How Yield Relative to 6% Affects Bond Futures CTD Ranking
Summary
The document explains how yield relative to the 6% conversion-factor benchmark can change which bond is cheapest to deliver in a futures contract. A conversion factor is the bond’s clean price at a 6% yield. To compare delivery candidates at a market yield, divide each bond’s price at that yield by its conversion factor and select the minimum adjusted price.
A first-order expansion around 6% shows that the ranking depends on the sign of the yield difference and on each bond’s price sensitivity relative to its price. That sensitivity ratio is the negative of modified duration, subject to duration and sign conventions. Thus the yield regime can reverse which duration profile is favored; the conversion factor itself remains the price at the fixed benchmark. The approximation may not settle the ranking when candidates are close or higher-order effects matter. In such cases, frequent CTD switching signals uncertainty, and the delivery option should be considered rather than relying on a single stable CTD.
Key ideas
- A conversion factor is the clean bond price at the benchmark yield of 6%.
- CTD selection compares each bond’s market-yield price divided by its conversion factor.
- A first-order expansion links the adjusted-price ranking to modified duration and the sign of the yield deviation from 6%.
- The preferred duration profile can differ above and below the benchmark yield.
- Close rankings and frequent switches make a single identifiable CTD less reliable and raise delivery-option uncertainty.
Tags
Full text
# CTD and conversion factors above and below 6%
# CTD and conversion factors above and below 6%
I'm trying to deepen my understanding of how cheapest-to-deliver (CTD) bonds and conversion factor (CF) behave under different market conditions. Specifically, I want to analyze the impact of bond maturity and coupon rates on CTD and CF, particularly when coupon rates are:
- Above 6%
- Below 6%
I know that CF is used to standardize bond prices to a common value for delivery in futures contracts. However, I want to be clear on the following:
- How do maturity and coupon rates (above or below 6%) influence the CTD bond selection?
- Are these regimes' behaviours correct?
| YTM Regime | CTD char. | Conversion Factor Behavior for the CTD |
| YTM > 6% | High duration, high maturity, low coupon | High CF |
| YTM < 6% | Short duration, short maturity, high coupon | High CF |
I think the cf is the price of a unit notional bond at 6% ytm. As such it should increase with maturity - and cannot see how that can have differing behaviours above and below 6%? Is there a mistake in the above table?
## Answer by Andrea (score 2)
https://quant.stackexchange.com/a/81528
Assume all bonds have the same yield $y$; we know that the Conversion Factor $C_i$ of each bond is its clean price when $y=0.06$
$C_i=P_i(0.06)$ ($P$ being clean price)
In order to rank bonds (when deciding which bond you should deliver), you scale each price as $\frac{P_i(y)}{C_i}$, and take the minimum.
Taylor expand around 6% and call $\delta y=y-0.06$, and $\Delta_i=\frac{\partial P_i}{\partial y}$
$\frac{P_i(y)}{C_i} \sim \frac{P_i(0.06)+\Delta_i \, \delta y}{P_i(0.06)}=1+\delta y \frac{\Delta_i(0.06)}{P_i(0.06)} \sim \delta y \frac{\Delta_i(y)}{P_i(y)}$
You can see now, the different behaviour above and below 6%, since the sign of $\delta y$ changes. The exact $yield$ at which you compute the $\Delta_i$ is not so relevant, and it is lost in the higher order terms.
What is $\frac{\Delta_i}{P_i}$? It is the negative of the so called Modified Duration https://en.wikipedia.org/wiki/Duration_(finance)#Modified_duration
Remember to factor in all sign conventions and navigate the different duration definitions.
However, if you find that the ranking is very unstable (e.g. higher order terms matter, or bonds very close, frequent switch), then the whole concept of a single identifiable CTD breaks down, and you might want to accept there is uncertainty and price in the optionality.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.