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in BUS 4027W - Actuarial Risk Management by (820 points)

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?

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The critical difference is that the memo accounts for the swapping of the notional at maturity. (Your answer would come out very close, at 6.44%). In a plain vanilla foreign currency swap, the principal amounts are exchanged, unlike in a plain vanilla IRS. Hence, if the notional is exchanged at maturity, the valuation of the floating leg can be valued like a floating-rate bond (LHS), and the fixed leg can be valued like a fixed-rate bond (RHS).

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