Formula for forward price of bond

What is the formula for the forward price of a bond (assuming there are coupons in the interim period, and that the deal is collateralised)

Please also prove it with an arbitrage cashflow scenario analysis!

I suppose it is like fwd = spot - pv coupons) × (1+ repo × T ), I am not certain at what rate to pv the coupons.

Answer

Amazingly, there are several different methods for computing bond forward price – the underlying ideas are the same (forward price = spot price - carry), but the computational details differ a bit based on market convention.

Let's start with the basics. Assume between now ($t_0$) and the forward settlement date $t_2$, the bond makes a coupon payment at time $t_1$. Now consider the following series of trades:

• Today, a trader buys a bond at a price of $P + AI_0$ (spot clean price + spot accrued interest).


• To fund the purchase, the trader enters into a $t_1$-year term repo agreement at a repo rate of $r$. More specifically, he/she sells the repo by borrowing $P + AI_0$ and delivering the bond as collateral.


• At time $t_1$ (coupon payment date), the repo balance is $(P + AI_0)(1 + rt_1)$ and the trader receives a coupon payment of $c / 2$ for being the owner of the bond.



• The trader re-enters into another repo agreement that spans from $t_1$ to $t_2$ on a principal of $(P + AI_0)(1 + rt_1) - c/2$. This new loan, combined with the coupon payment of $c/2$, allows the trader to retire the old repo loan without putting up any additional capital.


• Finally, at time $t_2$, the trader gets back the bond and repays the repo loan along with interest from $t_1$ to $t_2$: $$\left((P + AI_0)(1 + rt_1) - \frac{c}{2}\right) \bigl(1 + r(t_2-t_1)\bigr) .$$

These trades are economically no different from buying the bond forward at time $t_2$. Therefore, the forward clean price for settlement at $t_2$ must be

$$F(t_2) = (P + AI_0)(1 + rt_1)\bigl(1 + r(t_2-t_1)\bigr) - \frac{c}{2}\bigl(1 + r(t_2-t_1)\bigr) - AI_{t_2}.$$

The method above is known as the Compounded Method. In the US Treasury market (and most international bond markets), a small approximation is made. Recall for small $rt$, we have

$$(1 + rt_1)(1+r(t_2-t_1))\approx 1 + r(t_1+t_2-t_1) = 1 + rt_2,$$ we therefore have the Proceeds Method: $$F(t_2) = (P + AI_0)(1 + rt_2) - \frac{c}{2}\bigl(1 + r(t_2-t_1)\bigr) - AI_{t_2}.$$

The Proceeds Method is for all intents and purposes the standard/default way of pricing bond forwards. There's also the "Simple" and "Scientific" methods, but these are rarely used.