# Implied repo rate from carry component

Carry is coupon income + pull-to-par - financing cost. Pull to par is derived as ytm-coupon. So carry can be rewritten as ytm - financing costs. Carry cash value is the current dirty price minus the cash flows in period x discounted at the current yield to maturity. So let's say you have a bond with annual coupon $$2.5$$, $$5$$ years remaining maturity and a ytm of $$1\%$$. The dirty price of this bond is $$104.7826$$. The dirty price for the cashflows in 6m time is $$105.5656$$. So an increase of $$0.782945$$.Ytm for 6m equals a value of $$104.7826 \cdot 1.5\% \cdot 0.5 = 0.78587$$. Can you say that the difference of $$-0.00293$$ is the implied repo value which implied a repo rate of $$- 0.00293/104.7826/0.5 = -0.0058\%$$?

I've never seen Implied Repo defined like this but there's a way to check whether your reasoning is correct. The implied repo is called implied repo for a reason: take your favorite Repo pricer of choice, load the current CTD bond with delivery date equal to the future's expiration (e.g. first delivery date) and plug the IR rate as the repo cost. If the calculated Forward price is equal to the bond future price divided by the CTD conversion factor, then your methodology is correct.

The implied repo rate is simply the return that you get from selling the future and buying the CTD bond. Ignoring interim coupons:

$$IR = \frac{CF\times P_{fut} - P_{bond} + (a_2-a_1)}{(P_{bond}+a_1)t}$$

where $$a_1$$ is the accrued at the spot and $$a_2$$ the accrued at the delivery date, $$CF$$ is the conversion factor and $$t$$ the day count fraction from spot to delivery date.