*directional*trades: how much money do you make (or sadly, lose) if something happens (such as a yield change)? Directional trades are fun to talk about -- and easy to analyse -- but unless you have an amazing forecasting record, not a great way to make money in the short term. In order to control your risk, you need to do

*relative value*trades -- buy one or more instruments, and sell one or more others.

In particular, if we are interested in breakeven inflation, we are interested in a relative value trade (unless you enter into an inflation swap). As soon as we no longer just buying (or selling) one instrument, we run into the oft-repeated question: what is the hedging ratio? (The

*hedging ratio*is the relative sizes of the instrument positions involved in the trade.) The correct answer to this question is: what are you trying to accomplish? For relative value trades involving linkers and nominal instruments, we could have either a larger or smaller position in the linker, depending upon what you are trying to do.

Even if you are not interested in structuring fixed income relative value trades, understanding this concept will help you better understand fixed income analysis produced by sell side research analysts, and possibly understand some empirical behaviour of breakeven inflation.

**Note:**

*This article is an unedited first draft of ideas that will make their way into an upcoming report on breakeven inflation. This report is expected to be more technical than my previous output, and so there may be more quantitative examples injected into the text before it is completed. My objective here is to get all my ideas sketched out, and I will then chop out the loose bits of logic as part of the editing process.*

## Background: DV01

The usual way of discussing the price (or return) sensitivity for bonds in introductory texts is to use the bond duration. However, duration is an inadequate (if not downright wrong) way to measure fixed income portfolio risk. The easy way to measure risk is to use the dollar value of a basis point, or DV01. Since not all investors are dollar-based, we typically write DV01.For a position in an instrument, the usual definition is to define the DV01 as the change in the local currency (e.g., dollar) value of the position if all interest rates increase by one basis point. For vanilla bonds, the relationship yield up/price down implies that the DV01 is a negative number. (Some systems might flip the sign, since looking at negative numbers all of the time gets annoying.) However, the DV01 of an instrument can be positive, such as paying fixed in a vanilla interest rate swap.

The downside of the DV01 is that the measure is dependent upon the size of the position. This makes it hard to describe the scale of positions. For a retail investor, a DV01 of $1,000 might seem dizzying -- losing (about) $100,000 if rates move 1%! Conversely, for a big institutional investor, a DV01 of $1,000 is laughably small.

The advantage of DV01 over modified duration as a risk measure is that it applicable to derivative instruments. For vanilla bonds, we can use the modified duration and the change in yields to approximate the percentage return of the position. For an on-market swap, the NPV of the position is zero, and so any profit or less represents an infinite (negative) return. Duration analysis also a complete mess if we try to apply it to index-linked/nominal spread trades.

One should note that the relationship between price changes and yield changes is not exactly linear: the change in value for a 10 basis point move is not exactly equal to 10 times the DV01. The DV01/modified duration changes slowly as a function of yield, an effect that is described by the convexity. However, interest rates have to move a lot for convexity effects to show up; you just need to periodically refresh hedge ratios. It would be fairly unusual for convexity effects to reverse the sign between the true profitability of a trade, and the approximation generated by multiplying the spread change by the DV01, and adding in carry. (This breaks down for instruments with embedded options. As a result, instruments with optionality are referred to as non-linear instruments, and fixed income chatter will use

*optionality*and

*convexity*interchangeably.)

With respect to index-linked bonds, the beauty of the DV01 is that it converts the yield sensitivity to a current dollar amount. This is unlike other analytics for such bonds (for the Canadian linker model, at least), which are expressed in real terms.

## Hedge Ratio and Carry

Most relative value analysis will start off with a spread chart: the yield of one instrument versus another. The analyst will then come up with a story why that spread will go up or down, possibly by using the highly advanced technique of drawing some lines through the spread history.However, the spread movement is not enough to tell us about the profitability of spread trades for holding periods that extend beyond the current trading day. We need to incorporate the carry of trade, which is the interest cost/gain that you make solely based on holding the position.

Take an example where one bond has an interest rate sensitivity ten times the other. A $10,000 position in the long maturity bond has a DV01 of $10, while a $100,000 position in the short maturity also has a DV01 of $10.

We cannot just look at the fact that the long maturity bond has a yield of 100 basis points more than the short to assume that buying the long and selling the short on a DV01 neutral basis has a positive carry. Since we are selling short $100,000 and only buying $10,000, we are implicitly stuck with a $90,000 dollar investment in cash -- which we assume has a DV01 of zero. If the short maturity bond has a sufficient yield pickup over cash, the interest cost on the short position is greater than what is received from the long legs.

Since inflation-linked bond notional amounts are indexed to CPI -- which can achieve very high annualised returns over short periods -- the net exposure to inflation for breakeven trades is highly dependent upon the hedge ratio.

## Hedge Ratios for Linker-Nominal Trades

There are three plausible hedge ratios that one can use for trades between conventional bonds and linkers.- Maturity/Market Value matched.
- DV01 matched.
- Empirical DV01 matched.

I discuss these in turn.

## Maturity/Market Value Matched

If we believe that the economic breakeven inflation at a maturity point is too high or too low relative to our expectations, and we have the capacity to hold to maturity, we want to match maturity and market value amounts of holdings (at the time of trade entry).

(Note that market value will diverge from notional amount as a result of the bonds trading away from par, and the result of previous inflation indexation.)

If we assume that both instruments are zero coupon (and make simplifying assumptions about yield convention), such a trade structure will break even if held to maturity and realised annualised inflation matches the quoted spread between the two instruments.

*(Note: this is the definition of economic breakeven; the economic breakeven will differ slightly from the instrument spread as a result of various deviations from the simplified mathematical quote conventions.)*
The key is that the discounted value of the instruments are equal both at trade inception and maturity if realised inflation exactly hits economic breakeven inflation. At an instant before maturity, the instruments are converging to a zero maturity, and the DV01 for both will equal zero. If the yields are unchanged, we will see that the DV01 of the two positions will also converge. However, unless the economic breakeven is zero, the starting position has a net DV01 that is non-zero. If the economic breakeven inflation rate is positive, the nominal yield is greater than the indexed yield, and the inflation linked bond position will have a DV01 with greater magnitude than the conventional bond position. That is, the position is not DV01-matched at inception.

Since you are not DV01-matched at inception, some traders will argue that this is the "wrong way" to hedge; you need to DV01 match at inception (described next). However, doing it that way means that you are no longer locking in the relationship between economic breakevens and realised inflation.

In the real world, things are complicated by a few factors. Firstly, bonds have coupons, and so the future value of the index-linked bond depends upon the path of inflation, not just the annualised rate to maturity. The second issue is that maturities may not be matched, creating a gap that is exposed to the full force of CPI seasonality -- which is large. Finally, there are complications with the quote convention, and the fact that CPI indexation is done with a lag (which means that the initial path of indexation is fixed).

## DV01 Matched

In this structure, you buy/sell bonds so that the index-linked leg has the opposite DV01. Once again, if we assume a simple zero coupon example, if expected inflation is positive, the unit DV01 of a index-linked bond is greater. This means that we would have a smaller market value position in the index-linked bond than the conventional.

If we continue to DV01-match the position to maturity, and quoted yields are unchanged, the implication is that we need to keep buying the index-linked bond to keep up with the decay of the position DV01. This means that we will underperform if inflation is greater than the economic breakeven during the beginning of the trade, as we had less market value held of the index-linked bonds. Therefore, we can no longer compare the economic breakeven of inflation and realised inflation to determine the profitability of the trade.

The way to keep this straight is to realise that there are two interpretations of the yield difference between a maturity-matched conventional and index-linked bond.

- It is an approximation of the true economic breakeven; if we want to trade the economic breakeven versus realised inflation, we market value match.
- It is a spread between two instruments, and we are just trading the spread in the same way we trade other spreads. There is no interpretation with regards to realised inflation to maturity; this is a short-term trading concept.

## Empirical Matching

Finally, there is a school of thought that argues that both of the previous approaches are the wrong answer for true relative value trading. Both approaches give a hedging ratio that results in a position that has embedded directionality in practice.

The argument is as follows. One can observe that index-linked bond yields move less than conventional yields during severe market moves. (I discussed this concept in an earlier article.) Basically, breakeven inflation is directional with interest rates. If we want to be truly non-directional, we need to put on hedges based on empirical hedging ratios.

*(This is similar to the idea behind using Principal Component Analysis factors to weighting butterfly trades.)*
I often heard that we could assume a 2:1 ratio for conventional/indexed yield moves.

*(As a disclaimer, I believe I put out proprietary research making such a claim.)*That is, a matched-maturity index-linked bond yield moves 50% as much as the conventional bond in a big interest rate move. (There are a lot of theoretical issues around that claim; the linked article introduces them.)
If we believe that claim, that means that the conventional DV01 has to be 50% of the index-linked DV01. (This is obviously way different than the DV01-matched position.) Once again, we will no longer be locking in an economic breakeven; rather, we are trading the relative attractiveness of the asset classes.

Under the assumption of a positive inflation breakeven, we can have have these hedging ratios implying either an equal market values (first method), a smaller index-linked market value (DV01-matched), or a greater index-linked market value (empirical DV01 matching).

*You pays your money, and you takes your chances.*
My intuition is that empirical DV01-matching is a self-fulfilling prophecy. If relative value trading is dominated by traders who believe in the same hedging ratio, any yield shifts generated by directional traders that do not conform to the empirical hedging ratio will generate profits/losses to the relative value traders on the two sides of the trade. Profit-taking activity would tend to push the yields back to the relationship implied by the empirical hedge ratio.

As I discussed in the previous article, such arguments are incompatible with anchored inflation expectations (which are also an empirical feature of modern breakevens). The net result is that someone with a purely fundamentalist approach to analysing breakevens will have a hard time interpreting breakeven inflation changes. What may appear to be "unanchored inflation expectations" or "changing risk premia" may be just the result of market participants following an empirical DV01 hedging strategy. As a result, market movements are very useful for generating excitement among economists about unusual market behaviour, when the markets are just following a simple behavioural pattern.

## Concluding Remarks

You need to know what you want to do first, then decide on what hedging ratio gets you there afterwards.

## Appendix: Inflation Swaps

Entering into an inflation swap is one way to literally lock in the relationship between realised inflation and a market-implied expectation. If that is the only leg to your trade, there is no hedging ratio. However, if you are hedging an inflation swap with bonds, you are back to worrying about hedging ratios.

(c) Brian Romanchuk 2018

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