Suppose you have a perpetuity with the first payment of $1,449 made at time t=1.

Question

Suppose you have a perpetuity with the first payment of $1,449 made at time t=1. Each successive payment is increased by 0.8%. (so the payment stream will look like X, X(1+k), X(1+k)2, ...with the first payment at t=1). Find the value of the modified convexity at an annual effective rate of i = 5.1%. Round your answer to three decimal places. You must construct a price function and the derivative definition of convexity to calculate the answer. Failing to do so will result in a score of 0 points.

Please show work no excel

Explanation / Answer

P(i)=1449/(1+i)+1449*(1+0.8%)/(1+i)^2+1449*(1+0.8%)^2/(1+i)^3...........

=1449/(1+i)*(1/(1-(1+0.8%)/(1+i)))

=1449/(i-0.8%)

P'(i)=-1449/(i-0.8%)^2

P''(i)=1449*2/(i-0.8%)^3

Modified convexity=P''(i)/P(i)=(1449*2/(i-0.8%)^3)/(1449/(i-0.8%))=2/((i-0.8%)^2)

At i=5.1%, modified convexity=2/(4.3%^2)=1081.66577

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