Question

5) a) What is the present value of $40 earned 2-years from now if compounding was semi-annual and the interest rate is annually 3%? A "black box"just paid $20, which is expected to grow by 3% when the interest rate is 7% forever, what is the present value of this "black box"? b) c) What is the future value of an annuity due with a $15 cash flow, 4% annual interest with quarterly compounding three-years from now? d) If the monthly interest rate is 0.5%, then what is the effective annual rate? What is the stated annual rate?

Please refrain from using excel thanks!

Answer 1

a)

Rate = 3% / 2 = 1.5%

Number of periods = 2 *2 = 4

Present value = Future value / ( 1 + r)ⁿ

Present value = 40 / ( 1 + 0.015)⁴

Present value = 40 / 1.061364

Present value = $37.69

b)

Present value = Cash flow1 / interest rate - growth rate

Present value = (20 * 1.03) / 0.07 - 0.03

Present value = 20.6 / 0.04

Present value = $51.5

c)

Number of periods = 3 * 4 = 12

Rate = 4% / 4 = 1%

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

Future value of annuity due = (1 + 0.01) * 15 *[(1 + 0.01)¹² - 1] / 0.01

Future value of annuity due = 1.01 * 15 * 12.682503

Future value of annuity due = $192.14

d)

Effective annual rate = (1 + monthly rate)ⁿ - 1

Effective annual rate = (1 + 0.005)¹² - 1

Effective annual rate = 0.0617 or 6.17%

Stated annual rate = 0.5% * 12

Stated annual rate = 6%