### Short end vs. Long end

As you come across short rate interest rate mean-reversion models i.e. Hull-White model

that has a time-dependent long term rate and time-dependent short term rate, there are short term policy rates that **prices expectations of changes in policy rate** that transmits through the entire curve, as well as the long term rate that **prices the outlook for inflation or future interest rate decisions, economic growth, supply-and-demand (issuance, reinvestment) and investors’ general attitude toward risk**. The model parameters are calibrated by fitting the model to near-risk-free market rates (OIS, Libor) outputting that the risk-neutral rate probability distribution.

**Short end (<1y):** From economics, we learn that exchange rate and interest rate are “shock absorbers” substitutes to stabilise foreign and domestic price pressures respectively (Only rates act as “credit regulators”). So you understand why the Fed needs to wait for domestic price pressures to build up to a point until they hike short term rates, which they need not if the external environment are deflationary i.e. oil prices are falling which exports counter-inflationary pressures to oil-import dependent countries, or that her trading partners exchange rates are appreciating against her currencies causing imported inflation, or that everyone is taking an accommodative stance on their domestic interest rate. **In summary, short end rate depends on domestic and external price pressures that are usually fueled by leveraged transactions provided by credit from banks.**

**Long end (have extended pass 5y belly due to QE, think Japan’s yield-curve-control):** Generally speaking, it likens the long term outlook of asset managers who allocate their core assets strategically to be either under- or over-weight to debts that provide a steady cash flow or assets whose cash flow depends on the economy. If he thinks that inflation will not pick up and rise to the targeted expectation set by the central bank in the long run i.e. 2% growth in price level implies stable growth and spending and hence employment and livelihood, which in turns means that business owners will reduce their expense and CAPEX in light of excess production capacity such as human capital, machinery or storable inventories, which causes the downstream flow of earnings and hence their expense too to tighten. Overall, this means a slower growth and depressed earnings expectations for assets, resulting in a depressed risk appetite that’s happy with a steady flow of cash flow that guarantees a smaller but stable return to his portfolio. However, this has been complicated with the large balance sheets of central banks debt/asset purchase programmes that artificially suppress the long term expected return. **Given the issuance-reinvestment technicalities and illiquidity of long term debts, as well as the subjectivity of long term expectations that looks beyond present hard data, the long end is the tricky part of the curve.**

*Note: An inverted curve either means that the country is very likely to have a near term crisis i.e. sovereign default on its maturing short term debt, such that near term risk premium is high. Or that the country is going to slump into a phase of low inflation, slow growth and depressed earnings expectations season. *

Briefly speaking, at the long end, there is uncertainty on where yields might be for the unforseeable future. That said, investors demand compensation for holding a higher duration risk on the long end. This is called **term premium **which is embedded into the long end pricing. TP does not account for real rate or inflation expectations and hence is calculated after offsetting those expectations. TP only compensates for uncertainty over fiscal and monetary policy mix, as well as actual possible inflation shock from expectations that’s different from what the current market is pricing in. However, there are still debates on whether the FNY model accurately captures this premium, as small changes to the starting point of calculation and holding periods does significantly change the magnitude and positivity of the premium. Think of it, it should be directly related to difference of 2 maturities sensitivity to the compounded maturity weighted average one factor YTM,or difference in duration. Usually, the 10y-5y spread is much smaller than 5y-2y spread as most yield volatility comes from the short end that’s within 5 years, and since 5 years is the age of a typical business cycle. So if 10-2 = 10-5 + 5-2 is flattening, it could be that policy mix such as timing of a change from dovish to hawish is less uncertain and higher yield is attracting capital flow which compress the term premium at the long end so 5-2 spread is decreasing. In other words, short end is as much as sensitive to long end now that uncertainty have been eased as plans of QT and hiking cycle have been revealed. **If you think, all that should be out there is already priced in and there’s no more alpha left to capture, think again. There’s still this unsettled pricing for uncertainty. **

If you want to keep your risk exposure small, you need not take a directional exposure with outright spot. You need access to forwards and futures though. **Instead, you can take a speculative view on the shape of the swaps curve**. *Note: Libor will be decommission by 2021*.

### Swap Curve

Let’s suppose you found a barbell portfolio where the 2 pivoting maturity gives the appropriate convexity volatility you are comfortable, or you are trading this on some technicality reason e.g. demand-supply schedule delayed due to policy stumbling blocks.

Short end maturity is S. Short end maturity is L. In swaps, we are concerned with the **fixed** leg only, that’s the leg we are paying or receiving. *Once we lock in the present rate for both shorter term and longer term legs on contract (i.e. pay fixed 6% longer end, receive fixed 5% shorter end, so that initial swap contract is valued at zero, *

*PV(spread trade) = 0 since each leg PV(swap 6%) – PV(swap 5%) = PV(spread trade), using PV full valuation avoid convexity approx error for large swap rate changes… in this case, the counterparty pays you 1% per annum*

*), if we*In other words, if we expect that maturity bucket of rate to rise along time, we pay to fix the relatively lower rate now so that we can receive higher in the future. If the contract maturity is long enough, it is likely to be partitioned into blocks of 3m swaps or FRAs, but need not be rolled from one contract to another, in which if the new issuance is very liquid and has a high bid-to-cover ratios, the on-run yield could be lower, where a roll adjusted returns is needed each time its carried forward.

**assume the forward swap rate is realised**and the longer end forward swap rate is moving up less than shorter end forward swap rate is moving up i.e. flattening, we**expect**to end up receiving >>>5% + fixed 1% and paying >5%, so we are in green.**Contract notional amount** always refer to the longest maturity swap contract since its more sensitive to swap rates. Whenever the investor is buying (selling), paying (receiving), lifting (hitting), taking (giving) or putting on a **steepener** (flattener), they are **paying** fixed on the **longest **swap, **receiving** fixed on the **shortest** swap. Selling a steepener aka buying a flattener. **Size of hedge exposure** is DV01(t) adjusted. The **fixed swap rate** **spread** you lock in on entering contract is the yield spread between the 2 legs of different maturities, expressed in bp. Given swaps curves are **typically upward sloping** due to a positive **term premium**, it is conventional to quote a price of a spread trade (swap rate) as a positive number i.e. the longest minus the shortest maturity swap rate. *Quotes are longest maturity swap rate and the fixed swap rate spread.*

**Bull steepener**–**receive nearer**swap rate,**pay further**swap rate**Bull flattener**–**pay nearer**swap rate,**receive further**swap rate**Bear steepener**–**receive nearer**swap rate,**pay further**swap rate**Bear flattener**–**pay nearer**swap rate,**receive further**swap rate

*Note: trading the curve is market-neutral (DV01 or delta) and P/L is only concern with the change in the shape of the swap curve, regardless if it is bullish or bearish on the underlying constant-maturity-treasury price. *

### Swap Fly

Besides having a view on the **steepness** of the swap curve, a trade on a view on the **curvature** of curve can be expressed with 3 maturities swap rates, the middle maturity called the ‘**belly of the fly’ **and the shortest and longest maturity as** the wings**.

A **butterfly trade **has** 3 legs**. Common example of maturities are **2y5y10y in ratios -1:+2:-1**. Its like short a **bullet** bond portfolio, long a **barbell** bond portfolio. Entering a contract to fix the present rate for these 3 legs, the price of this trade or the fixed swap spread is 2*PV(swap at belly) – PV(swap at shortest) – PV(swap at longest).

Similarly, this is delta-neutral trade as the notional amount of (typically) the belly is DV01 adjusted to determine the wings notional amount. DV01(t2) x Notional$(t2) / **2** / DV01(**t1**) = Notional$(**t1**) where t1<t2<t3. *Quotes are the belly swap rate, PV(either shortest or longest leg) and the fixed butterfly spread swap rate. Here, we calculate the implied swap rate from given PV and we can find the remaining wing PV and swap rate. *If the investor is **selling** (buying), paying (receiving), lifting (hitting), or taking (giving) a **butterfly**, then they are **paying** (receiving) **fixed** on the **belly** leg. And doing the opposite on the wings.

**Betting on a humped curve**– pay 2 times notional belly swap rate, receive shortest and longest swap rate**Betting on an inverted humped curve**– vice versa… Note: humped belly need not be 5y, can be customised. In other words, a crisis need not be in the next 5y that they are pricing in a higher yield for that duration, a hurdle in which if resolved, could mean a reverse of hump

Here are the corollaries that fermatslastspreadsheet have effectively been using:

Corollary 1: in the world of rates we refer to the price of bonds.

Corollary 2: in the world of swaps we refer to the price of libors.

The principle can be applied to every trade you ever look at, and you *should try* to apply it to every trade you look at. For example:

**if you are pricing a steepener trade are you buying or selling the spread?****if you are pricing an options trade are you buying or selling volatility?****if you are pricing a rates trade are you long or short the market?****if you price an asset swap are you buying or selling the basis?**

### Carry & Roll

However, there are scenarios where the curve or fly trade did not turn out as expected but you are still in green.

Carry is **guaranteed** and equals to the receiving payment over the holding period known at current time. That is, carry have no curve assumptions. We assume no counterparty default risk so that the debt is held to maturity and carry is earned.

Roll is **uncertain** subjected to assumptions (**not an expectation**) that the forward curve remains unchanged over time and the forward rates are realised as time pass i.e. future curve being equal to today’s curve, SR[0,5y] equals to SR[0,4.5y] in 6m time. **Long bond** positions or **receiver** positions typically have **positive** Roll + Carry return. Roll can be thought of as a buffer against market movements, but neither promises the receiver swap holder anything.

Note that **since Roll is contingent on static curve, whereas Carry is not**, Roll and Carry are **addictive**. Thus, we typically use Roll + Carry.

Consider two **receiver swaps (long bonds)**:

- Euribor
**5y swap**with a swap rate at SR[0,5y] - Euribor
**6m forward-starting 5y swap**with a swap rate at SR[6m,5y]. That is, in 6 months’ time, it becomes SR[0,5y]

**Notation:**

*F(Start Date, Maturity Date) is Fixing. Start Date not zero means forward starting. *

*SR(S, M) is Swap Rate. If S > 0, its a forward starting swap rate. If S = 0, its spot rate. *

*Swp(S, M, K) is a fixed-fixed rate swap, where K is the fixed rate. *

**Data:**

*spot SR[0,5y] = 1.023%*

*spot SR[0,4.5y] = 0.928%*

*forward-starting SR[6m, 5y] = 1.195%*

*forward-starting SR[6m, 4.5y] = 1.1015%*

*Euribor 6m = F[0,6m] = 0.319%*

*DV01(Swp(6m,4.5y)) = 4.45. Recap: DV01 = V*Duration/10,000*

*PV(0,6m) = 99.95% discount*

While there’s no carry in a forward starting swap SR[6m,5y] since there’s no certain payments (on the floating nearer leg 12m ahead that will only be fixed the next 6m), there’s a guaranteed carry for **spot** swap SR[**0**,5y] since their nearer leg (e.g. front month) have already been fixed i.e. 6m Euribor F[0,6m]. Carrying duration will be 1/2 since its 6m/1y i.e have to check day count convention.

That is, **for spot swaps**,

Carry (6m, FV) = ( SR[0,5y] – F[0,6m] ) * carrying duration = (1.023% – 0.319%)/2

However, the above is a **future value in terms of rates not scaled with DV01 for absolute returns (i.e. bps adjustment to swap price rather than SR)**. We can discount to PV with PV(0, 6m) and scale to absolute returns with DV01(Swp(6m,4.5y)). Here DV01 adjustment to absolute return is used and convexity is ignored as time increment of 6m is small.

That is, Carry(6m) = **99.95%** * (1.023% – 0.319%) / **4.45** = +7.9bps (**Usable for Non-Addictive Roll+Carry scenario. To discuss later**)

Alternatively, (**Addictive scenario**) we can first roll down the forward curve along the neared fixed 6m before determining the carry i.e. 2nd payment to 3rd payment (6m to 12m), 3rd payment to 4th payment (12m to 18m), .. so on and forth for longer carry settlement periods.

Roll is defined as the number implied by freezing the current curve, and reducing the maturity of the swap tail or letting part of the forward period elapse. That is, the PV will converge to FV. However, freezing the curve means roll is accurate **contingent on an unchanged nearest FV in the forward curve** e.g. FX Tomorrow/Next Deposit Rate, ON forward points.

In this case, we entered into a SR[0,5y] = 1.023% at present. Between spot and next 3m forward swap rate SR[0,4.5y] at 0.928%, implied 3m forward points is (positive since backwardation) +9.5bps (1 bps = 0.0001). The daily roll would be +9.5bps / days in that 6m. This concept is prevalent in any contract with a forward or futures curve i.e. you can see forward points in FX spot against its forwards, VIX (or ES as proxy) against VX (weeklies) futures, CLA cash-settled spot against next month CL1 futures contract. Just imagine that for a USDCHF, the forward curve is in backwardation and the longer the tenor of carrying FX, the larger the + implied forward points. The longer you carry USDCHF, your opening price at PV will be adjusted lower to FV and that will increase your long position P/L. That’s why backwardation curve represents positive carry.

For 5y spot swap, Roll(6m) = 1.023% – 0.928% = +9.5bps

For 6m5y forward-starting swap,

We can**linearly approximate**absolute carry as +7.9bps. Again, this is only for

**standard**roll where roll period is before the next fixing period.

We can **linearly** approximate **absolute** carry (No need scaling with DV01 or discounting to PV) with

Note that **by rolling down 6m, SR[0,5y] rolls to SR[6m,4.5y]** and that is used to minus against spot swap rate SR[0,5y] to determine carry(6m).

By the second (addictive) method, we get the same result.

Carry(6m) = 1.1015% – 1.023% = +7.9bps

**However, such addictive approximation is possible only if the roll period is equal or less than the payment frequency of the swap**, i.e. the next maturity. Say **6m** roll on **1y** forward starting 5y swap. Otherwise, its a **non-standard** roll & carry i.e. settlement period in excess of fixing period (**Non-Addictive Scnario**). For example, look at 12m carry+roll on a spot swap that pays every 6m, or say 6m roll on a 3m5y swap. Why? For 3m5y, F[0,6m] is fixed and known, but F[3m,9m] fix is floating. That said, determining carry using the linear approximation after first rolling the curve cannot be used since the roll only capture F[0,6m] and miss F[3m,9m].

*Say 1y roll + carry = { SR[0,5y] – SR[0,4y] } + { SR[1y,4y] – SR[0,5y] } *

*= SR[1y,4y] – SR[0,4y]*

*Above, Carry is using linear approximation formula. *

*That said, SR[1y,4y] which derived from the linear approximation carry assumption also calculates a roll contingent on static curve, that is, Roll(1y) = SR[0,5y] – SR[0,4y] to roll SR[0,5y] down by 1y to a forward-starting SR[1y,4y]*

*Problem is that forward starting SR[1y,4y] and spot SR[0,4y] are non-addictive because both roll and carry are contigent on the same assumption i.e. Roll(1y) = SR[0,5y] – SR[0,4y]. *

That said, **Non-Addictive Roll+Carry scenario **is preferred if roll period is in excess of payment frequency of the swap.

To sum up, Roll(6m) + Carry(6m) = 9.5bps + 7.9bps = +17.4bps

**For forward starting swaps**, there’s no carry, and roll simply entails reducing the forward period. This avoids the hassle of non-addictive roll+carry but forego a guarantee carry.

That is, Roll(6m) = SR[6m,5y] – SR[0,5y] = 1.195% – 1.023% = +17.2bps

Note that forward starting swap SR[6m,5y] expires (date when no more trade allowed) and matures (settlement date) in 5.5y whereas spot swap SR[0,5y] expires and matures in 5y.

An **example** is a 2s5s 6m forward-starting flattener.

Recap: **Flattener receives the further **maturity (5y), **pay the nearer maturity** (2y) i.e. receive 6m5y, pay 6m2y in (DV01-neutral) ratio 1:2.48 (2.48 since 2y less sensitive to rates than 5y) with notional amount settled in the longest maturity 6m5y or 5.5 years. Risk basis is then the DV01 of 6m5y swap (4.93)

**Pay => – Roll**

**Receive => + Roll**

For the **DV01-neutral** 2s5s i.e. **long 100m notional of 5y swap, short 248m notional of 2y swap**

Roll(6m) = Roll on 5y – Roll on 2y

= ( SR[6m,5y] – SR[0,5y] ) – ( SR[6m,2y] – SR[0,2y] )

= (1.195% – 1.023%) – (0.632%-0.498%) = +3.7bps in terms of rate (or yield)

Since both pay and receive legs are on a consistent risk basis (DV01-neutral), roll are addictive.

To calculat the P/L, convert the roll in terms of rate to absolute returns by multiplying with DV01 of 6m5y swap, that is,

P/L = 100m * 3.7bps * 4.93 = $182,400

In summary, the 10y-2y yield spread stands at +85bp as highlighted by veteren Bill Gross. It has flatten the last 4 months that actual inflation fell short of Fed inflation target. Not to forget, debt ceiling issue is around the corner and presents a near term hurdle for the US sovereign to overcome, that said, it was one of the contributors to a flatter curve. The curve is away from recession (inversion) by only 3 25bp hikes, which market has not been pricing into the OIS and FFR futures. September will be closely watched for whether the incumbent party is able to convince the Democrats to suspend it’s limits again or raise the ceiling, as well as the Fed’s decision to reveal more details on the QT besides the series of cap of 5bil that could be insuffice to steepen the curve.

**References:**