VRM 13: Modeling Non-Parallel Term Structure Shifts and Hedging

Principal components analysis finds structure inside a group of correlated series. Applied to interest rates, it rewrites the daily changes in rates of several maturities as combinations of a few underlying movements called factors. Each factor is itself a term structure movement, a fixed pattern saying how far every rate shifts.

Three properties define the output. Every daily movement in the data is a linear combination of the factors, the factors are uncorrelated, and the first two or three account for the overwhelming share of what was observed. The number of factors equals the number of rates, since a change in n rates can be written as a combination of n factors by solving n simultaneous equations.

Factor loadings and factor scores

The Federal Reserve Board publishes daily Treasury rates at eight maturities, from one year out to thirty, listed in the table below. Running the analysis on daily changes in those rates over January 2008 to December 2019 produces eight factors, ranked by importance.

A factor loading is the amount by which one rate moves for one unit of a given factor. Under the first factor, one unit moves the one-year rate by -0.134 basis points and the two-year rate by -0.266, with the rest of the curve going the same way. A factor score counts how many units of a factor a given day’s change contains, so scores vary day by day while loadings stay fixed. The change in the jth rate on a given day is then:

Signs carry no information on their own. Every loading for the first factor is negative, so one unit of it pushes all rates down. Reversing the whole column leaves the model unchanged, because one unit of the reversed factor does what minus one unit of the original did. The reversed loadings, below, climb from 0.134 at one year to a peak of 0.432 at seven years and ease back to 0.364 at thirty.

Source: Federal Reserve Board daily Treasury data.

Importance is measured by the standard deviation of a factor score across the sample: a factor whose score swings widely drives much of what is observed. For the Treasury data the standard deviations of the first three factor scores are 13.58, 4.66 and 2.32.

Because the factors are uncorrelated, the variances of the factor scores add to the total variance of all rate movements, so each factor’s share reads off directly. The first factor accounts for about 85% of the variance in Treasury rate changes, the first three for about 97.6%.

What each of the three factors looks like

The first factor is a shift in which all rates move in the same direction by approximately, though not exactly, the same amount. It is the closest thing in the data to a parallel shift, which is why a DV01 hedge is useful without being sufficient.

The second factor is a change of slope. One unit of it pushes the rates from one to five years down and the rates from seven years outward up, which is a steepening of the term structure, or a flattening once the sign is reversed.

The third factor is a bowing. One unit of it lifts the two shortest and the two longest rates while pushing the four in between down, so the curve becomes more or less humped through the middle.

Under the first two factors the one-year rate barely shifts next to the rest of the curve. One-year rates sat very low through this period and did not move much.

DV01 is the change in a portfolio’s value when every spot rate rises by one basis point. Key rate analysis splits it into several numbers, each anchored to a chosen point on the curve. Take the two-year, five-year and ten-year spot rates as those points, called key rates.

Each key rate shift raises its own rate by one basis point, leaves the other key rates untouched, and interpolates in a straight line between them. The two-year shift holds a full basis point out to two years, then tapers to nothing at five. The five-year shift climbs from nothing at two years to a peak at five and tapers back to nothing at ten. The ten-year shift climbs from nothing at five years to a peak at ten and holds there for longer maturities. The three heights add to one basis point at every maturity, so together they partition the parallel shift.

Sensitivities to these shifts are called partial 01s, or key rate 01s, normally written KR01s. The first KR01 is the reduction in a portfolio’s value under the two-year shift, the second under the five-year shift, the third under the ten-year shift. Since the shifts add to a parallel shift, so do the value changes:

An investor who neutralises all three is protected against much more than a parallel move: the hedge covers each shape and every combination of them.

KR01s are built from a portfolio’s sensitivities to the individual spot rates: take the value change for a one-basis-point move in each spot rate on its own, then weight those changes by how far each rate travels under each shift.

One number of 2,412.9 says nothing about where along the curve the risk lives, while the three KR01s put almost two thirds of it on the ten-year shift.

Neutralising three key rate exposures takes three hedging instruments, and their positions come from three simultaneous equations, one per shift: the portfolio KR01 plus the weighted sum of the instrument KR01s must equal zero.

The idea generalises: when n term structure movements are considered, n simultaneous equations determine the positions that neutralise all of them, whether the movements are key rate shifts, bucket shifts or factors.

What bank regulators require

Regulators require banks to analyse portfolio risk using ten KR01s, for one-basis-point shifts in ten spot rates: three months and six months at the front, then one, two, three and five years, and at the long end ten, 15, 20 and 30 years. The shifts are tapered as described earlier, so they sum to the DV01. Banks are not asked to drive them to zero, but to combine them with the standard deviations of, and correlations between, the ten rates:

Here the exposures are the ten KR01s. If the change in portfolio value can be treated as normal, value at risk and expected shortfall follow directly, and both feed capital for market risk under the new Basel Committee rules and initial margin for derivative transactions not cleared through a central counterparty. Hedging is rarely perfect, and the same expression measures what is left once a hedge is on.

The three factors from a principal components analysis are also worth hedging, because they were extracted from what the curve actually did. To hedge with them, build the same exposure table with the factors in place of the key rate shifts and solve three simultaneous equations.

That hedge covers the three factor shapes and any combination of them, such as three units of the first factor, minus two of the second and one of the third. If the ten years of history are representative, it covers about 97.6% of the variance in term structure movements. The factor shapes involve more rates than the tapered key rate shifts, and are not linear combinations of them, so a key rate hedge leaves gaps that a factor hedge closes.

Why uncorrelated factors make the volatility calculation simple

Every cross term in the general variance expression carries a correlation, and the factors are uncorrelated, so those terms vanish. Write sigma for the standard deviation of a factor score and f for the change in portfolio value under one unit of it:

The key rate shifts above were defined as changes to spot rates, but they can equally describe changes to par yields, with each shift reading change in par yield. The three then add up to a yield-based DV01, and the sensitivities are KR01s with respect to par yields.

The advantage is that the hedge falls out immediately. Suppose one-basis-point increases in the two-year, five-year and ten-year par yields raise a portfolio value by 20, 30 and 35 respectively. The portfolio is hedged with positions in par yield bonds of those three maturities whose yield-based DV01s are 20, 30 and 35. No simultaneous equations are needed.

Par yields are also the natural quantity to shift. The spot rate term structure is derived from the prices of whichever instruments trade most actively, and in fixed income markets those are par yield or close-to-par yield bonds, so the spot curve can change only when a par yield changes. To find the exposure to one par yield, change it, recompute the spot rates by bootstrapping, revalue the portfolio, and take the difference. Swap traders do the same with swap rates, though overnight index swap discounting brings a second term structure into the calculation.

Bucketing spot rates instead of choosing key rates

A bucketing approach divides the maturity spectrum into segments called buckets, then shifts every spot rate inside one bucket by one basis point and leaves the rest alone. With buckets of 0 to 2 years, 2 to 10 years and 10 to 30 years, the three effects on portfolio value again add to the DV01.

The contrast with key rate analysis is the shape of the shift: a key rate shift tapers away from its own rate, while a bucket shift moves every rate in a range by a full basis point and stops at the boundary. Banks use bucketing heavily in asset-liability management, usually with more than three buckets. A bank is reasonably well hedged when, for each bucket, a one-basis-point increase reduces the value of assets and of liabilities by about the same amount, fixed-rate loans being the major interest-rate-sensitive asset and fixed-rate term deposits the major such liability.

The bucketing idea applies just as well to forward rates. Suppose forward rates are calculated for six-month periods and the buckets run 0 to 2 years, 2 to 5 years and 5 to 10 years. Shifting the first bucket raises the four six-month forward rates that start at 0, 6, 12 and 18 months. The second covers six periods, starting at 24, 30, 36 and running on to 54 months, and the third covers everything from 60 months outward. The decrease in portfolio value from a one-basis-point increase in all the forward rates in a bucket is a forward bucket 01, and these sum to the DV01 calculated by changing forward rates.

Spot rates are built from forward rates, so the effect of any shift can be computed directly. With every rate expressed as a six-month rate under semi-annual compounding, the N-year spot rate R relates to the forward rate f covering t to t plus 0.5 years by:

The first bucket carries the largest 01 and the last the smallest, which is general rather than an accident of these numbers: an early forward rate sits inside the discount factor of every later cash flow, while a late one touches only the cash flows beyond it. Swap traders find these measures natural, since swaps are built out of forward rate agreements priced off current forward rates, and forward bucket 01s also suit swaptions.

Set beside a KR01, two differences stand out. A KR01 tapers away from one spot rate or par yield, reaching zero at the next key rate, whereas a forward bucket 01 shifts every forward rate in a range equally and none outside it. And a KR01 works on the rates used to discount cash flows, a forward bucket 01 on the building blocks those rates are made from. Both decompose the DV01, and both are hedged by solving simultaneous equations.

Every measure in this chapter is a currency sensitivity per basis point. Each has a duration counterpart, which states a percentage sensitivity with interest rates measured as decimals. Duration relates to DV01 by:

The identical conversion turns any component 01 into a component duration, and it applies without change to spot rate KR01s, par yield KR01s, bucketed 01s and forward bucket 01s:

For the three-year bond above, the 0 to 1 year forward bucket duration is 10,000 multiplied by 0.0102 and divided by 105.6014, which is 0.97.

Source: values computed for the bond in Example 4. The share column is derived here.

The three durations add to 2.74, the bond’s total duration measured against forward rates, split into the contribution of each bucket.