The title of this article is deliberately silly, but there are times when I just need to be deliberately silly. I’m not going to explain why people (like me, sigh) decided not to cut their bond allocations to zero before the recent bond disaster, but rather how to judge or even calculate returns based on the most current market data bond – constant maturity. yield series.
(This post is a bit rushed, as I once again have family visiting from out of town…)
Constant Return Series
The US Federal Reserve (along with most other developed central banks) publishes a daily table of constant maturity yield series – the H.15 report. In other words, for different types of fixed income instruments (such as non-financial commercial papers), there are rates displayed by maturity (1 month, 2 months, etc.). These deadlines are clear deadlines, not things like “9 years, 7 months and 3 days” if you were to use issued securities. (Swaps are naturally their own maturities since the reference durations are those usually quoted.)
Depending on the source, there are different ways to calculate a constant maturity series.
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Simply use the yield of the reference bond at that maturity. The problem is that the return increases when we move to a new benchmark.
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Fit a yield curve for all bonds, then read the fitted curve to that maturity. (My preference, but you can get benchmark effects, and the fitted curve can be very different from the benchmark yield, which drives some people crazy.)
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Adjustment based on bonds in this maturity range (which Fed H.15 uses), which keeps the adjustment closer to the benchmark yield.
Yield quotes can be considered to correspond to the yield of an even coupon bond (link to primer on even coupon bonds). In summary, a nominal coupon bond is a bond whose yield matches the coupon rate, resulting in the bond being priced at par. For example, if the nominal coupon rate for the 10-year yield is 4%, then a 10-year bond with a 4% coupon will have a price of $100 (and a yield of 4%).
So how do you lose money?
Although government bonds are often described as “risk-free,” this actually refers to government bonds. without credit risk — sovereign states with floating currencies are effectively immune from any involuntary default for financial reasons. (Some people disagree with this assessment, but it is incorrect.) Although payments are guaranteed, you still face potential capital losses on the bond before maturity.
As a reminder, increasing yield → decreasing price, and vice versa. When you purchase a bond, you can calculate the internal rate of return of payments based on the initial payment price and, modulo certain fixed income yield conventions, that is, the quoted yield that corresponds to this price. The easiest way to see how this works is to play with the bond price function that is commonly built into spreadsheet software.
If bond yields rise after the bond is purchased, this implies a higher discount rate and therefore the corresponding price of the bond falls.
The correct way to calculate returns is to use certain bond pricing routines. However, we can approximate the return on holding a bond by:
Yield = (interest earned over the period — “carry”) + (change in yield) × (modified duration).
The first element is relatively simple: simply multiply the initial return by the fraction of a year represented by the holding period. This is a bit annoying for daily calculations (since the holding period changes between weekends, weekdays and holidays). There is an issue that bond yield prices follow funny conventions, but simply dividing the yield by 12 for a monthly holding period will be a “good enough” approximation in most cases.
The second term is more delicate. There are a few variations of “duration” used in fixed income (and appear in statistics published by bond funds), but the duration is modified. (or effective duration, which is equivalent to the modified duration for government bonds without options) is the first-order sensitivity of bond yields to changes in yield (i.e., the first term in the Taylor series of capital gains versus yield). Note that this is an approximation since the modified duration changes as the yield changes – so we need higher order terms to correctly calculate yields under large yield changes.
As noted, the modified duration depends on the bond yield and its characteristics (maturity, coupon), but is generally less than the maturity in years (decreasing as yields increase). As of this writing, the 10-year Treasury benchmark has an effective duration of 7.91. (The S&P US Treasury Bond Current 10-Year Index is an index that tracks the 10-year benchmark). The approximation suggests that we would lose 7.91% if the 10-year yield increased by 1% (100 basis points). However, the losses would be less for such a large movement – the second order term of the Taylor series (“convexity”) would reduce the loss. However, this would still be enough to wipe out more than a year of income.
I’ll then simply leave the exercise of observing yield changes on the 10-year Treasury yield chart H.15 above to get an idea of how much money the hapless longs have lost since 2020. (L (previously linked S&P index gives a more accurate view.)
Calculate yields with H.15 data
This is a fairly standard exercise of calculating total bond returns based on constant maturity yield data from the central bank using the formula above. (One reason for this is that bond index data is normally quite expensive – only a handful of deep-pocketed investors are actually interested in this data.) If we properly account for the change in duration as returns change (the simplest version is to take the average of the start and end durations of each period), the results will be close to an index return (which abstracts from transaction costs). However, there will be inevitable gaps due to missing security data.
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When a new benchmark bond is issued, it typically trades at a premium to previously issued bonds. If the “constant maturity” series uses only the benchmark yield, then there will be a variation in yields due to the benchmark change that does not match the yield variations of other bonds. If constant maturity uses an adjustment procedure, this effect is less but still present. Since this effect occurs every time the benchmark index changes and the error is in the same direction, the error will accumulate over time with the proxy return being lower than the index returns.
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There is normally a slope towards the curve. For example, we could have a period where the 9-year yield is systematically 20 basis points lower than the 10-year yield. This means that bond yields are “moving down” the curve, and there will be about 20 basis points of missed capital gains per year if we just look at the 10-year point.
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Benchmarks often offer favorable funding rates in the repo market. Although this is important for investors with access to the repo market, index calculations will ignore this effect.
We cannot infer the magnitude of these effects from constant maturity yield data alone, but they generally result in approximate yields that are lower than index yields. However, if we compare the returns of other asset classes (equities, cash), these calculation flaws are not so significant given the wide differences in returns between asset classes. They would be greater if a comparison of long-term returns was made.