Separating Carry, Pull to Par, and Roll Down for a Premium Bond
Summary
The document examines how to distinguish coupon and funding carry from a premium bond’s price change as it approaches maturity, and from yield-curve roll down. Its example considers a five-year, 5% coupon bond priced at 110 with funding at 2.5%. The accepted answer first recalculates the yield implied by the stated price and cash flows, finding it differs from the assumed 3%. It then reprices the bond one year later at the same yield with one fewer year to maturity. Combining that value with the coupon received and financing cost yields a modest positive net effect in the example, rather than treating the full coupon-minus-funding amount as spread cushion.
The response distinguishes known cash-flow carry from pull-to-par price movement; roll down is a further effect that depends on how the bond’s maturity point moves along an unchanged yield curve. The source’s answers disagree on whether the initial carry figure is pure carry, and its simplified forward-price language is not fully consistent across responses. The example therefore illustrates the components, not a universal calculation rule.
Key ideas
- Coupon income minus funding cost measures the cash-flow carry described in the example.
- A premium bond’s value can decline as its maturity shortens, creating a pull-to-par effect.
- The accepted answer recomputes yield from the bond’s price and coupon before estimating the one-year value.
- Roll down depends on the yield curve and is separate from coupon carry and pull to par.
- The source includes conflicting explanations, so its numerical example should not be treated as a universal convention.
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# Carry and Rolldown of a Premium bond
# Carry and Rolldown of a Premium bond
I'm hoping that you may help me understand how the pull to par of a premium bond impacts the carry and roll calculations over a year.
I understand that carry = Coupon income - cost of funds and that the forward price = Spot-carry
If a 5y bond paying a 5% coupon was priced at 110 to yield 3% with a cost of funds of 2.5%, I would say that the carry of the position is 5 - 110*0.025 = 225bp. This tells me that the 1y forward price must be 107.75. A lot of traders will take 225/5 = 25bp and say that there is 25bp of spread cushion over the year before the trade breaks even. I can see doing that on a par bond but am not sure that it applies on a premium bond.
Now, the pull to par on this 5y bond is going to be roughly 2 points a year (coup-ytm). Does this mean that out of the 2.25% carry, 2% is from rolling to par and 25bp is true carry?
## Answer by ZRH (score 5, accepted)
https://quant.stackexchange.com/a/44117
To determine the yield, you need to solve the following equation ($R$ being the yield, $N=5$ in your example):
$P_{bond}=\frac{100}{(1+R)^{N}}+\sum_{i=1}^{N}\frac{\mathit{coupon}}{(1+R)^{i}}$
For $P_{bond}=110$ and $\mathit{coupon}=5$, this results in a yield of `2.83%` and not `3%`, as stated above.
The pull to par of the bond would be determined via revaluing the bond after 1 year with the above formula as a 4-year bond with `5%` coupon (still assuming the `2.83%` yield determined above), resulting in `108.11`.
Therefore, the overall balance looks as follows:
- Cost of funding for 1 year the purchase price: `110*2.5%=2.75`
- Coupon received at `T=1: 5`
- Value of bond after 1 year (now a `5%` 4-yr bond): `108.11`
Makes for a new value of the position (after paying interest) of `108.11+5-2.75=110.36`. So, net interest effect equals `5-2.75=2.25`, or in yield terms `2.25/110=2.05%` (gain), and pull-to-par loss equals `(110-108.11)/110=1.72%` (loss). Net effect is thus 2.05%-1.72%=33bp. With your simplified formula you would have said it should be `225bp`.
As for the forward price, this would be the `108.11` price that the bond would have in a year's time, discounted back to today at the yield, i.e. `105.14`.
## Answer by VanillaCall (score 0)
https://quant.stackexchange.com/a/44118
No, the 225p is your pure carry. This is the portion related to known cash flows. You know exactly what your coupon earned is and what your repo financing costs are.
The pull to par effect is separate. If your bond is currently priced at 110 at 3% yield, then you would basically price what the bond would yield assuming the same 110 price but one year shorter. This will give you the pull to par effect.
The roll down effect assumes a static environment where the yield curve is unchanged. In one year, your 5 year bond will become a 4 year bond. If the 4 year point is currently yielding 2%, this means the yield curve is upwards sloping so your bond will roll down from 3% to 2%.
In sum, you will have pull to par plus roll down effect.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.