# Calculating bond forward rate/price

What's the difference these two methods on calculating the bond forward rate/price. First of all I'm assuming forward rate is the same as forward price in this context, if this assumption is false, please tell me how they are different.

This method uses the the spot price and repo rate:

This method only uses two spot rates:

Thanks!

The first method is how you actually calculate the forward price of a specific bond. You need to use the repo rate for that bond as the financing rate inside the calculation.

The second method is a quick way of estimating bond forward yields, but it is not something you can execute in practice. For example, if you try to lock in the yield , 5yrs from today, of a 10year zero coupon bond , the method assumes you will buy a 10year ZCB and sell a 5yr ZCB. But during the first 5years, the financing costs of being long the 10yr and short the 5yr do not perfectly offset. You have to borrow the 5yr bond in order to short it, for which you provide cash collateral receiving some interest rate. This may not match the cost of financing the purchase of the 10yr.

The second method is more appropriate for interest rate derivatives , whose implicit funding rates do offset perfectly. It is also commonly found in textbooks about bonds, but as I said it is an approximation.

Last comment : for a specific bond, there is a one to one correspondence between forward price and forward yield. It is just simple bond math. If you know one, you know the other.

A forward rate is not the same as a forward price.

A forward price is the price you need to pay at time $$t$$ to receive (purchase) an asset at a future date $$T$$. This forward price can be derived from no-arbitrage arguments and is, in its simplest form, given by $$F_t=S_te^{r(T-t)}.$$ You can of course incorporate coupons (or dividends), accrued interests and consider discrete compounding which replaces $$e^{r(T-t)}$$ by $$(1+R)^{T-t}$$. The time to maturity $$T-t$$ may be computed in any day-time convention.

A (simply-compounded) forward rate is the interest rate known at time $$t$$ which you need to pay to borrow money for a future time period $$[T_a,T_b]$$. It can be derived by no-arbitrage from spot rates and reads as $$F(t,T_a,T_b) = \frac{1}{T_b-T_a}\cdot\left(\frac{P(t,T_a)}{P(t,T_b)}-1\right).$$ Here, $$P(t,T)$$ denotes the time-$$t$$ price of a default-free zero-coupon bond maturing at time $$T$$ with face value one.

To put it simply, the difference is that an interest rate forward can be replicated with a long and short position on deposits with different maturities, hence you can obtain it with just the spot rates.

However, for a forward bond price (or yield), that can also be obtained with a long and short position on bonds with different maturities, you have to take into account the repo rate of the short bond.

Hope this helps...

I think that the formula of $$F(t,T_a,T_b)$$ is misleading. The ratio $$P(t,T_b)/P(t,T_a)$$ should be $$P(t,T_a)/P(t,T_b)$$ instead.