Bobl spread is 53.1bp, we are 3 months away from mar18 delivery, and a client blasts “what do you see as carry and roll for OE asw?”.
Here are my notes on the mechanics of the calculation as well as some thoughts on its limitations.
Published on 12 Dec 2017 (updated)
Back to basics
Carry and roll-down are often used interchangeably, but are fundamentally different.
Carry is the PNL resulting from the income and costs of running a position over a certain horizon, regardless of the mark-to-market. Note this is technically a cash amount.
A simple example is the carry from renting a flat: the carry is the difference between the rent (income) and the outflows such as mortgage, tax, etc (costs). As mentioned, this is a cash amount, say $2k per month; but it might be helpful to express it in relation to the asset we are inspecting: for example, say the value [not the cost price!] of the flat is $500k, then the annual carry in percentage of the asset value is (2k * 12) / 500k = 4.8%.
Similarly, in the rates world, it is more convenient to express the carry not in actual cash terms, but in relation to the yield of the instruments we are inspecting.
In the case of the swap, the carry comes from the difference between the current swap rate and the first fixing rate. It is preferable to express it in terms of annualised basis points running, so the difference in rates is multiplied by the day count basis and then divided by forward Modified Duration.
In the case of a bond, it is the difference between the yield-to-maturity and the repo or funding rate, then again multiplied by the day count basis and dividend by the forward Modified Duration. Note that using the coupon instead of yield-to-maturity would lead to a misleading result: take for example the carry of a 10% coupon 1y bond trading at 110.
More conveniently, as the forward is itself priced under the no-arbitrage assumption by using the spot price and the carry, we can derive the carry by relying on common analytics that calculate the forward: for a bond, carry is the difference between the forward yield and the spot yield and, similarly for a swap, the carry [let’s pick a 3y swap] equals
$$ \text{3m carry for 3y swap:}\quad 3\text{m fwd }2\text{y}9\text{m swap } – \text{ spot }3\text{y swap} $$
Roll-down is the mark-to-market of a position resulting from the passage of time, assuming that the shape of the curve remains unchanged. This represents only one of the possible scenarios and constitutes therefore a very strong and arbitrary assumption about the fate of the position.
The concept of the roll-down is a bit less intuitive. I like making the example of the depreciation of a car: say the price of a brand new car is $25k, but in the secondary market the 1 year old used car of the very same model trades at $20k, so you can indicate that the 1y roll-down is -$5k.
For the swap, the calculation is straightforward:
$$ \text{3m roll-down for 3y swap:}\quad \text{ spot } 3\text{y swap } – \text{ spot }2\text{y}9\text{m swap} $$
Observe that, when combining the 2 equations above to obtain the carry and roll-down for swaps, the spot-3y terms cancel out, so
$$ \text{carry + roll-down for the 3y swap:}\quad 3\text{m fwd }2\text{y}9\text{m swap } – \text{ spot }2\text{y}9\text{m swap} $$
It can get more complicated though. Say we are looking at wine. Each vintage has its own idiosyncrasies. The value of the 2016 vintage might trade at premium vs both the 2015 and the 2017 vintages just because of how the weather ended up being in 2016. We cannot reasonably suggest that in 1y time the 2017 vintage will roll-down to the current value of the 2016 vintage.
The wine framework applies more to the bond curve, where each bond (vintage) has its own idiosyncrasies. These unique traits might never vanish because the bond cash flows are not fungible, while on the other hand swap cash flows are. So, in the case of a bond, the 3-month roll is the difference between the current spot yield and the yield of a proxy bond with similar characteristics but 3-month-shorter duration.
Bobl asset swap in numbers
The bobl asset swap (or OE asw) is defined as the spread in basis points between i) the yield of the swap starting on the delivery date of the OE future and ending on the maturity date of the cheapest-to-deliver (CTD) and ii) the future-implied forward yield of the CTD. So a positive asw means the future trades rich vs swap.
Buying a bond future is equivalent to buying the underlying cheapest-to-deliver (CTD) at a certain forward price and locking the funding at the implied repo rate. For this reason, we can improperly talk about the carry of a future as the difference between the future-implied forward yield of the CTD and the spot yield of the CTD, even though technically being long a future only provides mark-to-market exposure and therefore should not have any carry.
To answer the client’s question, we could calculate the carry and roll-down (C&R) of both the future’s leg and swap’s leg separately. However, as the future asset swap (asw) is quoted with matched maturities, we only really need a run of the German spot asset swap curve. The CTD is highlighted in red.
We also need to specify a horizon for the calculation. We should pick the 3-month horizon because it exactly matches the delivery date of the future. It is rather hard to define the carry and roll-down of a future for a horizon that goes beyond the future expiry.
The future’s matched-maturity asw represents the CTD asw out of the delivery date, so the 3-month carry is the spread between the CTD spot asw and the OEH8 asw. With our numbers,
$$ \text{Carry:}\quad 53.7 \text{bp} \; – \; 53.1\text{bp} = 0.6\text{bp for 3 months.} $$
For the roll-down, we need an estimate of the spot asw level for a proxy bond with similar characteristics to the CTD, but with 3-months shorter maturity. Same as before, we only need a run of the spot asw curve.
However, note that we should make sure we disregard structurally different types of bonds, such as the whole DBR curve or historically rich/cheap paper, because their valuation would not be representative of the proxy bond we want to construct. Compared to the swap calculation, the roll-down analysis of a bond is somewhat subjective. In this case, I choose to believe that the OBL oct22 will simply roll towards the OBL apr22, hence
$$ \text{Roll-down:}\quad 55.8 \; – \; 53.7 = 2.1\text{bp over 6 months} \rightarrow 1.0\text{bp for 3 months.} $$
Interestingly, the roll-down hypothesis forces to zero the value of any optionality, including the short option deriving from being long a bond futures position. This option rarely goes in-the-money anyway, but it could.
The limitations of the roll-down measure
The assumption that the curve remains unchanged through time is very strong and cannot be hedged.
The textbook theory on interest rates term structure indicates that the shape of the curve is determined by a combination of i) interest rate expectations, ii) liquidity preference/risk premium, and iii) preferred habitat. By definition, the roll-down assumption ignores any interest rate expectation: it assumes that the forwards will not be realised.
At times, this assumption can be totally misplaced. E.g., if the FOMC curve is fully pricing in a 25bp hike for tomorrow, the roll-down of receiving the FOMC meeting dates would be 25bp over 1 day, which is a silly consideration for this trade. This trap is extreme and therefore easy to spot, but it is a bit trickier when discussing the roll of the AUD 1y1y vs USD: how much is interest rate expectation and how much is risk premium on the curve?
Similarly, when calculating the 1y roll-down of the OE asw, how much is risk premium and how much is expectations of continued QE?
Appendix: the Australian example
The AUD market has its own jargon and conventions. The asset swap is easier to quote, but less straightforward to analyse.
The exchange only offers two liquid bond futures: YM in 3y and XM in 10y. The futures trade in 
3y and 10y are also where the activity in swap spreads, improperly called EFP or Exchange-For-Physical, is concentrated. The EFP is defined as the spread between the spot-starting generic swap (3y or 10y) and the bond future’s implied yield (YM or XM respectively).
In light of this convention, the calculation of the C&R is particularly laborious because the start-end dates do not match and the shortcut of looking at the spot asw run cannot be used: C&R need to be calculated independently for both the swap leg and bond leg.
I noticed that a surprisingly common method is to work out the C&R on the swap leg (so far so good) and to ignore the future’s leg “because it is a forward price already and has no cash flows”. This method is very straightforward and (alas!) very wrong: it assumes that the swap curve remains unchanged through time while the bond’s forward will be fully realised.
Let’s work out the carry and roll-down of being long 3y EFP: we again assume we are 3 months away from expiry and the YMH8 is trading at 98.03 / 98.035, hence 1.9675% yield.
The swap leg is trivial. If we buy the 3y EFP, we pay the swap, so it’s a negative amount, roughly -5.1bp at the time of writing.
$$ \text{C&R for the 3y swap:}\quad 3\text{m fwd }2\text{y}9\text{m swap } – \text{ spot }2\text{y}9\text{m swap} = -5.1 \text{bp for 3 months}$$
The future’s carry is the difference between the future’s yield (1.9675%) and the weighted spot yield of the basket’s bonds (1.9625%), so +0.5bp.
The roll-down is the difference between the spot yield of the basket and spot yield of a proxy basket with 3-months shorter maturity, which is constructed by identifying the yields of proxy bonds for every bond in the basket and then by taking the weighted average of the yields. By looking at the ACGB curve, I would say approx 3.5bp for 3 months. So,
$$ \text{C&R for the 3y future:}\quad 0.5 + 3.5 = 4.0 \text{bp for 3 months} $$
Putting it all together, we get
$$ \text{C&R for 3y EFP:}\quad -5.1 + 4.0 = -1.1 \text{bp for 3 months} $$
The data itself is not meant to be accurate, but only used for illustration purposes. Please read the full disclaimer.