Question

4.   Now consider the second alternative—5 annual payments of $2,000 each. Assume that the payments are made at the end of each year.
   a.   What type of annuity is this?

   b.   What is the future value of this annuity if the payments are invested in an account paying 10.0 percent interest annually?

   c.   What is the future value if the payments are invested with the First National Bank which offers semiannual compounding?

   d.   What size payment would be needed to accumulate $20,000 under annual compounding at a 10.0 percent interest rate?

   e.   What lump sum, if deposited today, would produce the same ending value as in Part b?

   f.   Suppose the payments are only $1,000 each, but are made every 6 months, starting 6 months from now. What would be the future value if the 10 payments were invested at 10.0 percent annual interest? If they were invested at the First National Bank which offers semiannual compounding?

   5.   Assume now that the payments are made at the beginning of each period. Repeat the analysis in Question 4. (just need this one, last question posted for data)

Answer 1

* All the following calculation are made considering that the payments made at the beginning of each period

a.   What type of annuity is this?

This type of annuity is known as Annuity Due. In Annuity Due the payment/receipts made at the beginning of the period

b.   What is the future value of this annuity if the payments are invested in an account paying 10.0 percent interest annually?

Future Value of Annuity = Annuity * (1 + i) * [( 1 + i)^n) - 1] / i

Future Value of Annuity = 2000 * (1 + 10%) * [( 1 + 10%)^5) - 1] / 0.10

Future Value of Annuity = 2200 * [1.61051 - 1] / 0.10

Future Value of Annuity = 2200 * 6.1051

Future Value of Annuity = $13431.22

c.   What is the future value if the payments are invested with the First National Bank which offers semiannual compounding?

Future Value of Annuity = Annuity * (1 + i/2) * [( 1 + i/2)^(n*2) - 1] / (i/2)

Future Value of Annuity = 2000 * (1 + 5%) * [( 1 + 5%)^10) - 1] / 0.05

Future Value of Annuity = 2100 * 0.6288946 / 0.05

Future Value of Annuity = 2100 * 12.57789

Future Value of Annuity = $26413.57

d. What size payment would be needed to accumulate $20,000 under annual compounding at a 10.0 percent interest rate?

Future Value of Annuity = Annuity * (1 + i) * [( 1 + i)^(n) - 1] / (i)

20000 = Annuity * (1 + 10%) * [( 1 + 10%)^(5) - 1] / (10%)

20000 = Annuity * (1 + 10%) * 6.1051

Annuity = 20000 / 6.7156

Annuity = $2978.14

e.   What lump sum, if deposited today, would produce the same ending value as in Part b?

Future value = Present Value * (1 + Interest)^Years

13431.22 = Present Value * (1 + 10%)^5

Present Value = $13431.22 / 1.61051 = $8339.73

f. Suppose the payments are only $1,000 each, but are made every 6 months, starting 6 months from now. What would be the future value if the 10 payments were invested at 10.0 percent annual interest? If they were invested at the First National Bank which offers semiannual compounding?

Future Value of Annuity = Annuity * (1 + i/2) * [( 1 + i/2)^(n*2) - 1] / (i/2)

Future Value of Annuity = 1000 * (1 + 5%) * [( 1 + 5%)^(10) - 1] / 0.05

Future Value of Annuity = $1050 * 0.6289 / 0.05

Future Value of Annuity = $13206.79