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Unformatted text preview: rT Fe P-= . The first and second derivatives are given by rT TFe dr dP--= and rT Fe T dr P d-= 2 2 2 . The convexity is defined as 2 2 2 1 T dr P d P C = = . Problem 4.12 First, we need to write down the equation for present value ∑ = + = n k k k m k m s e x PV 1 / )) / 1 ( ( ) ( λ Next we differentiate it: ∑ = +-= n k k k m k m s e x m k d dPV 1 / )) / 1 ( ( ) / ( ) ( Therefore ∑ = + =-n k k k k m s x m k PV d dPV PV 1 ) / 1 ( ) / ( ) ( 1 ) ( ) ( 1 Since we may interpret (k/m) as time, this is a weighted average of times where the weights are determined by the present value of cash flow k divided by the total present value of the cash flow stream....
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