(a) Consider an annuity that pays 1 every quarter for M years. In other words, the payment times are T = t + 1/4 , t + 2/4 , . . . ,( t + 4M/4) . Show that the value at present time t is

Vt = ((1-(1+r4/4)^(-4M))/(r4/4),

assuming the quarterly compounded interest rate has constant value r4.

(b) A fixed rate bond with notional N, coupon c, start date T0, maturity Tn, and term length is an asset that pays N at time Tn and coupon payments Nc at times Ti for i = 1, . . . , n, where Ti+1 = Ti + . It is equivalent to an annuity plus N ZCBs. Consider a fixed rate bond with notional N and coupon c that starts now, matures M years from now, and has quarterly coupon payments. Show that the value at present time t is

Vt = (cN/4) ((1 (1 + r4/4)^(4M))/(r4/4)) + N(1 + r4/4)^(4M), assuming the quarterly compounded interest rate has constant value r4.