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<p>The spread for a FTD is 395bp, while the spreads for its constituents, IBM, MS and APPLE is 120,100,200 respectively. 395<420, why is that? that is, why the FTD's constituents are riskier than itself? Can someone give me an intuitive explanation？</p>

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关键词：Default basket fault First FIRS default someone itself

FTD insures the first default name,the spread should be smaller than one covering all four stocks. Think about it in this way, if there are two banks, one issuing FTD underlying three names,another bank issues three CDS with the same underlyings. When the first default occurs, two banks has to pay the same amount of money. When the second occurs bank one is free of any obligation because of the feature of FTD, but bank two has its obiligation because the CDS is written individually on each name. So FTD gives less "credit insurance" to policy holder, it only cover the first default. That's why FTD spread is less than the sum of all underlying CDS.

In the extreme case when all underlyings of the FTD have perfect correlations with each other (one defaults then all default) then buying one CDS is equivalent to buying FTD, so the price or the spread of FTD should be the lowest no matter how many names it covers (it costs less becasue it insures less randomness). On the other side of the spectrum where there is no correlations among all underlyings, in this case FTD will costs you a lot (it insures more randomness). But the spread of FTD will never exceed the sum of all CDS spread as I explained in the beginning. In your case the theoretical spread of FTD will be in the range of (200,420), how expensive FTD is depends on correlations, less correlation less cost. In today's financial crisis when correlations go up, CDS spead rockets while FTD climb up as well but relatively slower than amount of  aggregation. You will see the spread for the most risky name is much more approaching the spread for its FTD.

[此贴子已经被作者于2009-5-22 2:13:06编辑过]

I say the FTD cannot be lower than 200.

[此贴子已经被作者于2009-5-23 1:04:19编辑过]

以下是引用bmlf_001在2009-5-21 20:18:00的发言：

The spread for a FTD is 395bp, while the spreads for its constituents, IBM, MS and APPLE is 120,100,200 respectively. 395<420, why is that? that is, why the FTD's constituents are riskier than itself? Can someone give me an intuitive explanation？

大于420或者小于200都是arbitrage

恩，用概率的交并补就好

假设 IBM, MS and APPLE的违约概率为P(A),P(B),P(C)，对应的spread为120,100,200

max(P(A),P(B),P(c))<=P（FTD）=P(A or B or C)<=P(A)+P(B)+P(C)，而spread和违约概率是单调对应的，所以就得到irvingy的那个答案 

以下是引用垃圾树在2009-5-22 1:53:00的发言：

恩，用概率的交并补就好

假设 IBM, MS and APPLE的违约概率为P(A),P(B),P(C)，对应的spread为120,100,200

max(P(A),P(B),P(c))<=P（FTD）=P(A or B or C)<=P(A)+P(B)+P(C)，而spread和违约概率是单调对应的，所以就得到irvingy的那个答案 

我完全同意你和irvingy的分析，我一开始是这么想的，FTD只保护其中一个credit，但如果我单独买single name的，他们可能一起default，所以当然更risky。但有一个问题，考虑FTD的settle方式，比如FTD的notional是20m，再假设recovery rate都为40%，无论其中哪一个credit default了，protection seller都需要支付12m；

如果分别买这个三个公司的single name CDS，say IBM 10m，MS 5m，APPLE 5m，只有当他们一起default的时候seller才支付12m 与FTD相同，其余任何时候损失都会较少，不知道你们怎么看？

[此贴子已经被作者于2009-5-22 4:13:52编辑过]

我没太看懂你的意思,如果你是认为前面的no arbitrage的区间不成立的话,你不应该和三个single name CDS的相加来比较.另外recovery rate只要相等,是多少都不影响分析

max(P(A),P(B),P(c))<=P（FTD）=P(A or B or C)<=P(A)+P(B)+P(C)

式子中第一个不等式是用单个名义本金和FTD相同的single name CDS分别和FTD相比得到,比如20m single name和20m的FTD，不然我就可以买一个FTD来保护我single nameCDS seller的头寸，而且收益绝对是正的。第二个不等式不能用三个不同比例的single name相加，它相当于给一个三个标的违约交集的限制，这时候的操作应该是：比如现在第二个不等式不成立，那我就卖20m的FTD，然后直接买3个20m的single name来保护。由于第二个不等式不成立，这时候期初的花费就小于零了。后面那个不等式的操作不需要本金匹配，而是要求spread值匹配。

以下是引用垃圾树在2009-5-22 11:17:00的发言：

我没太看懂你的意思,如果你是认为前面的no arbitrage的区间不成立的话,你不应该和三个single name CDS的相加来比较.另外recovery rate只要相等,是多少都不影响分析

max(P(A),P(B),P(c))<=P（FTD）=P(A or B or C)<=P(A)+P(B)+P(C)

式子中第一个不等式是用单个名义本金和FTD相同的single name CDS分别和FTD相比得到,比如20m single name和20m的FTD，不然我就可以买一个FTD来保护我single nameCDS seller的头寸，而且收益绝对是正的。第二个不等式不能用三个不同比例的single name相加，它相当于给一个三个标的违约交集的限制，这时候的操作应该是：比如现在第二个不等式不成立，那我就卖20m的FTD，然后直接买3个20m的single name来保护。由于第二个不等式不成立，这时候期初的花费就小于零了。后面那个不等式的操作不需要本金匹配，而是要求spread值匹配。

en~,I'm convinced, many thanks.

極端假設,如果Default Correlation是1的話,要麼不Default,要麼就三個一起Default. 在''三個一起Default''的這種情況下,FTD其實就等於是對"組合裡面最容易Default的公司"的Default Protection, 因為只要這個公司Default, 其他的公司都會Default. 所以FTD價錢就應該是Portfolio裡面Spread最高的公司的CDS Spread, 在此例裡面就是APPLE的Spread 200.

如果Default Correlation是0的話, FTD就跟買了三個CDS一樣,價錢就應該是Spread的和420.

所以FTD的價錢就是 200<=FTD<=420, 395嘛,就表明現在的Default Correlation蠻高囉.
不知道這樣解釋有夠intuivtive嗎?