4. At time t = 0, Karl buys three perpetuities which pay according to the following (i.) Perpetuity X pays 1 at times 2, 4, 6, 8, 10, 12, (ii.) Perpetuity Y pays 2 at times 3, 6, 9, 12, 15, .... (iii.) Perpetuity Z pays 3 at times 6, 12, 18, 24, 30, Additionally suppose Perpetuity X has an annual effective interest rate of i> 0, Perpetuity Y has an annual effective interest rate of 2i and Perpetuity Z has an annual effective interest rate of 3i. 1 (a) What proportion of the total present value is accounted for by Perpetuity X? (b) What proportion of the total present value is accounted for by Perpetuity Y? (c) What proportion of the total present value is accounted for by Perpetuity Z? (d) Verify that (a)-(c) sum to one. 4. At time t = 0, Karl buys three perpetuities which pay according to the following (i.) Perpetuity X pays 1 at times 2, 4, 6, 8, 10, 12, (ii.) Perpetuity Y pays 2 at times 3, 6, 9, 12, 15, .... (iii.) Perpetuity Z pays 3 at times 6, 12, 18, 24, 30, Additionally suppose Perpetuity X has an annual effective interest rate of i> 0, Perpetuity Y has an annual effective interest rate of 2i and Perpetuity Z has an annual effective interest rate of 3i. 1 (a) What proportion of the total present value is accounted for by Perpetuity X? (b) What proportion of the total present value is accounted for by Perpetuity Y? (c) What proportion of the total present value is accounted for by Perpetuity Z? (d) Verify that (a)-(c) sum to one