Consider a deferred perpetuity that pays you $500,000 every 6 months, however, the first payment happens only 30 months (i.e., 2.5 years from now). Assume all spot rates are equal to each other and denote them by r.

a) Write down the value of such perpetuity today P(r) as a function of r. (Do not forget that rates are semi-annually compounded and $500,000 is paid every 6 month)

b) Find Duration of this perpetuity D(r) today as a function of r. Find the value D(r) for r=8%. Note: if you cannot find the exact formula for D(r), you can still receive 0.5 points for finding the value D(0.08) by using appropriate approximation.

c) Lets denote the original perpetuity by S1. Consider three other perpetuities: S2 that pays $500,000 every six months starting one year from, S3 that pays $1,000,000 every six months starting 1 year from now, and S4 that pays $250,000 every six months starting 1 year from now. Let D1, D2, D3, and D4 be durations of these perpetuities. Assume all spot rates are equal to each other but not necessarily equal to 8%. Rank the Durations of these four perpetuities from lowest to highest (if some or all of them are equal, make sure to say so too). No explanation is necessary.