versus upside gains. Agents who placegreater weight on downside risk demand additional compensationfor holding stocks with high sensitivities to downside marketmovements. We show that the cross section of stock returns reflectsa downside risk premium of approximately 6% per annum. Stocksthat covary strongly with the market during market declineshave high average returns. The reward for beasring downsiderisk is not simply compensation for regular market beta, noris it explained by coskewness or liquidity risk, or by size,value, and momentum characteristics. (JEL C12, C15, C32, G12)     

Downside Risk in Practice

Although investors associate risk with negative outcomes and downside fluctuations, modern portfolio theory does not. For investors, volatility per se is not necessarily bad; volatility below a benchmark is. A stock that magnifies the market's fluctuations is not necessarily bad; one that magnifies the market's downside swings is. Even Harry Mar‐kowitz, the father of modern portfolio theory, viewed downside risk as a better way to assess risk than the “mean‐variance” framework that he ultimately proposed and that has since become the standard. This article highlights the shortcomings of traditional measures of risk (the standard deviation and beta), introduces the concept of downside risk, and discusses two measures of it—the “semideviation” and “downside beta.” It also discusses the use of such measures in asset pricing models to estimate required returns on equity. Data from a few well‐known companies are used to illustrate that the cost of equity based on downside risk can be substantially different from that based on the CAPM. The article concludes with a brief discussion of risk‐adjusted returns and a comparison of the traditional method of calculating such returns with both the Sharpe ratio and its counterpart in a downside risk framework, the Sortino ratio. The appendix demonstrates how to calculate these risk measures in Excel.     

Downside Risk in Emerging Markets

This paper investigates the relation between downside risk and expected returns on the aggregate stock market in an international context. Nonparametric and parametric value at risk are used as measures of downside risk to determine the existence of a risk-return trade-off. For emerging markets, fixed effects panel data regressions provide evidence for a significantly positive relationship between monthly expected market returns and downside risk. This result is robust after controlling for aggregate dividend yield and price-to-fundamental ratios. The relationship between expected returns and downside risk is weaker for developed markets and vanishes when control variables are included in the specification.     

Downside Risk and Investment Choice

This paper develops an optimal investment strategy for individuals concerned with avoiding the possibility of realizing returns below a predetermined target level within a prescribed period of time. Assuming a Brownian motion process, a model is developed which allows computation of the exact probability of failure. The algorithm and associated comparative statics with respect to the mean and standard deviation of returns, target return, time horizon, and risk-free rate of return are likely to have many useful practical applications.     

Strong Increases in Downside Risk Aversion

A ranking of risk preferences is of economic interest insofar as it leads to unambiguous comparative statics predictions, and for this to be the case, the ranking must be a strict partial ordering. The ranking by greater risk aversion meets this demand at the second order, and yields a variety of well-known predictions concerning the effect of greater risk aversion on demands for insurance and risky assets, among many other applications. There has been less success at the third order, where ranking preferences by aversion to downside risk has not produced a strict partial ordering. The problem is that account has not been taken of the fact that an increase in downside risk aversion must induce changes in risk aversion as well. We propose a definition of stronger downside risk aversion that does yield a strict partial ordering by requiring a nested increase in both second- and third-order risk aversion, so that v is more strongly downside risk averse than u if v is more risk averse and more downside risk averse than u. We demonstrate that v being more strongly downside risk averse than u is characterized by v never liking any change in the probability distribution for y that induces a third-order stochastic dominance deterioration in the distribution for u(y). We apply the definition to obtain intuitive comparative statics predictions in the precautionary saving problem, and relate the definition to alternatives proposed in the literature.     

Portfolio Diversification Effects of Downside Risk

Risk managers use portfolios to diversify away the unpricedrisk of individual securities. In this article we compare thebenefits of portfolio diversification for downside risk in casereturns are normally distributed with the case of fat-taileddistributed returns. The downside risk of a security is decomposedinto a part which is attributable to the market risk, an idiosyncraticpart, and a second independent factor. We show that the fat-tailed-baseddownside risk, measured as value-at-risk (VaR), should declinemore rapidly than the normal-based VaR. This result is confirmedempirically.     

Risk Aversion,Downside Risk Aversion and Paying for Stochastic Improvements

This paper considers the relationship between risk preferences and the willingness to pay for stochastic improvements. We show that if the stochastic improvement satisfies a double-crossing condition, then a decision maker with utility v is willing to pay more than a decision maker with utility u, if v is both more risk averse and less downside risk averse than u. As the condition always holds in the case of self-protection, the result implies novel characterizations of individuals’ willingness to pay to reduce the probability of loss. By establishing a general result on the correspondence between an individual's willingness to pay, and his optimal purchase of stochastic improvement when there is a given relationship between stochastic improvements and the amount paid for them, we further show that all results on the willingness to pay can be applied directly to characterize the conditions under which a more risk averse individual will optimally choose to buy more stochastic improvement. Generalizations of existing results on optimal choice of self-protection can be obtained as corollaries.     

Downside Risk Management of a Defined Benefit Plan Considering Longevity Basis Risk

To control downside risk of a defined benefit pension plan arising from unexpected mortality improvements and severe market turbulence, this article proposes an optimization model by imposing two conditional value at risk constraints to control tail risks of pension funding status and total pension costs. With this setup, we further examine two longevity risk hedging strategies subject to basis risk. While the existing literature suggests that the excess-risk hedging strategy is more attractive than the ground-up hedging strategy as the latter is more capital intensive and expensive, our numerical examples show that the excess-risk hedging strategy is much more vulnerable to longevity basis risk, which limits its applications for pension longevity risk management. Hence, our findings provide important insight on the effect of basis risk on longevity hedging strategies.     

经济下行、PPP风险来源与机制设计

自金融危机之后,PPP模式在全球范围内遭到了广泛的质疑,曾经被视为发展公共基础设施建设重要手段的PPP模式沦为各国地方政府沉重的财政负担.为此,我们通过分析经济下行情境下PPP模式在欧盟国家的风险来源,研究了中国现行PPP模式机制设计中的风险识别与分配机制.结果 显示,现阶段物有所值定性评价、地方财政压力测算与风险分配框架受到地方政府强大内生性需求的影响,在经济下行情境中一部分潜在的风险就很容易被忽视:(1)物有所值定性评价结果高度一致,政府官员背景评审专家影响了最终评价结果;(2)地方财政承受能力论证测算以经济繁荣时期的财政增长率为基准,低估了经济下行情境中PPP项目财政支出对于财政预算的压力;(3)风险分配框架不合理,倾向于由政府与社会资本共同承担,将前期风险识别与谈判成本转嫁到了后期再协商阶段.上述发现有助于揭示我国PPP模式推广过程中的机制缺陷,为提高PPP项目抵御未来不确定性的能力,进一步规范中国PPP模式在公共基础设施领域中的应用提供了指引.     

行业趋同度、投资者情绪与股市下跌风险

行业趋同度及投资者情绪可以反映市场交易热度.用格兰杰因果网络计算市场的行业趋同度,指数换手率、波动率及BEYR代表投资者情绪,研究这些指标对指数收益及股市下跌风险的影响发现:4个指标对指数收益的预测能力并不稳健,但是对股市下跌风险却有显著的预测能力;用4个指标构造合成因子,发现合成因子对股市下跌风险的预测能力更为显著;利用合成因子进行择时,可以有效避免市场极端的下跌行情,因子择时策略的夏普比率达到0.53,远超指数买入持有策略的0.14.综合来看,行业趋同度及投资者情绪代表的交易热度能够对股市下跌风险提前预警,从而带来投资收益的显著提高.     

A Method of Calculating the Downside Risk by Multivariate Nonnormal Distributions

A method of calculating the downside risk by fitting multivariate nonnormal distributions to financial data is proposed. Firstly, maximum likelihood method by using the random numbers of the Pearson distribution system are introduced. The rates of returns of the stock index are fitted to the multivariate nonnormal distributions by this method. Secondly, the cases of calculating the downside risk by the standard deviation, the percentile of historical simulation method and this method, are compared.     

Analysing Perceived Downside Risk: the Component Value‐at‐Risk Framework

Multinational companies face increasing risks arising from external risk factors, e.g. exchange rates, interest rates and commodity prices, which they have learned to hedge using derivatives. However, despite increasing disclosure requirements, a firm's net risk profile may not be transparent to shareholders. We develop the ‘Component Value‐at‐Risk (VaR)’ framework for companies to identify the multi‐dimensional downside risk profile as perceived by shareholders. This framework allows for decomposing downside risk into components that are attributable to each of the underlying risk factors. The firm can compare this perceived VaR, including its composition and dynamics, to an internal VaR based on net exposures as it is known to the company. Any differences may lead to surprises at times of earnings announcements and thus constitute a litigation threat to the firm. It may reduce this information asymmetry through targeted communication efforts.     

Downside risk and asset pricing

We analyze if the value-weighted stock market portfolio is stochastic dominance (SD) efficient relative to benchmark portfolios formed on size, value, and momentum. In the process, we also develop several methodological improvements to the existing tests for SD efficiency. Interestingly, the market portfolio seems third-order SD (TSD) efficient relative to all benchmark sets. By contrast, the market portfolio is inefficient if we replace the TSD criterion with the traditional mean–variance criterion. Combined these results suggest that the mean–variance inefficiency of the market portfolio is caused by the omission of return moments other than variance. Especially downside risk seems to be important for explaining the high average returns of small/value/winner stocks.     

Downside risk of international stock returns

This paper investigates the downside risk exposure of international stock returns in 14 major industrialized economies around the world. For the period 1975–2010, we find that differences in returns on value and growth portfolios can be rationalized by assets’ reagibilities to market’s downside shocks. International value stocks are particularly sensitive to market’s permanent downside shocks, while international growth stocks are particularly sensitive to market’s temporary downside shocks. In line with recent evidence for the US, risk associated with unfavorable changes in market’s cash-flow innovations carries a premium which is pervasive and statistically significant.     

Asymmetric Volatility,Skewness, and Downside Risk in Different Asset Classes: Evidence from Futures Markets

This study examines the cross‐sectional variation of futures returns from different asset classes. The monthly returns are positively correlated with downside risk and negatively correlated with coskewness. The asymmetric volatility effect generates negatively skewed returns. Assets with high coskewness and low downside betas provide hedges against market downside risk and offer low returns. The high returns offered by assets with low coskewness and high downside betas are a risk premium for bearing downside risk. The asset pricing model that incorporates downside risk partially explains the futures returns. The results indicate a unified risk perspective to jointly price different asset classes.     

Downside risk and the size of credit spreads

We investigate why spreads on corporate bonds are so much larger than expected losses from default. Systematic factors make very little contribution to spreads, even if higher moments or downside effects are taken into account. Instead we find that sizes of spreads are strongly related to idiosyncratic-risk factors: not only to idiosyncratic equity volatility, but even more to idiosyncratic bond volatility and idiosyncratic bond value-at-risk. Idiosyncratic bond volatility helps to explain spreads because it reflects not just the distribution of firm value but is also a proxy for liquidity risk. Idiosyncratic bond value-at-risk adds to this by capturing the left-skewness of the firm-value distribution. We confirm our results both for the initial 1997-2004 sample period and also out of sample for 2005-2009, which includes the sub-prime crisis. Overall, credit spreads are large because they incorporate a large risk premium related to investors’ fears of extreme losses.     

The Downside of Asset Screening for Market Liquidity

This paper explores the tension between asset quality and market liquidity. I model an originator who screens assets whose cash flows are later sold in secondary markets. Screening improves asset quality but gives rise to asymmetric information, hindering trade of the asset cash flows. In the optimal mechanism (second‐best), costly retention of cash flows is essential to implement asset screening. Market allocations can feature too much or too little screening relative to second‐best, where too much screening generates inefficiently illiquid markets. Furthermore, the economy is prone to multiple equilibria. The optimal mechanism is decentralized with two tools: retention rules and transfers.     

Tradeoffs for Downside Risk-Averse Decision-Makers and the Self-Protection Decision

In addition to risk aversion, decision-makers tend to be also downside risk averse. Besides the usual size for risk trade-off, this allows several other trade-offs to be considered. The decision to increase the level of self-protection generates five trade-offs each involving an unfavourable downside risk increase and an accompanying beneficial change. Five stochastic orders that correspond to these trade-offs are defined, characterised and used to prove comparative static theorems that provide information concerning the self-protection decision. The five stochastic orders are general in nature and can be applied in any decision model where downside risk aversion is assumed.