Question

8. You are scheduled to receive annual payments of $60,000 for each of the next 20 years. The annual rate of return is 8 percent. What is the difference in the future value in year 20 if you receive these payments at the beginning of each year rather than at the end of each year?

9. You make the following deposits for the next five years into an investment account. All deposits are made at the end of the year and the first deposit occurs one year from now. No more deposits are made after year 5. You will leave all the money in the account until year 30. If you earn 10 percent annual return for the first five years and 8 percent annual return for all subsequent years, how much will you have in the account at the end of year 30?

            ___________________________

             Year        Deposit

            ___________________________

            1           $40,000

            2           $50,000

            3           $60,000

            4           $70,000

            5           $80,000

           _____________________________

10. A bond has 10 years remaining to maturity. The annual coupon rate of the bond is 8 percent. The bond makes annual coupon payments. The next coupon payment occurs one year from today. The par value (or face value) of the bond is $1,000. If the annual yield to maturity (or the required rate of return) on the bond is 6 percent, find the current bond price (or bond value).

Answer 1

8.

FV of annuity = P x [(1+r) n -1/r]

P = Periodic payment = $ 20,000

r = Periodic interest rate = 0.08

n = Number of periods = 20

FV = $ 60,000 x [(1+ 0.08)20 -1/0.08]

   = $ 60,000 x [(1.08)20 -1/0.08]

   = $ 60,000 x [(4.66095714384931 -1)/0.08]

   = $ 60,000 x (3.66095714384931/0.08)

= $ 60,000 x 45.7619642981163

= $ 2,745,717.85788698 or $ 2,745,717.86

FV of annuity due =(1+r) x P x [(1+r) n -1/r]

                               = (1+0.08) x $ 2,745,717.85788698

                               = 1.08 x $ 2,745,717.85788698

                              = $ 2,965,375.28651794 or $ 2,965,375.29

Difference in FV of ordinary annuity and annuity due

                              = $ 2,965,375.29 - $ 2,745,717.86

                              = $ 219,657.43

9.

Computation of FV of 5 deposits in year 5th:

Year	Deposits(C)	Computation of FV Factor	FV Factor @ 10 % (F)	FV (C x F)
1	$40,000	(1+0.1)4	1.4641	$58,564
2	50,000	(1+0.1)3	1.331	66,550
3	60,000	(1+0.1)2	1.21	72,600
4	70,000	(1+0.1)1	1.1	77,000
5	80,000	(1+0.1)0	1	80,000
			Total FV	$ 354,714

As per compound interest accumulated value can be computed as:

A = P x (1+r) n

    = $ 354,714 x (1+ 0.05)25

    = $ 354,714 x (1.05)25

    = $ 354,714 x 3.38635494089939

    = $ 1,201,187.50650618 or $ 1,201,187.51

At the end of 30 years account will have $ 1,201,187.51

10.

Bond price = C x 1 – (1+r)-n/r + F/(1+r) n

F = Face value = $ 1,000

n = Number periods to maturity = 10

C = Periodic coupon payment = Face value x Coupon rate/Annual coupon frequency

= $ 1,000 x 0.08/1 = $ 1,000 x 0.08 = $ 80

r = Rate of return = 0.06

Bond price = $ 80 x [1 – (1+0.06)-10]/ 0.06 + $ 1,000/ (1+0.06) -10

                   = $ 80 x [1 – (1.06)-10]/ 0.06 + $ 1,000 x (1.06) -10

                   = $ 80 x [(1 – 0.558394776915118)/0.06] + $ 1,000 x 0.558394776915118

                   = $ 80 x (0.441605223084882/0.06) + $ 558.394776915118

                   = $ 80 x 7.3600870514147 + $ 558.394776915118

                   = $ 588.806964113176 + $ 558.394776915118

                    = $ 1,147.201741028294 or $ 1,147.20

Current price of bond is $ 1,147.20