收益率曲線形態的動力學：實證證據、經濟學解釋和理論基礎

著：Antti Ilmanen, Raymond Iwanowski 譯：徐瑞龍

The Dynamics of the Shape of the Yield Curve: Empirical Evidence, Economic Interpretations and Theoretical Foundations

收益率曲線形態的動力學：實證證據、經濟學解釋和理論基礎

INTRODUCTION

引言

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.

咱們如何解釋給定日期收益率曲線的形狀（陡峭程度和曲率）？收益率曲線隨着時間的推移如何演變？在本報告中，咱們研究了兩個關於收益率曲線行爲的通常問題。咱們早先的報告顯示，市場的收益率預期、債券風險溢價和凸度誤差決定了收益率曲線的形狀。如今咱們討論與這三個決定因素在影響曲線陡峭程度和曲率中的相對做用有關的各類經濟學假說和經驗證據。咱們還討論了描述收益率曲線隨時間變化如何演化的期限結構模型，並總結了相關的經驗證據。

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.

曲線陡峭程度或斜率的關鍵決定因素是市場的收益率預期和債券風險溢價。徹底預期假說假設陡峭程度的全部變化反映了市場的收益率變化預期，而風險溢價假說假設陡峭程度的變化僅反映債券風險溢價的變化。實際上，收益率預期和風險溢價共同影響曲線斜率。歷史證據代表，高於平均水平的債券回報，以及非上漲的長期收益率，更可能在異常陡峭的收益率曲線以後出現。這種證據與徹底預期假說不一致，可能反映了時變的債券風險溢價。或者，證據可能表明非理性的投資者行爲，以及長期收益率對通貨膨脹或貨幣政策新聞的反應遲鈍。

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 bearish environment; and the insurance characteristic of a positively convex position.

早期研究中收益率曲線曲率的決定因素較少受到關注。彷佛曲率主要隨着市場的曲線形變預期而變化。曲線變平的預期使得收益率曲線更加上凸（隆起），而且曲線變陡的預期使得它較少上凸或甚至下凸（向下隆起）。這看起來彷佛不可能，然而，平均看來上凸的收益率曲線由系統的曲線變平的預期產生。更有可能的是，它反映了凸度誤差和槓鈴、子彈組合之間明顯的回報差別。若是平均來看凸度誤差是收益率曲線形狀呈上凸的惟一緣由，則能夠預期槓鈴組合的凸度優點剛好抵消子彈組合的收益率優點，在這種狀況下，久期匹配的槓鈴組合和子彈組合將具備相同的預期回報。歷史證據代表：從長遠來看，子彈組合比久期匹配的槓鈴組合得到的回報略高。也就是說，風險溢價曲線關於久期看起來是上凸的而不是線性的。咱們討論了合理的解釋，即對於投資者總的來講，能夠接受槓鈴組合的預期回報低於子彈組合：槓鈴組合有較低的回報波動率（相同久期狀況下）;在熊市環境中，作平頭寸跑贏市場的趨勢和正凸度頭寸的保險特徵。

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).

關於第二個問題，咱們描述了與評估各類期限結構模型相關的收益率曲線行爲的一些經驗特徵。這些模型關於短時間收益率（均值迴歸程度）的預期路徑、風險溢價的做用、非預期收益率部分的行爲（收益率的波動率是否隨時間、期限或收益率水平而變化）、影響收益率的因子數量的假設各有所不一樣。例如，收益率曲線平行偏移的簡單模型與沒有均值迴歸，債券風險溢價不隨時間變化的模型一致。對於不一樣債券，曲線平行偏移的假設意味着收益率的波動率期限結構是平的，並且收益率變更是徹底相關的（所以由一個因素驅動）。

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.

經驗證據代表，短時間收益率表現出至關緩慢的均值迴歸，風險溢價隨時間而變化，收益率波動率隨時間變化（與回報水平部分相關），基點收益率波動率的期限結構一般是倒掛或隆起的，並且這個收益率變化並不徹底相關，可是兩個或三個因素能夠解釋收益率曲線95%-99%的波動。

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.

在附錄A中，咱們總結了關於期限結構模型的一系列文獻，並將其與本系列中描述的框架相關聯。事實證實，許多流行的期限結構模型容許將收益率分解爲收益率預期部分、風險溢價部分和凸度部分。然而，期限結構模型在對債券關係的分析中更加一致，由於它們具體說明了少數系統因素如何影響整個收益率曲線。相比之下，咱們的方法能夠分析每一個債券的預期回報、收益率和收益率波動率。在附錄B中，咱們討論了多因子期限結構模型和現代資產訂價模型中風險溢價的理論決定因素。

HOW SHOULD WE INTERPRET THE YIELD CURVE STEEPNESS?

如何解釋收益率曲線的陡峭程度

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.

收益率曲線的陡峭程度主要反映了市場的收益率預期和債券風險溢價，由於第三個決定因素——凸度誤差，只有在曲線長期端纔是重要的。特別陡峭的收益率曲線可能反映了收益率上漲的廣泛預期，異常高的債券風險溢價或二者的某種組合。相反，倒掛的收益率曲線可能反映了對收益率降低的預期，負的債風險溢價或二者的某種組合。

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.

咱們能夠將關於曲線形狀的結論類比到關於遠期收益率的結論。當收益率曲線向上傾斜時，長期債券比無風險短時間債券具備收益率優點，而遠期收益率隱含收益率上漲。隱含的遠期收益率曲線顯示將來收益率的盈虧平衡水平，經過資產損失抵消長期債券的收益率優點，這將使全部債券得到相同的持有期回報。

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).

因爲預期不可觀察，咱們不能肯定收益率預期和風險溢價的相對做用。檢查兩個極端假設多是有用的，即遠期收益率僅反映市場的收益率預期或僅反映債券風險溢價。若是徹底預期假說成立，則遠期收益率反映市場的收益率預期，隱含的收益率曲線變化極可能會實現（即收益率上漲趨向於跟隨向上傾斜的曲線，收益率降低每每會致使倒掛的曲線）。相比之下，若是風險溢價假說成立，則隱含收益率曲線變化不可能實現，平均來看，高收益率債券得到滾動收益率優點（即高的債券超額回報傾向於跟隨向上傾斜曲線和低的債券超額回報傾向於致使倒掛的曲線）。

Empirical Evidence

實證證據

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.¹ 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.

爲了評估上述假說，咱們將隱含的遠期收益率變化（與遠期收益率曲線的陡峭程度成比例）與隨後收益率變化的平均實現和債券超額回報進行比較。在圖1中，咱們顯示了（i）平均即期收益率曲線形狀；（ii）三個月內遠期收益率隱含的不一樣期限即期收益率變更的平均值；（iii）隨後三個月內實現的收益率變更的平均；（iv）（ii）和（iii）之間的差或遠期收益率的平均「預測偏差」；以及（v）三個月內隱含收益率變更與實現收益率變更之間的相關係數估計。咱們在1968年1月至1995年12月之間使用重疊的月度數據，而且特地選擇一個長的中性區間，其中開始和結束的收益率曲線很是類似。

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.

圖1顯示，平均來講，遠期收益率隱含着收益率上漲，特別是在短時間端內，這僅僅是由於收益率曲線趨向於向上傾斜。然而，平均而言抵消長期債券收益率優點的收益率變更並無實現，這致使正的預測偏差。咱們未發表的分析代表，這個結論在比三個月更長的期間和不一樣子樣本（包括平坦和陡峭的收益率曲線環境）上保持成立。遠期收益率傾向於隱含着太高的收益率上漲幅度，這一事實多是因爲正的債券風險溢價的影響。

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 are quite weak; they are not statistically significant.²

圖1的最後一行顯示，隱含的遠期收益率變化（或遠期收益率曲線的陡峭程度）與後續實現的收益率變化的相關性估計爲負。這些估計代表，若是有的話，收益率每每會向遠期收益率隱含的相反方向變更。直觀上來講，平均來看長期收益率的小幅下滑跟隨着向上傾斜的曲線，從而增長了長期債券的收益率優點（而不是抵消）。相反，平均而言小幅度的收益率增加跟隨倒掛的曲線。1980年代和90年代的大牛市發生在收益率曲線向上傾斜的時期，而1970年代的大熊市在曲線倒掛時發生。然而，咱們強調，圖1中的負相關性至關弱，沒有統計學意義。

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.³

許多市場參與者認爲，債券風險溢價隨着時間的推移不發生變化，而且曲線陡峭程度的變化反映了市場收益率預期的變化。然而，圖1和許多早期研究中的實證證據與這種傳統智慧相矛盾。歷史上，陡峭的收益率曲線與隨後的債券超額回報更加相關，而不是債券收益率的增加。

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 subsequent rate changes; 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.

人們可能會認爲，圖1中的歷史證據再也不可信。也許今天的投資者能更好地預測收益率變更，部分緣由是他們能夠經過易於交易的工具（如歐洲存款期貨）更有效地表達本身的觀點。一些軼事證據支持這一觀點：與早期的收益率曲線倒掛不一樣，最近的倒掛（1989年和1995年）發生以後很快出現收益率降低。若是市場參與者實際上正在變成更好的預測者，則子時段分析應該代表隱含的遠期收益率變化已經成爲後續收益率變化更好的預測因子。也就是說，最近樣本中隱含和實現的收益率變化之間的滾動相關性應該比以前高。在圖2中，咱們繪製了這種滾動相關性，代表在過去十年中估計的相關性有所增長。

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.

在圖3中，咱們比較了歐元美圓期貨和國庫券在1987-95年期間的預測能力。歐洲存款期貨市場的平均預測偏差小於國債市場，反映了歐洲存款的即期收益率曲線更平坦的形狀（也多是最短時間的國庫券系統的「高估」預測值）。相比之下，隱含和實現的收益率變化之間的相關性代表，國債遠期預測將來收益率變化略好於歐洲存款期貨。與圖1（長樣本期）的相關性的比較代表，前端的國債遠期的預測能力隨着時間的推移變得更好。對於三月期收益率，這種相關性從-0.04上升到0.45，而對三年期收益率，這種相關性從-0.13上升到0.01。所以，與圖1中的數據相比，最近的證據與徹底預期假說更爲一致，可是這些關係很是弱，判斷隱藏的相關性是否真的發生了變化爲時尚早。不管如何，即便最近的相關性也代表長於一年的債券每每會得到滾動收益率。

Figure 3 Evaluating the Implied Eurodeposit and Treasury Forward Yield Curve’s Ability to PredictActual Rate Changes, 1987-95

Interpretations

解釋

The empirical evidence in Figure 1 is clearly inconsistent with the pure expectations hypothesis.⁴ 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 are negatively related; 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).

圖1中的實證證據顯然與徹底預期假說不一致。一個可能的解釋是，曲線陡峭程度主要反映了時變的風險溢價，這種影響足夠抵消曲線陡峭程度和收益率預期之間本來的正相關性。也就是說，若是市場須要高風險溢價，目前的長期收益率將會變得更高，曲線比單純的收益率預期隱含的更爲陡峭，長期債券的收益率最初必須上漲得很高，才能經過其高收益率及其預期收益率降低所帶來的資本回報提供所要求的債券回報。在這種狀況下，收益率預期和風險溢價是負相關的；陡峭的曲線預測高風險溢價和長期收益率降低。這能夠解釋1994年春季美國收益率曲線前端的陡峭（可是在政策收緊緻使收益率曲線平坦化的早期狀況下，並非這樣）。

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."

長期平均債券風險溢價是正的（參見本系列的第3部分和本報告中的圖11），但可預測性證據代表債券風險溢價是時變的，而不是恆定的。爲何債券風險溢價隨時間而變化？通常來講，資產的風險溢價反映了風險的大小和風險的市場價格（詳見附錄B）。兩個決定因素隨着時間的推移可能會波動，並致使可預測性。它們可能隨着收益率水平（依賴收益率水平的波動率），或市場方向（不對稱的波動率或風險厭惡），或經濟情況而變化。例如，債券回報的週期性模式可能反映了風險規避水平中的財富依賴性變化，即「恐懼和貪婪的循環」。

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.

圖4顯示了債券回報和收益率曲線陡峭程度典型的商業週期行爲：債券回報高，收益率曲線在波谷附近陡峭；債券回報低，收益率曲線在峯值附近平坦或倒掛。這些反週期模式可能反映了貨幣政策對經濟通貨膨脹的動態反應，以及時變的風險溢價（高風險厭惡和「壞時期」要求的風險溢價，反之亦然）。圖4的構造說明，若是債券傾向於得到其滾動收益率，則兩條線徹底契合。然而，該圖表顯示，債券每每會從波谷附近的收益率降低中得到額外的資本回報（超出滾動收益率），以及波峯附近的收益率上漲中產生資本損失。所以，實現的債券回報與收益率曲線的陡峭程度相關，而且還與經濟活動水平有關。

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.⁵

這些實證結果啓發了以下觀點，時變的債券風險溢價隨着收益率曲線的陡峭程度和其餘一些變量而變化。在本系列的第4部分中，咱們顯示，收益率曲線陡峭程度和實際債券收益率，加上近期股票和債券市場表現的度量，可以預測的月度債券超額回報高達10%。也就是說，債券回報是部分可預測的。對於季度或年度頻率的數據，可預測的部分甚至更大。

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.

若是市場參與者理性，債券回報可預測性應反映債券風險溢價的時變性。當債券風險溢價異常高時（因爲債券風險過大，或由於投資者過於規避風險）能夠預測債券回報也會高。若是長期債券的供應量增長使收益率曲線更加陡峭，並增長所要求的債券回報，債券風險溢價也可能會很高。另外一種解釋是系統性的預測錯誤致使可預測性。若是遠期收益率真的反映了市場的收益率預期（沒有風險溢價），這些預期是不合理的。

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).⁶

當收益率曲線向上傾斜時，它們每每過高，當曲線倒掛時，它們過低。市場彷佛重複了昂貴的錯誤，只能經過不試圖預測收益率變更來避免。這種不合理的行爲與市場有效性不一致。什麼樣的預期錯誤能夠解釋收益率曲線形狀和隨後的債券回報之間觀察到的模式？一個解釋是市場對通貨膨脹消息或貨幣政策行動速度的預期延遲反應。例如，若是利好的通貨膨脹消息下降了目前的短時間收益率，但對將來收益率的預期反應遲緩，收益率曲線變得向上傾斜，隨後債券回報變高（好消息的影響充分反映在收益率預期和長期收益率）。

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.⁷ 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.

因爲預期不可觀察，咱們永遠不知道回報可預測性在多大程度上反映了時變的債券風險溢價和系統預測偏差。學術研究人員試圖開發模型，將可預測性解釋爲所要求回報的理性變化。然而，收益率波動率和其餘明顯的風險度量彷佛幾乎沒有預測將來債券回報的能力。相比之下，預期回報中觀察到的反週期模式代表了風險規避水平的理性變化，儘管它們也能夠反映市場情緒的非理性變化。使用調查數據表明市場預期的研究得出結論，風險溢價和非理性預期有助於回報可預測性。

Investment Implications

投資實踐

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.⁸ 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.

若是債券的預期回報隨時間而變化，則歷史平均回報包含關於將來回報的信息，與當前經濟環境的指標（如當前收益率曲線中的信息）相比較少。原則上，遠期收益率結構中的信息是固定收益投資者研究的核心問題之一。若是遠期收益率（通過凸度誤差調整）僅反映市場的收益率預期，若是這些預期是無偏見的（平均來看將會實現），則全部政府債券策略將具備相同的短時間預期回報。追求收益率的活動（如收斂交易和相對價值交易）將是浪費時間和交易成本的。以前的實證證據代表，狀況並不是如此：債券回報是部分可預測的，長期來看追求收益率的策略是有利可圖的。然而，因爲債券回報的可預測部分小並且不肯定，所以能夠將其餘預測變量與收益率組合使用或進行分散化投資。

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-run average returns; 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.

在實踐中，關鍵問題可能不在於遠期收益率是否反映收益率預期或風險溢價，而是實際回報可預測性是否存在，誰應該利用它。若是遠期收益率（通過凸度誤差調整）反映了無偏的收益率預期，則不存在可預測性。若是可預測性存在而且是由系統性錯誤的預期引發的，每一個人均可以利用它。若是存在可預測性，是由債券風險溢價的合理變更引發的，只有一些投資者能利用機會增長長期平均回報；許多其餘人會在「壞時期」中找到更高的預期回報，而不是爲更大的風險或更高的風險規避水平提供公平的補償。只有風險中性的投資者和不典型的投資者，他們的風險感知和風險承受能力與市場上的人不一樣步，他們會想要利用任何利潤機會，而這些投資者不會關心理性的風險溢價或市場的系統性預測錯誤是否會帶來這些機會。

HOW SHOULD WE INTERPRET THE YIELD CURVE CURVATURE?

如何解釋收益率曲線的曲率

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.

市場曲線形變預期、波動率預期和預期回報結構決定了收益率曲線的曲率。收益率曲線平坦化的預期意味着久期中性的多槓鈴-空子彈組合的預期利潤，傾向於使收益率曲線上凸（所以，這些頭寸的收益率劣勢抵消了其曲線平坦化帶來的預期利潤）。高波動率的預期增長了槓鈴-子彈組合的凸度價值和預期利潤，再次致使了收益率曲線上凸的形狀。最後，相對於短時間和長期債券（槓鈴組合），市場對中期債券（子彈組合）要求的高回報使收益率曲線更上凸。相反，對於收益率曲線變陡或低波動率的預期，以及對子彈組合要求的低迴報，甚至可使收益率曲線下凸。

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?

在本節中，咱們分析收益率曲線的曲率，並重點關注兩個關鍵問題：（1）三個決定因素在改變曲率中的重要性分別是多少？和（2）爲何長期平均來看收益率曲線的形狀是上凸的？

Empirical Evidence

實證證據

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.

一些較早的研究代表，收益率曲線的曲率與市場的波動率預期密切相關，推測是因爲凸度誤差。然而，咱們的實證分析代表，曲率更傾向於隨着市場的曲線形變預期而不是波動率預期變化。收益率曲線的曲率大體緊隨曲線的陡峭程度變化，可能反映了均值迴歸的收益率預期。

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).

圖5繪製了當近期歷史中收益率曲線最陡峭、最倒掛以及最平坦的國債即期收益率曲線。該圖代表，歷史上低的短時間收益率與陡峭的收益率曲線和高曲率（上凸）相關聯，而歷史上高的短時間收益率與倒掛的收益率曲線和負曲率（下凸）相關聯。

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.

圖6中收益率水平、曲線陡峭程度和曲率月度變化的相關矩陣證明了這些關係。陡峭程度與短時間收益率水平呈負相關（但與長期收益率水平幾乎無關），反映出陡峭程度與牛陡和熊平而不是牛平和熊鬥之間更高的相關性。然而，咱們專一於陡峭程度變化和曲率變化之間的高相關性（0.79）。這種關係有一個很好的經濟邏輯。咱們的曲率能夠看做是作陡曲線頭寸的收益率Carry，即久期加權的子彈-槓鈴組合（作多三年期零息債券和作空等量的三月期零息債券和5.75年期零息債券）。若是市場參與者具備均值迴歸的收益率預期，那麼長期來看，他們預期收益率曲線將恢復到必定的平均形狀（略向上傾斜）。而後，很是陡峭的曲線與後續曲線平坦化的預期和作陡頭寸的資本損失相關。鑑於預期的資本損失，這些頭寸須要提供初始的收益率補償，這致使了上凸（隆起）的收益率曲線形狀。相反，異常平坦或倒掛的收益率曲線與市場對隨後曲線陡峭的預期和作陡頭寸的資本回報有關。鑑於預期的資本回報，這些頭寸能夠提供初始收益率損失，這會產生一個下凸（向下隆起）的收益率曲線。

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 (1979-81, 1989-90 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.

圖7示出了咱們的曲線陡峭程度和曲率之間的緊密聯繫。上述均值迴歸的收益率預期是這種模式的一個可能解釋。陡峭收益率曲線的時期（1980年代中期和90年代初）與高曲率相關，所以，對於作陡頭寸而言，大量的收益率補償可能抵消了隨後收益率曲線變平的預期損失。相比之下，平坦或倒掛曲線的時期（1979-81,1989-90和1995）與低曲率甚至下凸相關。所以，槓鈴組合相對於久期匹配的子彈組合有收益率和凸度優點，當預期收益率曲線朝向其正常形狀而變陡峭時，能夠抵消其預期的損失。

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).

曲線陡峭程度均值迴歸的預期影響了大部分收益率曲線的曲率。此外，曲線前端的曲率有時反映了市場對近期貨幣政策行動及其對曲線陡峭程度影響的強烈觀點。歷史上，美聯儲等央行已經試圖經過逐步調整收益率來平滑收益率行爲。這種收益率平滑政策使中央銀行的行爲部分可預測，並致使短時間收益率行爲正的自相關性。所以，若是中央銀行最近開始放鬆（收緊）貨幣政策，能夠合理地預期貨幣寬鬆政策（緊縮）將繼續，曲線將變陡峭（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 volatility estimates; 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).

在早期的文獻中，收益率曲線曲率主要與波動率水平有關。Litterman，Scheinkman 和 Weiss 指出（《Volatility and the Yield Curve》，Journal of Fixed Income，1991），較高的波動率應使收益率曲線更加上凸（因爲凸度效應），而且收益率曲線曲率和國債期貨期權的隱含波動率之間存在着密切的關係。然而，圖8顯示，曲率和波動率之間的關係僅存在於研究的樣本期間（1984-88）。有趣的是，1980年代中期沒有發生經濟衰退，收益率曲線變化同步性至關高，變平或變陡的預期可能至關薄弱。這種關係在1984-88年度以前和以後不成立，尤爲是在近期的經濟衰退時期，這時美聯儲活躍，市場理性的預期曲線形變。例如，1981年的收益率波動率很是大，但收益率曲線是下凸的（向下隆起），見圖5和圖13。彷佛市場對將來曲線形變的預期是收益率曲線曲率的重要決定因素，而不是其波動率預期（凸度誤差）。咱們測算的曲率與曲線陡峭程度的相關性約爲0.8，而與期權隱含波動率的相關性約爲0.1。所以，基於收益率曲線曲率的隱含波動率估計與基於期權價格的隱含波動率並不密切相關。使用收益率曲線形狀導出隱含波動率可致使負的波動率估計，這種不合理的結果發生在簡單的模型中，當曲線變陡峭的預期使得收益率曲線向下隆起時（見本系列的第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.⁹ 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.

如今咱們轉到第二個問題：「爲何收益率曲線的長期平均形狀是上凸的？」圖9顯示，在28年的樣本週期內，平均到期和即期收益率曲線均呈上凸。回想一下，上凸形意味着，平均來看遠期收益率隱含着收益率曲線變平（這將抵消中期債券相對於久期匹配的槓鈴組合的初始收益率優點）。圖10顯示，平均而言，隱含的平坦化並無被充分實現。絕不奇怪，變平和變陡傾向於隨着時間的推移而逐漸消失，但上凸即期收益率曲線的形狀已經至關持久。事實上，隱含和實現的曲線平坦化之間存在顯着的正相關，但圖10中的平均預測偏差揭示了過於隱含平坦化的誤差。當咱們將樣本依據陡峭與平坦或收益率上升與降低的情形分解成子樣本時，這一結論是成立的。

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.

圖10顯示，平均來講，曲線平坦化引發的資本回報並未抵消槓鈴組合的收益率劣勢（相對於久期匹配的子彈組合）。更合理的可能性是，槓鈴組合的凸度優點抵消了其收益率劣勢。咱們能夠經過檢查凸度對實際回報的影響來評估這種可能性。經驗證據代表，凸度優點不足以抵消收益率劣勢（參見本系列第5部分的圖12）。或者，咱們能夠檢查歷史平均回報的形狀，由於實現的回報應該反映凸度優點。這種凸度效應固然是對典型收益率曲線形狀的部分解釋，但只有久期匹配的槓鈴組合和子彈組合具備相同的預期回報，纔是惟一的效果。一樣地，若是債券風險溢價隨久期線性增加，久期匹配的槓鈴組合和子彈組合的平均回報在長時間的中性時期應該是相同的（由於槓鈴組合的凸度優點剛好抵消了他們的收益率劣勢）。第3部分圖1的平均回報曲線形狀以及第5部分圖11中的平均槓鈴-子彈組合回報代表，子彈組合比久期匹配的槓鈴組合具備較高的長期預期回報。咱們還指出1968-95年期間合成零息債券頭寸的歷史表現：4年期零息債券的平均年化月度回報爲9.14%，而久期匹配的槓鈴組合的平均回報逐漸降低（3-5年期組合爲9.05%，2-6年期組合爲9.00%，1-7年期組合爲8.87%）。整體而言，考慮到樣本中平均來看曲線平坦的比例爲零，收益率曲線典型的上凸形態可能反映了凸度誤差和債券風險溢價曲線的上凸形態，而不是系統的曲線變平預期。

Figure 11 Average Treasury Maturity-Subsector Returns as a Function of Return Volatility

Interpretations

解釋

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.

曲線形變預期和凸度誤差對收益率曲線形狀的影響很容易理解，但債券風險溢價曲線的上凸形態更使人困惑。在本小節中，咱們探討爲何子彈組合應該比久期匹配的槓鈴組合具備微弱的預期回報優點。一個可能的答案是，久期不是有意義的風險度量。然而，咱們發現即便做爲回報波動率的函數平均回報也是上凸的，這代表須要一個多因子風險模型。咱們首先詳細討論各類基於風險的解釋，而後考慮觀察到的均值迴歸模式的替代「技術性」解釋。

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.

全部單因子期限結構模型認定預期回報隨着債券對風險因子的敏感性而線性增加。由於這些模型假設債券回報徹底相關，因此預期回報應隨回報波動率線性增長（不管風險因子如何）。然而，只有全部債券具備相同的基點收益率波動率，債券久期才與回報波動率成比例。平均回報-久期曲線的上凸形狀也許是由（i）預期回報和回報波動率之間的線性關係引發的；（ii）回報波動率與久期之間的上凸關係，反過來又反映了一個倒掛或隆起的收益率波動率期限結構（見圖15）。直覺上，實際回報波動率與久期之間的上凸關係將使槓鈴組合比久期匹配的子彈組合更具防守（看跌）性。槓鈴組合的回報波動率只是其成分回報波動率的加權平均值（給定完美的相關性）；所以，槓鈴組合的波動率將低於久期匹配的子彈組合。

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 arithmetic average returns; the geometric average return curves would be even more concave.¹⁰

圖13和14將證實大部分時間內收益率波動率的經驗性期限結構是倒掛或隆起的。所以，也許槓鈴組合和子彈組合具備相等的回報波動率（而不是相等的久期）才具備相同的預期回報。然而，事實證實，即便咱們繪製平均回報關於歷史回報波動率的變化，子彈組合的回報優點仍然存在。圖11顯示了國債各類期限投資組合的歷史平均回報（做爲回報波動率的函數）。平均回報基於1968年1月至1995年12月和1986年4月至1995年3月的兩個相對中性的時期。咱們仍然發現平均收益率曲線有一些上凸。注意，經過繪製算術平均回報，咱們以保守的方式展現上凸形狀；幾何均值收益率曲線將更加上凸。

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，單因子期限結構模型假定債券回報徹底相關。單因子資產訂價模式更爲通常化。他們假定實現的債券回報只受一個系統風險因子的影響，但也包含一個特定於債券的剩餘風險成分（可使個別債券回報不徹底相關）。因爲債券特定風險易於分散，所以只有系統風險才能在市場上獲得回報。所以，預期回報關於回報波動率中對應系統風險的部分是線性的。這種區別對於政府債券來講不是很重要，由於它們的債券特定風險很小。若是咱們僅繪製平均收益率關於系統波動率的關係，圖11中前端將略低，由於較大部分短時間國庫券的回報波動率是因資產而異。然而，平均收益率曲線的總體形狀將保持上凸。

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.

凸度誤差和收益率波動率的期限結構部分解釋了平均收益率曲線的上凸形狀，但非線性預期收益率曲線彷佛是一個額外的緣由。圖11代表，預期回報關於回報波動率有些上凸。也就是說，長期債券的所要求的回報要低於單因子模型所隱含的。較中期現金流而言，長期現金流的一些有利特性使得市場願意接受單位回報波動率上較低的預期超額回報。除了收益率水平的風險，咱們須要第二個風險因子以解釋這種模式。此外，這種模式可能會告訴咱們關於第二個因子的性質和風險溢價的可能符號。咱們接下來討論第二個因子的兩個流行選項，收益率波動率和收益率曲線陡峭程度。咱們在附錄B中進一步討論風險溢價的理論決定因素。

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.¹¹ 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.

波動率做爲第二個因子能夠解釋觀察到的模式，若是市場參與者整體上偏好保險類型或「作多波動率」的回報。即便不嵌入期權的政府債券也有一個期權特徵，由於它們的價格-收益率曲線是下凸的形狀。如本系列第5部分所述，凸度價值隨着債券的凸度和收益率波動率的感知水平而增長。若是波動率風險在預期回報中沒有「訂價」（也就是說，若是全部「delta中性」期權頭寸都得到零風險溢價），收益率劣勢應該剛好抵消較長期債券的凸度優點。然而，圖11中平均收益率曲線的上凸形狀代表，受益於較高波動率頭寸的預期回報低於受制於較高波動率影響的頭寸。雖然證據薄弱，但咱們發現負的波動率風險價格在直覺上是有吸引力的。國債市場參與者可能特別反對高波動率狀態的損失，或者他們可能更喜歡保險型（有偏）的回報，以致於他們接受較低的長期回報。所以，長期債券的低預期回報可能反映出許多投資者給予正凸度的高價值。然而，因爲短債券表現出很小的凸度，所以須要其餘因素來解釋收益率曲線前端的曲率。

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 recession hedges either; the yield curves typically have been flat or inverted at the beginning of a recession and have steepened during it (see also Figure 4).¹² Nonetheless, flattening positions typically have been profitable in a rising rate environment; thus, they have been reasonable hedges against a bear market for bonds.

做爲第二個因子的收益率曲線陡峭程度（或者短時間和長期收益率做爲兩個因素）能夠解釋觀察到的模式，只有當投資者看重時，作平曲線的頭寸纔會有利可圖。咱們不認爲曲線陡峭程度自己就是投資者擔憂的風險因子，但它可能傾向於與更根本的因素相吻合。回想一下，平均來看上凸的收益率曲線代表，自融資的作平曲線頭寸具備負的預期回報，由於它們對長期收益率（回報波動率的回報較低）比對短時間或中期收益率（回報波動率的回報較高）更敏感。理論上這個負的風險溢價能夠是合理的，若是作平交易是對「壞時期」特別好的對衝。當被問及什麼是壞時期時，學術界的答案是利潤的高邊際效用時期，而從業者的回答多是嚴重衰退期或熊市。關於這個問題的經驗證據是混合的。很明顯，長期債券在通貨緊縮的衰退期（1930年代的美國，1990年代的日本）中表現良好。然而，當可預測和實現的債券超額回報爲負數時，它們在1970年代的困境中表現不佳。自二次大戰以來，美國長期債券表現與股市表現呈正相關，儘管在1987年10月的股市崩盤期間債券成爲良好的對衝。如今轉向作平頭寸，這些衰退對衝表現並很差，收益率曲線一般在經濟衰退開始時已經平坦或倒掛，並在其期間陡峭（參見圖4）。儘管如此，作平的頭寸一般在收益率上漲的環境中是有利可圖的。所以，他們對債券市場進行了合理的對衝。

We conclude that risk factors that are related to volatility or curve steepness could perhaps explain the concave shape of the average return curve —— but these are not the only possible explanations. "Technical" or "institutional" explanations include the value of liquidity (the ten-year note and the 30-year bond have greater liquidity and lower transaction costs than the 11-29 year bonds, and the on-the-run bonds can earn additional income when they are "special" in the repo market), institutional preferences (immunizing pension funds may accept lower yield for "riskless" long-horizon assets, institutionally constrained investors may demand the ultimate safety of one-month bills at any cost, fewer natural holders exist for intermediate bonds), and the segmentation of market participants (the typical short-end holders probably tolerate return volatility less well than do the typical long-end holders, which may lead to a higher reward for duration extension at the front end).¹³

咱們得出結論，與波動率或曲線陡峭程度相關的風險因子可能解釋了平均回報曲線的上凸形狀，但這並非惟一可能的解釋。「技術性」或「制度性」解釋包括流動性的價值（10年期債券和30年期債券的流動性更大，交易成本比11-29年期債券低，當活躍債券在回購市場被「特殊」對待時，能夠賺取額外收入）、機構偏好（養老基金可能會接受較低迴報的「無風險」長期資產，制度上受限制的投資者可能要求不惜成本的保障一月期國庫券的最終安全，中期債券存在較少的天然持有人）、市場參與者的分割（典型的短時間持有者可能不如長期持有者更能忍受回報波動率，這可能會致使在曲線前端延長久期能得到更高回報）。

Investment Implications

投資應用

Bullets tend to outperform barbells in the long run, although not by much. It follows that as a long-run policy, it might be useful to bias the investment benchmarks and the core Treasury holdings toward intermediate bonds, given any duration. In the short run, the relative performance of barbells and bullets varies substantially —— and mainly with the yield curve reshaping. Investors who try to "arbitrage" between the volatility implied in the curvature of the yield curve and the yield volatility implied in option prices will find it very difficult to neutralize the inherent curve shape exposure in these trades. An interesting task for future research is to study how well barbells' and bullets' relative short-run performance can be forecast using predictors such as the yield curve curvature (yield carry), yield volatility (value of convexity) and the expected mean reversion in the yield spread.

從長遠來看，子彈組合每每跑贏槓鈴組合，雖然不算太多。所以，做爲長期策略策，在任何期限內將投資基準和持有的核心國債偏向於中期債券多是有用的。在短時間內，槓鈴組合和子彈組合的相對錶現差別很大，主要是收益率曲線形變致使。投資者試圖在收益率曲線曲率隱含的波動率與期權價格中隱含的收益率波動率之間「套利」，將很難中和這些交易中固有的曲線形狀敞口。將來研究的一個有趣的任務是研究如何使用預測因子，諸如收益率曲線曲率（收益率 Carry）、收益率波動率（凸度價值）和利差的預期均值迴歸，來預測槓鈴組合和子彈組合的相對短時間表現。

HOW DOES THE YIELD CURVE EVOLVE OVER TIME?

收益率曲線如何隨時間變化

The framework used in the series Understanding the Yield Curve is very general; it is based on identities and approximations rather than on economic assumptions. As discussed in Appendix A, many popular term structure models allow the decomposition of forward rates into a rate expectation component, a risk premium component, and a convexity bias component. However, various term structure models make different assumptions about the behavior of the yield curve over time. Specifically, the models differ in their assumptions regarding the number and identity of factors influencing interest rates, the factors' expected behavior (the degree of mean reversion in short rates and the role of a risk premium) and the factors' unexpected behavior (for example, the dependency of yield volatility on the yield level). In this section, we describe some empirical characteristics of the yield curve behavior that are relevant for evaluating the realism of various term structure models.¹⁴ In Appendix A, we survey other aspects of the term structure modeling literature. Our literature references are listed after the appendices; until then we refer to these articles by author's name.

《理解收益率曲線》系列中使用的框架很是通用，是基於肯定性和近似而不是經濟假設。如附錄A所述，許多流行的期限結構模型容許將遠期收益率分解爲收益率預期部分、風險溢價部分和凸度誤差部分。然而，各類期限結構模型對收益率曲線隨時間的行爲作出不一樣的假設。具體來講，這些模型的差別在於影響收益率的因子的數量和特性、因子的預期行爲（短時間收益率均值迴歸的程度和風險溢價的做用）以及因子的非預期行爲（例如，收益率波動率對收益率水平的依賴）。在本節中，咱們描述與評估與各類期限結構模型相關的若干收益率曲線行爲經驗特徵。在附錄A中，咱們對期限結構模型文獻的其餘方面進行了綜述。咱們的參考文獻列在附錄以後，咱們以做者的名字來引用這些文章。

The simple model of only parallel shifts in the spot curve makes extremely restrictive and unreasonable assumptions —— for example, it does not preclude negative interest rates.¹⁵ In fact, it is equivalent to the Vasicek (1977) model with no mean reversion. All one-factor models imply that rate changes are perfectly correlated across bonds. The parallel shift assumption requires, in addition, that the basis-point yield volatilities are equal across bonds. Other one-factor models may imply other (deterministic) relations between the yield changes across the curve, such as multiplicative shifts or greater volatility of short rates than of long rates. Multi-factor models are needed to explain the observed imperfect correlations across bonds —— as well as the nonlinear shape of expected bond returns as a function of return volatility that was discussed above.

在即期收益率曲線上只有平行偏移的簡單模型，構成限制極大和不合理的假設，例如，它不排除負收益率。事實上，它至關於沒有均值迴歸的 Vasicek（1977）模型。全部單因子模型意味着收益率變化在債券之間徹底相關。平行偏移假設另外要求基點收益率波動率在債券之間是相等的。其餘單因子模型可能存在着曲線上的收益率變化之間的其餘（肯定性）關係，如可乘性偏移或短時間收益率波動率大於長期收益率。須要多因子模型來解釋觀察到的債券之間的不徹底相關性，以及做爲回報波動率函數的預期債券回報的非線性形狀。

Time-Series Evidence

時間序列證據

In our brief survey of empirical evidence, we find it useful to first focus on the time-series implications of various models and then on their cross-sectional implications. We begin by examining the expected part of yield changes, or the degree of mean reversion in interest rate levels and spreads. If interest rates follow a random walk, the current interest rate is the best forecast for future rates —— that is, changes in rates are unpredictable. In this case, the correlation of (say) a monthly change in a rate with the beginning-of-month rate level or with the previous month's rate change should be zero. If interest rates do not follow a random walk, these correlations need not equal zero. In particular, if rates are mean-reverting, the slope coefficient in a regression of rate changes on rate levels should be negative. That is, falling rates should follow abnormally high rates and rising rates should succeed abnormally low rates.

在咱們對實證證據的簡短綜述中，咱們發現應該首先關注各類模型的時間序列應用，而後是橫截面上的應用。咱們首先檢查收益率變化的預期部分，或收益率水平和利差的均值迴歸程度。若是收益率服從隨機遊走，現行收益率是對將來收益率的最佳預測，即收益率變更是不可預測的。在這種狀況下，（例如）月度收益率變更與月初收益率水平或上月收益率變更的相關性應爲零。若是收益率不遵循隨機遊走，這些相關性沒必要等於零。特別是，若是收益率是均值迴歸的，則收益率變化關於收益率水平迴歸的係數應爲負。也就是說，收益率降低應該跟隨異常高的收益率水平，收益率上漲應該跟隨異常低的收益率水平。

Figure 12 shows that interest rates do not exhibit much mean reversion over short horizons. The slope coefficients of yield changes on yield levels are negative, consistent with mean reversion, but they are not quite statistically significant. Yield curve steepness measures are more mean-reverting than yield levels. Mean reversion is more apparent at the annual horizon than at the monthly horizon, consistent with the idea that mean reversion is slow. In fact, yield changes seem to exhibit some trending tendency in the short run (the autocorrelation between the monthly yield changes are positive), until a "rubber-band effect" begins to pull yields back when they get too far from the perceived long-run mean. Such a long-run mean probably reflects the market's views on sustainable real rate and inflation levels as well as a perception that a hyperinflation is unlikely and that negative nominal interest rates are ruled out (in the presence of cash currency). If we focus on the evidence from the 1990s (not shown), the main results are similar to those in Figure 12, but short rates are more predictable (more mean-reverting and more highly autocorrelated) than long rates, probably reflecting the Fed's rate-smoothing behavior.

圖12顯示，收益率在短時間內並無表現出很大的均值迴歸。收益率變化關於收益率水平的迴歸係數爲負，符合均值迴歸，但不統計顯著。收益率曲線陡峭程度比收益率水平更具均值迴歸性。均值迴歸在年度水平上比月度水平更明顯，這與均值迴歸緩慢的觀點一致。事實上，收益率變化彷佛在短時間內呈現出一些趨勢（月度收益率變更之間的自相關是正的），直到「橡皮筋效應」開始將遠離長期平均值的收益率拉回到平均水平。這樣一個長期的平均值可能反映了市場對當前持續性的實際收益率和通貨膨脹水平的見解，以及惡性通貨膨脹不大可能發生，而且不存在負的名義收益率（在現金貨幣存在的狀況下）。若是咱們專一於1990年代的證據（未顯示），主要結果與圖12類似，但短時間收益率比長期收益率更可預測（更明顯的均值迴歸和更高的自相關性），可能反映了美聯儲的收益率平滑行爲。

Figure 12 Mean Reversion and Autocorrelation of US Yield Levels and Curve Steepness, 1968-95

Moving to the unexpected part of yield changes, we analyze the behavior of (basis-point) yield volatility over time. In an influential study, Chan, Karolyi, Longstaff, and Sanders (1992) show that various specifications of common one-factor term structure models differ in two respects: the degree of mean reversion and the level-dependency of yield volatility. Empirically, they find insignificant mean reversion and significantly level-dependent volatility —— more than a one-for one relation.¹⁶ Moreover, they find that the evaluation of various one-factor models' realism depends crucially on the volatility assumption; models that best fit US data have a level-sensitivity coefficient of 1.5. According to these models, future yield volatility depends on the current rate level and nothing else: High yields predict high volatility. Another class of models —— so called GARCH models —— stipulate that future yield volatility depends on the past volatility: High recent volatility and large recent shocks (squared yield changes) predict high volatility. Brenner, Harjes and Kroner (1996) show that empirically the most successful models assume that yield volatility depends on the yield level and on past volatility. With GARCH effects, the level-sensitivity coefficient drops to approximately 0.5. Finally, all of these studies include the exceptional period 1979-82 which dominates the results (see Figure 13). In this period, yields rose to unprecedented levels —— but the increase in yield volatility was even more extraordinary. Since 1983, the US yield volatility has varied much less closely with the rate level.¹⁷

轉移到收益率變化的非預期部分，咱們分析（基點）收益率波動率隨時間的變化行爲。Chan，Karolyi，Longstaff 和 Sanders（1992）在一個有影響力的研究中代表，常見的單因子期限結構模型在兩個方面有所不一樣：均值迴歸的程度和波動率對收益率水平的依賴。經驗上，他們發現均值迴歸是微不足道的而波動率顯著依賴收益率水平。此外，他們發現，評估各類單因子模型在很大程度上取決於波動性假設，最適合美國數據的模型收益率水平敏感係數爲1.5。根據這些模型，將來收益率波動率僅僅取決於當前的收益率水平，因此，高收益率預示高波動率。另外一類模型，即所謂的 GARCH 模型，規定將來收益率波動率取決於過去的波動率：近期波動率較大和近期的大幅震盪（平均收益率變化）預示高波動率。Brenner，Harjes 和 Kroner（1996）代表，經驗上最成功的模型假設收益率波動率取決於收益率水平和過去的波動率。考慮到 GARCH 效應，收益率水平敏感度係數降至約0.5。最後，全部這些研究都包括了1979-82年這一特殊時期，並主導着結果（見圖13）。在這一時期，收益率上升到史無前例的水平，收益率波動率的增長更是巨大。自1983年以來，美國的收益率波動率與收益率水平關係不大。

Figure 13 24-Month Rolling Spot Rate Volatilities in the United States

A few words about the required bond risk premia. In all one-factor models, the bond risk premium is a product of the market price of risk, which is assumed to be constant, and the amount of risk in a bond. Risk is proportional to return volatility, roughly a product of duration and yield volatility. Thus, models that assume rate-level-dependent yield volatility imply that the bond risk premia vary directly with the yield level. Empirical evidence indicates that the bond risk premia are not constant —— but they also do not vary closely with either the yield level or yield volatility (see Figure 2 in Part 4). Instead, the market price of risk appears to vary with economic conditions, as discussed above Figure 4. One point upon which theory and empirical evidence agree is the sign of the market price of risk. Our finding that the bond risk premia increase with return volatility is consistent with a negative market price of interest rate risk. (Negative market price of risk and negative bond price sensitivity to interest rate changes together produce positive bond risk premia.) Many theoretical models, including the Cox-Ingersoll-Ross model, imply that the market price of interest rate risk is negative as long as changing interest rates covary negatively with the changing market wealth level.

關於債券風險溢價要說幾句話。在全部單因子模型中，債券風險溢價是風險市場價格（假定爲不變）和債券風險量的乘積。風險與回報波動率成正比，大體是久期和收益率波動率的乘積。所以，假設收益率水平依賴的收益率波動率模型意味着債券風險溢價與收益率水平直接相關。經驗證據代表，債券風險溢價不是恆定的，但它們也不會隨收益率水平或收益率波動率而變化（見第4部分，圖2）。相反，風險的市場價格彷佛隨着經濟情況而變化，如上圖4所述。理論和實證證據一致的一點是風險市場價格的符號。咱們發現，債券風險溢價隨回報波動率的增長而與負的收益率風險市場價格一致（負的風險市場價格和負的債券價格對收益率變更的敏感性共同產生正的債券風險溢價）。許多理論模型，包括 Cox-Ingersoll-Ross 模型，都認爲收益率風險的市場價格爲負，只要收益率隨着市場財富水平的變化而反向變化。

Cross-Sectional Evidence

橫截面證據

We first discuss the shape of the term structure of yield volatilities and its implications for bond risk measures and later describe the correlations across various parts of the yield curve. The term structure of basis-point yield volatilities in Figure 14 is steeply inverted when we use a long historical sample period. Theoretical models suggest that the inversion in the volatility structure is mainly due to mean-reverting rate expectations (see Appendix A). Intuitively, if long rates are perceived as averages of expected future short rates, temporary fluctuations in the short rates would have a lesser impact on the long rates. The observation that the term structure of volatility inverts quite slowly is consistent with expectations for very slow mean reversion. In fact, after the 1979-82 period, the term structure of volatility has been reasonably flat —— as evidenced by the ratio of short rate volatility to long rate volatility in Figure 13. The subperiod evidence in Figure 14 confirms that the term structure of volatility has recently been humped rather than inverted. The upward slope at the front end of the volatility structure may reflect the Fed's smoothing (anchoring) of very short rates while the one- to three-year rates vary more freely with the market's rate expectations and with the changing bond risk premia.

咱們首先討論收益率波動率的期限結構形狀及其對債券風險度量的影響，而且稍後描述收益率曲線的各個部分之間的相關性。當咱們使用漫長的歷史樣本週期時，圖14中基點收益率波動率的期限結構迅速倒掛。理論模型代表，波動率結構的倒掛主要是因爲均值迴歸的收益率預期（見附錄A）。直觀地說，若是長期收益率被視爲預期將來短時間收益率的平均水平，短時間收益率的暫時波動對長期收益率的影響較小。波動性期限結構至關緩慢倒掛的觀察結果與很是緩慢的均值迴歸的預期一致。實際上在1979-82年期間以後，波動率的期限結構至關平坦，這由圖13所示的短時間波動率與長期波動率的比率證實。圖14中的子樣本證據證明了波動率的期限結構最近已經不倒掛了。波動率結構前端的向上傾斜可能反映了美聯儲對很是短時間的收益率的平滑（固定），而一年到三年的收益率更爲自由地隨市場收益率預期和債券風險溢價的變化而變化。

Figure 14 Term Structure of Spot Rate Volatilities in the United States

The nonflat shape of the term structure of yield volatility has important implications on the relative riskiness of various bond positions. The traditional duration is an appropriate risk measure only if the yield volatility structure is flat. We pointed out earlier that inverted or humped yield volatility structures would make the return volatility curve a concave function of duration. Figure 15 shows examples of flat, humped and inverted yield volatility structures (upper panel) —— and the corresponding return volatility structures (lower panel). The humped volatility structure reflects empirical yield volatilities in the 1990s, while the flat and inverted volatility structures are based on the Vasicek model with mean reversion coefficients of 0.00, 0.05, and 0.10. The model's short-rate volatility is calibrated to match that of the three-month rate in the 1990s (77 basis points or 0.77%). It is clear from this figure that the traditional duration exaggerates the relative riskiness of long bonds whenever the term structure of yield volatility is inverted or humped. Moreover, the relative riskiness will be quite misleading if the assumed volatility structure is inverted (as in the long sample period in Figure 14) while the actual volatility structure is flat or humped (as in the 1990s).

收益率波動率期限結構的非平坦形狀對各類債券頭寸的相對風險具備重要意義。只有當收益率波動率結構平坦時，傳統的久期纔是適當的風險度量。咱們之前指出，倒掛或隆起的收益率波動率結構將使回報波動率曲線成爲久期的上凸函數。圖15示出了平坦、隆起和倒掛的收益率波動率結構（上圖）和相應的回報波動率結構（下圖）的實例。隆起的波動率結構反映了1990年代的經驗收益率波動率，而平坦和倒掛的波動率結構基於 Vasicek 模型，均值迴歸係數分別爲0.00，0.05和0.10。該模型的短時間波動率被校準爲與1990年代的三月期收益率（77個基點或0.77%）相一致。從這個數字能夠看出，只要收益率波動率的期限結構倒掛或隆起，傳統的久期就會誇大長期債券的相對風險。此外，相對風險將會誤導投資者，若是假設的波動率結構倒掛（如圖14中的長期樣本期間），而實際波動率結構平坦或隆起（如1990年代）。

Figure 15 Basis-Point Yield Volatilities and Return Volatilities for Various Models

Historical analysis shows that correlations of yield changes across the Treasury yield curve are not perfect but are typically very high beyond the money market sector (0.82-0.98 for the monthly changes of the two, to 30-year on-the-run bonds between 1968-95) and reasonably high even for the most distant points, the three-month bills and 30-year bonds (0.57). Thus, the evidence is not consistent with a one-factor model, but it appears that two or three systematic factors can explain 95%-99% of the fluctuations in the yield curve (see Garbade (1986), Litterman and Scheinkman (1991), Ilmanen (1992)). Based on the patterns of sensitivities to each factor across bonds of different maturities, the three most important factors are often interpreted as the level, slope and curvature factors.¹⁸

歷史分析顯示，國債收益率曲線上收益率變化的相關性不是徹底的，但一般很是高於貨幣市場（1968之1995年之間，2到30年期債券收益率的月度相關性爲0.82-0.98），即便是期限間隔最遠的兩點，三月期國庫券和三十年期債券收益率的相關性也是至關高的（0.57）。所以，證據與單因子模型不一致，但彷佛兩個或三個系統因子能夠解釋收益率曲線的95%-99%波動（參見 Garbade（1986），Litterman 和 Scheinkman（1991），Ilmanen（1992））。根據不一樣期限債券對每一個因子的敏感度模式，三個最重要的因子一般被解釋爲水平、斜率和曲率因子。

APPENDIX A. A SURVEY OF TERM STRUCTURE MODELS¹⁹

期限結構綜述

A vast literature exists on quantitative modeling of the term structure of interest rates. Because of the large number of these models and the fact that the use of stochastic calculus is needed to derive these models, many investors view them as inaccessible and not useful for their day-to-day portfolio management. However, investors use these models extensively in the pricing and hedging of fixed-income derivative instruments and, implicitly, when they consider such measures as option-adjusted spreads or the delivery option in Treasury bond futures. Furthermore, these models can provide useful insights into the relationships between the expected returns of bonds of different maturities and their time-series properties. It is important that investors understand the assumptions and implications of these models to choose the appropriate model for the particular objective at hand (such as valuation, hedging or forecasting) and that the features of the chosen model are consistent with the investor's beliefs about the market. Although the models are developed through the use of stochastic calculus, it is not necessary that the investor have a complete understanding of these techniques to derive some insight from the models. One goal of this section is to make these models accessible to the fixed-income investor by relating them to risk concepts with which he is familiar, such as duration, convexity and volatility.

關於收益率期限結構的量化模型已經出現了大量文獻。因爲這些模型數量衆多，並且須要使用隨機分析來推導這些模型，許多投資者認爲它們是沒法理解的，對於他們的平常投資組合管理毫無用處。然而，投資者在固定收益衍生品的訂價和對衝中普遍使用這些模型，而且在考慮期權調整利差或國債期貨交割期權等度量是也隱含地使用這些模型。此外，這些模型能夠爲不一樣期限債券的預期回報與其時間序列特徵之間的關係提供有用的看法。投資者必須瞭解這些模型的假設和影響，覺得特定用途（例如估值、對衝或預測）選擇適當的模型，而且所選模型的特徵要與投資者對市場的信念相一致。雖然這些模型經過使用隨機分析構造出來，但投資者並不須要先對這些技術有徹底的瞭解再從模型中得出一些看法。本節的一個目標是使固定收益投資者能夠將這些模型與他熟悉的風險概念相關聯，例如久期、凸度和波動率。

Equation (1) in Part 5 of this series gives the expression of the percentage change in a bond's price (\(\Delta P/P\)) as a function of changes in its own yield (\(\Delta y\)).

本系列第5部分中的方程式（1）給出了債券價格變更百分比（\(\Delta P/P\)）做爲自身收益率變化（\(\Delta y\)）函數的表達式。

\[ 100 * \Delta P/P \approx -Duration * \Delta y + 0.5 * Convexity * (\Delta y)^2. \tag{1} \]

This expression, which is derived from the Taylor series expansion of the price-yield formula, is a perfectly valid linkage of changes in a bond's own yield to returns and expected returns through traditional bond risk measures such as duration and convexity.

這一表達式從價格-到期收益率公式的泰勒級數展開獲得，將債券自身到期收益率變化與回報和預期回報經過傳統的債券風險度量（如久期和凸度）完美的聯繫起來。

One problem with this approach is that every bond's return is expressed as a function of its own yield. This expression says nothing about the relationship between the return of a particular bond and the returns of other bonds. Therefore, it may have limited usefulness for hedging and relative valuation purposes. One must impose some simplifying assumptions to make these equations valid for cross-sectional comparisons. In particular, more specific assumptions are needed for the valuation of derivative instruments and uncertain cash flows. Of course, the marginal value of more sophisticated term structure models depends on the empirical accuracy of their specification and calibration.

這種方法的一個問題是每一個債券的回報都表示爲其自身到期收益率的函數。這種表示不涉及特定債券回報與其餘債券回報之間的關係。所以，對於對衝和衡量相對價值來講用途可能有限。必須強加一些簡化的假設，使這些方程對於橫向比較是有效的。特別是，對衍生品的估值和不肯定的現金流須要更具體的假設。固然，更復雜的期限結構模型的邊際價值取決於其具體形式和校準的經驗準確性。

Factor Model Approach

因子模型方法

Term structure models typically start with a simple assumption that the prices of all bonds can be expressed as a function of time and a small number of factors. For ease of explanation, the analysis is often restricted to default-free bonds and their derivatives. We first discuss one-factor models which assume that one factor (\(F_t\))²⁰ drives the changes in all bond prices and the dynamics of the factor is given by the following stochastic differential equation:

期限結構模型一般從簡單的假設開始，即全部債券的價格能夠表達爲時間和少數因子的函數。爲了便於解釋，分析每每限於無違約債券及其衍生品。咱們首先討論單因子模型，假設一個因子（\(F_t\)）驅動全部債券價格的變化，因子的動態變化由如下隨機微分方程給出：

\[ \frac{dF}{F} = m(F,t)dt + s(F,t)dz \tag{2} \]

where F can be any stochastic factor such as the yield on a particular bond or the real growth rate of an economy, dt is the passage of a small (instantaneous) time interval, and dz is Brownian motion (a random process that is normally distributed with a mean of 0 and a standard deviation of \(\sqrt{dt}\)). The letter "d" in front of a variable can be viewed as shorthand for "change in". Equation (2) is an expression for the percentage change of the factor which is split into expected and unexpected parts. The "drift" term \(m(F,t)dt\) is the expected percentage change in the factor (over a very short interval dt). This expectation can change as the factor level changes or as time passes. In the unexpected part, \(s(F,t)\) is the volatility of the factor (also dependent on the factor level and on time) and dz is Brownian motion. For now, we leave the expression of the factors as general, but various one-factor models differ by the specifications of \(m(F,t)\) and \(s(F,t)\).

其中 F 能夠是任何隨機因子，如一個特定債券的收益率或一個經濟體的實際增加率，dt 是一個小的（瞬時的）時間間隔，dz 是布朗運動（一個服從正態分佈的隨機過程，平均值爲0，標準誤差爲\(\sqrt{dt}\)）。變量前面的字母「d」能夠被視爲「變化」的縮寫。方程（2）將因子百分比變化分解爲預期和非預期部分。「漂移」項\(m(F,t)dt\)是因子的預期百分比變化（在很是短的時間間隔dt內）。這個預期能夠隨着因子水平或時間的變化而改變。在非預期部分，\(s(F,t)\)是因子的波動率（也取決於因子水平和時間），dz是布朗運動。如今，咱們將這些因子的表達式通常化，各類單因子模型根據\(m(F,t)\)和\(s(F,t)\)的具體形式而有所不一樣。

Let the price at time t of a zero-coupon bond which pays $1 at time T be expressed as \(P_i(F,t,T)\). Because F is the only stochastic component of \(P_i\), Ito's Lemma —— roughly, the stochastic calculus equivalent of taking a derivative —— gives the following expression for the dynamics of the bond price:

假定在時間 T 支付$1的零息債券在時間 t 的價格表示爲\(P_i(F,t,T)\)。由於 F 是\(P_i\)的惟一隨機成分，因此根據 Ito 引理——大體至關於定義在隨機分析上的導數，給出瞭如下債券價格動態的表達式。

\[ \frac{dP_i(F,t,T)}{P_i} = \mu_i dt + \sigma_i dz \tag{3} \]

where \(\mu_i = \frac{\partial P_i}{\partial t} \frac{1}{P_i} + \frac{\partial P_i}{\partial F} \frac{1}{P_i} m(F,t)F + \frac{1}{2} \frac{\partial^2 P_i}{\partial F^2} \frac{1}{P_i} s(F,t)^2F^2\), and \(\sigma_i = \frac{\partial P_i}{\partial t} \frac{1}{P_i} s(F,t)F\).

其中\(\mu_i = \frac{\partial P_i}{\partial t} \frac{1}{P_i} + \frac{\partial P_i}{\partial F} \frac{1}{P_i} m(F,t)F + \frac{1}{2} \frac{\partial^2 P_i}{\partial F^2} \frac{1}{P_i} s(F,t)^2F^2\)，而且\(\sigma_i = \frac{\partial P_i}{\partial t} \frac{1}{P_i} s(F,t)F\)。

In this framework, Ito's Lemma gives us an expression for the percentage change in price of the bond over the time dt for a given realization of F at time t. \(\mu_i\) is the expected percentage change in the price (drift) of bond i over the period dt and \(\sigma_i\) is the volatility of bond i.

在這個框架下，Ito引理給出了 t 時刻給定實現的 F 狀況下債券價格變化百分比的表達式。\(\mu_i\)是債券 i 在時間 dt 內的價格預期百分比變化（漂移），\(\sigma_i\)是債券 i 的波動率。

The unexpected part of the bond return depends on the bond's "duration" with respect to the factor (its factor sensitivity)²¹ and the unexpected factor realization. The return volatility of bond i (\(\sigma_i\)) is the product of its factor sensitivity and the volatility of the factor.

債券回報的非預期部分取決於債券的「久期」關於因子（因子敏感度）和非預期因子的實現。債券 i 的回報波動率（\(\sigma_i\)）是其因子敏感度和因子波動率的乘積。

Equation (3) shows that the decomposition of expected returns in Part 6 of this series is very general. The expected part of the bond return over dt is given by the expected percentage price change \(\mu_i\) because zero-coupon bonds do not earn coupon income. Consider the three components of the expected return. (1) The first term is the change in price due to the passage of time. Because our bonds are zero-coupon bonds, this change (accretion) will always be positive and represents a "rolling yield" component; (2) The second term is the expected change in the factor (mF) multiplied by the sensitivity of the bond's price to changes in the factor. This price sensitivity is like "duration" with respect to the relevant factor; and (3) The third term comprises of the second derivative of the price with respect to changes in the factor and the variance of the factor. The second derivative is like "convexity" with respect to the factor.

方程（3）代表本系列第6部分預期回報的分解很是通常化。由於零息債券不會得到票息收入，債券收益率的預期部分由 dt 時間內的預期價格變更百分比\(\mu_i\)給出。考慮預期回報的三個組成部分。（1）第一項是因爲時間的推移而致使的價格變更。由於咱們的債券是零息債券，因此這種變化（增值）將永遠是正的，表明着「滾動收益率」的部分；（2）第二項是因子（mF）的預期變化乘以債券價格對因子變化的敏感度。這個價格敏感度至關於相關因子的「久期」；和（3）第三項包括關於因子變化和因子方差的價格二次導數。二次導數至關於因子的「凸度」。

Suppose we specify the factor F to be the yield on bond i (\(y_i\)). Then, the expected change in price of bond i over the short time period (dt) is given by the familiar equation that we developed in the previous parts of this series:

假設咱們將因子 F 指定爲債券 i 的收益率（\(y_i\)）。那麼，短時間內（dt）債券 i 的價格預期變化是由本系列前面部分所獲得的方程給出的：

\[ \begin{aligned} E\left(\frac{dP_i(y,t,T)}{P_i} \right) &= \mu_i dt = \frac{\partial P_i}{\partial t} \frac{1}{P_i} + \frac{\partial P_i}{\partial y_i} \frac{1}{P_i}E(\Delta y_i) + \frac{1}{2}\frac{\partial^2 P_i}{\partial y_i^2} \frac{1}{P_i} variance(\Delta y_i)\\ &= \textit{Rolling Yield}_i - Duration_i*E(\Delta y_i) + \frac{1}{2}Convexity_i*variance(\Delta y_i) \end{aligned} \tag{4} \]

where \(\Delta y_i\) is the change in the yield of bond i. We can also use Equation (4) to similarly link the factor model approach to the decompositions of forward rates made in the previous parts of this series. It can be shown (for "time-homogeneous" models) that the instantaneous forward rate T periods ahead equals the rolling yield component. Therefore, we rewrite Equation (4) in terms of the forward rate as follows:

其中\(\Delta y_i\)是債券 i 收益率的變化。咱們也可使用等式（4）將因子模型方法與本系列前面部分中的遠期收益率分解類比。能夠顯示（對於「時齊」模型），即T年期瞬時遠期收益率等於滾動收益率份量。所以，咱們根據遠期收益率重寫等式（4）以下。

\[ \begin{aligned} f_{T, T+dt} &= \frac{\partial P_i}{\partial t} \frac{1}{P_i} = E\left(\frac{dP_i}{P_i} \right) - \frac{\partial P_i}{\partial y_i} \frac{1}{P_i}E(\Delta y_i) - \frac{1}{2}\frac{\partial^2 P_i}{\partial y^2_i} \frac{1}{P_i} variance(\Delta y_i) \\ &= \textit{Expected Yield}_i + Duration_i*E(\Delta y_i) - \frac{1}{2}Convexity_i*variance(\Delta y_i) \end{aligned} \tag{5} \]

The expected return term can be further decomposed into the risk-free short rate and the risk premium for bond i. Thus, forward rates can be decomposed into the rate expectation term (drift), a risk premium term and a convexity bias (or a Jensen's inequality) term. Other term structure models contain analogous but more complex terms.

預期回報項能夠進一步分解爲無風險短時間收益率和債券 i 的風險溢價。所以，遠期收益率能夠分解爲收益率預期（漂移）、風險溢價和凸度誤差（或Jensen不等式）。其餘期限結構模型包含相似但更復雜的成分。

Unfortunately, by defining the one relevant factor to be the bond's own yield, Equation (4) only holds for bond i. For any other bond j, the chain rule in calculus tells us that

不幸的是，由於將一個相關因子限定在債券自身收益率上，方程（4）僅適用於債券 i。對於任何其餘債券 j，結論中的鏈式規則告訴咱們

\[ -\frac{\partial P_i}{\partial y_i} \frac{1}{P_i} = \frac{\partial P_i}{\partial y_j} \frac{\partial y_j}{\partial y_i} \frac{1}{P_i} \text{ which equals } Duration_j \text{ only if } \frac{\partial y_j}{\partial y_i} = 1 \tag{6} \]

Therefore, in a one-factor world where \(y_i\) represents the relevant factor, Equation (4) only holds for bonds other than bond i if all shifts of the yield curve are parallel. While this observation suggests that more sophisticated term structure models are needed for derivatives valuation, it does not deem useless the framework developed in this series. In particular, this framework is valuable in applications such as interpreting yield curve shapes and forecasting the relative performance of various government bond positions. Such forecasts are not restricted to parallel curve shifts if we predict separately each bond's yield change (or if we predict a few points in the curve and interpolate between them). The problem with using maturity-specific yield and volatility forecasts is that the consistency of the forecasts across bonds and the absence of arbitrage opportunities are not explicitly guaranteed.

所以，在單因子世界中\(y_i\)表明相關因子，若是收益率曲線的全部變更是平行的，則式（4）適用於債券 i 之外的債券。雖然這一觀察結果代表衍生品估值須要更復雜的期限結構模型，但在本系列中制定的框架並不是是無用的。特別地，這個框架在解釋收益率曲線形狀和預測各類政府債券頭寸相對錶現的應用中是有價值的。若是咱們單獨預測每一個債券的收益率變化（或者若是咱們預測曲線中的幾個點並在它們之間插值），那麼這種預測並不侷限於平行的曲線偏移。使用特按期限收益率和波動率預測的問題是，債券之間預測的一致性和無套利機會沒有明確的保證。

Arbitrage-Free Restriction

無套利約束

For the time being, we return to the world where the factor, F, is unspecified and the change in price (return) of any bond i is given by Equation (3). How should bonds be priced relative to each other? The first term of Equation (3) is deterministic —— that is, we know today what the value of this component will be at the end of time t+dt. However, the value of second term is unknown until the end of time t+dt. In fact, in this one-factor framework, this is the only unknown component of any bond's returns. If we can form a portfolio whereby we eliminate all exposure to the one stochastic factor, then the return on the portfolio is known with certainty. If the return is known with certainty, then it must earn the riskless rate r or else arbitrage opportunities would exist.

回到問題上來，其中因子 F 是未指定的，而且任何債券 i 的價格（回報）變化由公式（3）給出。債券如何相對於彼此訂價？方程（3）的第一項是肯定性的，也就是說，咱們今天知道在時間t+dt結束時這個部分的值。然而，第二項的值直到時間t+dt結束是未知的。事實上，在這個單因子框架下，這是任何債券回報中惟一未知的組成部分。若是咱們能夠構造一個投資組合消除這個隨機因素的全部風險，那麼投資組合的回報是肯定的。若是回報是肯定的，那麼它必須得到無風險收益率 r ，或者套利機會將會存在。

It also follows that in our one-factor world, the ratio of expected excess return over the return volatility must be equal for any two bonds to prevent arbitrage opportunities. This relation must hold for all bonds or portfolios of bonds, and in Equation (7) below is the value of this ratio, often known as the "market price of the factor risk".

一樣，在咱們的單因子世界中，爲防止套利機會存在，任何兩個債券的預期超額回報與回報波動率的比率必須相等。這種關係必須適用於全部債券或債券組合，下面的方程（7）是該比率的價值，一般稱爲「因子風險的市場價格」。

\[ \frac{\mu_1 - r}{\sigma_1} = \frac{\mu_2 - r}{\sigma_2} = \lambda \tag{7} \]

where r is the riskless short rate

其中 r 是無風險短時間收益率。

Solutions of the Term Structure Models for Bond Prices

期限結構模型關於債券價格的解

Combining Equations (3) and (7) leads to the following differential equation:

組合方程（3）和（7）獲得如下微分方程。

\[ \frac{\partial P_1}{\partial t} + \frac{\partial P_1}{\partial F}(mF -\lambda s) + \frac{1}{2} \frac{\partial^2 P_1}{\partial F^2} s^2 F^2 = r P_1 . \tag{8} \]

This differential equation is solved to obtain bond prices and derivatives of bonds. Virtually all of the existing one-factor term-structure models are developed in this framework.²² The next step is to impose a set of boundary conditions specific to the instrument that is being priced and then solve the differential equation for \(P(F,t,T)\). One boundary condition for zero-coupon bonds is that the price of the bond at maturity is equal to par (\(P(F,T,T)=100\)). Another example of a boundary condition is that the value of a European call option on bonds, at the expiration of the option, is given by \(C(t,T,K) = \max[P(F,t,T)-K,0]\). Various term structure models differ in the definition of the relevant factor and the specification of its dynamics. Specifically, the one-factor models differ from each other in how the variable F and the functions m(F,t) (factor drift) and s(F,t) (factor volatility) are specified. Different specifications lead to distinct solutions of Equation (8) and distinct implications for bond prices and yields. In the rest of this section, we will analyze one such specification to give the reader an intuitive interpretation of these models, and then we qualitatively discuss the trade-offs between various popular models.

解這個微分方程能夠得到債券價格和債券衍生品。幾乎全部現有的單因子期限結構模型都是在這個框架下開發的。下一步是對金融產品施加一組特定的邊界條件，而後求解\(P(F,t,T)\)的微分方程。零息債券的一個邊界條件是債券在到期時的價格等於面值（\(P(F,T,T)=100\)）。邊界條件的另外一個例子是在期權到期時，債券上的歐式看漲期權的價值由\(C(t,T,K) = \max[P(F,t,T)-K,0]\)給出。各類期限結構模型在相關因子的定義和其動態結構上有所不一樣。具體來講，單因子模型在如何規定變量 F 和函數m(F,t)（因子漂移）和s(F,t)（因子波動率）之間彼此不一樣。不一樣的規定致使方程（8）的不一樣解，對債券價格和收益率有明顯的影響。在本節的其他部分，咱們將分析一個規定用來爲讀者直觀地解釋這些模型，而後定性地討論如何在各類流行模型之間權衡。

One Example: The Vasicek Model

示例：Vasicek 模型

Many of the existing term-structure models begin by specifying the one stochastic factor that affects all bond returns as the riskless interest rate (r) on an investment that matures at the end of dt. One of the earliest such model developed by Vasicek (1977) took this approach and specified the dynamics of the short rate as follows:

許多現有的期限結構模型首先指定影響全部債券回報的一個隨機因子，做爲在 dt 到期的投資的無風險收益率（r）。最先由Vasicek（1977）開發的一個模型採用了這種方法，並指出短時間收益率的動態結構以下：

\[ dr = k(l-r)dt + s dz. \tag{9} \]

This fits in the framework of Equation (2) if F is defined to be r and \(m(r,t) = (k(l-r))/r\) and \(s(r,t)=s/r\). The second term indicates that the short rate is normally distributed with a constant volatility of s which does not depend on the current level of r. The basis-point yield volatility is the same regardless of whether the short rate is equal to 5% or 20%. The drift term requires some interpretation. In the Vasicek model, the short rate follows a mean-reverting process. This means that there is some long-term mean level toward which the short rate tends to move. If the current short rate is high relative to this long-term level, the expected change in the short-rate is negative. Of course, even if the expected change over the next period is negative, we do not know for sure that the actual change will be negative because of the stochastic component. In Equation (9), l is the long-term level of the short rate and k is the speed of mean-reversion. If k = 0, there is no mean-reversion of the short rate. If k is large, the short rate reverts to its long-term level quite quickly and the stochastic component will be small relative to the mean reversion component.

若是 F 定義爲 r，而且\(m(r,t) = (k(l-r))/r\)，\(s(r,t)=s/r\)，則符合等式（2）的框架。第二項表示短時間收益率服從正態分佈，常波動率 s 不依賴於目前 r 的水平。不管短時間收益率是否等於5%或20%，基點收益率波動率是相同的。漂移項須要一些解釋。在 Vasicek 模型中，短時間收益率遵循均值迴歸過程。這意味着短時間收益率趨向於回到某個長期平均水平。若是目前的短時間收益率相對於長期水平較高，短時間收益率的預期變更將是負的。固然，即便下一期的預期變化是負的，因爲隨機因素，咱們不知道實際的變化是否將是負的。在等式（9）中，l 是短時間收益率的長期水平，k 是均值迴歸的速度。若是k = 0，則不存在短時間收益率的均值迴歸。若是 k 比較大，則短時間收益率會至關快地回覆到長期水平，隨機部分相對於均值迴歸部分將比較小。

This specification falls into a class of models known as the "affine" yield class. "Affine" essentially means that all continuously compounded spot rates are linear in the short rate. Many of the popular one-factor models belong to this class. For the affine term structure models, the solution of Equation (8) for zero-coupon bond prices is of the following form:

這種形式屬於所謂「仿射」收益率類的一類模型。「仿射」基本上意味着全部連續複利即期收益率在短時間內是線性的。許多受歡迎的單因子模型屬於這一類。對於仿射期限結構模型，等式（8）對零息債券價格的解以下。

\[ P(r,t,T) = e^{A(t,T) - B(t,T)r} \tag{10} \]

Typically, \(A(t,T)\) and \(B(t,T)\) are functions of the various parameters describing the interest rate dynamics such as k, l, s, and \(\lambda\). It is easy to show that the "duration" of the zero-coupon bond with respect to the short rate equals \(B(t,T)\). How does this duration measure differ from our traditional definition of duration with respect to a bond's own yield? For example, in the Vasicek model, the solution for B(t,T) is given by the following:

一般，\(A(r,t,T)\)和\(B(r,t,T)\)是描述收益率動態結構的各類參數（k、l、s 和\(\lambda\)）的函數。很容易顯示零息債券相對於短時間收益率的「久期」等於\(B(t,T)\)。這個久期衡量標準與傳統相對於債券自己收益率的久期有何不一樣？例如，在 Vasicek 模型中，\(B(t,T)\)的解以下：

\[ B(t,T) = \frac{1-e^{-k(T-t)}}{k} . \tag{11} \]

Therefore, the duration measure with respect to changes in the short rate is a function of the speed of mean-reversion parameter, k. As this parameter approaches 0, the duration of a bond with respect to changes in the short rate approaches the traditional duration measure with respect to changes in the bond's own yield. Without mean reversion, the Vasicek model implies parallel yield shifts, and Equation (4) holds. However, as the mean reversion speed gets larger, long bonds' prices are only slightly more sensitive to changes in the short rate than are intermediate bonds' prices because the impact of longer (traditional) duration is partly offset by the decay in yield volatility (see Figure 15). With mean reversion, long rates are less volatile than short rates. In this case, the traditional duration measure would overstate the relative riskiness of long bonds.

所以，關於短時間收益率變化的久期度量是均值迴歸速度參數 k 的函數。隨着這個參數接近0，關於短時間收益率變化的債券久期接近於傳統的相對於債券自己收益率的久期。若是沒有均值迴歸，Vasicek 模型意味着平行的收益率變化，等式（4）成立。然而，隨着均值迴歸速度變得愈來愈大，長期債券價格對短時間收益率的變化比中期債券價格稍微更敏感，由於（傳統的）長久期的影響被收益率波動率的衰退部分抵消了（見圖15）。由於均值迴歸，長期收益率波動率低於短時間收益率。在這種狀況下，傳統的久期度量會誇大長期債券的相對風險。

Comparisons of Various Models

不一樣模型的比較

Most of the one-factor term structure models that have evolved over the past 20 years are remarkably similar in the sense that they all essentially were derived in the framework that we described above. However, dissatisfaction with certain aspects of the existing technologies have motivated researchers in the industry and in academia to continue to develop new versions of term structure models. Four issues that have motivated the model-builders are:

• Consistency of factor dynamics with empirical observations;
• Ability to fit the current term structure and volatility structure;
• Computational efficiency; and
• Adequacy of one factor to satisfactorily describe the term structure dynamics.

在過去20年中大多數的單因子期限結構模型顯然是類似的，由於它們基本都上在咱們上面描述的框架中得出的。然而，對現有技術某些方面的不滿，促使行業和學術界的研究人員繼續開發新版本的期限結構模型。促成模型創建的四個問題是：

• 因子動態結構與經驗觀察的一致性；
• 擬合當前期限結構和波動率結構的能力；
• 計算效率；以及
• 單因子模型完善地描述期限結構動態的穩當性。

Differing Factor Specifications. Some of the one-factor models differ by the definition of the one common factor. However, the vast majority of the models assume that the factor is the short rate and the models differ by the specification of the dynamics of the factor.

改變因子的形式。一個單因子模型因一個共同因子的定義而有所不一樣。然而，絕大多數模型都認爲這個因子是短時間收益率，並且這些模型因這個因子的動態結構而異。

For example, the mean-reverting normally distributed process for the short rate that is used to derive the Vasicek model (Equation (9)) leads to features that many users find problematic. Specifically, nominal interest rates can become negative and the basis-point volatility of the short rate is not affected by the current level of interest rates. The Cox-Ingersoll-Ross model (CIR) is based on the following specification of the short rate which precludes negative interest rates and allows for level-dependent volatility:

例如，用於推導 Vasicek 模型（等式（9））的短時間收益率的均值迴歸正態分佈過程致使許多用戶發現一個問題。具體來講，名義收益率可能變爲負數，而且短時間收益率的基點波動率不受當前收益率水平的影響。Cox-Ingersoll-Ross 模型（CIR）基於如下短時間收益率的形式，排除了負收益率，並容許依賴收益率水平的波動率。

\[ dr = k(l-r)dt + s\sqrt{r} dz . \tag{12} \]

Because this model is a member of the affine yield class, the solution of the model is of the form shown in Equation (10). The function \(B(t,T)\), which represents the "duration" of the zero-coupon bond price with respect to changes in the short rate, is a complex function of the parameters k, l, s, and \(\lambda\). As in the Vasicek model, when the mean-reversion parameter is non-zero, the durations of long bonds with respect to changes in the short rate are significantly lower than the traditional duration.

由於這個模型是仿射收益率類的一個成員，因此模型的解是由式（10）所示。函數\(B(t,T)\)（表示零息債券價格相對於短時間收益率變化的「久期」）是參數 k、l、s 和\(\lambda\)的複雜函數。與 Vasicek 模型同樣，當均值迴歸參數爲非零值時，長期債券相對於短時間收益率變化的久期顯着低於傳統久期。

Chan, Karolyi, Longstaff and Sanders (CKLS, 1992) empirically compare the various models by noting that most of the one-factor models developed in the 1970s and 1980s are quite similar in that they define the one factor to be the short rate, r, and their dynamics are described by the following equation:

Chan，Karolyi，Longstaff 和 Sanders（CKLS，1992）經驗性地比較了各類模型，注意到在1970年代和80年代開發的大多數單因子模型是很是類似的，由於它們將單因子定義爲短時間收益率，而且它們的動態結構由如下等式描述：

\[ dr = k(l-r)dt + s r^\gamma dz . \tag{13} \]

The differences between the models are in their specification of k and \(\gamma\). For example, the Vasicek model has a non-zero k and \(\gamma = 0\). CIR also has a non-zero k and \(\gamma = 0.5\). We discuss the findings of CKLS and subsequent researchers in the section "How Does the Yield Curve Evolve Over Time?".

模型之間的差別在於 k 和\(\gamma\)的形式。例如，Vasicek 模型具備非零 k，且 \(\gamma = 0\)。CIR 模型也具備非零 k，且\(\gamma = 0.5\)。咱們在「收益率曲線如何隨時間變化」一節中討論了 CKLS 模型和研究人員的後續發現。

Fitting the Current Yield Curve and Volatility Structure. One of the problems that practitioners have with the early term structure models such as the original Vasicek and CIR models is that the parameters of the short-rate dynamics (k, l, s) and the market price of risk, \(\lambda\), must be estimated using historical data or by minimizing the pricing errors of the current universe of bonds. Nothing ensures that the market prices of a set of benchmark bonds matches the model prices. Therefore, a user of the model must conclude that either the benchmarks are "rich" or "cheap" or that the model is misspecified. Practitioners who must price derivatives from the model typically are not comfortable assuming that the market prices the benchmark Treasury bonds incorrectly.

擬合當前收益率曲線和波動率結構。從業者在使用早期結構模型（如原始的 Vasicek 和 CIR 模型）中發現的一個問題是，必須經過使用歷史數據或最小化當前債券的訂價錯誤來估計短時間收益率動態因子（k，l，s）和風險市場價格\(\lambda\)。沒有什麼能夠確保一組基準債券的市場價格與模型價格相一致。所以，模型的用戶必須得出結論，即基準是「高估的」或「低估的」，或者模型是錯誤的。假設市場對基準國債的估價不正確，那麼用模型進行衍生品訂價的從業者一般會感到擔心。

In 1986, Ho and Lee introduced a model that addressed this concern by specifying that the "risk-neutral" drift of the spot rate is a function of time. This addition allows the user to calibrate the model in such a way that a set of benchmark bonds are correctly priced without making assumptions regarding the market price of risk. Subsequently developed models address some shortcomings in the process implied by the Ho-Lee model (possibility of negative interest rates) or fit more market information (term structure of implied volatilities). Such models include Black-Derman-Toy, Black-Karasinski, Hull-White, and Heath-Jarrow-Morton. These models have become known as the "arbitrage-free" models, as opposed to the earlier "equilibrium" models. Our brief discussion does not do justice to these models; interested readers are referred to surveys by Ho (1994) and Duffie (1995).

1986年，Ho 和 Lee 介紹了一個解決這一擔心的模型，指出即期收益率的「風險中性」漂移是時間的函數。添加的這個條件容許用戶校準模型，使得一組基準債券的價格是正確的，而不對風險的市場價格作出假設。隨後的模型解決了 Ho-Lee 模型所隱含的隨機過程出現的一些缺陷（負收益率的可能性），或者使模型能夠適應更多的市場信息（隱含波動率的期限結構）。這些模型包括 Black-Derman-Toy、Black-Karasinski、Hull-White 和 Heath-Jarrow-Morton 模型。這些模型被稱爲「無套利」模型，而不是早期的「均衡」模型。咱們的簡短討論不對這些模型評判，有興趣的讀者請參考 Ho（1994）和 Duffie（1995）的綜述。

These arbitrage-free models represent the current "state of the art" for pricing and hedging fixed-income derivative instruments. One theoretical problem with these models is that they are time-inconsistent. The models are calibrated to fit the market data and then bonds and derivatives are priced with the implicit assumption that the parameters of the stochastic process remain as specified. However, as soon as the market changes, the model needs to be recalibrated, thereby violating the implicit assumption (see Dybvig (1995)). In reality, most practitioners find this inconsistency a small price to pay for the ability to calibrate the model to market prices.

這些無套利模型表明了當前固定收益衍生品訂價和對衝中的「發展水平」。這些模型的一個理論問題是它們是時間不一致的。模型被校準以適應市場數據，而後債券和衍生品的訂價隱含地假定隨機過程的參數保持當前水平。然而，一旦市場發生變化，模型就須要從新校準，從而違反了隱含的假設（參見Dybvig（1995））。實際上，大多數從業者發現這種不一致性是爲了將模型校準到市場價格而付出的代價。

Computational Efficiency. Some of the issues in choosing a model involve computational efficiency. For example, some of the models have the feature that the price of bonds and many derivatives on bonds have a closed-form solution, but others must be solved numerically by techniques such as Monte Carlo methods and finite differences. Because such techniques can be employed quite quickly, most practitioners do not feel that a closed-form solution is necessary. However, a closed-form solution allows a better understanding of the model and the sensitivities of the price to the various input variables.

計算效率。選擇模型涉及一些計算效率上的問題。例如，一些模型保證債券價格和債券上的衍生品具備閉式解，而其餘模型必須經過諸如蒙特卡羅方法和有限差分之類的技術進行數值求解。由於這樣的技術能夠很快地執行，因此大多數從業者並不以爲須要一個閉式解。然而，閉式解能夠更好地瞭解模型和價格對各類輸入變量的敏感度。

Many practitioners and researchers prefer the Heath-Jarrow-Morton model, which specifies the entire term structure as the underlying factor, because it provides the user with the most degrees of freedom in calibrating the model. However, the major shortcoming of this model is that, when implemented on a lattice (or tree) structure, the nodes of the lattice do not recombine. Therefore, the number of nodes grows exponentially as the number of time steps increase, rendering the time to obtain a price unacceptably long for many applications. Much of the recent research has been devoted to approximating this model to make it more computationally efficient.

許多從業者和研究人員喜歡使用 Heath-Jarrow-Morton 模型（它將整個期限結構指定爲潛在因子），由於它爲用戶校準模型提供最大的自由度。然而，該模型的主要缺點是，當在網格（或樹）結構上實現時，網格的節點不會重合。所以，隨着時間步長的增長，節點的數量呈指數增加，使得許多應用程序具備不可接受的過長計算時間。最近的許多研究已經致力於近似該模型，使其計算效率更高。

Extensions to Multi-Factor Models. Empirical analysis by Litterman and Scheinkman (1991), among others, shows that two or three factors can explain most of the cross-sectional differences in Treasury bond returns. A glance at the imperfect correlations between bond returns provides even simpler evidence of the insufficiency of a one-factor model. Yet, while multi-factor models, by definition, explain more of the dynamics of the term structure than a one-factor model, the cost of the additional complexity and computational time can be significant. In assessing whether a one-, two- or three-factor model is appropriate, the tradeoff is the efficiency gained in pricing and hedging because of the additional factors against these costs. For certain applications in the fixed-income markets, a one-factor model is adequate. For a systematic and detailed comparison of one-factor models vs two-factor models, see Canabarro (1995).

擴展到多因子模型。Litterman 和 Scheinkman（1991）的實證分析顯示，兩個或三個因素能夠解釋國債回報的大部分橫截面差別。債券回報之間不徹底的相關性，提供了揭示單因子模型不足的簡單證據。然而，雖然根據定義多因子模型比單因子模型解釋了更多的期限結構的動態特徵，可是額外的複雜性和計算時間的成本多是顯着的。在評估1、二或三因子模型是否合適時，須要權衡因爲添加因子帶來的成本與得到的訂價和對衝效率。對於固定收益市場的某些應用，單因子模型是足夠的。對於單因子模型與雙因子模型的系統詳細比較，參見 Canabarro（1995）。

The general framework in which a multi-factor term structure model is derived is similar to the one-factor model with the n factors specified in a similar manner as in Equation (2):

導出多因子期限結構模型的通常性框架相似於經過相似等式（2）的方式指定 n 個因子的單因子模型。

\[ \frac{dF_j}{F_j} = m_j(F_j, t)dt + s_j(F_j, t) dz_j . \tag{14} \]

where \(j = 1, \dots , n\) and the \(dz_j\)'s can be correlated with correlations given by \(\rho _{jk}(F,t)\).

其中\(j = 1, \dots , n\)，而且\(dz_j\)之間的相關性經過\(\rho _{jk}(F,t)\)給出。

For example, the Cox-Ingersoll-Ross model can be extended into a multi-factor model. To keep the analysis tractable, most term structure models define a small number of factors (n = 2 or 3). Some examples in the literature include the Brennan-Schwartz model, which specifies the two factors as a long and a short rate, the Brown-Schaefer model, which specifies the two factors as a long rate and the yield curve steepness, the Longstaff-Schwartz model, which specifies the two factors as a short rate and the volatility of the short rate, and the Duffie-Kan model, which specifies the factors as the yields on n bonds. A multi-factor version of Ito's Lemma provides the following expression for the return of bonds in the multi-factor world:

例如，Cox-Ingersoll-Ross 模型能夠擴展到多因子模型。爲了使分析易於處理，大多數期限結構模型定義了少許因子（n = 2或3）。文獻中的一些例子包括 Brennan-Schwartz 模型，將其兩個因子指定爲長期和短時間收益率，Brown-Schaefer 模型，將起兩個因子指定爲長期收益率和收益率曲線陡峭程度，Longstaff-Schwartz 模型，將其兩個因子指定爲短時間收益率和短時間收益率的波動率，以及 Duffie-Kan 模型，將其因子指定爲 n 個債券的收益率。Ito引理的多因子版本爲多因子世界中債券的回報提供瞭如下表達式：

\[ \frac{dP_i(F,t,T)}{P_i} = \mu_i dt + \sigma_i dz , \tag{15} \]

where \(\mu_i = \frac{\partial P_i}{\partial t} \frac{1}{P_i} + \sum_{j=1}^n \frac{\partial P_i}{\partial F_j} \frac{1}{P_i} m_j(F,t)F_j + \frac{1}{2}\sum_{j=1}^n \sum_{k=1}^n \frac{\partial^2 P_i}{\partial F_j \partial F_k} \frac{1}{P_i} s_j(F,t) s_k(F,t) \rho_{jk}(F,t)F_jF_k\) and \(\sigma_i = \sum_{j=1}^n \frac{\partial P_i}{\partial F_j} \frac{1}{P_i} s_jF_j\).

其中\(\mu_i = \frac{\partial P_i}{\partial t} \frac{1}{P_i} + \sum_{j=1}^n \frac{\partial P_i}{\partial F_j} \frac{1}{P_i} m_j(F,t)F_j + \frac{1}{2}\sum_{j=1}^n \sum_{k=1}^n \frac{\partial^2 P_i}{\partial F_j \partial F_k} \frac{1}{P_i} s_j(F,t) s_k(F,t) \rho_{jk}(F,t)F_jF_k\)而且\(\sigma_i = \sum_{j=1}^n \frac{\partial P_i}{\partial F_j} \frac{1}{P_i} s_jF_j\)。

While this expression may appear onerous, it is really a restatement of Equation (3). Qualitatively, Equation (15) simply states that the return on a bond can be decomposed in the multi-factor world as follows:

Return on bond i = expected return on bond i + unexpected return on bond i

where expected return on bond i =

• return on bond i due to the passage of time (rolling yield) -
• the sum of the "durations" with respect to each factor \(\times\) the expected realization of the factor +
• the value of all the convexity and cross-convexity terms,

and where unexpected return = the sum of the durations with respect to each factor \(\times\) the realization of the factors.

雖然這個表達可能顯得很雜亂，但其實是等式（3）的重述。等式（15）簡單地指出，債券的回報能夠在多因子世界中被分解以下：

債券的回報 = 預期回報 + 非預期回報

其中預期回報 =

• 時間推移產生的回報（滾動收益率） -
• 每一個因子的「久期」 \(\times\) 因子的預期實現 +
• 凸度價值以及交叉凸度價值，

其中非預期回報 =

• 每一個因子的「久期」 \(\times\) 因子的實現。

APPENDIX B. TERM STRUCTURE MODELS AND MORE GENERAL ASSET PRICING MODELS

期限結構與廣義資產訂價模型

In this Appendix, we link the return decomposition in Equation (15) to the broader asset pricing literature in modern finance, emphasizing the determination of bond risk premia. While term structure models focus on the expected returns and risks of only default-free bonds, asset pricing models analyze the expected returns and risks of all assets (stocks, bonds, cash, currencies, real estate, etc.).

在本附錄中，咱們將等式（15）中的回報分解與現代金融中普遍的資產訂價文獻聯繫起來，強調債券風險溢價的決定因素。期限結構模型側重於無違約債券的預期回報和風險，資產訂價模型分析了全部資產（股票、債券、現金、貨幣、房地產等）的預期回報和風險。

The traditional explanation for positive bond risk premia is that long bonds should offer higher returns (than short bonds) because their returns are more volatile.²³ However, a central theme in modern asset pricing models is that an asset's riskiness does not depend on its return volatility but on its sensitivity to (or covariation with) systematic risk factors. Part of each asset's return volatility may be nonsystematic or asset-specific. Recall that the realized return is a sum of expected return and unexpected return. Unexpected return depends (i) on an asset's sensitivity to systematic risk factors and actual realizations of those risk factors and (ii) on asset-specific residual risk. Expected return depends only on the first term because the second term can be diversified away. That is, the market does not reward investors for assuming diversifiable risk. Note that the term structure models assume that only systematic factors influence bond returns. This approach is justifiable by the empirical observation that the asset-specific component is a much smaller part of a government bond's return than a corporate bond's or a common stock's return.

債券風險溢價爲正的傳統解釋是，長期債券應該提供較高的回報（較短時間債券），由於它們的回報波動率更大。然而，現代資產訂價模型的中心主題是資產的風險並不取決於其回報波動率，而是其對系統性風險因子的敏感性（或相關性）。每一個資產的部分回報波動率多是非系統的或資產特定的。回想一下，實現的回報是預期回報和非預期回報的總和。非預期回報取決於（i）資產對系統性風險因子的敏感性和這些風險因子的實現；（ii）資產特定的剩餘風險。預期回報僅取決於第一項，由於第二項能夠實現分散化。也就是說，市場不會獎勵投資者承擔可分散風險。請注意，期限結構模型假設只有系統因子因素影響債券回報。這種作法是經過實證觀察證實的，相對於公司債券或普通股票，資產特定部分的回報僅是政府債券回報的一小部分。

The best-known asset pricing model, the Capital Asset Pricing Model(CAPM), posits that any asset's expected return is a sum of the risk-free rate and the asset's required risk premium. This risk premium depends on each asset's sensitivity to the overall market movements and on the market price of risk. The overall market is often proxied by the stock market (although a broader measure is probably more appropriate when analyzing bonds). Then, each asset's risk depends on its sensitivity to stock market fluctuations (beta). Intuitively, high-beta assets that accentuate the volatility of diversified portfolios should offer higher expected returns, while negative-beta assets that reduce portfolio volatility can offer low expected returns. The market price of risk is common to all investors and depends on the market's overall volatility and on the aggregate risk aversion level. Note that in a world of parallel yield curve shifts and positive correlation between stocks and bonds, all bonds would have positive betas —— and these would be proportional to the traditional duration measures. This is one explanation for the observed positive bond risk premia.

最著名的資產訂價模型——資本資產訂價模型（CAPM）假定任何資產的預期回報是無風險收益率和資產風險溢價之和。這種風險溢價取決於每種資產對市場總體走勢和風險的市場價格的敏感性。市場總體每每被股票市場所替代（儘管在分析債券時更普遍的替代選項可能更爲合適）。那麼，每一個資產的風險都取決於它對股市波動的敏感度（\(\beta\)）。直觀上，強調投資組合波動性的高\(\beta\)資產應提供更高的預期回報，而減小投資組合波動性的負\(\beta\)資產提供更低的預期回報。風險的市場價格對全部投資者都是相同的，而且取決於市場總體的波動率和整體風險規避水平。請注意，在收益率曲線平行偏移與股票和債券之間存在正相關性的世界中，全部債券都將具備正的\(\beta\)，而且與傳統的久期度量成正比。這是觀察到正的債券風險溢價的一個解釋。

In the CAPM, the market risk is the only systematic risk factor. In reality, investors face many different sources of risk. Multi-factor asset pricing models can be viewed as generalized versions of the CAPM. All these models state that each asset's expected return depends on the risk-free rate (reward for time) and on the asset's required risk premium (reward for taking various risks). The latter, in turn, depends on the asset's sensitivities ("durations") to systematic risk factors and on these factors' market prices of risk. These market prices of risk may vary across factors; investors are not indifferent to the source of return volatility. An example of undesirable volatility is a factor that makes portfolios perform poorly at times when it hurts investors the most (that is, when so-called marginal utility of profits/losses is high). Such a factor would command a positive risk premium; investors would only hold assets that covary closely with this factor if they are sufficiently rewarded. Conversely, investors are willing to accept a low risk premium for a factor that makes portfolios perform well in bad times. Thus, if long bonds were good recession hedges, they could even command a negative risk premium (lower required return than the risk-free rate).

在 CAPM 中，市場風險是惟一的系統性風險因子。實際上，投資者面臨着許多不一樣的風險來源。多因子資產訂價模型能夠看做是CAPM的通常化版本。全部這些模型都代表，每一個資產的預期回報取決於無風險收益率（對持有時間的獎勵）和資產風險溢價（對承擔各類風險的獎勵）。這反過來又取決於資產對系統性風險因子的敏感性（「久期」）和這些因子的風險市場價格。這些風險市場價格可能因因子而異，投資者對回報波動率的來源並不不聞不問。舉個例子，不良波動在損害絕大多數投資者時使投資組合表現不佳（即所謂的利潤/損失的邊際效用很高）。這樣一個因子將會帶來正的風險溢價，若是投資者想獲得足夠的回報，投資者將只能持有與這個因素密切相關的資產。相反，投資者願意接受低風險溢價，這是由於因子使投資組合在不利時期表現良好。所以，若是長期債券在衰退期是良好的對衝，他們甚至能夠接受負的風險溢價（回報比無風險收益率要低）。

The multi-factor framework provides a natural explanation for why assets' expected returns may not be linear in return volatility. One can show that expected returns are concave in return volatility if two factors with different market prices of risk influence the yield curve —— and the factor with a lower market price of risk has a relatively greater influence on the long rates. That is, if long bonds are highly sensitive to the factor with a low market price of risk and less sensitive to the factor with a high market price of risk, they may exhibit high return volatility and low expected returns (per unit of return volatility).

多因子框架提供了一個天然的解釋，爲何資產的預期回報關於回報波動率可能不是線性的。能夠看出，若是兩個具備不一樣風險市場價格的因子影響收益率曲線，則預期回報關於回報波動率是上凸的，而風險市場價格較低的因子對長期收益率的影響相對較大。也就是說，若是長期債券對風險市場價格較低的因子高度敏感，而且對具備風險市場價格較高的因子較不敏感，則可能表現出高回報波動率和低預期回報（單位回報波動率） 。

What kind of systematic factors should be included in a multi-factor model? By definition, systematic factors are factors that influence many assets' returns. Two plausible candidates for the fundamental factors that drive asset markets are a real output growth factor (that influences all assets but the stock market in particular) and an inflation factor (that influences nominal bonds in particular). The expected excess return of each asset would be a sum of two products: (i) the asset's sensitivity to the growth factor * the market price of risk for the growth factor and (ii) the asset's sensitivity to the inflation factor * the market price of risk for the inflation factor. However, these macroeconomic factors cannot be measured accurately; moreover, asset returns depend on the market's expectations rather than on past observations. Partly for these reasons, the term structure models tend to use yield-based factors plausibly —— as proxies for the fundamental economic determinants of bond returns.

多因子模型應包括什麼樣的系統因子？根據定義，系統因子是影響許多資產回報的因子。兩個推進資產市場的基本因子的合理候選是實際產出增加因子（影響全部資產，而非僅限股票市場）和通貨膨脹因子（影響名義債券）。每一個資產的預期超額回報將是兩個乘積的總和：（1）資產對增加因子的敏感性*增加因子的風險市場價格和；（2）資產對通貨膨脹因子的敏感性*通貨膨脹因子的風險市場價格。可是，這些宏觀經濟因素沒法準確測量。此外，資產回報取決於市場的預期，而不是過去的觀察。部分地因爲上述這些緣由，期限結構模型傾向於把基於收益率的因子做爲債券回報基本經濟決定因素的指代。

REFERENCES

參考文獻

Previous Parts of the Series Understanding the Yield Curve and Related Salomon Brothers Research Pieces

《理解收益率曲線》系列與相關文獻

• Ilmanen, A., Overview of Forward Rate Analysis (Part 1), May 1995.
• Ilmanen, A., Market‘s Rate Expectations and Forward Rates (Part 2), June 1995.
• Ilmanen, A., Does Duration Extension Enhance Long-Term Expected Returns? (Part 3), July 1995.
• Ilmanen, A., Forecasting US Bond Returns (Part 4), August 1995.
• Ilmanen, A., Convexity Bias and the Yield Curve (Part 5), September 1995.
• Ilmanen, A., A Framework for Analyzing Yield Curve Trades (Part 6), November 1995.
• Iwanowski, R., A Survey and Comparison of Term Structure Models, January 1996 (unpublished).

Surveys and Comparative Studies

綜述與比較研究

• Campbell, J., Lo, A. and MacKinlay, A. C., "Models of the Term Structure of Interest Rates," MIT Sloan School of Management Working Paper RPCF-1021-94, May 1994.
• Canabarro, E., "When Do One-Factor Models Fail?," Journal of Fixed Income, September 1995.
• Duffie, D., "State-Space Models of the Term Structure of Interest Rates," Stanford University Graduate School of Business Working Paper, May 1995.
• Dybvig, P., "Bond and Bond Option Pricing Based on the Current Term Structure," Washington University Working Paper, November 1995.
• Fisher, M., "Interpreting Forward Rates," Federal Reserve Board Working Paper, May 1994.
• Ho, T.S.Y., "Evolution of Interest Rate Models: A Comparison," Journal of Derivatives, Summer 1995.
• Tuckman, B., Fixed-Income Securities, John Wiley and Sons, 1995.

Equilibrium Term Structure Models

均衡期限結構模型

• Brennan, M. and Schwartz, E., "An Equilibrium Model of Bond Pricing and a Test of Market Efficiency," Journal of Financial and Quantitative Analysis, 1982, v17(3).
• Brown, R. and Schaefer, S., "Interest Rate Volatility and the Shape of the Term Structure," London School of Business IFA Working Paper 177, October 1993.
• Cox, J., Ingersoll, J., and Ross, S., "A Theory of the Term Structure of Interest Rates," Econometrica, 1985, v53(2).
• Longstaff, F. and Schwartz, E., "Interest Rate Volatility and the Term Structure: A Two-Factor General Equilibrium Model," Journal of Finance, 1992, v47(4).
• Vasicek, O., "An Equilibrium Characterization of the Term Structure," Journal of Financial Economics, 1977, v5(2).

Arbitrage-Free Term Structure Models

無套利期限結構模型

• Black, F., Derman, E., and Toy, W., "A One-Factor Model of Interest Rates and Its Application To Treasury Bond Options," Financial Analysts Journal, 1990, v46(1).
• Black, F. and Karasinski, P., "Bond and Option Pricing When Short Rates are Lognormal," Financial Analysts Journal, 1991, v47(4).
• Duffie, D. and Kan, R., "A Yield Factor Model of Interest Rates," Stanford University Graduate School of Business Working Paper, August 1995.
• Heath, D., Jarrow, R., and Morton, A., "Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contigent Claims Valuation," Econometrica, v60(1).
• Ho, T.S.Y., and Lee, S., "Term Structure Movements and Pricing Interest Rate Contingent Claims," Journal of Finance, v41(5).
• Hull, J. and White, A., "Pricing Interest Rate Derivative Securities," Review of Financial Studies, 1990, v3(4).

Empirical Evidence

實證

• Brenner, R., Harjes, R., and Kroner, K., "Another Look at Models of the Short-Term Interest Rate," Journal of Financial and Quantitative Analysis (forthcoming).
• Chen, K., Karolyi, G. A., Longstaff, F., and Sanders, A., "An Empirical Comparison of Alternative Models of the Short-Term Interest Rate," Journal of Finance, 1992, v47(3).
• Garbade, K., "Modes of Fluctuations in Bond Yields —— An Analysis of Principal Components," Bankers Trust Company Topics in Money and Securities Markets #20, June 1986.
• Ilmanen, A., "How Well Does Duration Measure Interest Rate Risk?," Journal of Fixed Income, 1992, v1(4).
• Litterman, R. and Scheinkman, J., "Common Factors Affecting Bond Returns," Journal of Fixed Income, 1991, v1(1).
• Litterman, R. and Scheinkman, J., and Weiss. L., "Volatility and the Yield Curve," Journal of Fixed Income, 1991, v1(1).

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1. Another way to get around the problem that the market's rate expectations are unobservable is to assume that a survey consensus view can proxy for these expectations. Comparing the forward rates with survey-based rate expectations indicates that changing rate expectations and changing bond risk premia induce changes in the curve steepness (see Figure 9 in Part 2 of this series and Figure 4 in Part 6).
   解決市場的收益率預期不可觀測問題的另外一種方法是，假設調查獲得的一致觀點能夠表明這些預期。將遠期收益率與基於調查的收益率預期相比較代表，變化的收益率預期和債券風險溢價會引發曲線陡峭程度的變化（見本系列第2部分圖9和第6部分圖4）。↩

2. The deviations from the pure expectations hypothesis are statistically significant when we regress excess bond returns on the steepness of the forward rate curve. Moreover, as long as the correlations in Figure 1 are zero or below, long bonds tend to earn at least their rolling yields.
   當咱們對債券超額回報關於遠期收益率曲線陡峭程度作迴歸分析時，與徹底預期假說的誤差具備統計顯著性。此外，只要圖1中的相關性爲零或低於零，那麼長期債券每每至少能夠得到滾動收益率。↩

3. Figure 7 in Part 2 shows that the forwards have predicted future excess bond returns better than they have anticipated future yield changes. Figures 2-4 in Part 4 show more general evidence of the forecastability of excess bond returns. In particular, combining yield curve information with other predictors can enhance the forecasts. The references in the cited reports provide formal evidence about the statistical significance of the predictability findings.
   第二部分的圖7顯示，遠期收益率預測將來債券超額回報比預期的將來收益率變化更好。第四部分的圖2-4顯示了更通常的債券超額回報可預測性的證據。特別是，將收益率曲線信息與其餘預測變量相結合能夠提升預測效果。所引用報告中的參考文獻提供了關於可預測性結果統計顯著性的正式證據。↩

4. However, some other evidence is more consistent with the expectations hypothesis than the short-run behavior of long rates. Namely, long rates often are reasonable estimates of the average level of the short rate over the life of the long bond (see John Campbell and Robert Shiller: "Yield Spreads and Interest Rate Movements: A Bird's Eye View," Review of Economic Studies, 1991).
   然而，其餘一些證據與長期收益率的短時間行爲相比，更符合預期假說。也就是說，長期收益率一般是對長期債券存在期內短時間收益率平均水平的合理估計（參見 John Campbell 和 Robert Shiller：《Yield Spreads and Interest Rate Movements: A Bird's Eye View》，Review of Economic Studies，1991）。↩

5. Our forecasting analysis focuses on excess return over the short rate, not the whole bond return. We do not discuss the time-variation in the short rate. The nominal short rate obviously reflects expected inflation and the required real short rate, both of which vary over time and across countries. From an international perspective, nominally riskless short-term rates in high-yielding countries may reflect expected depreciation and/or high required return (foreign exchange risk premium). In such countries, yield curves often are flat or inverted; investors earn a large compensation for holding the currency but little additional reward for duration extension.
   咱們預測分析的重點是超額回報超太短期收益率的部分，而不是整個債券回報。咱們不討論短時間收益率的時變性。名義短時間收益率顯然反映了預期的通貨膨脹率和所要求的實際短時間收益率，這二者都隨着時間和各國的不一樣而變化。從國際角度來看，高收益國家名義無風險短時間收益率可能反映了預期的貶值和（或）對高回報的要求（外匯風險溢價）。在這樣的國家，收益率曲線每每是平坦的或倒掛的; 投資者在持有貨幣方面得到很大的報酬，但在增長久期時額外得到一些報酬。↩

6. See Kenneth Froot's article "New Hope for the Expectations Hypothesis of the Term Structure of Interest Rates," Journal of Finance, 1989, and Werner DeBondt and Mary Bange's article "Inflation Forecast Errors and Time Variation in Term Premia," Journal of Financial and Quantitative Analysis, 1992.
   參見 Kenneth Froot 的文章——《New Hope for the Expectations Hypothesis of the Term Structure of Interest Rates》，Journal of Finance，1989，以及 Werner DeBondt 和 Mary Bange 文章——《Inflation Forecast Errors and Time Variation in Term Premia》，Journal of Financial and Quantitative Analysis，1992↩

7. Other explanations to the apparent return predictability include "data mining" and "peso problem." Data mining or overfitting refers to situations in which excessive analysis of a data sample leads to spurious empirical findings. Peso problems refer to situations where investors appear to be making systematic forecast errors because the realized historical sample is not representative of the market's (rational) expectations. In the two decades between 1955 and 1975, Mexican interest rates were systematically higher than the US interest rates although the peso-dollar exchange rate was stable. Because no devaluation occurred within this sample period, a statistician might infer that investors' expectations were irrational. This inference is based on the assumption that the ex post sample contains all the events that the market expects, with the correct frequency of occurrence. A more reasonable interpretation is that investors assigned a small probability to the devaluation of peso throughout this period. In fact, a large devaluation did occur in 1976, justifying the earlier investor concerns. Similar peso problems may occur in bond market analysis, for example, caused by unrealized fears of hyperinflation. That is, investors appear to be making systematic forecast errors when in fact investors are rational and the statistician is relying on benefit of hindsight. Similar problems occur when rational agents gradually learn about policy changes, and the statistician assumes that rational agents should know the eventual policy outcome during the sample period. However, while peso problems and learning could in principle induce some systematic forecast errors, it is not clear whether either phenomenon could cause exactly the type of systematic errors and return predictability that we observe.
   對明顯的收益可預測性的其餘解釋包括「數據挖掘」和「比索問題」。數據挖掘或過分擬合是指對數據樣本的過分分析致使虛假實證結果的狀況。比索問題是指投資者彷佛出現了系統性預測偏差的狀況，由於已實現的歷史樣本並不表明市場（理性）的預期。從1955年到1975年的20年間，雖然比索匯率穩定，但墨西哥收益率系統性地高於美國的收益率。因爲在這個樣本期內沒有發生貶值，統計學家可能會推斷投資者的預期是不合理的。這個推斷是基於這樣一個假設，即過後樣本包含市場預期的全部事件，並具備正確的發生頻率。一個更合理的解釋是投資者在這段時間內認定比索貶值機率比較小。事實上，1976年確實發生了大幅貶值，證實了投資者早期的擔心。相似的比索問題可能發生在債券市場分析中，例如，對未實現的惡性通貨膨脹擔心。也就是說，當事實上投資者是理性的，而統計學家則是後見之明的，投資者彷佛出現了系統性的預測偏差。當理性代理人逐漸瞭解政策變化，而統計學家認爲理性代理人在樣本期間應該知道最終政策的結果時，也會出現相似的問題。然而，儘管比索問題和學習原則上可能致使一些系統性的預測偏差，但目前尚不清楚這兩種現象是否會致使咱們所觀察到的系統性錯誤的類型和回報的可預測性。↩

8. We provide empirical justification to a strategy that a naive investor would choose: Go for yield. A more sophisticated investor would say that this activity is wasteful because well-known theories —— such as the pure expectations hypothesis in the bond market and the unbiased expectations hypothesis in the foreign exchange market —— imply that positive yield spreads only reflect expectations of offsetting capital losses. Now we remind the sophisticated investor that these well-known theories tend to fail in practice.
   咱們爲一個幼稚的投資者會選擇的策略提供了經驗證實：追求收益率。一個更老練的投資者會說，這種行爲是浪費，由於衆所周知的理論——好比債券市場的徹底預期假說和外匯市場上的無偏預期假說——意味着正的利差只反映了抵消資本損失的預期。如今咱們提醒老練的投資者，這些著名的理論在實踐中每每是失敗的。↩

9. Our discussion will focus on the concavity of the spot curve. Some authors have pointed out that the coupon bond yield curve tends to be concave (as we see in Figure 9) and have tried to explain this fact in the following way: If the spot curve were linearly upward-sloping and the par yields were linearly increasing in duration, the par curve would be a concave function of maturity because the par bonds' durations are concave in maturity. However, this is only a partial explanation to the par curve's concavity because Figure 9 shows that the average spot curve too is concave in maturity/duration.
   咱們的討論將集中在即期曲線的凸度。一些做者指出，付息債券收益率曲線每每是上凸的（如圖9所示），並試圖用如下方式解釋這一事實：若是即期曲線線性向上傾斜而且到期收益率關於久期線性增加，因爲債券久期關於期限是上凸的，所以到期收益率曲線將成爲期限的上凸函數。然而，這只是對曲線凸度的一個部分解釋，由於圖9顯示到期曲線在期限/久期上也是上凸的。↩

10. Here is another way of making our point: If short rates are more volatile than long rates, a duration-matched long-barbell versus short-bullet position would have a negative "empirical duration" or beta (rate level sensitivity). That is, even though the position has zero (traditional) duration, it tends to be profitable in a bearish environment (when curve flattening is more likely) and unprofitable in a bullish environment (when curve steepening is more likely). This negative beta property could explain the lower expected returns for barbells versus duration-matched bullets, if expected returns actually are linear in return volatility. However, the concave shapes of the average return curves in Figure 11 imply that even when barbells are weighted so that they have the same return volatility as bullets (and thus, the barbell-bullet position empirically has zero rate level sensitivity), they tend to have lower returns.
    若是短時間收益率比長期收益率更具波動性，那麼久期匹配的作多槓鈴組合-作空子彈組合頭寸會有一個負的「經驗久期」或\(\beta\)（收益率水平敏感性）。也就是說，即便頭寸（傳統）久期爲零，但在看跌的環境中（此時曲線變平時更有可能）每每有利可圖，而在看漲的環境中（此時曲線變陡更有可能）每每不是。若是預期回報實際上關於回報波動率是線性的，那麼這種負\(\beta\)的性質能夠解釋槓鈴組合與久期匹配子彈組合的較低預期回報。然而，圖11中平均回報曲線的上凸形狀意味着，即便當槓鈴組合被加權使得它們具備與子彈組合相同的回報波動率（而且所以槓鈴-子彈組合經驗上的收益率敏感度爲零），它們傾向於具備較低的回報。↩

11. Some market participants prefer payoff patterns that provide them insurance. Other market participants prefer to sell insurance because it provides high current income. Based on the analysis of Andre Perold and William Sharpe ("Dynamic Strategies for Asset Allocation," Financial Analysts Journal, 1989), we argue that following the more popular strategy is likely to earn lower return (because the price of the strategy will be bid very high). It is likely that the Treasury market ordinarily contains more insurance seekers than income-seekers (insurance sellers), perhaps leading to a high price for insurance. However, the relative sizes of the two groups may vary over time. In good times, many investors reach for yield and don't care for insurance. In bad times, some of these investors want insurance —— after the accident.
    一些市場參與者更喜歡爲他們提供保險的支付模式。其餘市場參與者願意出售保險，由於提供了高額的現金收入。根據 Andre Perold 和 William Sharpe（《Dynamic Strategies for Asset Allocation》，Financial Analysts Journal，1989）的分析，咱們認爲遵循越流行的策略越可能得到低迴報（由於策略價格的出價將會很高）。國債市場上尋求保險的一方比尋求收入的一方（出售保險）要多，也許致使了保險價格高昂。可是，兩方的相對規模可能會隨着時間而變化。在好的時候，許多投資者尋求收益率而不關心保險。在不景氣的時候，這些投資者中的一些人想要在事故發生後尋求保險。↩

12. Another perspective may clarify our subtle point. Long bonds typically perform well in recessions, but leveraged extensions of intermediate bonds (that are duration-matched to long bonds) perform even better because their yields decline more. Thus, the recession-hedging argument cannot easily explain the long bonds' low expected returns relative to the intermediate bonds —— unless various impediments to leveraging have made the long bonds the best realistic recession-hedging vehicles.
    另外一個觀點可能會澄清咱們的微妙之處。長期債券一般在經濟衰退時表現良好，但加槓桿的中期債券（與長期債券久期匹配）表現更好，由於其收益率降低更多。所以，經濟衰退對衝的論點不能輕易解釋長期債券相對於中期債券較低的預期回報，除非各類槓桿障礙使長期債券成爲現實中最好的衰退對衝工具。↩

13. Simple segmentation stories do not explain why arbitrageurs do not exploit the steep slope at the front end and the flatness beyond two years and thereby remove such opportunities. A partial explanation is that arbitrageurs cannot borrow at the Treasury bill rate; the higher funding cost limits their profit opportunities. These opportunities also are not riskless. In addition, while it is likely that supply and demand effects influence maturity-specific required returns and the yield curve shape in the short run, we would expect such effects to wash out in the long run.
    簡單的市場分割理論並不能解釋爲何套利者不利用曲線前端的陡峭和曲線超過兩年後部分的平坦，從而消除這種機會。部分解釋是套利者不能以國庫券收益率借款， 較高的資金成本限制了他們的獲利機會。這些機會也不是無風險的。另外，雖然短時間內供求關係極可能會影響期限特定要求的回報和收益率曲線的形狀，但長期來看，咱們預期這種效應會被沖淡。↩

14. We provide empirical evidence on the historical behavior of nominal interest rates. This evidence is not directly relevant for evaluating term structure models in some important situations. First, when term structure models are used to value derivatives in an arbitrage-free framework, these models make assumptions concerning the risk-neutral probability distribution of interest rates, not concerning the real-world distribution. Second, equilibrium term structure models often describe the behavior of real interest rates, not nominal rates.
    咱們提供名義收益率歷史行爲的經驗證據。這個證據與評估在一些重要的狀況下的期限結構模型沒有直接的關係。首先，當利用期限結構模型在無套利框架下對衍生品進行訂價時，這些模型假設考慮風險中性機率分佈下的收益率，而不考慮真實世界的分佈。其次，均衡期限結構模型一般描述實際收益率的行爲，而不是名義收益率。↩

15. Moreover, a model with parallel shifts would offer riskless arbitrage opportunities if the yield curves were flat. Duration-matched long-barbell versus short-bullet positions with positive convexity could only be profitable (or break even) because there would be no yield giveup or any possibility of capital losses caused by the curve steepening. However, the parallel shift model would not offer riskless arbitrage opportunities if the spot curves were concave (humped) because the barbell-bullet positions' yield giveup could more than offset their convexity advantage.
    此外，若是收益率曲線平坦，則容許曲線平行移動的模型將提供無風險的套利機會。久期匹配的多槓鈴-空子彈組合具備正的凸度，進而只會有利可圖（或收支平衡），由於沒有收益率損失或曲線變陡峭致使資本損失的可能性。然而，若是即期曲線是上凸的（隆起的），那麼容許曲線平行移動的模型不會提供無風險的套利機會，由於槓鈴-子彈頭寸的收益率損失可能會超過它們的凸度優點。↩

16. As shown in Equation (13) in Appendix A, the short rate volatility in many term structure models can be expressed as proportional to \(r^\gamma\) where is the coefficient of volatility's sensitivity on the rate level. For example, in the Vasicek model (additive or normal rate process), \(\gamma = 0\), while in the Cox-Ingersoll-Ross model (square root process), \(\gamma = 0.5\). The Black-Derman-Toy model (multiplicative or lognormal rate process) is not directly comparable but \(\gamma \approx 1\). If \(\gamma = 0\), the basis-point yield volatility (\(Vol(\Delta y)\)) does not vary with the yield level. If \(\gamma = 1\), the basis-point yield volatility varies one-for-one with the yield level —— and the relative yield volatility (\(Vol(\Delta y / y)\)) is independent of the yield level (see Equation (13) in Part 5 of this series).
    如附錄A中的等式（13）所示，許多期限結構模型的短時間收益率波動率能夠表示爲與\(r^\gamma\)（波動率對收益率水平的敏感度）成正比。例如，在 Vasicek 模型（可加或正態收益率過程）中，\(\gamma = 0\)，而在 Cox-Ingersoll-Ross 模型（平方根過程）中，\(\gamma = 0.5\)。Black-Derman-Toy 模型（可乘或對數正態收益率過程）不能直接比較，但\(\gamma \approx 1\)。若是\(\gamma = 0\)，則基點收益率波動率（\(Vol(\Delta y)\)）不隨收益率水平而變化。若是\(\gamma = 1\)，則基點收益率波動率與收益率水平一一對應地變化，相對收益率波動率（\(Vol(\Delta y / y)\)）與收益率水平無關（見本系列第5部分的等式（13））。↩

17. When we estimate the coefficient (yield volatility's sensitivity to the rate level —— see Equation (13) in Appendix A) using daily changes of the three-month Treasury bill rate, we find that the coefficient falls from 1.44 between 1977-94 to 0.71 between 1983-94. Moreover, when we reestimate the coefficient in a model that accounts for simple GARCH effects, it falls to 0.37 and 0.17, suggesting little level-dependency. (The GARCH coefficient on the past variance is 0.87 and 0.95 in the two samples, and the GARCH coefficient on the previous squared yield change is 0.02 and 0.03.) GARCH refers to "generalized autoregressive conditional heteroscedasticity," or more simply, time-varying volatility. GARCH models or other stochastic volatility models are one way to explain the fact that the actual distribution of interest rate changes have fatter tails than the normal distribution (that is, that the normal distribution underestimates the actual frequency of extreme events).
    當咱們使用三月期國庫券收益率的每日變化來估計係數（收益率波動率對收益率水平的敏感性，參見附錄A中的等式（13）），咱們發現該係數從1977-94年的1.44降低到1983-94年的0.71。此外，當咱們在一個考慮 GARCH 效應的模型中從新考慮係數時，係數分別將降低到0.37和0.17，這代表很小的收益率水平依賴性。（兩個樣本中過去方差的 GARCH 係數分別爲0.87和0.95，過去平方收益率變化的 GARCH 係數分別爲0.02和0.03）。GARCH 指的是「廣義自迴歸條件異方差性」，或者更簡單的說是時變的波動率。GARCH 模型或其餘隨機波動率模型是解釋收益率變化的實際分佈比正態分佈（即正態分佈低估極端事件的實際頻率）更加厚尾的一種方式。↩

18. Principal components analysis is used to extract from the data first the systematic factor that explains as much of the common variation in yields as possible, then a second factor that explains as much as possible of the remaining variation, and so on. These statistically derived factors are not directly observable —— but we can gain insight into each factor by examining the pattern of various bonds' sensitivities to it. These factors are not exactly equivalent to the actual shifts in the level, slope and curvature. For example, the level factor is not exactly parallel, as its shape typically depends on the term structure of yield volatility. In addition, the statistically derived factors are uncorrelated, by construction, whereas Figure 6 shows that the actual shifts in the yield curve level, slope and curvature are not uncorrelated.
    主成分分析用於從數據中提取儘量多地解釋收益率常見變化的首個系統性因子，而後用第二個因子儘量多地解釋剩餘變化，以此類推。這些統計推導出來的因子並不可直接觀察，可是咱們能夠經過考察各類債券對它的敏感性的模式來了解每一個因子。這些因素並不徹底等同於水平、斜率和曲率的實際變化。例如，水平因子並不徹底平行，由於其形狀一般取決於收益率波動率的期限結構。此外，統計推導的因子是不相關的、經過構造的，而圖6顯示，收益率曲線水平、斜率和曲率的實際變化不是不相關的。↩

19. This Appendix is an abbreviated version of Iwanowski (1996), an unpublished research piece that is available upon request. In this survey, we mention several term structure models; a complete reference list can be found after the appendices.
    本附錄是 Iwanowski（1996）未發表研究成果的縮寫版本，可根據要求提供。在此次綜述中，咱們將提到幾個期限結構模型；附錄以後能夠找到完整的參考文獻列表。↩

20. The subscript refers to the realization of factor F at time t. For convenience, we subsequently drop this subscript.
    下標是指在時間 t 實現的因子 F。爲了方便，咱們隨後去掉這個下標。↩

21. At this point, we refer to "duration" in quotes to signify that this is a duration with respect to the factor and not necessarily the traditional modified Macaulay duration.
    這裏，咱們對「久期」加上引號用來講明是相對於因子的久期，而不是傳統的修正 Macaulay 久期。↩

22. This is also the framework in which the Black-Scholes model to price equity options is developed.
    爲股票期權訂價的 Black-Scholes 模型也在此框架內。↩

23. One problem with this explanation is that short positions in long-term bonds are equally volatile as long positions in them; yet, the former earn a negative risk premium. Stated differently, why would borrowers issue long-term debt that costs more and is more volatile than short-term debt? The classic liquidity premium hypothesis offered the following "institutional" answer: Most investors prefer to lend short (to avoid price volatility) while most borrowers prefer to borrow long (to fix the cost of a long-term project or to ensure continuity of funding). However, we focus above on the explanations that modern finance offers.
    這個解釋的一個問題是，長期債券的空頭頭寸與多頭頭寸波動性至關。然而，前者具備負的風險溢價。換句話說，爲何借款人發行的長期債務成本更高，且比短時間債務更具波動性？ 傳統的流動性溢價假說提供瞭如下「制度性」答案：大多數投資者傾向借出短時間（以免價格波動），而大多數借款人更願意借入長期（以肯定長期項目的成本或確保資金的連續性）。可是，咱們重點關注現代金融理論提供的解釋。↩