# 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?

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:

1. Cost of funding for 1 year the purchase price: 110*2.5%=2.75
2. Coupon received at T=1: 5
3. 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.

• Thanks. Concerning the no arbitrage forward price, the formula forward px = spot-carry must really only apply to a bond that starts in the future, not a bond that actually ages over the carry period as it doesn't take into account the pull to par. – Wadstk Feb 18 '19 at 22:32
• Im saying, pull to par is known income in a sense, so that carry = (coupon income - financing rate) - pull to par – Wadstk Feb 18 '19 at 23:41
• If you look at this answer by @Helin quant.stackexchange.com/questions/25329/… you will find some useful info. "[carry] is basically coupon income + pull-to-par - financing cost, in yield terms", and "forward price = spot price – carry" – Alex C Feb 19 '19 at 1:06
• I have seen that. It is really the only time i have ever seen pull to par added to the basic carry = CI-financing cost equation. Using this equation, then, I get a arbitrage free forward price of 110-(5-1.9-2.75) = 109.65 which is silly. No one would buy a forward contract on this bond at that price given pull to par – Wadstk Feb 19 '19 at 1:25
• You can estimate carry using (Spot Yield - Repo)/DV01. Where is the pull to par effect in this method then? – VanillaCall Feb 20 '19 at 2:02

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.

• i dont see a 5% 5yr bond pricing at 110, if the yield is 3% – ZRH Feb 18 '19 at 22:24
• I'm just using the OP's hypothetical example for discussion purposes of the pull to par and carry effects – VanillaCall Feb 18 '19 at 22:44