Why a Bond’s Implied Repo Rate Is Below Its Term Repo Rate

Summary

The document explains why the implied repo rate for a bond in a futures contract is below that bond’s term repo rate. The key comparison is between the futures price and the bond’s forward price for delivery, adjusted by the conversion factor. To rule out arbitrage, the futures price must be lower than the adjusted forward price based on the bond’s actual repo rate. Because the forward price increases with the repo rate, the implied rate consistent with the futures price must be lower than the actual term repo rate.

The explanation gives a simple relationship: the forward price is modeled as the bond’s clean price grown at the repo rate over the time to delivery. It answers the question about why the spread is negative and how it relates to the cost of delivery options and cheapest-to-deliver analysis. The treatment is concise and relies on a simplified price relationship; it does not explore market conventions, financing details, or cases where assumptions behind that relationship may need adjustment.

Key ideas

• The futures price must sit below the bond’s forward price adjusted by its conversion factor to avoid arbitrage.
• The bond forward price increases as the repo rate increases.
• The implied repo rate is therefore below the bond’s term repo rate in the stated setup.
• The spread between those rates can be used to compare delivery-option costs across bonds.

Tags

# Difference between implied repo and term repo always negative?


# Difference between implied repo and term repo always negative?












From The Treasury Bond Basis:

> A somewhat better guide to finding the cheapest to deliver is the spread or difference between a bond's implied repo rate and its own term repo rate. The difference, which would be negative for all bonds, is a measure of the cost of the strategic delivery options. And the bond for which this number is least negative (largest algebraically) is the cheapest to deliver.

I feel like I may be confusing myself here, but I thought if the yield curve is upward sloping, then term repo will be less than the implied repo (because the short rate is less than the yield you locked in with the cash and carry) and hence why it's positive carry. But this doesn't seem to be the case here—what am I missing?

## Answer by Andrea (score 0)

https://quant.stackexchange.com/a/81056

Remember that a Bond Futures price must be lower than all bond forward prices (to delivery) adjusted for the conversion factor (or there would be arbitrage).

$Fut = \frac{FWD_i(IRR_i)}{CF_i} < \frac{FWD_i(R_i)}{CF_i}$

where $FWD$ is the forward price of the bond as function of the repo rate (real or implied). This is an increasing function and so this implies

$IRR_i < R_i$

The $FWD$ function is basically $e^{R \cdot T} B$ where $B$ is the clean price and $T$ the time to the futures delivery.

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.