1. Time Value of Money I [10 points] We know that the present value of an annuity and perpetuity are given by PV=rC[1(1+r)T1],PV=rC where C,T and r denote the periodic cash flow, number of periods and periodic rate respectively: Let's prowe them. (a) 2 points] Present value of an annuity is by definition PV=1+rC+(1+r)2C++(1+r)TC Prove: PV in the form above could be rewrite as PV=C(a+a2+I+aT),wherea=1+r1 (b) [2 points ] Denote AT=a+a2++aT. Therefore, aAT=a2+a3++aT+1. Manipulate with aAT and AT to prove AT=a1a(aT1) (c) [2 points ) Based on the result of AT=a1a(aT1), rewrite the result in (a) by PV=CAT (d) [2 points ] Based on the result (c), plug in a=1+r1, and prove the PV of an annuity is PV=rC[1(1+r)T1] (e) [2 points ] Assuming r>0, by taking limit on one parameter in (d), prove the PV of a perpetuity PV=rC