# Implied repo rate and slope of the yield curve

In page 34 of "Treasury Bond Basis" (Third Edition) by Burghardt et al, it says:

If the yield curve has a positive slope, carry for someone who is long bonds and short futures is positive. Every day that goes by is money in the bank. The implied repo rates simply confirm this.

Why is this the case?

Up to that point in the book, the implied repo rate (IRR) is defined (simple version) as:

IRR = (Futures Invoice Price / Bond Purchase Price - 1) x (360 / n)

where n is number of days to delivery.

It seems to be a function of coupon and actual term repo rate (i.e. carry), or at least it's not clear from this formula how the slope of yield curve impacts IRR.

E.g. if we have a bond with coupon=5% and meanwhile a negatively sloped yield curve with spot rate starting at 0%. Then carry should be actually positive, but using the conventional wisdom of looking at yield curve will say negative carry.

## 2 Answers

Think he means for holding onto your basis for longer in a positive sloped curve the irr should increase. This is just because you will carry positively. Carry is measured versus your repo.

If your term repo is 5.5% with a coupon of 5% you will carry negatively to term. Your futures invoice price is a function of accrued interest since last coupon. You should compare irr to repo to get an idea of the gain versus cost of holding bonds and delivering into the future at term

I think it just means the implied repo rate is higher than the actual repo rate. Explanation: if you are long bonds and short futures, what do you have? Well assuming that the bond is the CTD and the net basis is zero, then your economics are zero after the futures has expired since you just deliver the bond. All you have left is the economics between now and expiration, which is that you are receiving the implied repo rate (fixed) and you are paying the daily repo on your bond. If the curve is upward sloping , then you have a daily gain.