# Valuing interest rate swaps

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In the 2018 Test 4 a question asks to show that the standard 3 year swap rate is 6.45% if we have the following 12 month forward rates: Year 1: 5%, Year 2: 7%, Year 3: 7.5%.

In finding the answer they solve for $$i$$ such that $$5v+7v^2+107.5v^3=100$$. Why do they do this? To me, it would make sense to solve for $$x$$ such that

$$5v_{5\%}+7v_{7\%}v_{5\%}+7.5v_{7.5\%}v_{7\%}v_{5\%} = x(v_{5\%}+v_{7\%}v_{5\%}+v_{7.5\%}v_{7\%}v_{5\%})$$

Am I misinterpreting the question?