全网唯一研究利率必备!​远期收益率分析概述(2)

远期收益率分析概述（2）

在本节中，我们描述一些影响远期收益率期限结构，或更一般地说，影响收益率曲线形状的经济因素。三个主要的影响因素是市场的收益率预期、债券风险溢价（不同债券预期回报的差）和所谓的凸度偏差。事实上，这三个分量完全决定了收益率曲线。我们将在后续的报告中揭示，一年期远期收益率和一年期即期收益率之间的差异大约等于即期收益率的预期变化、债券风险溢价和凸度偏差的总和。我们首先讨论每个因素如何影响曲线形状，然后分析它们的组合产生的影响。

原文：In this section, we describe some economic forces that influence the term structure of forward rates or, more generally, the yield curve shape. The three main influences are the market’s rate expectations, the bond risk premia (expected return differentials across bonds) and the so-called convexity bias. In fact, these three components fully determine the yield curve; we will show in later reports that the difference between each one-year forward rate and the one-year spot rate is approximately equal to the sum of an expected spot rate change, a bond risk premium and the convexity bias. We first discuss how each component influences the curve shape, and then we analyze their combined impact.

很明显，市场对未来收益率变化的预期是收益率曲线形状的重要决定因素。例如，陡峭向上倾斜的曲线可能表明市场预期美联储近期将采取紧缩政策或通货膨胀上升。然而，假设不同期限债券收益率的差异仅反映市场对收益率预期的观点可能太过局限。众所周知的完全预期假说就有这样的极端含义。完全预期假说认为所有政府债券具有相同的近期预期回报（作为名义上无风险的短期债券），因为风险中性交易者的逐利行为消除了债券间的预期回报差异。如果所有收益率比短期收益率高的债券产生的资本损失抵消其收益率优势，则近期预期回报将相等。当市场预期债券收益率增加时，当前期限结构变得向上倾斜，因此任何长期债券的收益率优势和预期资本损失（由于预期收益率增长）完全相互抵消。换句话说，如果投资者预期他们的长期债券投资将因收益率上升而失去价值，那么他们需要更高的初始收益率作为久期增加的补偿。相反，预期收益率下降和资本回报将使当前长期债券收益率低于短期收益率，期限结构出现倒挂。

原文：It is clear that the market’s expectation of future rate changes is an important determinant of the yield curve shape. For example, a steeply upward-sloping curve may indicate that the market expects near-term Fed tightening or rising inflation. However, it may be too restrictive to assume that the yield differences across bonds with different maturities only reflect the market’s rate expectations. The well-known pure expectations hypothesis has such an extreme implication. The pure expectations hypothesis asserts that all government bonds have the same near-term expected return (as the nominally riskless short-term bond) because the return-seeking activity of risk-neutral traders removes all expected return differentials across bonds. Near-term expected returns are equalized if all bonds that have higher yields than the short-term rate are expected to suffer capital losses that offset their yield advantage. When the market expects an increase in bond yields, the current term structure becomes upward-sloping so that any long-term bond’s yield advantage and expected capital loss (due to the expected yield increase) exactly offset each other. In other words, if investors expect that their long-term bond investments will lose value because of an increase in interest rates, they will require a higher initial yield as a compensation for duration extension. Conversely, expectations of yield declines and capital gains will lower current long-term bond yields below the short-term rate, making the term structure inverted.

同样的逻辑，正（负）的初始利差抵消预期资本损失（回报）以使近期预期回报相等，这也适用于包括久期中性策略在内的债券组合。举一个例子，从两年期和十年期之间收益率曲线的平坦化盈利：卖出一个单位的两年期债券，购买一个依久期加权的十年期债券并把剩余的回报作为“现金”持有（相当于非常短期的债券）。给定典型的上凸曲线（作为久期的函数），曲线在变平的过程中产生负的Carry。只有当曲线变平的程度足以抵消负Carry的影响时，交易才会有利可图。隐含的远期收益率表示了曲线需要变平多少（两年与十年的利差收窄）才能使交易保持收支平衡。

原文：The same logic —— that positive (negative) initial yield spreads offset expected capital losses (gains) to equate near-term expected returns —— also holds for combinations of bonds, including duration-neutral yield curve positions. One example is a trade that benefits from the flattening of the yield curve between two- and ten-year maturities: selling a unit of the two-year bond, buying a duration-weighted amount of the ten-year bond and putting the remaining proceeds from the sale to “cash” (very short-term bonds). Given the typical concave yield curve shape (as a function of duration), such a curve flattening position earns a negative carry.7 The trade will be profitable only if the curve flattens enough to offset the impact of the negative carry. Implied forward rates indicate how much flattening (narrowing of the two-to ten-year spread) is needed for the trade to break even.

与市场对未来收益率水平的预期影响当前收益率曲线陡峭程度的方式相同，市场对收益率曲线未来陡峭程度的预期会影响当前收益率曲线的曲率。如果市场预期曲线变平，做平交易的负Carry将增加（以抵消预期的资本回报），使当前的收益率曲线更加上凸（弯曲）。图2说明了这些点。这副图根据不同的收益率预期，绘制了付息债券的收益率与其久期的关系，或等效地，零息债券收益率与其期限的关系。忽略债券风险溢价和凸度偏差，如果市场预期曲线的水平或斜率没有变化，当前的收益率曲线将是水平的。如果市场预期明年的收益率同时上涨但形状不变，当前的收益率曲线将呈线性增长（作为久期的函数）。如果市场预期收益率上升并且曲线变平，当前的收益率曲线将是向上倾斜和上凸的（作为久期的函数）。

原文：In the same way as the market’s expectations regarding the future level of rates influence the steepness of today’s yield curve, the market’s expectations regarding the future steepness of the yield curve influence the curvature of today’s yield curve. If the market expects more curve flattening, the negative carry of the flattening trades needs to increase (to offset the expected capital gains), making today’s yield curve more concave (curved). Figure 2 illustrates these points. This figure plots coupon bonds’ yields against their durations or, equivalently, zeros yields against their maturities, given various rate expectations. Ignoring the bond risk premia and convexity bias, if the market expects no change in the level or slope of the curve, today’s yield curve will be horizontal. If the market expects a parallel rise in rates over the next year but no reshaping, today’s yield curve will be linearly increasing (as a function of duration). If the market expects rising rates and a flattening curve, today’s yield curve will be increasing and concave (as a function of duration).8

完全预期假说中的一个关键假设是，所有不同期限的政府债券都具有相同的预期回报。不过，许多理论和实证证据表明债券的预期回报不同。我们将债券风险溢价定义为长期债券的预期回报超过债券无风险回报的部分。正的债券风险溢价将倾向于使收益率曲线向上倾斜。然而，各种理论对于债券风险溢价的符号（+/-）、决定因素和恒定性（随时间）的看法不同。经典流动性溢价假说认为大多数投资者不喜欢资产价格的短期波动，这些投资者持有长期债券除非长期债提供正的风险溢价作为其更大的波动率的补偿。还有一些现代资产定价理论表明，债券风险溢价应该随债券的久期、回报的波动率或与市场财富的协方差而增加。相比之下，优先偏好假说（preferred habitat hypothesis）认为风险溢价可能随久期而减少，长久期负债持有人可能认为长期债券是无风险资产，并且对持有短期资产要求更高的预期回报。虽然学术分析侧重于研究风险相关的溢价，但市场从业者通常强调其他导致收益率曲线上预期回报不同的因素。这些因素包括不同市场之间的流动性差异、制度限制和供求关系影响。我们使用术语“债券风险溢价”广泛地涵盖债券所有的预期回报差异，包括那些与风险无关的因素引起的回报差异。

原文：A key assumption in the pure expectations hypothesis is that all government bonds, regardless of maturity, have the same expected return. In contrast, many theories and empirical evidence suggest that expected returns vary across bonds. We define the bond risk premium as a longer-term bond’s expected one-period return in excess of the one-period bond’s riskless return. A positive bond risk premium would tend to make the yield curve slope upward. However, various theories disagree about the sign (+/-), the determinants and the constancy (over time) of the bond risk premium. The classic liquidity premium hypothesis argues that most investors dislike short-term fluctuations in asset prices; these investors will hold long-term bonds only if they offer a positive risk premium as a compensation for their greater return volatility. Also some modern asset pricing theories suggest that the bond risk premium should increase with a bond’s duration, its return volatility or its covariance with market wealth. In contrast, the preferred habitat hypothesis argues that the risk premium may decrease with duration; long-duration liability holders may perceive the long-term bond as the riskless asset and require higher expected returns for holding short-term assets. While academic analysis focuses on risk-related premia, market practitioners often emphasize other factors that cause expected return differentials across the yield curve. These include liquidity differences between market sectors, institutional restrictions and supply and demand effects. We use the term “bond risk premium” broadly to encompass all expected return differentials across bonds, including those caused by factors unrelated to risk.

美国国债的历史数据提供了债券风险溢价经验行为的证据。例如，近几十年国债收益率曲线90％的时间内呈现上升形态，可能反映了债券的正风险溢价的影响。较历史收益率而言，历史平均收益率提供了更多预期回报因期限不同的直接证据。尽管债券回报的周度和月度的大幅波动大多是不可预见的，但收益率意外上升和下跌的影响应在长样本期内清除掉。因此，各不同期限的历史平均回报应该反映了长期预期回报。

原文：Historical data on US Treasury bonds provide evidence about the empirical behavior of the bond risk premium. For example, the fact that the Treasury yield curve has been upward sloping nearly 90% of the time in recent decades may reflect the impact of positive bond risk premia. Historical average returns provide more direct evidence about expected returns across maturities than do historical yields. Even though weekly and monthly fluctuations in bond returns are mostly unexpected, the impact of unexpected yield rises and declines should wash out over a long sample period. Therefore, the historical average returns of various maturity sectors should reflect the long-run expected returns.

图3示出了做为平均久期函数的经验平均回报曲线，并且将其与一个理论预期回报曲线对比，其中理论预期回报曲线随久期线性增加。图3中不同久期的理论债券风险溢价通过年化预期回报与无风险一月期国库券（曲线上的最左点）年化回报之间的差来度量。同样，不同久期的经验债券风险溢价是通过历史平均债券回报超过一月期国库券年化回报的部分来衡量的。历史经验表明，债券风险溢价关于久期不是线性的，曲线的前端迅速增加，而两年后增加速度要慢得多。上凸的形状可以反映养老基金和其他长久期负债持有人对长期债券的需求。

原文：Figure 3 shows the empirical average return curve as a function of average duration and contrasts it to one theoretical expected return curve, one that increases linearly with duration. The theoretical bond risk premia are measured in Figure 3 by the difference between the annualized expected returns at various duration points and the annualized return of the riskless one-month bill (the leftmost point on the curve). Similarly, the empirical bond risk premia are measured by the historical average bond returns at various durations in excess of the one-month bill.9 Historical experience suggests that the bond risk premia are not linear in duration, but that they increase steeply with duration in the front end of the curve and much more slowly after two years. The concave shape may reflect the demand for long-term bonds from pension funds and other long-duration liability holders.

第三个影响收益率曲线的因素——凸度偏差，可能是最不为人所知的。不同的债券具有不同的凸度特征，并且不同期限的凸度差异可以产生（抵消）收益率差异。特别地，长期零息债券具有非常高的凸度（参见图4的上部），这倾向于降低其收益率。凸度偏差是指凸度差异对收益率曲线形状的影响。

原文：The third influence on the yield curve —— the convexity bias —— is probably the least well known. Different bonds have different convexity characteristics, and the convexity differences across maturities can give rise to (offsetting) yield differences. In particular, long-term zeros exhibit very high convexity (see top panel of Figure 4), which tends to depress their yields. Convexity bias refers to the impact these convexity differences have on the yield curve shape.

凸度与债券价格-收益率关系中的非线性部分密切相关。所有非可赎回债都有正的凸度，收益率下降对应的价格上升程度要比收益率上升产生的多一点。所有其他条件相同的情况下，正凸度是一个理想的特征，因为它增加了债券的回报（相对于没有凸度时的回报），无论收益率是上升还是下降。因为对于给定的收益率，正凸度能提高债券的表现，所以高凸度债券的收益率往往低于具有相同久期但凸度低的债券。换句话说，如果投资者预期通过增加凸度来提高回报，他们会接受较低的收益率。投资者主要对预期回报感兴趣，这些高凸度债券可以在较低的收益率水平下提供给定的预期回报。

原文：Convexity is closely related to the nonlinearity in the bond price-yield relationship. All noncallable bonds exhibit positive convexity; their prices rise more for a given yield decline than they fall for a similar yield increase. All else being equal, positive convexity is a desirable characteristic because it increases a bond’s return (relative to return in the absence of convexity) whether yields go up or down —— as long as they move somewhere. Because positive convexity can only improve a bond’s performance (for a given yield), more convex bonds tend to have lower yields than less convex bonds with the same duration.11 In other words, investors tend to demand less yield if they have the prospect of improving their returns as a result of convexity. Investors are primarily interested in expected returns, and these high-convexity bonds can offer a given expected return at a lower yield level.

凸度与债券价格-收益率关系中的非线性部分密切相关。所有非可赎回债都有正的凸度，收益率下降对应的价格上升程度要比收益率上升产生的多一点。所有其他条件相同的情况下，正凸度是一个理想的特征，因为它增加了债券的回报（相对于没有凸度时的回报），无论收益率是上升还是下降。因为对于给定的收益率，正凸度能提高债券的表现，所以高凸度债券的收益率往往低于具有相同久期但凸度低的债券。换句话说，如果投资者预期通过增加凸度来提高回报，他们会接受较低的收益率。投资者主要对预期回报感兴趣，这些高凸度债券可以在较低的收益率水平下提供给定的预期回报。

原文：Convexity is closely related to the nonlinearity in the bond price-yield relationship. All noncallable bonds exhibit positive convexity; their prices rise more for a given yield decline than they fall for a similar yield increase. All else being equal, positive convexity is a desirable characteristic because it increases a bond’s return (relative to return in the absence of convexity) whether yields go up or down —— as long as they move somewhere. Because positive convexity can only improve a bond’s performance (for a given yield), more convex bonds tend to have lower yields than less convex bonds with the same duration.11 In other words, investors tend to demand less yield if they have the prospect of improving their returns as a result of convexity. Investors are primarily interested in expected returns, and these high-convexity bonds can offer a given expected return at a lower yield level.

通过绘制即期收益率曲线和一年期远期收益率曲线（所有债券具有8%的预期回报并且短期收益率也为8％），图4的下半部分显示了凸度对曲线形状的影响。没有债券风险溢价和预期收益率变化的情况下，可以预期这些曲线将维持在8％的水平。然而，曲线加速下降，因为需要较低的收益率以抵消较长期债券的凸度优势，从而使债券的近期预期回报相等。短期债券几乎没有凸度，因此在曲线的前端几乎没有凸度偏差，但是凸度在长久期端对曲线形状具有显着的影响。凸度偏差是收益率曲线呈现典型的上凸形态的主要原因之一（因为曲线变平的趋势或在长久期端的倒挂）。

原文：The lower panel of Figure 4 illustrates the pure impact of convexity on the curve shape by plotting the spot rate curve and the curve of one-year forward rates when all bonds have the same expected return (8%) and the short-term rates are expected to remain at the current level. With no bond risk premia and no expected rate changes, one might expect these curves to be horizontal at 8%. Instead, they slope down at an increasing pace because lower yields are needed to offset the convexity advantage of longer-duration bonds and thereby to equate the near-term expected returns across bonds.12 Short-term bonds have little convexity; therefore, there is little convexity bias at the front end of the yield curve, but convexity can have a dramatic impact on the curve shape at very long durations. Convexity bias can be one of the main reasons for the typical concave yield curve shape (that is, for the tendency of the curve to flatten or invert at long durations).

凸度的价值随着收益率变化程度的大小而增加。因此，波动率增加会使收益率曲线形状更上凸（弯曲），并加大在不同凸度债券（久期匹配的付息债券-零息债券和杠铃-子弹组合）之间的利差。

原文：Figure 4. Convexity and the Yield Curve

The value of convexity increases with the magnitude of yield changes. Therefore, increasing volatility should make the overall yield curve shape more concave (curved) and widen the spreads between more and less convex bonds (duration-matched coupon bonds versus zeros and barbells versus bullets).13

当然，所有三种因素在同时影响债券收益率，这使得从总体上解释收益率曲线形状的任务变得相当困难。陡峭向上倾斜的曲线可以反映市场对收益率上升的预期抑或是增高的风险溢价。高度弯曲的曲线（即高曲率）可以反映市场对曲线变平或高波动性（使凸度更有价值）的预期，甚至反映风险溢价曲线的上凸。

原文：Of course, all three forces influence bond yields simultaneously, making the task of interpreting the overall yield curve shape quite difficult. A steeply upward-sloping curve can reflect either the market’s expectations of rising rates or high required risk premia. A strongly humped curve (that is, high curvature) can reflect the market’s expectations of either curve flattening or high volatility (which makes convexity more valuable), or even the concave shape of the risk premium curve.

理论上，收益率曲线可以完全分解为预期、风险溢价和凸度偏差三部分。实际上，精确分解是不可能的，因为三个分量随时间变化，并且不能直接观察到，必须估计。即使不可能进行精确分解，本报告和后续报告中的分析也能为投资者提供解释各种收益率曲线形状的框架。这些报告将描述预期收益率、风险溢价和凸度偏差的行为，显示它们如何影响曲线，并使用历史数据测算其影响的程度。此外，我们对早期文献的研究和我们新的实证工作将评估哪些理论和市场传言是正确的（与数据一致），哪些是错误的。主要结论如下：

原文：In theory, the yield curve can be neatly decomposed into expectations, risk premia and convexity bias. In reality, exact decomposition is not possible because the three components vary over time and are not directly observable but must be estimated.14 Even though an exact decomposition is not possible, the analysis in this and subsequent reports should give investors a framework for interpreting various yield curve shapes. These reports will characterize the behavior of rate expectations, risk premia and convexity bias; show how they influence the curve; and evaluate the magnitude of their impact using historical data. Furthermore, our survey of earlier literature and our new empirical work will evaluate which theories and market myths are correct (consistent with data) and which are false. The main conclusions are as follows:

我们经常听到“远期收益率显示市场对未来收益率的预期”。然而，这种说法只有在没有债券风险溢价存在且凸度偏差非常小的时候才是真的。如果目标是推断一到两年后的预期短期收益率，这时的凸度偏差非常小，以至于可以忽略。相比之下，我们的实证分析表明，债券风险溢价对短期收益率的影响是重要的。因此，如果远期收益率用于推断市场对近期收益率的预期，则应该从远期收益率中减去债券风险溢价，否则市场对收益率的预期将严重被高估。

原文：We often hear that “forward rates show the market’s expectations of future rates.” However, this statement is only true if no bond risk premia exist and the convexity bias is very small.15 If the goal is to infer expected short-term rates one or two years ahead, the convexity bias is so small that it can be ignored. In contrast, our empirical analysis shows that the bond risk premia are important at short maturities. Therefore, if the forward rates are used to infer the market’s near-term rate expectations, some measures of bond risk premia should be subtracted from the forwards, or the estimate of the market’s rate expectations will be strongly upward biased.

传统的期限结构理论假定零风险溢价（完全预期假说）或非零但恒定的风险溢价（流动性溢价假说、优先偏好假说），这与历史数据不一致。根据完全预期假说，向上倾斜的曲线预测长期收益率的增长，从而资本损失抵消长期债券的收益率优势。然而，实证证据表明，曲线向上倾斜时，长期收益率的小幅下降将增加长期债券的收益率优势。收益率曲线越陡，预期债券风险溢价就越高。这一发现明显违反完全预期假说，反而支持关于时变风险溢价的假设。

原文：The traditional term structure theories assume a zero risk premium (pure expectations hypothesis) or a nonzero but constant risk premium (liquidity premium hypothesis, preferred habitat hypothesis) which is inconsistent with historical data. According to the pure expectations hypothesis, an upward-sloping curve should predict increases in long-term rates, so that a capital loss offsets the long-term bonds’ yield advantage. However, empirical evidence shows that, on average, small declines in long-term rates, which augment the long-term bonds’ yield advantage, follow upward-sloping curves. The steeper the yield curve is, the higher the expected bond risk premia. This finding clearly violates the pure expectations hypothesis and supports hypotheses about time-varying risk premia.

比于上述传统理论，现代期限结构模型制定的约束假设更少。然而，许多流行的单因子模型假设，具有相同久期的债券获得相同的预期回报。这种假设意味着无论凸度多少，久期中性的头寸获得相同的预期回报（因为收益率的劣势恰好抵消了任何凸度的优势）。然而，如果市场为正凸度的保险特性给出高定价，则凸度大的头寸将获得较低的预期回报。我们对久期中性的杠铃-子弹组合的实证表现分析显示，从长远来看，杠铃组合的表现倾向于略微落后于子弹组合。

原文：Modern term structure models make less restrictive assumptions than the traditional theories above. Yet, many popular one-factor models assume that bonds with the same duration earn the same expected return. Such an assumption implies that duration-neutral positions with more or less convexity earn the same expected return (because a yield disadvantage exactly offsets any convexity advantage). However, if the market values very highly the insurance characteristics of positively convex positions, more convex positions may earn lower expected returns. Our analysis of the empirical performance of duration-neutral barbell-bullet trades will show that, in the long run, barbells tend to marginally underperform bullets.

回想一下，如果完全预期假说成立，所有债券都有相同的近期预期回报。特别地，向上倾斜的收益率曲线反映了加息和资本损失的预期，并且凸度被定价，使得收益率劣势恰好抵消凸度优势。在这样的世界中，收益率不反映价值，没有有利的交易，除非投资者具有真正优秀的预测能力，否则积极的管理无法增加价值。幸运的是，现实世界不太像这个理论世界。经验证据（在本系列报告的第2-4部分）表明，债券的预期回报是不同的。主要原因可能是大多数投资者表现出风险厌恶，并偏好其他资产特征。此外，投资者的行为可能不总是完全理性的。因此，收益率反映价值，并且某些相对价值交易有利可图。

原文：Recall that if the pure expectations hypothesis holds, all bond positions have the same near-term expected return. In particular, an upward-sloping yield curve reflects expectations of rising rates and capital losses, and convexity is priced so that a yield disadvantage exactly offsets the convexity advantage. In such a world, yields do not reflect value, no trades have favorable odds and active management can add value only if an investor has truly superior forecasting ability. Fortunately, the real world is not quite like this theoretical world. Empirical evidence (presented in parts 2-4 of this series of reports) shows that expected returns do vary across bonds. The main reason is probably that most investors exhibit risk aversion and preferences for other asset characteristics; moreover, investor behavior may not always be fully rational. Therefore, yields reflect value and certain relative value trades have favorable odds.

上一节提供了一个用于研究收益率期限结构形状的框架。在本节中，我们描述实际应用——在收益率曲线交易中使用远期收益率的不同方式。第一种方法需要强烈的主观观点和对预测能力的信心。

原文：The previous section provided a framework for thinking about the term structure shapes. In this section, we describe practical applications —— different ways to use forward rates in yield curve trades. The first approach requires strong subjective rate views and faith in one’s forecasting ability.

远期收益率显示了盈亏平衡的未来收益率和利差的路径。这个路径为主动投资组合管理者看待收益率曲线和预测收益率路径提供了一个明确的标准。它直接显示了对交易盈利的影响。例如，如果预计收益率上升幅度将超过远期收益率，管理者应该看跌。然而，如果他预期收益率上升幅度小于远期收益率的幅度（即小于抵消正Carry对应的利差），他应该看多。如果管理者的预测是正确的，那么当前头寸将是有利可图的。相比之下，即使管理者对收益率的预测是正确的，如果他们预期债券收益率上升，但忽视远期收益率的分析，依然会发现他们的头寸亏损，因为头寸产生负的Carry。

原文：The forward rates show a path of break-even future rates and spreads. This path provides a clear yardstick for an active portfolio manager’s subjective yield curve scenarios and yield path forecasts. It incorporates directly the impact of carry on the profitability of the trade. For example, a manager should take a bearish portfolio position only if he expects rates to rise by more than what the forwards imply. However, if he expects rates to rise by less than what the forwards imply (that is, by less than what is needed to offset the positive carry), he should take a bullish portfolio position. If the manager’s forecast is correct, the position will be profitable. In contrast, managers who take bearish portfolio positions whenever they expect bond yields to rise —— ignoring the forwards —— may find that their positions lose money, because of the negative carry, even though their rate forecasts are correct.

远期收益率作为盈亏平衡收益率的一个积极方面是它不依赖于关于预期、风险溢价或凸度偏差的假设。规则很简单，如果实现远期收益率，所有头寸将获得相同的回报。如果收益率上升超过远期收益率对应的部分，空头是有利可图，多头亏损。如果收益率上升小于远期收益率对应的部分，情况则相反。类似的陈述适用于任何利差和相关头寸，例如做平曲线的头寸。

原文：One positive aspect about the role of forward rates as break-even rates is that it does not depend on assumptions regarding expectations, risk premia or convexity bias. The rules are simple. If forward rates are realized, all positions earn the same return. If yields rise by more than the forwards imply, bearish positions are profitable and bullish positions lose money. If yields rise by less than the forwards imply, the opposite is true. Similar statements hold for any yield spreads and related positions, such as curve-flattening positions.

图5显示了1995年3月31日“Salomon Brothers 国债模型”测算的美国国债票面收益率曲线和提前三个月与12个月的票面收益率曲线。如果我们认为远期收益率只反映市场对收益率的预期，这些曲线告诉我们，市场预期收益率将上升，曲线在明年趋于平缓。或者，隐含收益率曲线上升可能反映债券风险溢价，隐含收益率曲线变平可能反映凸度价值。无论如何，远期收益率曲线反映了盈亏平衡水平。

原文：Figure 5 shows the US par yield curve and the implied par curves three months forward and 12 months forward based on the Salomon Brothers Treasury Model as of March 31, 1995. If we believe that forward rates only reflect the market’s rate expectations, a comparison of these curves tells us that the market expects rates to rise and the curve to flatten over the next year. Alternatively, the implied yield rise may reflect a bond risk premium and the implied curve flattening may reflect the value of convexity. Either way, the forward yield curves reflect the break-even levels between profits and losses.

远期收益率结构中的信息可以以几种方式表示。图5客观地提供了未来某个期间的盈亏平衡曲线，投资者可以将其同自己对未来收益率曲线的主观预测进行对比。投资者会基于宏观经济预测或关于美联储财政紧缩速度的主观观点判断收益率走势，这时将固定期限的盈亏平衡收益率的未来路径（而不是整个曲线）与自己的预测进行对比显得非常有用。作为一个例子，图6显示了1995年3月底以来三月期收益率的盈亏平衡路径。为了加深理解，图表还包含过去八年三月期收益率的历史路径，以及1994年年底开始的三月期收益率盈亏平衡路径，那时国债市场的看空情绪更加强烈。

原文：The information in the forward rate structure can be expressed in several ways. Figure 5 is useful for an investor who wants to contrast his subjective view of the future yield curve with an objective break-even curve at some future horizon. Another graph may be more useful for an investor who wants to see the break-even future path of any constant-maturity rate (instead of the whole curve) and contrast it with his own forecast, which may be based on a macroeconomic forecast or on the subjective view about the speed of Fed tightening. As an example, Figure 6 shows such a break-even path of future three-month rates at the end of March 1995.16 To add perspective, the graph also contains the historical path of the three-month rate over the past eight years and the break-even path of the future three-month rates at the end of 1994 when the Treasury market sentiment was much more bearish.

另一个使用远期收益率的方法比前面的方法涉及更少的主观判断。一个简单的例子，远期收益率曲线可用于识别便宜期限。远期收益率中反常高的部分比即期收益率或到期收益率更明显，因为后者是远期收益率的平均化。

原文：The other ways to use forwards require less subjective judgment than the first one. As a simple example, the forward rate curve can be used to identify cheap maturity sectors visually. Abnormally high forward rates are more visible than high spot or par rates because the latter are averages of forward rates.

图7显示一个现实的例子，这个十年开始时票面收益率曲线非常平坦（虽然在到期收益率曲线不平的情况下，远期收益率可能同样有用）。即使票面收益率曲线几乎是水平的（票面收益率的变动幅度在25个基点内），三月期年化远期收益率的变动范围几乎是200个基点，因为远期收益率曲线放大了不同期限的价格比。高远期收益率表示六年期和12年期债券相对便宜，低远期收益率表示四年期和九年期债券的价格被高估。

原文：Figure 7 shows one real world example from the beginning of this decade when the par yield curve was extremely flat (although forwards may be equally useful when the par curve is not flat). Even though the par yield curve was almost horizontal (all par yields were within 25 basis points), the range of three-month annualized forward rates was almost 200 basis points because the forward rate curve magnifies the cheapness/richness of different maturity sectors. High forward rates identify the six-year sector and the 12-year sector as cheap, and low forward rates identify the four-year sector and the nine-year sector as expensive.17

一旦投资者识别了一个具有异常高远期收益率的期限（例如，9到12年），他可以通过做多在期末（12年）到期的债券和做空在期初（9年）到期的债券发掘被低估的价值。如果市值相等的上述债券被交易，那么这种组合的风险敞口是收益率走高和曲线变陡。可以构建更复杂的交易策略（例如，通过以适当的权重做空9年期和15年期债券同时做多12年期债券），以保持水平和斜率中性。远期收益率曲线中的波动和扭曲反映了部分期限的债券被暂时低估，当远期收益率曲线变得更平坦并且被低估的债券修复时交易将获得资本回报（除非收益率走高并降低头寸回报）。在图7的示例中，这样的“修复”实际上在接下来的三个月中发生了。

原文：Once an investor has identified a sector with abnormally high forward rates (for example, between nine and 12 years), he can exploit the cheapness of this sector by buying a bond that matures at the end of the period (12 years) and by selling a bond that matures at the beginning of the period (nine years). If equal market values of these bonds are bought and sold, the position is exposed to a general increase in rates and a steepening yield curve. More elaborate trades can be constructed (for example, by selling both the nine-year and 15-year bonds against the 12-year bonds with appropriate weights) to retain level and slope neutrality. To the extent that bumps and kinks in the forward curve reflect temporary local cheapness, the trade will earn capital gains when the forward curve becomes flatter and the cheap sector richens (in addition to the higher yield and rolldown the position earns). In the example of Figure 7, such “richening” actually did happen over the next three months.

上面的例子中，远期收益率相当粗略地用于识别价值被低估的期限。使用远期收益率更正式的方法是为久期中性投资交易（如杠铃-子弹交易）构建识别价值低估的定量指标。我们首先介绍一些概念与市场方向性交易的例子。

原文：Above, forwards are used quite loosely to identify cheap maturity sectors. A more formal way to use forwards is to construct quantitative cheapness indicators for duration-neutral flattening trades, such as barbell-bullet trades. We first introduce some concepts with an example of a market-directional trade.

当收益率曲线向上倾斜时，长期债券的收益率优势相对于无风险短期债券提供了对收益率上升的缓冲。在某种意义上，当收益率曲线非常陡峭并且缓冲（正的Carry）大时，久期延长是“便宜的”。只有由于收益率上升造成的资本损失抵消了初始收益率优势，这些交易才会损失资金。此外，长期债券相对于短期债券的滚动收益率优势甚至大于其收益率优势。通过构造，一年期远期收益率等于n年期零息债券的滚动收益率（见附录C）。因此，它（远期收益率）是n年期零息债券滚动收益率优势的直接测量。（另一个与远期收益率相关的指标，即远期收益率所隐含的n-1年期即期收益率的变化，说明了收益率曲线必须变化多少以抵消这一优势，并在持有期使n年期零息债券的回报等于一年期零息债券）。

原文：When the yield curve is upward sloping, long-term bonds’ yield advantage over the riskless short-term bond provides a cushion against rising yields. In a sense, duration extensions are “cheap” when the yield curve is very steep and the cushion (positive carry) is large. These trades only lose money if capital losses caused by rising rates offset the initial yield advantage. Moreover, the longer-term bonds’ rolling yield advantages 18 over the short-term bond are even larger than their yield advantages. The one-year forward rate is, by construction, equal to the n-year zero’s rolling yield (see Appendix C). Thus, it is a direct measure of the n-year zero’s rolling yield advantage. (Another forward-related measure, the change in the n-1 year spot rate implied by the forwards, tells how much the yield curve has to shift to offset this advantage and to equate the holding-period returns of the n-year zero and the one-year zero.)

因为一年期远期收益率度量零息债券的近期预期回报，所以它们可被视为便宜期限的指标。使用这种指标不需要任何关于收益率的主观观点。相反，它需要一个由历史支持的信念，不变的收益率曲线是一个良好的基准。如果这是真的，当远期收益率曲线向上倾斜（向下倾斜）时，长期债券具有比短期债券更高的（更低的）近期预期回报。从长期来看，基于曲线形状动态调整投资组合久期的策略应该比恒定久期策略获得更高的平均回报。

原文：Because one-period forward rates measure zeros’ near-term expected returns, they can be viewed as indicators of cheap maturity sectors. The use of such cheapness indicators does not require any subjective interest rate view. Instead, it requires a belief, motivated by history, that an unchanged yield curve is a good base case scenario.19 If this is true, long-term bonds have higher (lower) near-term expected returns than short-term bonds when the forward rate curve is upward sloping (downward sloping). In the long run, a strategy that adjusts the portfolio duration dynamically based on the curve shape should earn a higher average return than constant-duration strategies.20

类似的分析适用于做平曲线交易。回想一下，当收益率曲线是久期的上凸函数时，任何久期中性的做平交易获得负Carry。收益率曲线的上凸程度（曲率）越高，表示做平交易越没有吸引力（较大的负Carry），远期收益率更加强烈的“隐含曲线的平坦化”（以抵消负Carry）。因此，对于收益率曲线交易，远期收益率隐含的利差变化是衡量曲线不同部分被低估程度的指标。如果隐含的变化处于历史高位，交易是昂贵的，反之亦然。

原文：Similar analysis holds for curve-flattening trades. Recall that when the yield curve is concave as a function of duration, any duration-neutral flattening trade earns a negative carry. Higher concavity (curvature) in the yield curve indicates less attractive terms for a flattening trade (larger negative carry) and more “implied flattening” by the forwards (which is needed to offset the negative carry). Therefore, the amount of spread change implied by the forwards is a useful cheapness indicator for yield curve trades at different parts of the curve. If the implied change is historically wide, the trade is expensive, and vice versa.

图8给出了最近情况的示例，其中发现做平交易非常昂贵。1994年12月底，国债收益率曲线在三个月期至两年期的区间内非常陡（扩大了200个基点），两年期至30年期区间内却相当稳定（扩大了20个基点）。高曲率表明强烈的做平预期——远期收益率意味着三月份两到三十年期利差的倒挂——或高波动率的预期（高凸度价值）。

原文：Figure 8 shows an example of a recent situation in which the flattening trades were extremely expensive. At the end of December 1994, the three-month to two-year sector of the Treasury curve was very steep (a spread of 200 basis points) and the two- to 30-year sector was quite flat (a spread of 20 basis points). The high curvature indicated strong flattening expectations —— forwards implied an inversion of the two- to 30-year spread by March —— or high expected volatility (high value of convexity).

当曲线变平的程度超过远期收益率隐含的量，或波动率突然增加时做多杠铃组合（30年期债券和三月期国库券），同时做空久期匹配的两年期子弹组合是有利可图的。纯粹基于收益率观察，两年期子弹组合（做陡）在横向比较（债券）和纵向历史比较（随着时间的推移）看来都被低估。有了后见之明，我们知道低估指标给出了正确的信号。两年期子弹组合在下个季度表现优于各种久期匹配的杠铃组合，因为它除了具有高的初始收益率和下滑回报优势之外，还获得了巨大的资本回报。截至3月底，曲线的前端已平坦化了108个基点，长端已经陡峭化了45个基点。图8通过绘制12月30日和3月31日的国债收益率曲线（作为久期的函数）显示了曲率的下降。在后面的报告中，我们将展示如何使用远期收益率分析来评估交易机会。

原文：The barbell (of the 30-year bond and three-month bill) over the duration-matched two-year bullet would become profitable only if the curve flattened even more than the forwards implied or if a sudden increase in volatility occurred. Purely on yield grounds, the two-year bullet (a steepening position) appeared cheap in an absolute comparison (across bonds) and in a historical comparison (over time). With the benefit of hindsight, we know that the cheapness indicator gave a correct signal. The two-year bullet outperformed various duration-matched barbell positions substantially over the next quarter as it earned large capital gains in addition to its high initial yield and rolldown advantage. By the end of March, the front end of the curve had flattened by 108 basis points and the long end had steepened by 45 basis points. Figure 8 illustrates the decline in curvature by plotting the Treasury on-the-run yield curves (as a function of duration) on December 30 and on March 31. In later reports, we will show how to use forward rate analysis to evaluate opportunities like this.

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