金融危機中相同到期期限的美國國債為何到期收益率不同？

11-16

更新：我們老師並沒有對這個case進行分析，我私下郵件問她，她也沒有具體回復，只是讓我參考同一節課的小組展示。
感謝每一個關注這個問題並提供了幫助的人。

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這是我們在一個hbs case中遇到的問題。
一個對沖基金經理在2008年11月觀察到兩張同樣是2015年8月15日到期的債券收益率不同，一張是1985年8月15日發的票面利率為10.625%，收益率3.61%；另一張2005年8月15日發的票面利率4.25%，到期收益率3.26%。它們的一個基點的美元價值也不同，前者是0.0741,後者是0.0625。
信息如下圖

這個收益率差在2008年3月的時候特別明顯超出了流動性差異能解釋的正常波動範圍（見下圖Exhibit2），那正是JP摩根收購Bear Stearns的時候。然後這個收益率之差不久後又消失了，直到6月又重新出現，之後這個差值一直擴大.在2008年9月雷曼破產的時候這個收益率差值達到了一個頂峰——10月7日時達到約35個基點。而聯邦政府在10月3日出台了問題資產救助計劃（Troubled Asset Relief Program).10月8日這個收益率只差又回落到零點附近。同一天聯邦政府為AIG提供了378億美元的貸款。所以這個收益率之差應該是和政府行動有關的，但如何解釋呢，有沒有辦法稍微量化地解釋？

但11月收益率差又上升到35個基點了。

由於新發行國債比舊國債流動性強引起的收益率差沒有那麼大，一般只有3?5個基點。
我們的問題是，這個利差是怎麼來的，是不是由於它們久期不同造成的？但是息票利率高的那個久期小，應該投資者要求的收益率更低才對啊……
還有，怎麼量化計算久期不同造成的收益率差？
下圖是2008年11月3日的美國國債的收益率曲線
收益率曲線給的只是maturity和yield的關係啊，現在我們研究的是它們為何maturity相同但是yield不同，收益率曲線有什麼用嗎？還是說，我們根據這個case給的信息構造yield理論即期利率曲線，看它與現實中國債收益率曲線的差距，看看有沒有套利機會？

Update :@陳烯慧,@Danny W,@尉然
Forgot to update.

This is the response from a sell-side trader:

Thanks
Lenny, don』t need a rates expert for this one… this was all at the height of
the crisis and the main
reason was that a lot of the high coupon bonds were deeply off the run and
during the crisis, aged bonds of the same maturity as existing bonds traded
quite wide due to liquidity issues. Hedge Funds had sold a lot of them in
Relative Value bets (which did not work out). To be clear, This is not
on-the-run 10y vs. off-the-run-10y. It』s more like on-the-run 10y vs.
really really old 30y that aged into the 10y basket. But yes, the short
answer is that it was indeed a liquidity issue.

Sorry I was at the library so I cannot type Chinese here.
I did some research on this question and would like to share my findings here.
(Update: Sorry I just found out that neither of the two bonds are really on the run at that time, but since one is relatively new and liquid. So maybe this explanation still works.)
This phenomenon is usually called "flight to liquidity". In normal time, both on/off the run treasury bonds can be used for financing in the Repo market rather interchangeably, although on the run bonds have a very little liquidity premium. While during crisis, people will strongly prefer on the run bonds due to the increasing macro uncertainty, which can be shown by the trading volume of these two bonds that day, as@Lenny Zhang has mentioned. Therefore magnify this liquidity premium significantly.

Actually similar phenomenon can also be observed during 1998 Ruble crisis, where the 30-yr T bond on the run premium rises from 4bps to 28bps.
The main reason for "10月8日這個收益率只差又回落到零點附近",I believe, is that the Fed lowered both discount rate and Fed Fund rate by 50bps the following day.
BTW,there are many papers talking about the on/off the run premium. Just google it.
But I still don"t get the point of the last figure (The term structure one). Hope someone can shed some light on that.
After your prof talked about this, please do share some thoughts with us.
Many thanks!

Sources:http://personal.lse.ac.uk/vayanos/WPapers/FQFLPR.pdf
Historical Changes of the Target Federal Funds and Discount Rates
http://www.rmi.nus.edu.sg/_files/events/paper/Bond%20Liquidity%20Premia.pdf
http://merage.uci.edu/~jorion/fixed/Sols-Ch5.pdf
=====================================================
Updates:
Since@Lenny Zhang mentioned that the indicators of trading liquidity of the two bonds are not available. I tried to get some clues from the Fed"s website. After reading the Press Release on Oct 08,2008 which is the date they lowered the interest rates and the testimony of Bernanke on Oct 20,2008 to U.S. House of representatives. I think the money market turmoil is the cause of this spread.
On Oct 20, Bernanke said "The financial turmoil intensified in recent weeks, as investors" confidence in banks and other financial institutions eroded and risk aversion heightened. Conditions in the interbank lending market have worsened, with term funding essentially unavailable. Withdrawals from prime money market mutual funds, which are important suppliers of credit to the commercial paper market, severely disrupted that market; and short-term credit, when available, has become much more costly for virtually all firms. "

Here is my interpretation. From his words we can infer that the money market credit was very tight at that time. Hence the demand for best collateral, namely on the run T-bonds, shot up. The Fed"s following actions like TARP, lower rates and increase currency swap lines with foreign central banks ease this demand. That"s why the spread went back a bit.

Unfortunately, I still can"t explain it quantitatively. Discussions are more than welcomed !!
Sources:
FRB: 2008 Testimony
FRB: Press Release--FOMC statement: Federal Reserve and other central banks announce reductions in policy interest rates--October 8, 2008

Updates 06/12/2013:
@Lenny Zhang There is a field called "Estimated Traded Volume" and it is available for the two bonds. It surprised me that the volume for old bond is always higher than the relatively new one. I"m confused now. What do you think?

先自問自答一下。我現在的思路：
1.在這種金融危機的大環境下（流動性高度緊張，信用危機），美國政府會不會回購國債，或者出台政策，影響利率，進而影響國債收益率？如果市場的預期是美國政府會這麼做的話，是不是構成這兩個國債的收益率有差別的原因？
2.根據法博齊的《債券市場分析與戰略》，定價一支債券時參考的必要收益率反應了
i.相對風險（Comparable Risk)
ii.可選（替代）投資（alternative or substitute investments）
會不會是因為兩支國債所承擔的其他風險（如再投資風險）不同導致人們對它們要求的收益率不同？例如，票面利率高的那支國債在金融危機下再投資風險更高？
3.是否應該用總收益率而不是到期收益率來考慮這兩支債券，總收益率=[未來的現金流總和/債券購買價格]^(1/n)-1 n是付息的期數。
4.債券市場定價時的一價定律在這時失效了。本題中是怎樣讓一價定律失效了呢？

5.現在我們知道了那個10.625%票面利率的國債的一個基點的美元價值更大，也就是它的價格波動性更小，是否可以由此量化計算出由於波動性不同造成的人們對它們要求的到期收益率的差值？但是不是應該人們更喜歡價格波動性更小的債券，對它要求的收益率更低嗎？

參考資料：維基百科中的一價定律http://zh.wikipedia.org/wiki/%E4%B8%80%E4%BB%B7%E5%AE%9A%E5%BE%8B

防範金融危機平息後美元與美國國債市場逆轉風險

金融危機衝擊下的中國國債利率期限結構分析

貢獻我的五毛。我個人感覺應該是repo market出問題了，off the run low liquidity bond在repo market上在錢多的時候，可能repo跟on the run 差幾個bps 甚至一樣，所以這個按照roll down 兩個券的長期價格差也就是這個repo差。但是在08年的時候，市場完全失靈，導致repo market只願意拿最liquid的bond（也能理解，因為誰都不知道明天借錢的人會不會倒閉，而倒閉之後拿著的沒有流動性的券，只怕就算90%discount也不一定會有人買，哪怕是ty），所以導致你拿這個off the run bond去repo，你可能要多花100-200 bps才能借到錢，那麼這種情況下，這兩個券差20-30bps，也沒法arbitrage，因為你賭的其實是最後事情會好的，repo market會重新function，這兩個券借錢的價格會converge。

on-the-run vs off-the-run. I think mostly liquidity driven.

同意 @Lenny Zhang 和 @尉然，主要應該是flight to liquidity的問題。
另一個可能的原因是high coupon的債券re-invest的風險更大，所以在利率波動大的時候要求高yield來補償。

猜測還是流動性不同的原因吧，新bond流動性高，當年longterm的人就用這個做arbitrage. 不過這個spread比較大，我覺得是市場不efficient的原因。至於為什麼不efficient,這個可能原因多了。就像有同學說的，可能是高頻交易的原因，具體的可能還要去查交易的tick data。

好像是這樣的，比如美國政府發行了一張一年期價值100的國債，利息為2%，到期了就可以去兌換102美元，現在你100塊買了，收益率就是2%，但是如果你急於用錢，90塊買了，那麼買家的收益率就是3%，大概是這樣，畢竟非專業