## 2008年4月2日星期三

### Fed up with the U.S. Treasury, long live ECB!

These days I have been doing some research on equity indexed annuities. For experiment purposes, I need U.S. zero rates for various maturities. Like many people, I am too lazy to bookmark the relevant web pages. So I googled 'us treasury yield' and found this 'Daily Treasury yield curve rates' page. Without suspicion, I thought they are the zero curves. I was wrong.

The U.S. Treasury actually does not make it clear what these yield rates mean. The only explanation that I can find is buried inside an FAQ web page that isn't linked from the above-mentioned yield curves page. Worse still, the explanation is not entirely clear, but to my understanding the following is what it means by the term 'yield curve'.

Put it simply, the yield curves the U.S. Treasury prepares are not zero curves but par yield curves. Also, it assumes semi-annual compounding. For maturities under six months, the Treasury hasn't stated what compounding convention is used, but judging from one of its answers to the FAQs, it seems that simple compounding is used for all maturities.

In other words, if we denote by D(t) the discount factor for a zero coupon bond with t years of maturity, and c(t) the par yield for maturity t, then for t = 1/n (n>=2) year, we have

c(t)*t*D(t) + 100*D(t) = 100 ...... (A)

meaning that we can calculate D(t) easily as 100/(c(t)*t+100). That is, the par yield is the zero rate in this case, albeit with various (monthly, quarterly or semi-annually) compounding conventions. For t=n/2 (where n is an integer), however, since semi-annual compounding is used, we have

(c(n/2)/2)*[D(1/2) + D(1) + ... + D(n/2)] + 100*D(n/2) = 100 ...... (B)

Note that the U.S. Treasury announces yield rates only for 11 different maturities, namely, 1 month, 3 months, 6 months and 1, 2, 3, 5, 7, 10, 20 and 30 years. So an important implication of equations (A) and (B) is that for t>1, one cannot back out the discount factor D(t) from the par yields, because the number of unknowns would become larger than the number of equations (D(1) though can still be inferred). In other words, in reporting the par yields instead of the zero rates, there is loss of information.

So, for those who need to know the zero rates, the U.S. Treasury website is pretty useless. Frankly, I don't see why the U.S. Treasury opts to provide the par yield curve but not the zero curves (one can derive the former from the latter but not conversely) . In contrast, the European Central Bank provides data for both the zero curve, the par yield curve as well as the forward curve.

U.S. Treasury sucks. ECB rocks!!!

(Updated on Apr 3: there are more reflections in the next blog entry.)