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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:

http://www.aplade.com/2016/12/17/forward-price-analysis/

This method only uses two spot rates:

https://www.investopedia.com/ask/answers/043015/how-do-i-convert-spot-rate-forward-rate.asp

Thanks!

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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 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 $$F(t,T_a,T_b) = \frac{1}{T_b-T_a}\cdot\left(\frac{P(t,T_b)}{P(t,T_a)}-1\right).$$ Here, $P(t,T)=\frac{1}{(1+r_T)^T}$ denotes the time $t$ price of a default-free zero-coupon bond maturing at time $T$ with face value one.

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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...

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