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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...!

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