汇添富可转换债券A(470058)基金净值_估值_行情走势—天天基金网基金名称：汇添富可转换债券A申购金额元壹仟壹佰贰拾万
根据您的申购金额和基金过往业绩，收益测算如下：近1年收益-- 元近6月收益-- 元近3月收益-- 元近1月收益728,600.00风险提示：收益测算仅供参考,过往业绩不预示未来表现。
基&金基金经理基金公司全球市场&&)基金类型：&&|&&中高风险：2.07亿元（）基金经理：等成 立 日：管 理 人：：交易状态：开放申购
购买手续费：0.80%&0.08%&1折金额：元手机也可以买基金 扫码下载手机版天天基金预计购买费用：0.80元(费率0.08%，节省0.72元。)
股票仓位测算并不能保证与真实仓位数据相同，仅供参考。实际股票持仓占净值比，每季度定期披露，请查阅基金资产明细表。
14.33% 债券名称 持仓占比 涨跌幅 相关资讯
拥抱全球机遇消费板块持续升级分享优势企业成长收益 活期宝随时取，最快1秒实时到账关联基金最高7日年化4.36%07-31活期宝累计销量超11000亿元定期宝短期理财就选定期宝7天4.38%14天4.28%30天4.43%60天4.83%日期单位净值累计净值日增长率
成立以来，总计分红次，拆分次
每份派现金0.1416元
每份派现金0.0700元 评级机构评级日期评级
★★ 18年2季度18年1季度17年4季度17年3季度17年2季度17年1季度16年4季度16年3季度
1247 | 1343
1148 | 1337
928 | 1243
四分位排名
2017年度2016年度2015年度2014年度2013年度2012年度2011年度2010年度
四分位排名
选择时间1月3月6月1年3年5年今年最大沪深300中证500上证指数深证成指中小板指创业板指PK当前基金选择指标:近3月排名近6月排名近1年排名选择指标:近3月排名近6月排名近1年排名
报告期 基金换手率84.81%38.40%40.54%18.14%基金换手率用于衡量基金投资组合变化的频率。截止至：
选择基金经理：累计任职时间：现任基金资产规模：截止至：--截止至：--任职时间基金经理任职天数任期回报
2年又110天
1年又235天
点击回复标题作者最新更新时间
全部日涨幅近1周更多
转债标的QFII重仓基金重仓
汇添富基金
基金名称关注度
郑重声明：以上信息（包括但不限于文字、数据及图表）均基于公开信息采集，相关信息并未经过本公司证实，本公司不保证该信息全部或者部分内容的准确性、真实性、完整性，不构成本公司任何推荐或保证。基金具体信息以管理人相关公告为准。投资者投资前需仔细阅读《基金合同》、《招募说明书》等法律文件，了解产品收益与风险特征。市场有风险，投资需谨慎。数据来源：东方财富Choice数据。&&&&&&|||||||||天天基金客服热线：95021&/&|客服邮箱：|人工服务时间：工作日 7:30-21:30 双休日 9:00-21:30郑重声明：。天天基金网所载文章、数据仅供参考，使用前请核实，风险自负。中国证监会上海监管局网址：CopyRight&&上海天天基金销售有限公司&&&&沪ICP证：沪B2-&&网站备案号：沪ICP备号-1
点击下载热门基金近6月收益24%抓住好"基"惠 买入费率全面1折个别基金除外Funsworld的K资产可以确保资产所有权吗？_百度知道
Funsworld的K资产可以确保资产所有权吗？
答题抽奖
首次认真答题后
即可获得3次抽奖机会，100%中奖。
夀夀很嚣张知道合伙人
夀夀很嚣张
擅长：暂未定制
当然可以啊，基于密码学的设计使K 资产只能被真实的拥有者转存或支付，确保了资产所有权与流通交易的匿名性。
为你推荐：
其他类似问题
个人、企业类
违法有害信息,请在下方选择后提交
色情、暴力
我们会通过消息、邮箱等方式尽快将举报结果通知您。平台高额返现，这真的会是“免费的午餐”？平台高额返现，这真的会是“免费的午餐”？喜马拉雅吴彦祖百家号来源丨喜马拉雅FM小赵说金融的精品课《金融说》编辑丨秋裤兜芝麻对于理财产品的不熟悉，近期又面临大量平台暴雷，每天只要刷微博，看微信，就会发现“这个平台暴雷了”，“那个平台老板跑路了”的新闻，如履薄冰的我们该如何选择？99%的理财产品都不熟悉，我们还能买吗？到底不熟悉的投资产品能不能买？这话如果用巴菲特的话来回答就是“不熟悉的产品当然就不要买”，可是人家是厉害的股神，懂得也多。那我们作为普通人其实对投资产品了解是非常少的，对于这些不熟悉的产品我们要怎么办？其实熟悉是一个过程，就像余额宝刚出来的时候我们也不知道是什么，随着用的人越来越多，就火了起来。可是很多人仍不清楚余额宝的本质是什么，只是因为余额宝是马云的，便大胆持有。余额宝的资金投向哪里？我们不关心也不知道，可是总有人会想要关心想要知道，所以这也是为什么余额宝的规模一直在被持续压制，监管部门也在不断地出新的政策，也就是希望它不要出现风险性事件。但是话说回来，就因为余额宝稳定，所以收益就低，更多的人倾向去买一些自己都不知道的投资或者理财产品，所以有的人踩了雷。有的人买了所谓的P2P，而你区别这些理财产品最主要的一个底层的逻辑就是,它把钱投向了哪里？谁在用这样的钱？余额宝是投资国债、银行储蓄等低风险产品，安全性高，当然收益就低。所谓的P2P,主要投资的是一些急需用钱的企业和个人，这些企业和个人有时候是没有办法在短期内通过银行渠道获得资金的，所以他们的信用等级比较低，产生坏账风险的可能性就会上升，因此他们借贷的成本高也就使得平台利率高的情况。平台暴雷，为什么理财机构纷纷倒台？近期平台频频暴雷，就连一些正规的理财机构都出现了问题，为什么会暴呢？第一类暴雷机构是兑付不了，或者被警察盯了好久直接被经侦部门介入，所以它一定会有一个引爆点，比如“唐小僧”。另一类是老板直接跑路，比如“米袋计划”。还有一类公布的项目大量逾期，最后只能分期兑付，直接停业对借款人进行催收，或者已经有部分的钱回到了账款里，然后慢慢再还完。以上是最近暴雷平台的几种主要表现，其实直接跑路的平台比例是在变小的，高返利的平台也没有那么多，问题的出现主要是因为存在“长短错配”的问题。另一个方面是这些平台资产流动性变差就导致不能及时的进行兑付，所以平台在资金的压力下只能宣布“斩息”。都是套路，“0元购”变“0元”的背后最近有一个爆掉的平台“联璧金融”，它之所以火起来是因为和一个卖路由器的“斐讯”捆绑在了一起。“斐讯”的商品明码标价，比如说一个路由器399元，但是对应有一个“K码”。购买路由器之后，“斐讯”会要求你去金融公司注册会员，然后激活K码，一个月之后就能拿回399元。当然后来“斐讯”又出了很多周边产品，净化器、扫地机器人等，价格也在不断的变高，返现时间也因为价格的上涨而拉长，最长返现时间达12个月。但不管怎么样就让你感觉花了这点钱白拿了一个扫地机器人或者路由器，唯一的代价是要去指定合作平台注册。新用户注册认证就可以获得K码，老用户如果想要不断的拿返现就需要投资，这种捆绑性的销售自然而然使得那些所谓的投资公司、理财机构、平台一起发展起来。通过买卖K码赚取差价的“中间商”也发展了起来，比如399元路由的K码，中间商以250到200的价格到手，新用户在一个月以后就可以得到399块钱的返现，获利就是150到200。这些人就通过收购大量的身份证去合作平台注册，赚取中间差价。随着身份证越收越少，平台越查越紧，懂得变通的这部分人开始大量收购一手路由器，一手K码，继续两边赚钱。你要玩这一套是要钱的，钱从哪里来？所以关键点还在合作的“联璧金融”。“联璧金融”的投资者和其他的“中间者”把钱借给这些人，然后这些人大量收购“斐讯”的路由器之后就拿着K码到平台激活，进行三个以上投资就能够返现。游戏进行到了这里，其实就变成了“击鼓传花”。“斐讯”之所以大量的出货，和大量囤货以及倒卖的“中间者”密不可分的。于是问题就来了，“中间者”变多的时候，真正的买家就会越来越少，越来越难找，所以两边注定会不再匹配，这个游戏迟早会玩完。下次再碰到什么“0元购”，别以为是贪到了便宜，实质上是为了给你一个低的门槛，然后从你口袋里掏更多的钱。世道险恶，希望大家都能够了解清楚这些所谓的套路关于主播：主播：小赵说金融，侃一侃财经，说一说金融、聊一聊工作，再扯一扯淡。不谈高深的知识，只说有价值、实用、听得懂的故事和资讯！专辑《金融说》，属于年轻人的理财和金融观，只说有趣、有深度、有启发的适合年轻人的赚钱方法论和理财技巧！！你的金钱你做主，把它放在最合适的地方，打开你未来发展的人生路！轻松学理财，开心过日子。关于喜马拉雅：随时随地，听我想听！4.7亿用户的共同选择，听段子听小说听新闻听音乐，就用喜马拉雅！本文由百家号作者上传并发布，百家号仅提供信息发布平台。文章仅代表作者个人观点，不代表百度立场。未经作者许可，不得转载。喜马拉雅吴彦祖百家号最近更新：简介:关注互联网、电影或文字。作者最新文章相关文章著：Antti Ilmanen, Raymond Iwanowski 译：徐瑞龙
The Dynamics of the Shape of the Yield Curve: Empirical Evidence, Economic Interpretations and Theoretical Foundations
收益率曲线形态的动力学：实证证据、经济学解释和理论基础
INTRODUCTION
How can we interpret the shape (steepness and curvature) of the yield curve on a given day? And how does the yield curve evolve over time? In this report, we examine these two broad questions about the yield curve behavior. We have shown in earlier reports that the market's rate expectations, required bond risk premia and convexity bias determine the yield curve shape. Now we discuss various economic hypotheses and empirical evidence about the relative roles of these three determinants in influencing the curve steepness and curvature. We also discuss term structure models that describe the evolution of the yield curve over time and summarize relevant empirical evidence.
我们如何解释给定日期收益率曲线的形状（陡峭程度和曲率）？收益率曲线随着时间的推移如何演变？在本报告中，我们研究了两个关于收益率曲线行为的一般问题。我们早先的报告显示，市场的收益率预期、债券风险溢价和凸度偏差决定了收益率曲线的形状。现在我们讨论与这三个决定因素在影响曲线陡峭程度和曲率中的相对作用有关的各种经济学假说和经验证据。我们还讨论了描述收益率曲线随时间变化如何演化的期限结构模型，并总结了相关的经验证据。
The key determinants of the curve steepness, or slope, are the market's rate expectations and the required bond risk premia. The pure expectations hypothesis assumes that all changes in steepness reflect the market's shifting rate expectations, while the risk premium hypothesis assumes that the changes in steepness only reflect changing bond risk premia. In reality, rate expectations and required risk premia influence the curve slope. Historical evidence suggests that above-average bond returns, and not rising long rates, are likely to follow abnormally steep yield curves. Such evidence is inconsistent with the pure expectations hypothesis and may reflect time-varying bond risk premia. Alternatively, the evidence may represent irrational investor behavior and the long rates' sluggish reaction to news about inflation or monetary policy.
曲线陡峭程度或斜率的关键决定因素是市场的收益率预期和债券风险溢价。完全预期假说假设陡峭程度的所有变化反映了市场的收益率变化预期，而风险溢价假说假设陡峭程度的变化仅反映债券风险溢价的变化。实际上，收益率预期和风险溢价共同影响曲线斜率。历史证据表明，高于平均水平的债券回报，以及非上涨的长期收益率，更可能在异常陡峭的收益率曲线之后出现。这种证据与完全预期假说不一致，可能反映了时变的债券风险溢价。或者，证据可能代表非理性的投资者行为，以及长期收益率对通货膨胀或货币政策新闻的反应迟钝。
The determinants of the yield curve's curvature have received less attention in earlier research. It appears that the curvature varies primarily with the market's curve reshaping expectations. Flattening expectations make the yield curve more concave (humped), and steepening expectations make it less concave or even convex (inversely humped). It seems unlikely, however, that the average concave shape of the yield curve results from systematic flattening expectations. More likely, it reflects the convexity bias and the apparent required return differential between barbells and bullets. If convexity bias were the only reason for the concave average yield curve shape, one would expect a barbell's convexity advantage to exactly offset a bullet's yield advantage, in which case duration-matched barbells and bullets would have the same expected returns. Historical evidence suggests otherwise: In the long run, bullets have earned slightly higher returns than duration-matched barbells. That is, the risk premium curve appears to be concave rather than linear in duration. We discuss plausible explanations for the fact that investors, in the aggregate, accept lower expected returns for barbells than for bullets: the barbell's lower return volatility (for the same duration); the tendency of a flattening position to outperform in a and the insurance characteristic of a positively convex position.
早期研究中收益率曲线曲率的决定因素较少受到关注。似乎曲率主要随着市场的曲线形变预期而变化。曲线变平的预期使得收益率曲线更加上凸（隆起），并且曲线变陡的预期使得它较少上凸或甚至下凸（向下隆起）。这看起来似乎不可能，然而，平均看来上凸的收益率曲线由系统的曲线变平的预期产生。更有可能的是，它反映了凸度偏差和杠铃、子弹组合之间明显的回报差异。如果平均来看凸度偏差是收益率曲线形状呈上凸的唯一原因，则可以预期杠铃组合的凸度优势恰好抵消子弹组合的收益率优势，在这种情况下，久期匹配的杠铃组合和子弹组合将具有相同的预期回报。历史证据表明：从长远来看，子弹组合比久期匹配的杠铃组合获得的回报略高。也就是说，风险溢价曲线关于久期看起来是上凸的而不是线性的。我们讨论了合理的解释，即对于投资者总的来说，可以接受杠铃组合的预期回报低于子弹组合：杠铃组合有较低的回报波动率（相同久期情况下）;在熊市环境中，做平头寸跑赢市场的趋势和正凸度头寸的保险特征。
Turning to the second question, we describe some empirical characteristics of the yield curve behavior that are relevant for evaluating various term structure models. The models differ in their assumptions regarding the expected path of short rates (degree of mean reversion), the role of a risk premium, the behavior of the unexpected rate component (whether yield volatility varies over time, across maturities or with the rate level), and the number and identity of factors influencing interest rates. For example, the simple model of parallel yield curve shifts is consistent with no mean reversion in interest rates and with constant bond risk premia over time. Across bonds, the assumption of parallel shifts implies that the term structure of yield volatilities is flat and that rate shifts are perfectly correlated (and, thus, driven by one factor).
关于第二个问题，我们描述了与评估各种期限结构模型相关的收益率曲线行为的一些经验特征。这些模型关于短期收益率（均值回归程度）的预期路径、风险溢价的作用、非预期收益率部分的行为（收益率的波动率是否随时间、期限或收益率水平而变化）、影响收益率的因子数量的假设各有所不同。例如，收益率曲线平行偏移的简单模型与没有均值回归，债券风险溢价不随时间变化的模型一致。对于不同债券，曲线平行偏移的假设意味着收益率的波动率期限结构是平的，而且收益率变动是完全相关的（因此由一个因素驱动）。
Empirical evidence suggests that short rates exhibit quite slow mean reversion, that required risk premia vary over time, that yield volatility varies over time (partly related to the yield level), that the term structure of basis-point yield volatilities is typically inverted or humped, and that rate changes are not perfectly correlated, but two or three factors can explain 95%-99% of the fluctuations in the yield curve.
经验证据表明，短期收益率表现出相当缓慢的均值回归，风险溢价随时间而变化，收益率波动率随时间变化（与回报水平部分相关），基点收益率波动率的期限结构通常是倒挂或隆起的，而且这个收益率变化并不完全相关，但是两个或三个因素可以解释收益率曲线95%-99%的波动。
In Appendix A, we survey the broad literature on term structure models and relate it to the framework described in this series. It turns out that many popular term structure models allow the decomposition of yields to a rate expectation component, a risk premium component and a convexity component. However, the term structure models are more consistent in their analysis of relations across bonds because they specify exactly how a small number of systematic factors influences the whole yield curve. In contrast, our approach analyzes expected returns, yields and yield volatilities separately for each bond. In Appendix B, we discuss the theoretical determinants of risk premia in multi-factor term structure models and in modern asset pricing models.
在附录A中，我们总结了关于期限结构模型的一系列文献，并将其与本系列中描述的框架相关联。事实证明，许多流行的期限结构模型允许将收益率分解为收益率预期部分、风险溢价部分和凸度部分。然而，期限结构模型在对债券关系的分析中更加一致，因为它们具体说明了少数系统因素如何影响整个收益率曲线。相比之下，我们的方法可以分析每个债券的预期回报、收益率和收益率波动率。在附录B中，我们讨论了多因子期限结构模型和现代资产定价模型中风险溢价的理论决定因素。
HOW SHOULD WE INTERPRET THE YIELD CURVE STEEPNESS?
如何解释收益率曲线的陡峭程度
The steepness of yield curve primarily reflects the market's rate expectations and required bond risk premia because the third determinant, convexity bias, is only important at the long end of the curve. A particularly steep yield curve may be a sign of prevalent expectations for rising rates, abnormally high bond risk premia, or some combination of the two. Conversely, an inverted yield curve may be a sign of expectations for declining rates, negative bond risk premia, or a combination of declining rate expectations and low bond risk premia.
收益率曲线的陡峭程度主要反映了市场的收益率预期和债券风险溢价，因为第三个决定因素——凸度偏差，只有在曲线长期端才是重要的。特别陡峭的收益率曲线可能反映了收益率上涨的普遍预期，异常高的债券风险溢价或两者的某种组合。相反，倒挂的收益率曲线可能反映了对收益率下降的预期，负的债风险溢价或两者的某种组合。
We can map statements about the curve shape to statements about the forward rates. When the yield curve is upward sloping, longer bonds have a yield advantage over the risk-free short bond, and the forwards "imply" rising rates. The implied forward yield curves show the break-even levels of future yields that would exactly offset the longer bonds' yield advantage with capital losses and that would make all bonds earn the same holding-period return.
我们可以将关于曲线形状的结论类比到关于远期收益率的结论。当收益率曲线向上倾斜时，长期债券比无风险短期债券具有收益率优势，而远期收益率隐含收益率上涨。隐含的远期收益率曲线显示未来收益率的盈亏平衡水平，通过资产损失抵消长期债券的收益率优势，这将使所有债券获得相同的持有期回报。
Because expectations are not observable, we do not know with certainty the relative roles of rate expectations and risk premia. It may be useful to examine two extreme hypotheses that claim that the forwards reflect only the market's rate expectations or only the required risk premia. If the pure expectations hypothesis holds, the forwards reflect the market's rate expectations, and the implied yield curve changes are likely to be realized (that is, rising rates tend to follow upward-sloping curves and declining rates tend to succeed inverted curves). In contrast, if the risk premium hypothesis holds, the implied yield curve changes are not likely to be realized, and higher-yielding bonds earn their rolling-yield advantages, on average (that is, high excess bond returns tend to follow upward-sloping curves and low excess bond returns tend to succeed inverted curves).
由于预期不可观察，我们不能确定收益率预期和风险溢价的相对作用。检查两个极端假设可能是有用的，即远期收益率仅反映市场的收益率预期或仅反映债券风险溢价。如果完全预期假说成立，则远期收益率反映市场的收益率预期，隐含的收益率曲线变化很可能会实现（即收益率上涨趋向于跟随向上倾斜的曲线，收益率下降往往会导致倒挂的曲线）。相比之下，如果风险溢价假说成立，则隐含收益率曲线变化不可能实现，平均来看，高收益率债券获得滚动收益率优势（即高的债券超额回报倾向于跟随向上倾斜曲线和低的债券超额回报倾向于导致倒挂的曲线）。
Empirical Evidence
To evaluate the above hypotheses, we compare implied forward yield changes (which are proportional to the steepness of the forward rate curve) to subsequent average realizations of yield changes and excess bond returns. In Figure 1, we report (i) the average spot yield curve shape, (ii) the average of the yield changes that the forwards imply for various constant-maturity spot rates over a three-month horizon, (iii) the average of realized yield changes over the subsequent three-month horizon, (iv) the difference between (ii) and (iii), or the average "forecast error" of the forwards, and (v) the estimated correlation coefficient between the implied yield changes and the realized yield changes over three-month horizons. We use overlapping monthly data between January 1968 and December 1995, deliberately selecting a long neutral period in which the beginning and ending yield curves are very similar.
为了评估上述假说，我们将隐含的远期收益率变化（与远期收益率曲线的陡峭程度成比例）与随后收益率变化的平均实现和债券超额回报进行比较。在图1中，我们显示了（i）平均即期收益率曲线形状；（ii）三个月内远期收益率隐含的不同期限即期收益率变动的平均值；（iii）随后三个月内实现的收益率变动的平均；（iv）（ii）和（iii）之间的差或远期收益率的平均“预测误差”；以及（v）三个月内隐含收益率变动与实现收益率变动之间的相关系数估计。我们在1968年1月至1995年12月之间使用重叠的月度数据，并且特意选择一个长的中性区间，其中开始和结束的收益率曲线非常相似。
Figure 1 Evaluating the Implied Treasury Forward Yield Curve’s Ability to Predict Actual Rate Changes, 1968-95
Figure 1 shows that, on average, the forwards imply rising rates, especially at short maturities —— simply because the yield curve tends to be upward sloping. However, the rate changes that would offset the yield advantage of longer bonds have not materialized, on average, leading to positive forecast errors. Our unpublished analysis shows that this conclusion holds over longer horizons than three months and over various subsamples, including flat and steep yield curve environments. The fact that the forwards tend to imply too high rate increases is probably caused by positive bond risk premia.
图1显示，平均来说，远期收益率隐含着收益率上涨，特别是在短期端内，这仅仅是因为收益率曲线趋向于向上倾斜。然而，平均而言抵消长期债券收益率优势的收益率变动并没有实现，这导致正的预测误差。我们未发表的分析表明，这个结论在比三个月更长的期间和不同子样本（包括平坦和陡峭的收益率曲线环境）上保持成立。远期收益率倾向于隐含着过高的收益率上涨幅度，这一事实可能是由于正的债券风险溢价的影响。
The last row in Figure 1 shows that the estimated correlations of the implied forward yield changes (or the steepness of the forward rate curve) with subsequent yield changes are negative. These estimates suggest that, if anything, yields tend to move in the opposite direction than that which the forwards imply. Intuitively, small declines in long rates have followed upward-sloping curves, on average, thus augmenting the yield advantage of longer bonds (rather than offsetting it). Conversely, small yield increases have succeeded inverted curves, on average. The big bull markets of the 1980s and 1990s occurred when the yield curve was upward sloping, while the big bear markets in the 1970s occurred when the curve was inverted. We stress, however, that the negative correlations in Figure 1 they are not statistically significant.
图1的最后一行显示，隐含的远期收益率变化（或远期收益率曲线的陡峭程度）与后续实现的收益率变化的相关性估计为负。这些估计表明，如果有的话，收益率往往会向远期收益率隐含的相反方向变动。直观上来说，平均来看长期收益率的小幅下滑跟随着向上倾斜的曲线，从而增加了长期债券的收益率优势（而不是抵消）。相反，平均而言小幅度的收益率增长跟随倒挂的曲线。1980年代和90年代的大牛市发生在收益率曲线向上倾斜的时期，而1970年代的大熊市在曲线倒挂时发生。然而，我们强调，图1中的负相关性相当弱，没有统计学意义。
Many market participants believe that the bond risk premia are constant over time and that changes in the curve steepness, therefore, reflect shifts in the market's rate expectations. However, the empirical evidence in Figure 1 and in many earlier studies contradicts this conventional wisdom. Historically, steep yield curves have been associated more with high subsequent excess bond returns than with ensuing bond yield increases.
许多市场参与者认为，债券风险溢价随着时间的推移不发生变化，并且曲线陡峭程度的变化反映了市场收益率预期的变化。然而，图1和许多早期研究中的实证证据与这种传统智慧相矛盾。历史上，陡峭的收益率曲线与随后的债券超额回报更加相关，而不是债券收益率的增长。
One may argue that the historical evidence in Figure 1 is no longer relevant. Perhaps investors forecast yield movements better nowadays, partly because they can express their views more efficiently with easily tradable tools, such as the Eurodeposit futures. Some anecdotal evidence supports this view: Unlike the earlier yield curve inversions, the most recent inversions (1989 and 1995) were quickly followed by declining rates. If market participants actually are becoming better forecasters, subperiod analysis should indicate that the implied forward rate changes have become better predictors of the sub that is, the rolling correlations between implied and realized rate changes should be higher in recent samples than earlier. In Figure 2, we plot such rolling correlations, demonstrating that the estimated correlations have increased somewhat over the past decade.
人们可能会认为，图1中的历史证据不再可信。也许今天的投资者能更好地预测收益率变动，部分原因是他们可以通过易于交易的工具（如欧洲存款期货）更有效地表达自己的观点。一些轶事证据支持这一观点：与早期的收益率曲线倒挂不同，最近的倒挂（1989年和1995年）发生之后很快出现收益率下降。如果市场参与者实际上正在变成更好的预测者，则子时段分析应该表明隐含的远期收益率变化已经成为后续收益率变化更好的预测因子。也就是说，最近样本中隐含和实现的收益率变化之间的滚动相关性应该比之前高。在图2中，我们绘制了这种滚动相关性，表明在过去十年中估计的相关性有所增加。
Figure 2 60-Month Rolling Correlations Between the Implied Forward Rate Changes and Subsequent Spot Rate Changes, 1968-95
In Figure 3, we compare the forecasting ability of Eurodollar futures and Treasury bills/notes in the 1987-95 period. The average forecast errors are smaller in the Eurodeposit futures market than in the Treasury market, reflecting the flatter shape of the Eurodeposit spot curve (and perhaps the systematic "richness" of the shortest Treasury bills). In contrast, the correlations between implied and realized rate changes suggest that the Treasury forwards predict future rate changes slightly better than the Eurodeposit futures do. A comparison with the correlations in Figure 1 (the long sample period) shows that the front-end Treasury forwards, in particular, have become much better predictors over time. For the three-month rates, this correlation rises from -0.04 to 0.45, while for the three-year rates, this correlation rises from -0.13 to 0.01. Thus, recent evidence is more consistent with the pure expectations hypothesis than the data in Figure 1, but these relations are so weak that it is too early to tell whether the underlying relation actually has changed. Anyway, even the recent correlations suggest that bonds longer than a year tend to earn their rolling yields.
在图3中，我们比较了欧元美元期货和国库券在1987-95年期间的预测能力。欧洲存款期货市场的平均预测误差小于国债市场，反映了欧洲存款的即期收益率曲线更平坦的形状（也可能是最短期的国库券系统的“高估”预测值）。相比之下，隐含和实现的收益率变化之间的相关性表明，国债远期预测未来收益率变化略好于欧洲存款期货。与图1（长样本期）的相关性的比较表明，前端的国债远期的预测能力随着时间的推移变得更好。对于三月期收益率，这种相关性从-0.04上升到0.45，而对三年期收益率，这种相关性从-0.13上升到0.01。因此，与图1中的数据相比，最近的证据与完全预期假说更为一致，但是这些关系非常弱，判断隐藏的相关性是否真的发生了变化为时尚早。无论如何，即使最近的相关性也表明长于一年的债券往往会获得滚动收益率。
Figure 3 Evaluating the Implied Eurodeposit and Treasury Forward Yield Curve’s Ability to PredictActual Rate Changes, 1987-95
Interpretations
The empirical evidence in Figure 1 is clearly inconsistent with the pure expectations hypothesis. One possible explanation is that curve steepness mainly reflects time-varying risk premia, and this effect is variable enough to offset the otherwise positive relation between curve steepness and rate expectations. That is, if the market requires high risk premia, the current long rate will become higher and the curve steeper than what the rate expectations alone would imply —— the yield of a long bond initially has to rise so high that it provides the required bond return by its high yield and by capital gain caused by its expected rate decline. In this case, rate expectations and risk premia ar the steep curve predicts high risk premia and declining long rates. This story could explain the steepening of the front end of the US yield curve in spring 1994 (but not on many earlier occasions when policy tightening caused yield curve flattening).
图1中的实证证据显然与完全预期假说不一致。一个可能的解释是，曲线陡峭程度主要反映了时变的风险溢价，这种影响足够抵消曲线陡峭程度和收益率预期之间原本的正相关性。也就是说，如果市场需要高风险溢价，目前的长期收益率将会变得更高，曲线比单纯的收益率预期隐含的更为陡峭，长期债券的收益率最初必须上涨得很高，才能通过其高收益率及其预期收益率下降所带来的资本回报提供所要求的债券回报。在这种情况下，收益率预期和风险溢价是负相关的；陡峭的曲线预测高风险溢价和长期收益率下降。这可以解释1994年春季美国收益率曲线前端的陡峭（但是在政策收紧导致收益率曲线平坦化的早期情况下，并不是这样）。
The long-run average bond risk premia are positive (see Part 3 of this series and Figure 11 in this report) but the predictability evidence suggests that bond risk premia are time-varying rather than constant. Why should required bond risk premia vary over time? In general, an asset's risk premium reflects the amount of risk and the market price of risk (for details, see Appendix B). Both determinants can fluctuate over time and result in predictability. They may vary with the yield level (rate-level-dependent volatility) or market direction (asymmetric volatility or risk aversion) or with economic conditions. For example, cyclical patterns in required bond returns may reflect wealth-dependent variation in the risk aversion level —— "the cycle of fear and greed."
长期平均债券风险溢价是正的（参见本系列的第3部分和本报告中的图11），但可预测性证据表明债券风险溢价是时变的，而不是恒定的。为什么债券风险溢价随时间而变化？一般来说，资产的风险溢价反映了风险的大小和风险的市场价格（详见附录B）。两个决定因素随着时间的推移可能会波动，并导致可预测性。它们可能随着收益率水平（依赖收益率水平的波动率），或市场方向（不对称的波动率或风险厌恶），或经济状况而变化。例如，债券回报的周期性模式可能反映了风险规避水平中的财富依赖性变化，即“恐惧和贪婪的循环”。
Figure 4 shows the typical business cycle behavior of bond returns and yield curve steepness: Bond returns are high and yield curves are steep near troughs, and bond returns are low and yield curves are flat/inverted near peaks. These countercyclic patterns probably reflect the response of monetary policy to the economy's inflation dynamics, as well as time-varying risk premia (high risk aversion and required risk premia in "bad times" and vice versa). Figure 4 is constructed so that if bonds tend to earn their rolling yields, the two lines are perfectly aligned. However, the graph shows that bonds tend to earn additional capital gains (beyond rolling yields) from declining rates near cyclical troughs —— and capital losses from rising rates near peaks. Thus, realized bond returns are related to the steepness of the yield curve and —— in addition —— to the level of economic activity.
图4显示了债券回报和收益率曲线陡峭程度典型的商业周期行为：债券回报高，收益率曲线在波谷附近陡峭；债券回报低，收益率曲线在峰值附近平坦或倒挂。这些反周期模式可能反映了货币政策对经济通货膨胀的动态反应，以及时变的风险溢价（高风险厌恶和“坏时期”要求的风险溢价，反之亦然）。图4的构造说明，如果债券倾向于获得其滚动收益率，则两条线完全契合。然而，该图表显示，债券往往会从波谷附近的收益率下降中获得额外的资本回报（超出滚动收益率），以及波峰附近的收益率上涨中产生资本损失。因此，实现的债券回报与收益率曲线的陡峭程度相关，并且还与经济活动水平有关。
Figure 4 Average Business Cycle Pattern of US Realized Bond Risk Premium and Curve Steepness, 1968-95
These empirical findings motivate the idea that the required bond risk premia vary over time with the steepness of the yield curve and with some other variables. In Part 4 of this series, we show that yield curve steepness indicators and real bond yields, combined with measures of recent stock and bond market performance, are able to forecast up to 10% of the variation in monthly excess bond returns. That is, bond returns are partly forecastable. For quarterly or annual horizons, the predictable part is even larger.
这些实证结果启发了如下观点，时变的债券风险溢价随着收益率曲线的陡峭程度和其他一些变量而变化。在本系列的第4部分中，我们显示，收益率曲线陡峭程度和实际债券收益率，加上近期股票和债券市场表现的度量，能够预测的月度债券超额回报高达10%。也就是说，债券回报是部分可预测的。对于季度或年度频率的数据，可预测的部分甚至更大。
If market participants are rational, bond return predictability should reflect time-variation in the bond risk premia. Bond returns are predictably high when bonds command exceptionally high risk premia —— either because bonds are particularly risky or because investors are exceptionally risk averse. Bond risk premia may also be high if increased supply of long bonds steepens the yield curve and increases the required bond returns. An alternative interpretation is that systematic forecasting errors cause the predictability. If forward rates really reflect the market's rate expectations (and no risk premia), these expectations are irrational.
如果市场参与者理性，债券回报可预测性应反映债券风险溢价的时变性。当债券风险溢价异常高时（由于债券风险过大，或因为投资者过于规避风险）可以预测债券回报也会高。如果长期债券的供应量增加使收益率曲线更加陡峭，并增加所要求的债券回报，债券风险溢价也可能会很高。另一种解释是系统性的预测错误导致可预测性。如果远期收益率真的反映了市场的收益率预期（没有风险溢价），这些预期是不合理的。
They tend to be too high when the yield curve is upward sloping and too low when the curve is inverted. The market appears to repeat costly mistakes that it could avoid simply by not trying to forecast rate shifts. Such irrational behavior is not consistent with market efficiency. What kind of expectational errors would explain the observed patterns between yield curve shapes and subsequent bond returns? One explanation is a delayed reaction of the market's rate expectations to inflation news or to monetary policy actions. For example, if good inflation news reduces the current short-term rate but the expectations for future rates react sluggishly, the yield curve becomes upward-sloping, and subsequently the bond returns are high (as the impact of the good news is fully reflected in the rate expectations and in the long-term rates).
当收益率曲线向上倾斜时，它们往往太高，当曲线倒挂时，它们太低。市场似乎重复了昂贵的错误，只能通过不试图预测收益率变动来避免。这种不合理的行为与市场有效性不一致。什么样的预期错误可以解释收益率曲线形状和随后的债券回报之间观察到的模式？一个解释是市场对通货膨胀消息或货币政策行动速度的预期延迟反应。例如，如果利好的通货膨胀消息降低了目前的短期收益率，但对未来收益率的预期反应迟缓，收益率曲线变得向上倾斜，随后债券回报变高（好消息的影响充分反映在收益率预期和长期收益率）。
Because expectations are not observable, we can never know to what extent the return predictability reflects time-varying bond risk premia and systematic forecast errors. Academic researchers have tried to develop models that explain the predictability as rational variation in required returns. However, yield volatility and other obvious risk measures seem to have little ability to predict future bond returns. In contrast, the observed countercyclic patterns in expected returns suggest rational variation in the risk aversion level —— although they also could reflect irrational changes in the market sentiment. Studies that use survey data to proxy for the market's expectations conclude that risk premia and irrational expectations contribute to the return predictability.
由于预期不可观察，我们永远不知道回报可预测性在多大程度上反映了时变的债券风险溢价和系统预测误差。学术研究人员试图开发模型，将可预测性解释为所要求回报的理性变化。然而，收益率波动率和其他明显的风险度量似乎几乎没有预测未来债券回报的能力。相比之下，预期回报中观察到的反周期模式表明了风险规避水平的理性变化，尽管它们也可以反映市场情绪的非理性变化。使用调查数据代表市场预期的研究得出结论，风险溢价和非理性预期有助于回报可预测性。
Investment Implications
If expected bond returns vary over time, historical average returns contain less information about future returns than do indicators of the prevailing economic environment, such as the information in the current yield curve. In principle, the information in the forward rate structure is one of the central issues for fixed-income investors. If the forwards (adjusted for the convexity bias) only reflect the market's rate expectations and if these expectations are unbiased (they are realized, on average), then all government bond strategies would have the same near-term expected return. Yield-seeking activities (convergence trades and relative value trades) would be a waste of time and trading costs. Empirical evidence discussed above suggests that this is not the case: Bond returns are partially predictable, and yield-seeking strategies are profitable in the long run. However, it pays to use other predictors together with yields and to diversify across various positions, because the predictable part of bond returns is small and uncertain.
如果债券的预期回报随时间而变化，则历史平均回报包含关于未来回报的信息，与当前经济环境的指标（如当前收益率曲线中的信息）相比较少。原则上，远期收益率结构中的信息是固定收益投资者研究的核心问题之一。如果远期收益率（经过凸度偏差调整）仅反映市场的收益率预期，如果这些预期是无偏见的（平均来看将会实现），则所有政府债券策略将具有相同的短期预期回报。追求收益率的活动（如收敛交易和相对价值交易）将是浪费时间和交易成本的。之前的实证证据表明，情况并非如此：债券回报是部分可预测的，长期来看追求收益率的策略是有利可图的。然而，由于债券回报的可预测部分小而且不确定，因此可以将其他预测变量与收益率组合使用或进行分散化投资。
In practice, the key question is perhaps not whether the forwards reflect rate expectations or risk premia but whether actual return predictability exists and who should exploit it. No predictability exists if the forwards (adjusted for the convexity bias) reflect unbiased rate expectations. If predictability exists and is caused by expectations that are systematically wrong, everyone can exploit it. If predictability exists and is caused by rational variation in the bond risk premia, only some investors should take advantage of the opportunities to enhance long- many others would find higher expected returns in "bad times" no more than a fair compensation for the greater risk or the higher risk aversion level. Only risk-neutral investors and atypical investors whose risk perception and risk tolerance does not vary synchronously with those of the market would want to exploit any profit opportunities —— and these investors would not care whether rationally varying risk premia or the market's systematic forecast errors cause these opportunities.
在实践中，关键问题可能不在于远期收益率是否反映收益率预期或风险溢价，而是实际回报可预测性是否存在，谁应该利用它。如果远期收益率（经过凸度偏差调整）反映了无偏的收益率预期，则不存在可预测性。如果可预测性存在并且是由系统性错误的预期引起的，每个人都可以利用它。如果存在可预测性，是由债券风险溢价的合理变动引起的，只有一些投资者能利用机会增加长期平均回报；许多其他人会在“坏时期”中找到更高的预期回报，而不是为更大的风险或更高的风险规避水平提供公平的补偿。只有风险中性的投资者和不典型的投资者，他们的风险感知和风险承受能力与市场上的人不同步，他们会想要利用任何利润机会，而这些投资者不会关心理性的风险溢价或市场的系统性预测错误是否会带来这些机会。
HOW SHOULD WE INTERPRET THE YIELD CURVE CURVATURE?
如何解释收益率曲线的曲率
The market's curve reshaping expectations, volatility expectations and expected return structure determine the curvature of the yield curve. Expectations for yield curve flattening imply expected profits for duration-neutral long-barbell versus short-bullet positions, tending to make the yield curve concave (thus, the yield disadvantage of these positions offsets their expected profits from the curve flattening). Expectations for higher volatility increase the value of convexity and the expected profits of these barbell-bullet positions, again inducing a concave yield curve shape. Finally, high required returns of intermediate bonds (bullets) relative to short and long bonds (barbells) makes the yield curve more concave. Conversely, expectations for yield curve steepening or for low volatility, together with bullets' low required returns, can even make the yield curve convex.
市场曲线形变预期、波动率预期和预期回报结构决定了收益率曲线的曲率。收益率曲线平坦化的预期意味着久期中性的多杠铃-空子弹组合的预期利润，倾向于使收益率曲线上凸（因此，这些头寸的收益率劣势抵消了其曲线平坦化带来的预期利润）。高波动率的预期增加了杠铃-子弹组合的凸度价值和预期利润，再次导致了收益率曲线上凸的形状。最后，相对于短期和长期债券（杠铃组合），市场对中期债券（子弹组合）要求的高回报使收益率曲线更上凸。相反，对于收益率曲线变陡或低波动率的预期，以及对子弹组合要求的低回报，甚至可以使收益率曲线下凸。
In this section, we analyze the yield curve curvature and focus on two key questions: (1) How important are each of the three determinants in changing the curvature over time?; and (2) why is the long-run average shape of the yield curve concave?
在本节中，我们分析收益率曲线的曲率，并重点关注两个关键问题：（1）三个决定因素在改变曲率中的重要性分别是多少？和（2）为什么长期平均来看收益率曲线的形状是上凸的？
Empirical Evidence
Some earlier studies suggest that the curvature of the yield curve is closely related to the market's volatility expectations, presumably due to the convexity bias. However, our empirical analysis indicates that the curvature varies more with the market's curve-reshaping expectations than with the volatility expectations. The broad curvature of the yield curve varies closely with the steepness of the curve, probably reflecting mean-reverting rate expectations.
一些较早的研究表明，收益率曲线的曲率与市场的波动率预期密切相关，推测是由于凸度偏差。然而，我们的实证分析表明，曲率更倾向于随着市场的曲线形变预期而不是波动率预期变化。收益率曲线的曲率大致紧随曲线的陡峭程度变化，可能反映了均值回归的收益率预期。
Figure 5 plots the Treasury spot curve when the yield curve was at its steepest and at its most inverted in recent history and on a date when the curve was extremely flat. This graph suggests that historically low short rates have been associated with steep yield curves and high curvature (concave shape), while historically high short rates have been associated with inverted yield curves and negative curvature (convex shape).
图5绘制了当近期历史中收益率曲线最陡峭、最倒挂以及最平坦的国债即期收益率曲线。该图表明，历史上低的短期收益率与陡峭的收益率曲线和高曲率（上凸）相关联，而历史上高的短期收益率与倒挂的收益率曲线和负曲率（下凸）相关联。
Figure 5 Treasury Spot Yield Curves in Three Environments
The correlation matrix of the monthly changes in yield levels, curve steepness and curvature in Figure 6 confirms these relations. Steepness measures are negatively correlated with the short rate levels (but almost uncorrelated with the long rate levels), reflecting the higher likelihood of bull steepeners and bear flatteners than bear steepeners and bull flatteners. However, we focus on the high correlation (0.79) between the changes in the steepness and the changes in the curvature. This relation has a nice economic logic. Our curvature measure can be viewed as the yield carry of a curve-steepening position, a duration-weighted bullet-barbell position (long a synthetic three-year zero and short equal amounts of a three-month zero and a 5.75-year zero). If market participants have mean-reverting rate expectations, they expect yield curves to revert to a certain average shape (slightly upward sloping) in the long run. Then, exceptionally steep curves are associated with expectations for subsequent curve flattening and for capital losses on steepening positions. Given the expected capital losses, these positions need to offer an initial yield pickup, which leads to a concave (humped) yield curve shape. Conversely, abnormally flat or inverted yield curves are associated with the market's expectations for subsequent curve steepening and for capital gains on steepening positions. Given the expected capital gains, these positions can offer an initial yield giveup, which induces a convex (inversely humped) yield curve.
图6中收益率水平、曲线陡峭程度和曲率月度变化的相关矩阵证实了这些关系。陡峭程度与短期收益率水平呈负相关（但与长期收益率水平几乎无关），反映出陡峭程度与牛陡和熊平而不是牛平和熊斗之间更高的相关性。然而，我们专注于陡峭程度变化和曲率变化之间的高相关性（0.79）。这种关系有一个很好的经济逻辑。我们的曲率可以看作是做陡曲线头寸的收益率Carry，即久期加权的子弹-杠铃组合（做多三年期零息债券和做空等量的三月期零息债券和5.75年期零息债券）。如果市场参与者具有均值回归的收益率预期，那么长期来看，他们预期收益率曲线将恢复到一定的平均形状（略向上倾斜）。然后，非常陡峭的曲线与后续曲线平坦化的预期和做陡头寸的资本损失相关。鉴于预期的资本损失，这些头寸需要提供初始的收益率补偿，这导致了上凸（隆起）的收益率曲线形状。相反，异常平坦或倒挂的收益率曲线与市场对随后曲线陡峭的预期和做陡头寸的资本回报有关。鉴于预期的资本回报，这些头寸可以提供初始收益率损失，这会产生一个下凸（向下隆起）的收益率曲线。
Figure 6 Correlation Matrix of Yield Curve Level, Steepness and Curvature, 1968-95
Figure 7 illustrates the close comovement between our curve steepness and curvature measures. The mean-reverting rate expectations described above are one possible explanation for this pattern. Periods of steep yield curves (mid-1980s and early 1990s) are associated with high curvature and, thus, a large yield pickup for steepening positions, presumably to offset their expected losses as the yield curve flattens. In contrast, periods of flat or inverted curves (89-90 and 1995) are associated with low curvature or even an inverse hump. Thus, barbells can pick up yield and convexity over duration-matched bullets, presumably to offset their expected losses when the yield curve is expected to steepen toward its normal shape.
图7示出了我们的曲线陡峭程度和曲率之间的紧密联系。上述均值回归的收益率预期是这种模式的一个可能解释。陡峭收益率曲线的时期（1980年代中期和90年代初）与高曲率相关，因此，对于做陡头寸而言，大量的收益率补偿可能抵消了随后收益率曲线变平的预期损失。相比之下，平坦或倒挂曲线的时期（9-90和1995）与低曲率甚至下凸相关。因此，杠铃组合相对于久期匹配的子弹组合有收益率和凸度优势，当预期收益率曲线朝向其正常形状而变陡峭时，可以抵消其预期的损失。
Figure 7 Curvature and Steepness of the Treasury Curve, 1968-95
The expectations for mean-reverting curve steepness influence the broad curvature of the yield curve. In addition, the curvature of the front end sometimes reflects the market's strong view about near-term monetary policy actions and their impact on the curve steepness. Historically, the Federal Reserve and other central banks have tried to smooth interest rate behavior by gradually adjusting the rates that they control. Such a rate-smoothing policy makes the central bank's actions partly predictable and induces a positive autocorrelation in short-term rate behavior. Thus, if the central bank has recently begun to ease (tighten) monetary policy, it is reasonable to expect the monetary easing (tightening) to continue and the curve to steepen (flatten).
曲线陡峭程度均值回归的预期影响了大部分收益率曲线的曲率。此外，曲线前端的曲率有时反映了市场对近期货币政策行动及其对曲线陡峭程度影响的强烈观点。历史上，美联储等央行已经试图通过逐步调整收益率来平滑收益率行为。这种收益率平滑政策使中央银行的行为部分可预测，并导致短期收益率行为正的自相关性。因此，如果中央银行最近开始放松（收紧）货币政策，可以合理地预期货币宽松政策（紧缩）将继续，曲线将变陡峭（平坦）。
In the earlier literature, the yield curve curvature has been mainly associated with the level of volatility. Litterman, Scheinkman and Weiss ("Volatility and the Yield Curve," Journal of Fixed Income, 1991) pointed out that higher volatility should make the yield curve more humped (because of convexity effects) and that a close relation appeared to exist between the yield curve curvature and the implied volatility in the Treasury bond futures options. However, Figure 8 shows that the relation between curvature and volatility was close only during the sample period of the study (1984-88). Interestingly, no recessions occurred in the mid-1980s, the yield curve shifts were quite parallel and the flattening/steepening expectations were probably quite weak. The relation breaks down before and after the 1984-88 period, especially near recessions, when the Fed is active and the market may reasonably expect curve reshaping. For example, in 1981 yields were very volatile but the yield curve was convex (inversely humped); see Figures 5 and 13. It appears that the market's expectations for future curve reshaping are more important determinants of the yield curve curvature than are its volatility expectations (convexity bias). The correlations of our curvature measures with the curve steepness are around 0.8 while those with the implied option volatility are around 0.1. Therefore, it is not surprising that the implied volatility estimates that are based on the yield curve curvature are not closely related to the implied volatilities that are based on option prices. Using the yield curve shape to derive implied volatility can result in negative
this unreasonable outcome occurs in simple models when the expectations for curve steepening make the yield curve inversely humped (see Part 5 of this series).
在早期的文献中，收益率曲线曲率主要与波动率水平有关。Litterman，Scheinkman 和 Weiss 指出（《Volatility and the Yield Curve》，Journal of Fixed Income，1991），较高的波动率应使收益率曲线更加上凸（由于凸度效应），并且收益率曲线曲率和国债期货期权的隐含波动率之间存在着密切的关系。然而，图8显示，曲率和波动率之间的关系仅存在于研究的样本期间（1984-88）。有趣的是，1980年代中期没有发生经济衰退，收益率曲线变化同步性相当高，变平或变陡的预期可能相当薄弱。这种关系在1984-88年度之前和之后不成立，尤其是在近期的经济衰退时期，这时美联储活跃，市场理性的预期曲线形变。例如，1981年的收益率波动率非常大，但收益率曲线是下凸的（向下隆起），见图5和图13。似乎市场对未来曲线形变的预期是收益率曲线曲率的重要决定因素，而不是其波动率预期（凸度偏差）。我们测算的曲率与曲线陡峭程度的相关性约为0.8，而与期权隐含波动率的相关性约为0.1。因此，基于收益率曲线曲率的隐含波动率估计与基于期权价格的隐含波动率并不密切相关。使用收益率曲线形状导出隐含波动率可导致负的波动率估计，这种不合理的结果发生在简单的模型中，当曲线变陡峭的预期使得收益率曲线向下隆起时（见本系列的第5部分）。
Figure 8 Curvature and Volatility in the Treasury Market, 1982-95
Now we move to the second question "Why is the long-run average shape of the yield curve concave?" Figure 9 shows that the average par and spot curves have been concave over our 28-year sample period. Recall that the concave shape means that the forwards have, on average, implied yield curve flattening (which would offset the intermediate bonds' initial yield advantage over duration-matched barbells). Figure 10 shows that, on average, the implied flattening has not been matched by sufficient realized flattening. Not surprisingly, flattenings and steepenings tend to wash out over time, whereas the concave spot curve shape has been quite persistent. In fact, a significant positive correlation exists between the implied and the realized curve flattening, but the average forecast errors in Figure 10 reveal a bias of too much implied flattening. This conclusion holds when we split the sample into shorter subperiods or into subsamples of a steep versus a flat yield curve environment or a rising-rate versus a falling-rate environment.
现在我们转到第二个问题：“为什么收益率曲线的长期平均形状是上凸的？”图9显示，在28年的样本周期内，平均到期和即期收益率曲线均呈上凸。回想一下，上凸形意味着，平均来看远期收益率隐含着收益率曲线变平（这将抵消中期债券相对于久期匹配的杠铃组合的初始收益率优势）。图10显示，平均而言，隐含的平坦化并没有被充分实现。毫不奇怪，变平和变陡倾向于随着时间的推移而逐渐消失，但上凸即期收益率曲线的形状已经相当持久。事实上，隐含和实现的曲线平坦化之间存在显着的正相关，但图10中的平均预测误差揭示了过于隐含平坦化的偏差。当我们将样本依据陡峭与平坦或收益率上升与下降的情形分解成子样本时，这一结论是成立的。
Figure 9 Average Yield Curve Shape, 1968-95
Figure 10 Evaluating the Implied Forward Yield Curve’s Ability to Predict Actual Changes in the Spot Yield Curve’s Steepness, 1968-95
Figure 10 shows that, on average, the capital gains caused by the curve flattening have not offset a barbell's yield disadvantage (relative to a duration-matched bullet). A more reasonable possibility is that the barbell's convexity advantage has offset its yield disadvantage. We can evaluate this possibility by examining the impact of convexity on realized returns over time. Empirical evidence suggests that the convexity advantage is not sufficient to offset the yield disadvantage (see Figure 12 in Part 5 of this series). Alternatively, we can examine the shape of historical average returns because the realized returns should reflect the convexity advantage. This convexity effect is certainly a partial explanation for the typical yield curve shape —— but it is the sole effect only if duration-matched barbells and bullets have the same expected returns. Equivalently, if the required bond risk premium increases linearly with duration, the average returns of duration-matched barbells and bullets should be the same over a long neutral period (because the barbells' convexity advantage exactly offsets their yield disadvantage). The average return curve shape in Figure 1, Part 3 and the average barbell-bullet returns in Figure 11, Part 5 suggest that bullets have somewhat higher long-run expected returns than duration-matched barbells. We can also report the historical performance of synthetic zero positions over the 1968-95 period: The average annualized monthly return of a four-year zero is 9.14%, while the average returns of increasingly wide duration-matched barbells are progressively lower (3-year and 5-year 9.05%, 2-year and 6-year 9.00%, 1-year and 7-year 8.87%). Overall, the typical concave shape of the yield curve likely reflects the convexity bias and the concave shape of the average bond risk premium curve rather than systematic flattening expectations, given that the average flattening during the sample is zero.
图10显示，平均来说，曲线平坦化引起的资本回报并未抵消杠铃组合的收益率劣势（相对于久期匹配的子弹组合）。更合理的可能性是，杠铃组合的凸度优势抵消了其收益率劣势。我们可以通过检查凸度对实际回报的影响来评估这种可能性。经验证据表明，凸度优势不足以抵消收益率劣势（参见本系列第5部分的图12）。或者，我们可以检查历史平均回报的形状，因为实现的回报应该反映凸度优势。这种凸度效应当然是对典型收益率曲线形状的部分解释，但只有久期匹配的杠铃组合和子弹组合具有相同的预期回报，才是唯一的效果。同样地，如果债券风险溢价随久期线性增长，久期匹配的杠铃组合和子弹组合的平均回报在长时间的中性时期应该是相同的（因为杠铃组合的凸度优势恰好抵消了他们的收益率劣势）。第3部分图1的平均回报曲线形状以及第5部分图11中的平均杠铃-子弹组合回报表明，子弹组合比久期匹配的杠铃组合具有较高的长期预期回报。我们还指出1968-95年期间合成零息债券头寸的历史表现：4年期零息债券的平均年化月度回报为9.14%，而久期匹配的杠铃组合的平均回报逐渐下降（3-5年期组合为9.05%，2-6年期组合为9.00%，1-7年期组合为8.87%）。总体而言，考虑到样本中平均来看曲线平坦的比例为零，收益率曲线典型的上凸形态可能反映了凸度偏差和债券风险溢价曲线的上凸形态，而不是系统的曲线变平预期。
Figure 11 Average Treasury Maturity-Subsector Returns as a Function of Return Volatility
Interpretations
The impact of curve reshaping expectations and convexity bias on the yield curve shape are easy to understand, but the concave shape of the bond risk premium curve is more puzzling. In this subsection, we explore why bullets should have a mild expected return advantage over duration-matched barbells. One likely answer is that duration is not the relevant risk measure. However, we find that average returns are concave even in return volatility, suggesting a need for a multi-factor risk model. We first discuss various risk-based explanations in detail and then consider some alternative "technical" explanations for the observed average return patterns.
曲线形变预期和凸度偏差对收益率曲线形状的影响很容易理解，但债券风险溢价曲线的上凸形态更令人困惑。在本小节中，我们探讨为什么子弹组合应该比久期匹配的杠铃组合具有微弱的预期回报优势。一个可能的答案是，久期不是有意义的风险度量。然而，我们发现即使作为回报波动率的函数平均回报也是上凸的，这表明需要一个多因子风险模型。我们首先详细讨论各种基于风险的解释，然后考虑观察到的均值回归模式的替代“技术性”解释。
All one-factor term structure models imply that expected returns should increase linearly with the bond's sensitivity to the risk factor. Because these models assume that bond returns are perfectly correlated, expected returns should increase linearly with return volatility (whatever the risk factor is). However, bond durations are proportional to return volatilities only if all bonds have the same basis-point yield volatilities. Perhaps the concave shape of the average return-duration curve is caused by (i) a linear relation between expected return and return volatility and (ii) a concave relation between return volatility and duration that, in turn, reflects an inverted or humped term structure of yield volatility (see Figure 15). Intuitively, a concave relation between the actual return volatility and duration would make a barbell a more defensive (bearish) position than a duration-matched bullet. The return volatility of a barbell is simply a weighted average of its constituents' return volatilities (given the perfect correlation); thus, the barbell's volatility would be lower than that of a duration-matched bullet.
所有单因子期限结构模型认定预期回报随着债券对风险因子的敏感性而线性增长。因为这些模型假设债券回报完全相关，所以预期回报应随回报波动率线性增加（无论风险因子如何）。然而，只有所有债券具有相同的基点收益率波动率，债券久期才与回报波动率成比例。平均回报-久期曲线的上凸形状也许是由（i）预期回报和回报波动率之间的线性关系引起的；（ii）回报波动率与久期之间的上凸关系，反过来又反映了一个倒挂或隆起的收益率波动率期限结构（见图15）。直觉上，实际回报波动率与久期之间的上凸关系将使杠铃组合比久期匹配的子弹组合更具防守（看跌）性。杠铃组合的回报波动率只是其成分回报波动率的加权平均值（给定完美的相关性）；因此，杠铃组合的波动率将低于久期匹配的子弹组合。
Figures 13 and 14 will demonstrate that the empirical term structure of yield volatility has been inverted or humped most of the time. Thus, perhaps a barbell and a bullet with equal return volatilities (as opposed to equal durations) should have the same expected return. However, it turns out that the bullet's return advantage persists even when we plot average returns on historical return volatilities. Figure 11 shows the historical average returns of various maturity-subsector portfolios of Treasury bonds as a function of return volatility. The average returns are based on two relatively neutral periods, January 1968 to December 1995 and April 1986 to March 1995. We still find that the average return curves have a somewhat concave shape. Note that we demonstrate the concave shape in a conservative way by graphing arithm the geometric average return curves would be even more concave.
图13和14将证明大部分时间内收益率波动率的经验性期限结构是倒挂或隆起的。因此，也许杠铃组合和子弹组合具有相等的回报波动率（而不是相等的久期）才具有相同的预期回报。然而，事实证明，即使我们绘制平均回报关于历史回报波动率的变化，子弹组合的回报优势仍然存在。图11显示了国债各种期限投资组合的历史平均回报（作为回报波动率的函数）。平均回报基于1968年1月至1995年12月和1986年4月至1995年3月的两个相对中性的时期。我们仍然发现平均收益率曲线有一些上凸。注意，通过绘制算术平均回报，我们以保守的方式展示上凸形状；几何均值收益率曲线将更加上凸。
As explained above, one-factor term structure models assume that bond returns are perfectly correlated. One-factor asset pricing models are somewhat more general. They assume that realized bond returns are influenced by only one systematic risk factor but that they also contain a bond-specific residual risk component (which can make individual bond returns imperfectly correlated). Because the bond-specific risk is easily diversifiable, only systematic risk is rewarded in the marketplace. Therefore, expected returns are linear in the systematic part of return volatility. This distinction is not very important for government bonds because their bond-specific risk is so small. If we plot the average returns on systematic volatility only, the front end would be slightly less steep than in Figure 11 because a larger part of short bills' return volatility is asset-specific. Nonetheless, the overall shape of the average return curve would remain concave.
如上所述，单因子期限结构模型假定债券回报完全相关。单因子资产定价模式更为一般化。他们假定实现的债券回报只受一个系统风险因子的影响，但也包含一个特定于债券的剩余风险成分（可以使个别债券回报不完全相关）。由于债券特定风险易于分散，因此只有系统风险才能在市场上得到回报。因此，预期回报关于回报波动率中对应系统风险的部分是线性的。这种区别对于政府债券来说不是很重要，因为它们的债券特定风险很小。如果我们仅绘制平均收益率关于系统波动率的关系，图11中前端将略低，因为较大部分短期国库券的回报波动率是因资产而异。然而，平均收益率曲线的整体形状将保持上凸。
Convexity bias and the term structure of yield volatility explain the concave shape of the average yield curve partly, but a nonlinear expected return curve appears to be an additional reason. Figure 11 suggests that expected returns are somewhat concave in return volatility. That is, long bonds have lower required returns than one-factor models imply. Some desirable property in the longer cash flows makes the market accept a lower expected excess return per unit of return volatility for them than for the intermediate cash flows. We need a second risk factor, besides the rate level risk, to explain this pattern. Moreover, this pattern may teach us something about the nature of the second factor and about the likely sign of its risk premium. We will next discuss heuristically two popular candidates for the second factor —— interest rate volatility and yield curve steepness. We further discuss the theoretical determinants of required risk premia in Appendix B.
凸度偏差和收益率波动率的期限结构部分解释了平均收益率曲线的上凸形状，但非线性预期收益率曲线似乎是一个额外的原因。图11表明，预期回报关于回报波动率有些上凸。也就是说，长期债券的所要求的回报要低于单因子模型所隐含的。较中期现金流而言，长期现金流的一些有利特性使得市场愿意接受单位回报波动率上较低的预期超额回报。除了收益率水平的风险，我们需要第二个风险因子以解释这种模式。此外，这种模式可能会告诉我们关于第二个因子的性质和风险溢价的可能符号。我们接下来讨论第二个因子的两个流行选项，收益率波动率和收益率曲线陡峭程度。我们在附录B中进一步讨论风险溢价的理论决定因素。
Volatility as the second factor could explain the observed patterns if the market participants, in the aggregate, prefer insurance-type or "long-volatility" payoffs. Even nonoptionable government bonds have an option like characteristic because of the convex shape of their price-yield curves. As discussed in Part 5 of this series, the value of convexity increases with a bond's convexity and with the perceived level of yield volatility. If the volatility risk is not "priced" in expected returns (that is, if all "delta-neutral" option positions earn a zero risk premium), a yield disadvantage should exactly offset longer bonds' convexity advantage. However, the concave shape of the average return curve in Figure 11 suggests that positions that benefit from higher volatility have lower expected returns than positions that are adversely affected by higher volatility. Although the evidence is weak, we find the negative sign for the price of volatility risk intuitively appealing. The Treasury market participants may be especially averse to losses in high-volatility states, or they may prefer insurance-type (skewed) payoffs so much that they accept lower long-run returns for them. Thus, the long bonds' low expected return could reflect the high value many investors assign to positive convexity. However, because short bonds exhibit little convexity, other factors are needed to explain the curvature at the front end of the yield curve.
波动率作为第二个因子可以解释观察到的模式，如果市场参与者总体上偏好保险类型或“做多波动率”的回报。即使不嵌入期权的政府债券也有一个期权特征，因为它们的价格-收益率曲线是下凸的形状。如本系列第5部分所述，凸度价值随着债券的凸度和收益率波动率的感知水平而增加。如果波动率风险在预期回报中没有“定价”（也就是说，如果所有“delta中性”期权头寸都获得零风险溢价），收益率劣势应该恰好抵消较长期债券的凸度优势。然而，图11中平均收益率曲线的上凸形状表明，受益于较高波动率头寸的预期回报低于受制于较高波动率影响的头寸。虽然证据薄弱，但我们发现负的波动率风险价格在直觉上是有吸引力的。国债市场参与者可能特别反对高波动率状态的损失，或者他们可能更喜欢保险型（有偏）的回报，以至于他们接受较低的长期回报。因此，长期债券的低预期回报可能反映出许多投资者给予正凸度的高价值。然而，由于短债券表现出很小的凸度，因此需要其他因素来解释收益率曲线前端的曲率。
Yield curve steepness as the second factor (or short rate and long rate as the two factors) could explain the observed patterns if curve-flattening positions tend to be profitable just when investors value them most. We do not think that the curve steepness is by itself a risk factor that investors worry about, but it may tend to coincide with a more fundamental factor. Recall that the concave average return curve suggests that self-financed curve-flattening positions have negative expected returns —— because they are more sensitive to the long rates (with low reward for return volatility) than to the short/intermediate rates (with high reward for return volatility). This negative risk premium can be justified theoretically if the flattening trades are especially good hedges against "bad times." When asked what constitutes bad times, an academic's answer is a period of high marginal utility of profits, while a practitioner's reply probably is a deep recession or a bear market. The empirical evidence on this issue is mixed. It is clear that long bonds performed very well in deflationary recessions (the United States in the 1930s, Japan in the 1990s). However, they did not perform at all well in the stagflations of the 1970s when the predictable and realized excess bond returns were negative. Since the World War II, the US long bond performance has been positively correlated with the stock market performance —— although bonds turned out to be a good hedge during the stock market crash of October 1987. Turning now to flattening positions, these have not been good rec the yield curves typically have been flat or inverted at the begi