## 2008年4月3日星期四

### More on yield curves

Yesterday I ranted on the decision of the U.S. Treasury to make public only the pay yield curves but not the zero curves. On a second thought, I realise that I have made a mistake.

You cannot (perfectly) back out zero rates from par yields, and vice versa
I claimed that it is not possible to back out the zero rates from the 11 par yields. This is correct. What I haven't realised, however, is that it is also impossible to back out these par yields from the zero rates that correspond to the same 11 maturities. This is simply because the Treasury's par yields are quoted on the basis of semi-annual interest payments, but given those 11 discount factors D(t), many of the discount factors D(n/2) are still unknown.

Yet if we know D(t) for t=1/12, 1/4, 1/2, 1, 2, 3, 5, 7, 10, 20 and 30, we can still compute the par yield c(t) for t=1/12, 1/4, 1/2, 1, 2, 3, provided that annual interest payments are assumed for bonds with maturities >= 1 year. So, in this sense, the zero rates are still more informative than the par yields.

There is another benefit of looking at the zero curve, namely, you can compare different yields directly. Commentators on newspapers used to talk about inverted yield curves, but they don't seem to realise that the yield curves they observe are par yield curves rather than zero curves. Now imagine that the zero curve is currently flat. For maturities of 1 month or 3 months, since the par yields are exactly the zero rates but with respectively monthly or quarterly compounding, they should be smaller than the 6-month par yield. Therefore the par yield curve will appear to have an upward slope, i.e. it will appear 'normal'.

By the same reasoning, if one day you are told that the (par) yield curve is inverted, then it is likely that the zero curve has already become inverted for some time.

Backing out zero rates from par yields, though, is numerically more stable
That said, there may be some merits for stating the par yields. Recall that when D(t) denotes the discount factor over [0,t], and c(t) denotes the par yield, we have

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

Now suppose we know only D(t) for 11 different values of t, and we want to determine c(n/2) for n=1, 2,... . Since

c(n/2) = 200*[1-D(n/2)]/[D(1/2) + D(1) + ... + D(n/2)] ...... (C)

and we don't know all values of D(t), we may first interpolate the whole zero curve D(t) using the 11 given values and then find c(n/2) using the formula (C). Unfortunately, by using interpolation we will introduce some noises in the values of D(t). The relative error in c(n/2) can be broken into two parts. The first part is contributed by the relative errors in D(t) in the denominator of the right hand side of formula (C). Given that the maximum relative error in D(t) is small, this should be less of a problem. The second part of the overall error comes from the relative error in D(n/2) in the numerator. This is more problematic. When D(n/2) is close to 1 (which is case when n is not large), a small relative error in D(n/2) will be magnified to a large relative error in the numerator of the right hand side of (C). To see this, let the relative error in D(n/2) be e. Then the relative error in 1-D(n/2) is given by

{[1-(1+e)*D(n/2)] / (1-D(n/2))} - 1 = -e*D(n/2)/(1-D(n/2)) ~ -e/(1-D(n/2)).

So, if only a few zero rates are given, backing out the par yield c(n/2) from the interpolated zero curve would become a numerically unstable procedure when n is small.

In contrast, suppose only a few par yields are given, and we want to back out D(n/2) from the interpolated par yield curve. Then we have

D(n/2) = {100 - (c(n/2)/2)*[D(1/2) + D(1) + ... + D((n-1)/2)] } / (100 + c(n/2)/2).

Again, the error contributed by the noise in the denominator is less of a problem, as c(n/2) is typically much smaller than 100. For the numerator, unless n is very large, the value D(1/2) + D(1) + ... + D((n-1)/2) and hence its product with c(n/2)/2 are both relatively small when compared to 100. Hence backing out the zero rates from the interpolated par yields seems to be a more stable operation than backing out par yields from the interpolated zero rates.

Still, why not offer the best of the two worlds?
So it seems that both yield curves have their own merits. As stating any one of them will result in some loss of information in the other, why doesn't the U.S. Treasury just announces both curves, or even simply make public the coefficients of the cubic splines that is used to generate the par yields? Any explanations on the Treasury's rationale are welcomed.