# What is the formula for the forward price of a inflation linked bond assuming there are coupons in the interim period and the deal is collateralised?

t2=forward settlement date

P=Spot clean price

AI0=Spot accrued interest

r=repo rate

t1=coupon payment date

AIt2= accrued interest of the forward settlement date

t0= now

Proceeds Method:

F(t2)=(P+AI0)(1+r*t2)−c2(1+r(t2−t1))−AIt2.

I want to understand how inflation affects this formula if we do this for a inflation linked bond.

• do we need to consider the coupons which are not falling in between settlement date and forward settlement date? Jun 26 '18 at 19:58
• Not explicitly. Those coupons are discounted into the linker's price. Jun 27 '18 at 9:04
• what does discounted into linker's price mean? Jun 27 '18 at 21:05
• I was being overly precise. What I meant is that the linker's price is the discounted present value of all future coupons and principal payments, so from that perspective, coupons not between spot date and forward date are implicitly accounted for. But you should just ignore this precision – you don't need to worry about coupons paid out after the forward date. Jun 28 '18 at 0:33

It's pretty much the same as a nominal bond, except cash flows need to be inflated. For example, here's the forward pricing formula for a Canadian-style linker, assuming one interim coupon payments:

$$\bigl(F(t_f) + AI_{t_f}\bigr) \frac{I(t_f)}{I_\text{base}} = (P + AI_{t_s})\cdot \frac{I(t_s)}{I_\text{base}}\cdot (1 + r \cdot t_f) - c\cdot \frac{I(t_c)}{I_\text{base}}\cdot\bigl(1 + r\cdot (t_f - t_c)\bigr),$$ where $t_f$ represents the forward settlement date, $t_s$ is the spot settlement date, $t_c$ is the coupon date, and $I(t)$ is the index ratio for time $t$.

Note that if all the index ratio terms are removed, you've got the nominal bond forward pricing formula.

If the indexed ratios corresponding to the coupon date and forward settlement dates are not known, you'll need a projection curve to impute them.

• what will be the formula for a imperial-style linker? Jun 27 '18 at 21:14
• is this not the formula which Bloomberg uses to calculate the forward price of a inflation linked bond? Jun 27 '18 at 21:22
• It's the same for any bond/linker – you just need to inflate all the cash flows into nominal terms accordingly. And yes, it's the formula BBG uses when you choose Proceeds method. Jun 28 '18 at 0:32
• inflate all the cashflows into nominal terms accordingly means to use the inflation curve accordingly or is there any other difference? Jun 28 '18 at 4:45
• The most common practice on the sell side is actually to use economists' inflation forecasts. You can also use inflation swap curve for a market-based gauge. Jun 28 '18 at 5:30