# Return of a bond held to maturity and realised forward rates

Let's assume that forward rates are realised as part of a carry-roll-down scenario.

The gross return of a bond under the realised forward assumption to maturity is:

$$\frac{c(1+f(2))(1+f(3))...(1+f(T))}{P_{0}(\mathbb{R}_{0})}+\frac{P_{T}(\mathbb{R}_{T})-P_{0}(\mathbb{R}_{0})}{P_{0}\mathbb({R}_{0})} = (1+f(1))(1+f(2))...(1+f(T))-1$$

I understand that the above is the return to a bond held to maturity under the assumption of realised forward rates, which is the same as rolling a $1 investment one period at a time under those forward rates (as per Tuckman P112). Now suppose the future short term rates end up being higher than the forward rates $$f(1), f(2)...$$. This means that we can invest$1 at higher short term rates than just holding the bond to maturity, but only when the assumption is that forward rates are realised?

I am confused as to what we are meant to be comparing and the thought process behind it, although I know it is meant to be intuitive...!