Implied repo rate calculation from Fabozzi

Question

I'm looking at the chapter Implied repo rate in Fabozzi's Fixed Income Handbook.

There it is defined as the return received by going long the basis, i.e. buying the cash bond (financing it with the repo rate to term) and shorting the futures, i.e. in simple terms (as outlined in the book)

$$ \frac{\texttt{cash in}-\texttt{cash out}}{\texttt{cash out}}\cdot\frac{360}{n}$$

Then they say for the exact return, the formula is as follow

$$ \frac{[(F\cdot CF) + A_e+I_c-(P+A_b)]\cdot 360}{d_1\cdot(P+A_b)-I_c\cdot d_2}$$

with

• $F$, the future price
• $C_f$ the conversion factor
• $A_e$ accrued interest of bond at the end
• $A_b$ accrued interest of bond at the beginning
• $I_c$ interim coupons
• $d_1$ number of days between settlement and actual delivery
• $d_2$ number of days between interim coupon and bond delivery
• $P$ clean price of bond

To finance the bond I have to pay the dirty price today, which is the $(P+A_b)$. The sell of the forward is equivalent to $F\cdot CF + A_e$. $I_c$ is simply the sum of all coupons I receive between the settlement and actual delivery for being long the cash bond. What I'm a bit confused of is the denominator. Why do we weight them differently? I would have gone for

$$ \frac{[(F\cdot CF) + A_e+I_c-(P+A_b)]}{(P+A_b)}\cdot\frac{360}{d_1}$$

Answer 1

We are going to this operation using borrowed money (via repo).

How much capital do you need to do this? How many dollars for how many years? At first thought you need to raise $(P+A_b$) dollars (the dirty price of the bonds) for $d_1/360$ years, but actually you need less because you will receive $I_c$ in cash when there are $d_2$ days left to go and can use that to (partially) repay your loan. So the dollars x years are $(P+A_b)\frac{d1}{360}−I_c \frac{d_2}{360}$.

The numerator represents what is left over in your account when the operation concludes, at the time the bonds are delivered vs futures and the loan is repaid. That is why there is no "timing" in the numerator. It represents the situation at the final date.

If you wanted to model this differently you could assume that the coupon is deposited in a separate bank account, where it earns interest $r$. Then you would have an expression similar to yours (same denominator but slightly different numerator): $$ \frac{[(F\cdot CF) + A_e+I_c(1+r\frac{d_2}{360})-(P+A_b)]}{(P+A_b)\frac{d_1}{360}}$$

(But I don't recommend this other method. It is cleaner both mathematically and in practice to use the coupon cash you receive to reduce the loan/denominator): $$ \frac{[(F\cdot CF) + A_e+I_c-(P+A_b)]}{(P+A_b)\frac{d_1}{360}-I_c\frac{d_2}{360}}$$