# Cost of Carry Bear Flattener

I was reading a report last week that

“the carry on a 2s5s gilt curve flattener is negative to the tune of 10bp over 6 months”

and I realised I have little understanding of this concept and how this cost is calculated since the only cost of carry I ever studied in was storage costs for commodities in futures formulae in college.

There was a surprising lack of literature I could find on curve flattener cost of carry but I did find this thread Roll down and Carry for 2/5 on the Wilmott Forums which gave a ballpark formula as

$$DV01*(2s5s slope-(5y1y-2y1y))$$

Ballpark formula is fine for me since this is just an intuition exercise. Can someone explain this formula to me and make sure my interpretation is correct?

1. DV01: Portfolio sensitivity to yield changes
2. 2s5s slope: the return from selling the 2yr position and buying the 5 yr in the bear flattening position
3. 5y1y/2y1y: the costs of buying and selling the positions in one years time when I exit the trade

Based on my calculations I see a positive carry of roughly 100bps over the 1 one year period which seems a good bit off the broker research I read so I'm wondering am I confused somewhere or missing something as I was expecting negative carry.

My DV01 is the average of a short gilt benchmark over the last two years and I calculated the rates one year from now by simply strapping the curve

$$5y1y: (1+y_5 )^5 (1+f_1 )^1=(1+y_6 )^6$$

$$2y1y: (1+y_2 )^2 (1+f_1 )^1=(1+y_3 )^3$$

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Just dawned on me this formula contains DV01 which is in dollar/currency format... need this on a relative rather than absolute basis (bps or % points) –  TylerDurden Aug 13 '14 at 13:39

In this context, I believe carry refers to the sum of "pure" carry + roll down.

Carry, in the most general sense, is the return of a position in a static world; i.e., assuming time is the only variable that is changing, what's your holding period return on a trade?

When you buy a bond, the "total carry" is the sum of

1) "Pure" carry – you get interest accrual and coupon payments. Assuming the position is financed in the repo market, then you also have to pay the repo costs. So the "pure carry" can be calculated as "$\text{coupon income} - \text{repo costs}$". This gives you the carry in dollar terms. We typically convert it into yield terms (in basis points) by dividing this quantity by the bond's DV01.

An alternative, in fact far more used way of computing pure carry, is "$\text{forward yield} - \text{spot yield}$". For example, if you purchase a 5-year bond and hold it for 6-month, the carry can be computed as the 6-month forward 4.5y yield, minus the current spot yield. This method is slightly more accurate then the first method described above, since it also accounts for the "pull-to-par" effect.

2) Rolldown – the yield curve is typically not flat. If it's upward sloping, yield will decline as time passes by. This is an additional source of static return. Rolldown is typically computed as the difference between the current yield, and the yield x-month later, assuming an unchanged yield curve.

To summarize, the total 6-month carry ("roll and carry" or "RD&C") for, say a 5-year bond is = (6-month forward 4.5y yield - 5y yield) + (5y yield - 4.5y yield) = 6m forward 4.5y yield - 4.5y yield.

In the case of a flattener, you simply compute the roll & carry for both legs, and then take the difference.

In your title, you mentioned "BEAR" flattener. Bear flatteners are typically structured using options. For example, you may buy a 6m2y payer, while selling a duration-weighted 6m5y payer. The carry of this trade would be much more complex - there will be carry from rolling down the yield curve, carry from time decay, and carry from changes in the vol surface.

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