Forecasting the Value-at-Risk of

今日读一篇：
Forecasting the Value-at-Risk of REITs using Realized Volatility Jump Models
使用已实现的波动率跳跃模型预测 REIT 的风险价值

DOI:
https://doi.org/10.1016/j.najef.2021.101426

Abstract
This paper examines jump risk in the time series of Real Estate Investment Trusts (REITs). Using high-frequency index-level and firm-level data, the econometric model in this paper integrates jumps into the volatility forecast by estimating jump augmented Heterogeneous Autoregressive (HAR) models of realized volatility. To assess the information value of these specifications, their forecasting accuracies for generating one-step ahead daily Value-at-Risk are also compared with other VaR specifications, including those generated from historical returns, bootstrap technique, and severity loss distribution.

摘要
本文研究了房地产投资信托基金 (REIT) 时间序列中的跳跃风险。 本文的计量经济模型使用高频指数级和公司级数据，通过估计已实现波动率的跳跃增强异质自回归 (HAR) 模型，将跳跃整合到波动率预测中。 为了评估这些规范的信息价值，还将其生成领先每日风险价值的预测精度与其他 VaR 规范进行比较，包括根据历史收益、引导技术和严重性损失分布生成的预测精度。

Introduction
One of the notable aftermaths of the 2007–2009 financial crises was the renewed interest in the impact of non-normal distributional features such as jumps, fat tails, and skewness on the dynamics of asset prices. In this paper, the effects of jumps on the performance of a portfolio of Real Estate Investment Trust (REIT) securities are examined. REITs are pass-through entities that own or invest primarily in real estate related assets.1 Shares of publicly-traded REITs trade on stock exchanges, just like the shares of other publicly traded companies. However, REITs differ from publicly listed companies in the sense that they are ownership investment in trusts, which manage investments in real estate assets. Assets held by REITs may include direct investment in commercial real estate, such as malls, office buildings, warehouses, and vacation resorts, or investment in commercial real estate mortgages and mortgage-backed securities.
介绍
2007-2009 年金融危机的显着后果之一是人们重新关注非正态分布特征（例如跳跃、厚尾和偏态）对资产价格动态的影响。 本文研究了跳跃对房地产投资信托（REIT）证券投资组合绩效的影响。 REIT 是主要拥有或投资于房地产相关资产的传递实体。1 公开交易的 REIT 股票在证券交易所交易，就像其他公开交易公司的股票一样。 然而，房地产投资信托基金与上市公司的不同之处在于，房地产投资信托基金是信托的所有权投资，管理房地产资产的投资。 REITs持有的资产可能包括对商业房地产的直接投资，例如购物中心、办公楼、仓库和度假村，或对商业房地产抵押贷款和抵押贷款支持证券的投资。

As an investable asset class, REITs are particularly attractive. The Nareit all equity REIT index outperformed the S&P 500 index in 15 of the past 25 years. REITs also have a significant presence in the capital market. At the end of 2019, there were 219 publicly-traded REITs, with a total market capitalization of over 1.3 trillion U.S. dollars. 2.9% of the sector weight for the S&P 500 index is also comprised of REIT stocks; exceeding the sector weightings for industries such as energy and basic materials. Also, over the past decade, the total net assets of real estate mutual funds grew from 44.1 billion U.S. dollars to 115.8 billion U.S. dollars (ICI 2020 factbook), to become the second-largest mutual fund sector in the U.S. today.2 Despite all of these attractive features, a recent study published by Fidelity Management and Research3 finds that actively managed fund investors are underexposed (relative to their benchmark) to REIT stocks. The reason for this anomaly may be that REITs are particularly susceptible to rare disaster risk events as underscored in Odusami (2021). Indeed, between 2007 and 2008, the two years that encompassed the housing-related financial crisis, the NAREIT all equity REITs index lost −47.99% of its value, compared to the S&P 500, which posted a 33.49% loss during the same period. Furthermore, at the onset of the virus-related pandemic crisis of 2020, most asset classes experienced significant losses in the immediate period, but quickly saw a rebound in value by the end of the year. Of the four sectors of the S&P 500 which had a negative return in 2020, REITs had the second-worst performance; posting a year-end return of −5.17%. Over the same period, the S&P 500 posted a total annual return of 18.43%.
作为一种可投资的资产类别，房地产投资信托基金尤其有吸引力。 Nareit 全股权 REIT 指数在过去 25 年中有 15 年跑赢了标准普尔 500 指数。 REITs 在资本市场上也占有重要地位。 截至2019年底，公开交易的REITs共有219只，总市值超过1.3万亿美元。 标准普尔 500 指数行业权重的 2.9% 也由 REIT 股票组成； 超过能源、基础材料等行业的行业权重。 此外，在过去十年中，房地产共同基金的总净资产从 441 亿美元增长到 1158 亿美元（ICI 2020 资料册），成为当今美国第二大共同基金领域。2 在这些吸引人的特征中，富达管理和研究公司最近发布的一项研究3发现，主动管理型基金投资者对房地产投资信托基金股票的投资不足（相对于其基准）。 造成这种异常的原因可能是 REIT 特别容易受到罕见灾害风险事件的影响，正如 Odusami（2021 年）所强调的那样。 事实上，在 2007 年至 2008 年期间，即与住房相关的金融危机爆发的两年间，NAREIT 全股权 REITs 指数损失了 -47.99% 的价值，而同期标准普尔 500 指数则损失了 33.49%。 此外，在 2020 年与病毒相关的大流行危机爆发时，大多数资产类别在短期内经历了重大损失，但到年底价值迅速反弹。 在标普 500 指数 2020 年出现负回报的四个板块中，REITs 表现第二差； 年终回报率为-5.17%。 同期，标准普尔 500 指数的年总回报率为 18.43%。

Jumps in the dynamics of asset returns have significant implications for asset pricing (Merton, 1976) and portfolio management (Branger et al, 2008). If jumps are widespread in the dynamics of asset prices, then their risk premia must account for both the diffusive and jump risk factors. Jump risks are particularly important because they cannot be diversified away; therefore, investors may demand large risk premia to hold assets with such risk. In the dynamics of asset returns, jumps allow for the impact of news with significant information value to quickly dissipate in the return process, but their effect on volatility is more persistent due to their impact on the diffusion process. For most risk managers, Value-at-Risk (also known as VaR) is an important tool for financial risk assessment and governance. VaR of a long position is the minimum loss the position can incur in a given time period, with
level of confidence.4 Since its introduction by the RiskMetrics Group in 1994, the use of VaR measures of market risk has increased exponentially because they have several appealing features.5 A crucial assumption when forecasting VaRs for portfolios is that the distribution of probable loss in the value of the portfolio will arise from “normal” market risk, as opposed to all possible market risks. Consequently, most VaR specifications do not typically account for acute tail risks, which are caused by the arrival of jump-inducing news into the financial markets. However, the consensus in the literature is that the scales of the higher-order moments seen in most financial data are much higher than expected for a normal distribution.6 Therefore, risk managers who rely on VaR measures that are based on normal distribution are ill-equipped to deal with significant and potentially damaging events that are rare in a normally distributed return series but appear to occur frequently in the real world.
资产回报动态的跳跃对资产定价（Merton，1976）和投资组合管理（Branger 等，2008）具有重大影响。 如果资产价格动态中普遍存在跳跃，那么其风险溢价必须同时考虑扩散风险因素和跳跃风险因素。 跳跃风险尤其重要，因为它们无法分散化。 因此，投资者可能会要求较大的风险溢价来持有具有此类风险的资产。 在资产收益动态中，跳跃使得具有重要信息价值的新闻的影响在收益过程中迅速消散，但由于其对扩散过程的影响，其对波动性的影响更加持久。 对于大多数风险管理者来说，风险价值（也称为VaR）是财务风险评估和治理的重要工具。 多头头寸的 VaR 是该头寸在给定时间段内可能产生的最小损失，其中
4 自 RiskMetrics Group 于 1994 年推出以来，市场风险 VaR 指标的使用呈指数级增长，因为它们具有几个吸引人的功能。5 预测投资组合的 VaR 时的一个关键假设是，可能损失的分布 投资组合的价值将来自“正常”市场风险，而不是所有可能的市场风险。 因此，大多数 VaR 规范通常不会考虑急性尾部风险，这种风险是由于金融市场出现跳跃性消息而引起的。 然而，文献中的共识是，大多数金融数据中看到的高阶矩的规模远高于正态分布的预期。6因此，依赖基于正态分布的 VaR 度量的风险管理者是有问题的。 - 能够处理重大且具有潜在破坏性的事件，这些事件在正态分布的回报序列中很少见，但在现实世界中似乎经常发生。

Given that REITs may be subject to rare events risk, quantifying the risk of REITs portfolios using a jump-augmented VaR framework is vital. Furthermore, an accurately specified VaR which reflects the jump risk of REITs assets will enhance risk governance and risk budgeting for portfolio managers with positions in REITs. A review of the most recent Form 13F regulatory filings shows that many banks and other covered financial institutions often hold significant positions in REIT stocks.7 For these institutions, incorporating jump risk in their VaR specifications for REITs assets will enhance the accuracy of their regulatory VaR reporting. Jump-augmented VaR models will also provide new insights for policymakers on the market risk of REIT companies, as well as a framework for intervention in periods of market turmoil. Overall, the above reasons yield compelling motivations for this paper.
鉴于 REITs 可能面临罕见事件风险，因此使用跳跃增强 VaR 框架量化 REITs 投资组合的风险至关重要。 此外，反映 REITs 资产跳跃风险的准确指定的 VaR 将增强持有 REITs 职位的投资组合经理的风险治理和风险预算。 对最新 13F 表格监管文件的审查显示，许多银行和其他涵盖的金融机构经常持有 REIT 股票的大量头寸。 7 对于这些机构来说，将跳跃风险纳入 REITs 资产的 VaR 规范中将提高其监管 VaR 的准确性 报告。 跳跃增强的 VaR 模型还将为政策制定者提供有关 REIT 公司市场风险的新见解，以及市场动荡时期的干预框架。 总的来说，上述原因为本文提供了令人信服的动机。

In this paper, two objectives are addressed. The first objective is to explore whether non-parametric models can detect jumps in high-frequency profit and loss (P/L) data of a portfolio of REIT securities using both firm-level and index-level data.8 The paper applies the statistical framework of the jump detection technique developed in Barndorff-Nielsen and Shephard, 2004, Barndorff-Nielsen and Shephard, 2006, (which is based on realized volatility and bipower variation measures) to identify jumps in the volatilities and returns of REIT securities. The paper also documents statistical artifacts of jumps in the dynamics of REIT returns; thereby motivating the second objective of the paper. Here, the paper explores whether jump-augmented volatility specifications can yield a more precise forecast of the daily VaRs for portfolios of REIT securities. The realized-volatility-jump (RVJ) specification presented in this study has distinct advantages over existing parametric specifications. First, the flexible form of the RVJ specification allows it to easily account for many of the important features of the distribution of REIT returns, such as conditional mean, variances, skewness, kurtosis, and leptokurtosis. Second, because the model delineates the latent news process into two distinct components, it can account for the contributions of each process to the overall risk of the portfolios. Third, the RVJ model yields daily jump probabilities; consequently, the model can distinguish VaRs in periods with high jump probabilities separately from VaRs in periods with low jump probabilities. To the best of our knowledge, this paper is the first to analyze the pertinent features of VaRs for REIT portfolios using this specific methodology and thus would provide a significant contribution to the literature.
本文提出了两个目标。 第一个目标是探索非参数模型是否可以使用公司层面和指数层面的数据来检测 REIT 证券投资组合的高频损益 (P/L) 数据的跳跃。8 本文应用了统计方法 Barndorff-Nielsen 和 Shephard，2004 年、Barndorff-Nielsen 和 Shephard，2006 年开发的跳跃检测技术框架（基于已实现的波动性和双幂变化测量），用于识别 REIT 证券波动性和回报的跳跃。 该论文还记录了 REIT 回报动态跳跃的统计假象； 从而激发了本文的第二个目标。 本文探讨了跳跃增强波动率规范是否可以对 REIT 证券投资组合的每日 VaR 进行更准确的预测。 本研究中提出的实现波动率跳跃（RVJ）规范比现有参数规范具有明显的优势。 首先，RVJ 规范的灵活形式使其能够轻松解释 REIT 收益分布的许多重要特征，例如条件均值、方差、偏度、峰度和尖峰度。 其次，由于该模型将潜在新闻过程划分为两个不同的组成部分，因此它可以解释每个过程对投资组合整体风险的贡献。 第三，RVJ模型产生每日跳跃概率； 因此，该模型可以将高跳跃概率时期的 VaR 与低跳跃概率时期的 VaR 区分开来。 据我们所知，本文是第一篇使用这种特定方法分析 REIT 投资组合 VaR 相关特征的论文，因此将为文献做出重大贡献。

In the empirical results, evidence of jumps and jump variations are highlighted at both the firm level and the index level. Jump frequencies are much higher at the index-level than at the firm level. Jump occurrences also appear to be more systematic at the index level than at the firm level. Furthermore, jumps were found to have large effects on the realized volatility of REIT returns. On average, jumps could account for up to half of the value of the daily realized volatilities observed in REIT securities. Concerning the forecast of future volatilities, evidence of mean reversion in the volatility process in the period after a jump incident was also found.
在实证结果中，公司层面和指数层面都突出了跳跃和跳跃变化的证据。 指数层面的跳跃频率比公司层面的跳跃频率高得多。 指数层面的跳跃事件似乎也比公司层面更加系统化。 此外，人们发现跳跃对 REIT 回报的已实现波动性有很大影响。 平均而言，跳跃可能占 REIT 证券每日实现波动率价值的一半。 关于未来波动率的预测，还发现了跳跃事件发生后一段时间内波动率过程均值回归的证据。

Moving on to assessing the market risk of portfolios of REIT securities, it is shown that including jumps in the underlying volatility models significantly improves the forecasting ability of the VaR specifications. The jump-augmented VaR specification dominates competing models in terms of the quality and the accuracy of the VaR forecasts. The results obtained from backtesting the jump-augmented volatility specification against competing models are strongly in favor of the jump-augmented volatility specifications. Furthermore, VaR forecasts obtained from the jump augmented volatility specifications are noticeably higher on trading days with high jump probabilities than days with low jump probabilities. Thus, providing statistically significant evidence that jumps are associated with higher values of REIT assets at risk.
接下来评估 REIT 证券投资组合的市场风险，结果表明，包含基础波动率模型的跳跃可以显着提高 VaR 规范的预测能力。 跳跃增强 VaR 规范在 VaR 预测的质量和准确性方面主导了竞争模型。 通过对竞争模型对跳跃增强波动率规范进行回溯测试获得的结果强烈支持跳跃增强波动率规范。 此外，从跳跃增强波动率规范获得的 VaR 预测在跳跃概率高的交易日明显高于跳跃概率低的交易日。 因此，提供了统计上显着的证据，表明跳跃与风险房地产投资信托资产的较高价值相关。

The rest of this paper is organized as follows. In Section 2, the jump detection framework employed in this research is briefly discussed. In Section 3, the REIT data used in this paper is described. Also discussed are the summary statistics obtained from the application of the jump detection methodology on the REIT data. Section 4 presents the framework of the jump augmented VaR specification and the results of a select group of VaR specifications. Section 5 presents the backtesting methodologies for the VaR specifications and their results. Section 6 provides some concluding remarks.
本文的其余部分安排如下。 在第二节中，简要讨论了本研究中采用的跳跃检测框架。 第 3 节描述了本文使用的 REIT 数据。 还讨论了通过对 REIT 数据应用跳跃检测方法获得的汇总统计数据。 第 4 节介绍了跳跃增强 VaR 规范的框架以及一组选定的 VaR 规范的结果。 第 5 节介绍了 VaR 规范的回溯测试方法及其结果。 第 6 节提供了一些结论性意见。

Section snippets
Estimating realized volatility and jumps
The jump detection method used in this paper is based on the methodology proposed in Barndorff-Nielsen and Shephard, 2004, Barndorff-Nielsen and Shephard, 2006. In the model, the logarithm of the intraday prices of REIT securities evolves in a continuous-time process which can be described by the standard jump-diffusion process shown below:
is a Poisson jump process with a jump intensity
and log jump size
, which are drawn from a normal distribution
Ñ
, and
章节片段
估计已实现的波动率和跳跃
本文使用的跳跃检测方法基于 Barndorff-Nielsen and Shephard, 2004、Barndorff-Nielsen and Shephard, 2006 提出的方法。在该模型中，REIT 证券日内价格的对数在连续时间内演变 过程可以用如下所示的标准跳跃扩散过程来描述：
是具有跳跃强度的泊松跳跃过程
和日志跳跃大小
，从正态分布中得出
Ñ
， 和

Data
The data used in this paper are fifteen-minute log differences in the prices of six REIT indices and six REIT stocks (including, two large-cap, two mid-cap, and two small-cap REITs).11 The data which
数据
本文使用的数据是 6 个 REIT 指数和 6 个 REIT 股票（包括两只大盘、两只中盘和两只小盘 REIT）价格的 15 分钟对数差值。 11

Accounting for realized volatility and jumps in VaR
Existing literature on VaR measurement suffers from several limitations and misspecification problems, which are due in part to the innate restrictions within each model, or from the assumptions, which govern their respective distribution functions.19
考虑已实现的波动性和 VaR 的跳跃
关于 VaR 测量的现有文献存在一些局限性和错误指定问题，部分原因是每个模型的固有限制，或者是控制各自分布函数的假设。 19

Backtesting VaR specifications
Under the Basel II Accord(Basel, 2009), covered financial institutions are required to hold capital reserves against market risks. The amount of reserve is determined by approved internal measurements of the maximum loss over 10 trading days at the 99% confidence level. Acceptable internal models are chosen based on the ability of the model to adequately predict the market risk of covered institutions through backtesting of the model’s output. There are no specific recommendations as to the
回测 VaR 规范
根据巴塞尔新协议（巴塞尔，2009），所涵盖的金融机构必须持有资本储备以应对市场风险。 准备金金额是根据经批准的内部测量确定的，在 99% 置信水平下对 10 个交易日内的最大损失进行测量。 可接受的内部模型是根据模型通过模型输出的回测充分预测所覆盖机构的市场风险的能力来选择的。 目前还没有具体的建议

Conclusions
The use of conditional distributions for estimating VaRs yield significant improvement over time-invariant specifications because of their innate flexibility. Conditional distributions rely on the forecast of the conditional variance of the underlying stochastic process. Thus, these types of specifications are better able to reflect market risk as they evolve. In this paper, the ability of non-parametric models to detect jumps in high-frequency REIT returns is investigated. The paper also
结论
由于其固有的灵活性，使用条件分布来估计 VaR 比时不变规范产生了显着的改进。 条件分布依赖于对基础随机过程的条件方差的预测。 因此，这些类型的规范能够更好地反映市场风险的演变。 本文研究了非参数模型检测高频 REIT 回报跳跃的能力。 论文还

Conclusion
The use of conditional distributions for estimating VaRs yield significant improvement over time-invariant specifications because of their innate flexibility. Conditional distributions rely on the forecast of the conditional variance of the underlying stochastic process. Thus, these types of
specifications are better able to reflect market risk as they evolve. In this paper, the ability of non- parametric models to detect jumps in high-frequency REIT returns is investigated. The paper also evaluates whether jump augmented models of realized volatility yield improved forecasts of the value at risk for portfolios of REIT securities. The null that jumps do not add new information to the subsumed HAR framework is rejected at the 5% level of significance for the REIT indices and in half of the REIT stocks examined. Furthermore, it is found that jumps and jump clustering are common phenomena in REIT returns and that their dynamics are best described by stochastic models, which incorporate conditional volatility and conditional jump coefficients.
结论
由于其固有的灵活性，使用条件分布来估计 VaR 比时不变规范产生了显着的改进。 条件分布依赖于对基础随机过程的条件方差的预测。 因此，这些类型的
随着市场的发展，规范能够更好地反映市场风险。 本文研究了非参数模型检测高频 REIT 回报跳跃的能力。 本文还评估了已实现波动率的跳跃增强模型是否可以改善对 REIT 证券投资组合风险价值的预测。 对于 REIT 指数和所检查的一半 REIT 股票，跳跃不会向所包含的 HAR 框架添加新信息的零值在 5% 的显着性水平上被拒绝。 此外，研究发现，跳跃和跳跃集群是 REIT 回报中的常见现象，并且它们的动态最好通过随机模型来描述，该模型包含条件波动率和条件跳跃系数。

Further, the paper also examines the effectiveness and the suitability of the augmented HAR model for the computation of VaRs for long positions in REITs by comparing the performances of the HAR-J VaR specifications with the performance of alternative model specifications. The goal is to evaluate the contributions of jumps to VaR forecasts and their violations or exceedances. In all, VaR specifications, which include jumps in the underlying volatility models performed significantly better than the unconditional VaR models.
此外，本文还通过比较 HAR-J VaR 规范的性能与替代模型规范的性能，检验了增强 HAR 模型用于计算 REIT 多头头寸 VaR 的有效性和适用性。 目标是评估跳跃对 VaR 预测的贡献及其违规或超出情况。 总而言之，VaR 规范（包括基础波动率模型的跳跃）的表现明显优于无条件 VaR 模型。

Highlights
 Jumps are salient features of the dynamics of REIT prices.
 Jumps are more systematic in REIT indices but more sporadic in REIT stocks.
 Adding jumps to VaR measures will improve their forecasting accuracy for REITs.
强调
 跳跃是 REIT 价格动态的显着特征。
 REIT 指数的上涨更为系统化，但 REIT 股票的上涨更为零星。
 增加 VaR 指标的跳跃将提高 REITs 的预测准确性。
Traditional REITs are more likely to benefits from jumps than specialty REITs.
传统房地产投资信托基金比专业房地产投资信托基金更有可能从跳跃中受益。