Question

Compute the present values of the following annuities first assuming that payments are made on the last day of the period and then assuming payments are made on the first day of the period: (Do not round intermediate calculations. Round your answers to 2 decimal places. (e.g., 32.16)) 1) 7-year, annual payment of $678.09, and YTM = 13%; 2) 13-year, annual payment of $7968.26, and YTM = 6%; 3) 23-year, annual payment of $20,322.93, and YTM = 4%; 4) 4-year, annual payment of $69,712.54, and YTM = 31%.

Answer 1

1)

Last day of the period:

Present value = Annuity * [1 - 1 / (1 + r)ⁿ] / r

Present value = 678.09 * [1 - 1 / (1 + 0.13)⁷] / 0.13

Present value = 678.09 * [1 - 0.425061] / 0.13

Present value = 678.09 * 4.42261

Present value = $2,998.93

First day of the period:

Present value of annuity due = (1 + r) * Annuity * [1 - 1 / (1 + r)ⁿ] / r

Present value of annuity due= (1 + 0.13) * 678.09 * [1 - 1 / (1 + 0.13)⁷] / 0.13

Present value of annuity due= 1.13 * 678.09 * [1 - 0.425061] / 0.13

Present value of annuity due= 1.13 * 678.09 * 4.42261

Present value of annuity due= $3,388.79

2)

Last day of the period:

Present value = Annuity * [1 - 1 / (1 + r)ⁿ] / r

Present value = 7968.26 * [1 - 1 / (1 + 0.06)¹³] / 0.06

Present value = 7968.26 * [1 - 0.468839] / 0.06

Present value = 7968.26 * 8.852683

Present value = $70,540.48

First day of the period:

Present value of annuity due = (1 + r) * Annuity * [1 - 1 / (1 + r)ⁿ] / r

Present value of annuity due= (1 + 0.06) * 7968.26 * [1 - 1 / (1 + 0.06)¹³] / 0.06

Present value of annuity due= 1.06 * 7968.26 * [1 - 0.468839] / 0.06

Present value of annuity due= 1.06 * 7968.26 * 8.852683

Present value of annuity due= $74,772.91

3)

Last day of the period:

Present value = Annuity * [1 - 1 / (1 + r)ⁿ] / r

Present value = 20,322.93 * [1 - 1 / (1 + 0.04)²³] / 0.04

Present value = 20,322.93 * [1 - 0.405726] / 0.04

Present value = 20,322.93 * 14.856842

Present value = $301,934.55

First day of the period:

Present value of annuity due = (1 + r) * Annuity * [1 - 1 / (1 + r)ⁿ] / r

Present value of annuity due= (1 + 0.04) *20,322.93 * [1 - 1 / (1 + 0.04)²³] / 0.04

Present value of annuity due = 1.04 * 20,322.93 * [1 - 0.405726] / 0.04

Present value of annuity due = 1.04 * 20,322.93 * 14.856842

Present value of annuity due = $314,011.94

4)

Last day of the period:

Present value = Annuity * [1 - 1 / (1 + r)ⁿ] / r

Present value = 69,712.54 * [1 - 1 / (1 + 0.31)⁴] / 0.31

Present value = 69,712.54 * [1 - 0.339559] / 0.31

Present value = 69,712.54 * 2.130456

Present value = $148,519.49

First day of the period:

Present value of annuity due = (1 + r) * Annuity * [1 - 1 / (1 + r)ⁿ] / r

Present value = (1+0.31) * 69,712.54 * [1 - 1 / (1 + 0.31)⁴] / 0.31

Present value = 1.31 * 69,712.54 * [1 - 0.339559] / 0.31

Present value = 1.31 * 69,712.54 * 2.130456

Present value = $194,560.54