MR 11 – Regression Hedging and Principal Component Analysis

Every hedge ratio quietly encodes a belief about how two yields move together. A DV01-neutral hedge encodes the simplest belief of all: when one yield rises by a basis point, the other rises by exactly a basis point too. The methods in this lesson replace that belief with an estimate drawn from historical data. That does not make them free of judgment, because the analyst still decides which rates to use, how many of them to include, and which stretch of history is relevant. Data-driven hedging is a different set of assumptions, not the absence of assumptions.

The running example comes from Tuckman and Serrat. On May 14, 2021, a market maker buys $100 million face amount of the Johnson & Johnson (JNJ) 2.450s of 09/01/2060 from a client and wants to neutralise the interest rate exposure. No Treasury bond with a 40-year maturity exists, so the closest available instrument is the Treasury 1.625s of 11/15/2050, which has roughly 30 years left to run. The table below lists that pair together with two shorter Treasuries that come back later in the lesson. DV01 is quoted per 100 of face amount, and the last column scales it up to a $100 million position so the sizes are easy to compare.

The DV01-neutral answer sets the dollar sensitivity of the Treasury short equal to that of the corporate long. The required Treasury face amount is the corporate face amount scaled by the ratio of the two DV01s.

That hedge works perfectly only if both yields move by the same number of basis points every day. Two features of this pair make that unlikely in practice. The JNJ bond trades at a credit spread over Treasuries, and that spread moves on its own. On top of that, a 40-year rate and a 30-year rate are related but not perfectly correlated, so even two risk-free bonds of these maturities would not shift in lockstep.

What the scatter plot of daily yield changes shows

Plotting the daily change in the JNJ yield against the daily change in the Treasury yield over roughly the four months ending on the trade date settles the matter. The cloud of points is wide. On some days the Treasury moved far more than the corporate, as with the point (-18.2, -11.1), where the Treasury yield fell 18.2 basis points and the JNJ yield fell only 11.1. On other days the order reversed, as with (-3.3, -7.7). There are even days such as (2.0, -2.2) on which the two yields went in opposite directions. Taken as a whole, the best-fitting line through the cloud is flatter than a 45-degree line, which means the corporate yield has typically moved by less than the Treasury yield.

So the scatter carries two messages. The average relationship is less than one for one, which argues for a smaller hedge than the DV01 ratio suggests, and the spread of points around the line is large, which means some risk will remain however the hedge is sized.

Choosing the estimation window

Using the most recent four months assumes that the recent past is the best guide to the next few weeks. Often that is reasonable. Sometimes, however, an earlier episode is a better match. If the market expects the Federal Reserve to start raising short-term rates soon, an older period in which the central bank was tightening may say more about future co-movements than a recent stretch in which policy was on hold or easing. Window length involves a trade-off of its own: a short window may contain too few observations to pin down the estimates, while a long one risks mixing in market behaviour that no longer applies.

A trader who has looked at the scatter plot wants two things: a hedge ratio that reflects the less than one-for-one relationship, and a measure of how far actual outcomes stray from that average relationship. Ordinary least-squares regression delivers both at once.

Write the daily change in the JNJ yield as the dependent variable and the daily change in the 30-year Treasury yield as the independent variable. The model links them with a constant, a slope and an error term.

Here alpha is the intercept, beta measures how many basis points the JNJ yield moves for each basis point move in the Treasury yield, and the error term captures everything the Treasury move does not explain on that day. The true parameters are unknown, so they are estimated from the data. Once estimates (written with hats) are in hand, the model gives a predicted JNJ yield change for any Treasury yield change, and the gap between what happened and what was predicted is the residual.

What least squares minimises and why it squares the errors

Least-squares estimation chooses the intercept and slope that make the sum of squared residuals over the sample as small as possible.

Squaring does two jobs. A residual of +3 and a residual of -3 cannot cancel each other out and masquerade as a perfect fit, because both contribute 9 to the sum. Large misses are also punished far more than small ones, since a residual of 4 adds sixteen times as much as a residual of 1.

The estimator comes with a set of conditions. The model must actually describe how the two yields relate; the errors must share one probability distribution through time; errors on different days must be independent of each other; and the errors must be uncorrelated with the independent variable. When these hold, least-squares estimates have four attractive statistical properties. They are linear, meaning each estimate is a linear function of the observed dependent variable. They are unbiased, so their expected value equals the true parameter. They are consistent, so with more and more data they converge on the true value. And they are efficient, which here means no other linear unbiased estimator has a smaller variance.

Any spreadsheet or statistics package will fit the model above. The output for the JNJ regression, estimated on 82 daily observations ending May 14, 2021, is summarised below. The interval column is derived from the printed estimates as each value plus or minus two standard errors.

Other output: 82 observations, R-squared 77.5%, standard error of the regression 2.00 basis points.

The fitted line drawn through the scatter plot is just the prediction equation with these estimates plugged in.

Slope and constant

The slope of 0.842 says that, on average, the JNJ yield has moved only 0.842 basis points for every basis point move in the Treasury yield. That is a long way from the parallel-shift assumption. The constant of 0.060 is close to zero, which is the usual outcome for regressions of yield changes on yield changes. There is also an economic reason to expect this: if the corporate yield drifted systematically up or down on days when the Treasury yield did not move at all, that drift would be hard to explain for long.

Standard errors and hypothesis tests

Each standard error measures how precisely its coefficient is pinned down. A range of two standard errors either side of an estimate captures the true value roughly 95% of the time. For the slope, 0.842 plus or minus 2 × 0.051 gives an interval from 0.740 to 0.944. That interval excludes 1, so the hypothesis that beta equals one, which is exactly the parallel-shift hypothesis, can be rejected at about 95% confidence. For the constant, 0.060 plus or minus 2 × 0.223 runs from -0.386 to 0.506. Zero lies inside, so the hypothesis that alpha equals zero cannot be rejected.

R-squared and correlation

An R-squared of 77.5% means the Treasury yield change accounts for 77.5% of the variance of the JNJ yield change. In a regression with a single explanatory variable, R-squared is the square of the correlation between the two variables, so the correlation here is the square root of 0.775, or 88.0%. Both figures sit well below 100%, which is a direct warning that a Treasury hedge cannot remove all of the interest rate risk in the corporate bond.

Standard error of the regression

The last statistic, the standard error of the regression, is in effect the standard deviation of the residuals, and it is quoted in the units of the dependent variable. At 2.00, it says that the typical daily miss when explaining JNJ yield changes with Treasury yield changes is about two basis points. Strictly, it is the square root of the residual sum of squares over n minus 2 (a sample standard deviation would use n minus 1), and with a constant in the model the residuals average zero. This figure turns out to be the key input for measuring the risk of the hedge.

Every number here is in-sample, so hedging with it assumes the coming weeks will resemble the fitting period.

Suppose, for the moment, that the JNJ yield always moves by exactly beta-hat basis points for each one basis point move in the Treasury yield. Let F denote face amounts (positive for long positions, negative for short) and DV01 the price sensitivity per 100 face. A one basis point rise in the Treasury yield then changes the value of the JNJ position by its dollar DV01 times beta-hat, and changes the value of the Treasury position by its own dollar DV01. The position is hedged when those two effects cancel.

Solving for the Treasury face amount gives the regression hedge.

Compare this with the DV01-neutral formula: the only change is the extra factor of beta-hat. When beta-hat is less than one, as it is here, the hedge is smaller.

Risk weights

Desks often quote hedges as risk weights: the DV01 of the hedge expressed as a fraction of the DV01 of the position being hedged. Rearranging the hedging condition shows that the risk weight of a single-variable regression hedge is simply the estimated slope.

A DV01-neutral hedge always has a risk weight of 100%, since its whole design is to match the two DV01s. The regression hedge here deliberately carries only 84.2% of the DV01 of the JNJ position on the Treasury side, reflecting the dampened response of the corporate yield.

Yields never follow the regression line exactly, so the argument that the hedge works because yields obey the model is not the strongest case for it. The stronger case comes from the profit and loss of the combined position. Over one day, each leg gains or loses its dollar DV01 times the change in its own yield, with a minus sign because a long position loses when its yield rises.

It can be shown that the Treasury face amount from the regression hedge is precisely the amount that minimises the variance of this P&L. So, if variance is accepted as the measure of risk, the regression hedge is the minimum-risk hedge for a given holding of the JNJ bond. That result holds whether or not the yields happen to fall on the fitted line on any particular day.

Standard deviation of the hedged P&L

Once the hedge is in place, the only part of the JNJ yield move left unhedged is the residual. The standard deviation of the hedged P&L is therefore the dollar DV01 of the position being hedged multiplied by the standard deviation of the residuals.

The intuition is direct. If the JNJ yield moves exactly 0.842 times the Treasury move, the hedged P&L that day is zero. If the residual is +1 basis point, the JNJ yield has risen one basis point more than the model predicted, and the position loses one DV01 of the JNJ bonds. A residual of -2 basis points produces a gain of twice that DV01. Hedged P&L therefore scales one for one with the residual, and its volatility scales with the volatility of the residuals.

Testing the hedge out of sample

Every statistic so far was measured on the same data used to fit the model. The real question is whether the relationship holds afterwards. Checking the JNJ and Treasury yield changes from May 17, 2021, to July 19, 2021, against the line fitted earlier shows the model holding up well: the out-of-sample residual standard error is 1.5 basis points, slightly better than the 2.00 measured in-sample.

A trader on May 14 cannot run that test, of course, because the data do not yet exist. Useful substitutes exist, though. The trader can check how the fitted model would have performed in a window before the estimation period, either the stretch immediately preceding it or an older episode thought to resemble the coming months. A model that performs poorly outside its fitting window signals unstable coefficients and, with them, an unreliable hedge.

Rates at different points on the curve are not perfectly correlated, and a single hedging bond may explain only part of the movement in the bond being hedged. Adding a second explanatory yield can capture more of it. A relative value trade on the Treasury curve shows why this matters.

Imagine a trader who thinks 20-year Treasury yields are too high, or equivalently 20-year prices too low, compared with the 10-year and 30-year sectors around them. Expressing that view requires buying the 20-year bond, but the way the purchase is hedged determines which risks the trader is really exposed to.

• An unhedged 20-year purchase loses money in a broad sell-off even if the 20-year bond does end up outperforming its neighbours. The view is about relative value, yet the result is dominated by the level of rates.
• Hedging with only a 10-year bond removes most of the exposure to the level of rates but leaves a curve bet. If the curve steepens, so that 30-year yields rise more than 20-year yields and 20-year yields rise more than 10-year yields, the long 20-year bond loses more than the short 10-year bond gains, and the trade can lose money despite the 20-year bond outperforming.
• Hedging with only a 30-year bond creates the mirror-image problem: a flattening of the curve can produce losses even when the view is right.

The standard structure is therefore a butterfly. The trader buys the 20-year bond and sells both the 10-year and the 30-year bonds. Both shorts protect against a general rise in rates, the 10-year short protects against flattening, and the 30-year short protects against steepening. The remaining question is how much of each wing to sell against, say, $100 million face of the 20-year bond.

The two-variable regression model

The bonds come from the earlier table: the 1.375s of 11/15/2040 serve as the 20-year bond, the 0.875s of 11/15/2030 as the 10-year bond, and the 1.625s of 11/15/2050 as the 30-year bond. The regression now has two slopes, one for each wing.

Least squares again picks the coefficients that minimise the sum of squared residuals, and the fitted coefficients give the predicted 20-year yield change.

Reading the estimates

Fitted on 82 daily observations from late January to May 14, 2021, the model produces the following. The interval column is each estimate plus or minus two standard errors.

Other output: 82 observations, R-squared 94.7%, standard error of the regression 1.15 basis points.

A one basis point rise in the 10-year yield is associated with a 0.465 basis point rise in the 20-year yield, holding the 30-year yield fixed, and a one basis point rise in the 30-year yield with a 0.669 basis point rise. Neither interval comes close to zero, so both wings add genuine explanatory power. The constant, as usual, is small relative to its standard error.

The fit is much tighter than in the JNJ example. R-squared is 94.7% against 77.5%, and the standard error is 1.15 basis points a day against 2.00. Two forces are at work: the model has two explanatory variables instead of one, and all three bonds are Treasuries, so no corporate credit spread is adding noise of its own.

The derivation follows the single-variable case. Start from the daily P&L of the three positions, each leg contributing minus its dollar DV01 times its own yield change.

Now assume the 20-year yield follows the fitted model, replace its change with the prediction equation, and collect the terms that multiply each wing’s yield change.

For P&L to be zero whatever the two wing yields do, each bracket must be zero on its own. Solving each bracket gives one face amount.

Rewritten as risk weights, each wing’s DV01 as a fraction of the 20-year DV01 equals its own slope coefficient.

Why the risk weights sum to more than 100%

The extra DV01 follows straight from the coefficients. If the 10-year and 30-year yields both rise by one basis point at the same time, the model predicts a 20-year move of 0.465 + 0.669 = 1.134 basis points. In other words, a parallel move in the two wings comes with a larger move in the body, so the wings need 13.4% more DV01 than the body to keep pace. A DV01-neutral butterfly would ignore this historical pattern and leave the position exposed whenever all three yields rose together.

Out-of-sample residuals

Running the fitted coefficients over May 17 to July 19, 2021, gives residuals that are, if anything, better behaved than those inside the fitting window. The standard error falls from 1.15 basis points in-sample to only 0.70 out-of-sample. That is a pleasant outcome but not one any trader should count on; coefficient stability needs checking each time.

Most practitioners regress yield changes on yield changes, as in every example so far. A minority regress the level of one yield on the level of another. With y as the dependent variable and x as the independent variable, the single-variable versions are:

If the least-squares assumptions are satisfied by the level model, they are satisfied by the change model too, and estimates from both are unbiased, consistent and efficient. The two can diverge when one assumption fails. If the error terms in either specification are not independent through time, the estimates from that specification remain unbiased and consistent but may no longer be efficient.

A thought experiment on serial correlation

Take a simple case with alpha equal to 0 and beta equal to 1, where x and y are the yields of two bonds. The x-bond yield is fixed at 5%. Yesterday, the y-bond yield was 1%, so yesterday’s error in the level model was -4%.

The level model predicts that today the y-bond yield will be 5%. That is implausible: a yield that stood at 1% yesterday is far more likely to be near 1% today than to have jumped four percentage points. Today’s error is therefore likely to be close to yesterday’s -4% rather than close to zero. Errors that persist like this are correlated over time, which is what serial correlation means, and it breaks the independence assumption in the level model.

The change model fares better on the first day but has its own problem. The x-bond yield did not change, so the change model predicts no change in the y-bond yield either, leaving it at 1%. That beats a sudden jump to 5%, but if the economic relationship really is one for one, the y-bond yield is more likely to drift upward from 1% toward 5% over time. Each day of that drift produces a positive change-model error, so its errors are also positive for a run of days and therefore serially correlated.

Neither the level model, which predicts an instant jump, nor the change model, which predicts no movement, matches the likely gradual convergence of the y-bond yield. A natural fix keeps the level relationship but gives the error term its own dynamics, so that each day’s error is a fraction of the previous day’s plus a fresh shock.

The parameter rho controls how quickly a gap from fair value decays. With rho close to one, mispricings linger; with rho close to zero, they vanish almost immediately and the level model with independent errors becomes a fair description.

So the autoregressive errors deliver the gradual convergence intuition suggested, pulling the y-bond yield from 1% toward 5% without an overnight jump. Estimating a level model with such errors needs techniques beyond ordinary least squares, found in standard econometrics texts.

Return to the JNJ trade. The first trader regressed JNJ yield changes on Treasury yield changes, obtained a slope of 0.842, and sold $100 million × (0.2124 / 0.1910) × 0.842 = $93.6 million of the Treasury against $100 million of the corporate bond. A second trader flips the regression around and explains Treasury yield changes with JNJ yield changes. That is a reverse regression, and it gives a different picture.

Using the reverse slope, the second trader treats $93.6 million of the Treasury as the position and hedges it with the JNJ bond. Applying the hedge formula with the roles swapped gives $93.6 million × (0.1910 / 0.2124) × 0.921 = $77.5 million of JNJ bonds. In risk-weight terms the two hedges look very different: with DV01 of the Treasury position quoted as a percentage of DV01 of the JNJ position, the original regression implies 84.2% while the reverse regression implies 1 / 0.921, or 108.6%.

Why the two hedges are not comparable

Neither hedge is wrong. They answer different questions, and the difference starts with which quantity the trader fixed in advance. The first trader fixed $100 million face of JNJ bonds, and that choice set the size of the risk: its hedged volatility is $100 million × 0.2124 / 100 × 2 basis points, roughly $425,000 a day. The second trader fixed $93.6 million of Treasuries, and the risk of that hedged position is $93.6 million × 0.1910 / 100 × 2.09, roughly $374,000 a day. Two positions with different amounts of risk are not two answers to the same problem.

Scaling does not fix this. Many combinations of JNJ and Treasury holdings deliver the same volatility. A scaled-up reverse hedge with $106.37 million of Treasuries and $88 million of JNJ bonds ($106.37 million × (0.1910 / 0.2124) × 0.921 = $88.1 million) has hedged volatility of $106.37 million × 0.1910 / 100 × 2.09, about $425,000, matching the original. Yet it is still a different trade. It does not hold the $100 million of JNJ bonds that the client sold, and if the two bonds have different expected return profiles, a portfolio with a different mix of them will have different expected returns too.

The conclusion is that the original regression hedge minimises the variance of a portfolio built around a fixed $100 million holding of JNJ bonds, which is exactly the market maker’s problem. A trade with a different objective, such as holding a fixed amount of Treasuries or targeting a fixed volatility with particular return characteristics, would be built differently, and the reverse regression is not a contradiction of the original but an answer to one of those other questions.

Regression hedging is ad hoc at the level of a desk: each application has its own bonds, variables and estimation window, so a trader running a one-month single-variable regression and a colleague running a six-month two-variable regression are using two different models of the term structure. Principal component analysis (PCA) instead gives one empirical description of how the whole curve evolves, applicable consistently across the book, with factors whose shapes are easy to interpret.

The data set

The illustration uses daily USD swap rates (fixed rates against three-month USD LIBOR) from June 1, 2020, to July 16, 2021. There are 13 series: annual terms from one to 10 years plus 15, 20 and 30 years. LIBOR swaps were being retired at the time, but a long enough history of liquid SOFR swap rates did not yet exist, so LIBOR swaps remain the natural sample for the exercise.

One way to summarise how these 13 rates move is the full set of variances and pairwise covariances (or correlations) of their daily changes. PCA offers an alternative: describe the same movements with 13 factors, each of which is a fixed pattern of changes across all 13 terms. A factor might, for instance, consist of a 0.2 basis point change in the one-year rate, 0.6 in the two-year rate, 1.2 in the three-year rate, and so on, up to 3.7 at 20 years and 3.8 at 30 years.

The three defining properties of principal components

• Component variances add up to the total of the rate variances, so between them the components carry all of the volatility in the data.
• The components are uncorrelated with one another, even though the underlying rates are highly correlated across terms.
• Subject to those two conditions, each component in turn takes the largest variance still available. The first component explains the biggest share of total variance, the second the next biggest, and so on down the list.

The construction would be a curiosity if all 13 components mattered equally. They do not. Across many markets and samples, the first three principal components of interest rates account for an overwhelming fraction of total variance. That empirical regularity means a risk manager can describe curve behaviour with the shapes and volatilities of just three factors instead of 13 variances and 78 covariances.

Each element of a principal component is the change, in basis points, that a one standard deviation move in that component produces at one term. The table lists the first three components of the USD swap data, alongside the volatility the three components jointly imply for each rate and the actual volatility of each rate in the sample.

The Total row is the square root of the sum of the 13 variances in each column.

The level component

A one standard deviation rise in the first component lifts every rate at once: 0.23 basis points at one year, 0.59 at two years, 3.44 at 10 years and 3.77 at 30 years. On a given day, the curve’s move might be best described as 1.5 standard deviations of this component, meaning 1.5 times each element is added to the corresponding rate. On another day it might be -0.75 standard deviations, meaning 0.75 times each element is subtracted. This factor is traditionally called the level component because it has usually looked like a roughly parallel shift across most terms. In this sample, though, it is far from flat between one and seven years, rising steeply before levelling off beyond about 10 years.

The slope component

A one standard deviation rise in the second component lowers the one-year rate by 0.16 basis points and the two-year rate by 0.51, barely moves the 10-year rate (up 0.07), and raises the 30-year rate by 1.35. Short rates fall while long rates rise, or the reverse when the component moves the other way, which is why it is called the slope component.

The short-rate component

The third component raises very short rates slightly, lowers intermediate rates slightly and raises long rates slightly. Its shape is why it is often labelled curvature. In this data, however, its most useful role is adding explanatory power at the short end of the curve, so it is called the short-rate component here.

Combining components and reading the short end

Any day’s curve change can be approximated as a combination of the three. For example, a day might consist of +1.5 standard deviations of level, -0.4 of slope and -1.8 of short rate; the end-of-day curve is then approximately the previous curve plus those multiples of each component’s elements. Applying this combination at the 10-year term gives 1.5 × 3.44 – 0.4 × 0.07 – 1.8 × (-0.23) = 5.160 – 0.028 + 0.414 = 5.546 basis points.

All three components are small at the very short end because those rates barely moved. The Federal Reserve had pledged to hold short-term rates down for a long time, so economic news showed up instead in intermediate and longer rates, through expectations about future policy. Before the 2007 to 2009 financial crisis the level component was typically hump-shaped, peaking near five years and then declining, which reflected a central bank that pegged the shortest rates but was expected to respond to economic news fairly soon.

Because the components are uncorrelated by construction, the volatility they jointly imply for a rate is the square root of the sum of the squared elements at that term. For the five-year rate:

Measured directly, five-year volatility also rounds to 2.54. With more decimal places, the ratio of the three-component volatility to the total volatility is 99.78%, so the first three components explain almost all of the movement in the five-year rate.

Variance shares by term

The next table breaks each term’s three-component variance into the share attributable to each component, and repeats the ratio of three-component volatility to total volatility. The terms are grouped by the number of factors a trader in that sector would need.

For the two-year rate, the shares are worked out with the components quoted to more decimal places than the earlier table, which avoids rounding confusion.

What the coverage numbers say

Three factors explain more than 99% of the variation of every rate with a term above three years, 97.47% for the two-year rate and 74.54% for the one-year rate. Thirteen rates are therefore described almost completely by three fixed patterns of change, which is possible only because changes across the curve are so highly correlated. The shortest rate is the exception, where even three components leave a noticeable gap.

The level component dominates overall, but the short-rate component and then the slope component matter at the short end, and the slope component matters again at the long end. These findings have direct consequences for risk management. A trader in eight-year to 10-year swaps, and arguably anywhere from seven to 15 years, can reasonably rely on one-factor tools such as DV01 hedging or a single-variable regression, because one factor describes those terms well. A trader in the three-year to six-year sector or at the long end probably needs two factors. A trader at the very short end may need all three.

Whole-curve totals and other currencies

The bottom row aggregates across terms. A sum of variances corresponds to no interesting portfolio, but it is a convenient summary: three components capture 99.84% of total volatility across the curve, and of that variance level supplies over 90%, slope about 8% and short rate under 1%.

Similar lessons emerge in other markets. Estimated over the same dates, GBP LIBOR swap components look almost identical to the USD ones in shape and size, except that the USD slope component is more volatile. Euribor swap components have shapes qualitatively like the other two but much lower volatility: the EUR level component flattens at a little above 2.5 basis points a day at long terms, whereas the USD and GBP level components flatten above 3.5. A plausible reason is that the European Central Bank was more determined than its US and UK counterparts to hold short-term rates down for an extended period.

PCA hedges are constructed in the same way as other multi-factor hedges. Price the portfolio off the current curve, shift the curve by each principal component in turn and reprice, use the price changes to compute the portfolio’s sensitivity to each component, and then choose hedging positions that set each of those sensitivities to zero.

The example repeats the 10s-20s-30s butterfly idea, this time in swaps. A trader believes the USD 20-year swap rate is too high relative to the 10-year and 30-year rates, so the trader receives fixed on the 20-year swap and pays fixed on the 10-year and 30-year swaps. Receiving fixed is economically similar to buying a bond, and paying fixed to selling one.

Let the trader receive fixed on 100 notional of the 20-year swap and on F¹⁰ and F³⁰ notional of the 10-year and 30-year swaps, with negative notionals meaning paying fixed. A one standard deviation move in a component changes each swap rate by that component’s element, and a swap on which the trader receives fixed loses value when its rate rises. The portfolio is neutral to the level and slope components when both of the following hold.

In the level equation, the first two terms are the value changes of the hedging swaps when the curve shifts by the level component, which raises the 10-year rate by 3.44 basis points and the 30-year rate by 3.77. The third term is the value change of the 20-year swap, whose rate rises by 3.73. The slope equation reads the same way with slope elements. If both equations hold for a one standard deviation move, they hold for a move of any size, since multiplying both sides by a constant leaves zero unchanged.

Most of the 20-year risk, 74%, is carried by the 30-year swap. The reason is that the 20-year swap’s exposures to both the level and the slope components look much more like those of the 30-year swap than those of the 10-year swap. The risk weights total 101.2%, so the hedge carries more DV01 than the position it protects. Risk weights always total 100% only when parallel shifts are assumed. Here a sizeable part of the hedge sits in 10-year swaps, which respond much less to the level and slope components than the 20-year swap does, so extra DV01 is needed to deliver the same factor exposure.

Residual exposure to the third component

With two hedging swaps, only two components can be neutralised. The exposure to the short-rate component is found by plugging its elements into the same structure, carrying one more significant digit to avoid rounding confusion.

Recomputing gives -0.01057 + 0.04554 – 0.02833 = 0.00664, which rounds to 0.007: less than one cent per 100 notional. A residual that small may not justify the cost of trading another swap. The view also matters: a relative value trader prefers to pay fixed only where rates look too low, and an extra hedging swap may sit outside that view.