d. Calculate the future sum of $1 comma 700, given that it will be held in the bank for 22 years and earn 8 percent compounded semiannually. e. What is an annuity due? How does this differ from an ordinary annuity? f. What is the present value of an ordinary annuity of $2 comma 600 per year for 8 years discounted back to the present at 15 percent? What would be the present value if it were an annuity due? g. What is the future value of an ordinary annuity of $2 comma 600 per year for 8 years compounded at 15 percent? What would be the future value if it were an annuity due? h. You have just borrowed $230 comma 000, and you agree to pay it back over the next 15 years in 15 equal end-of-year payments plus 14 percent compound interest on the unpaid balance. What will be the size of these payments? i. What is the present value of a perpetuity of $1 comma 900 per year discounted back to the present at 18 percent? j. What is the present value of an annuity of $1 comma 800 per year for 10 years, with the first payment occurring at the end of year 10 (that is, ten $1 comma 800 payments occurring at the end of year 10 through year 19), given a discount rate of 9 percent? k. Given a discount rate of 15 percent, what is the present value of a perpetuity of $1 comma 100 per year if the first payment does not begin until the end of year 10?
d.
Formula for compound interest:
A = P (1+r) n
A = Future value of deposit
P = Principal = $ 1,700
R = Periodic rate = 0.08/2 = 0.04 semiannually
n = Number of periods =22 x 2 = 44
A = $ 1,700 x (1+0.04)44
= $ 1,700 x (1.04)44
= $ 1,700 x 5.61651507832828
= $ 9,548.07563315807 or $ 9,548.08
Future value of deposit is $ 9,548.08
e.
Annuity due is a series of equal payment that occurs at the beginning of each period.
The only difference between annuity due and ordinary annuity is the time of payment.
For annuity due each payment occurs in the beginning of periods instead of end of periods as in case of ordinary annuity.
f.
Formula for ordinary annuity:
PV = P x [1-(1+r)-n/r]
P = Periodic cash flow = $ 2,600
r = Rate per period = 15 % or 0.15 p.a.
n = Numbers of periods = 8
PV = $ 2,600 x [1-(1+0.15)-8/0.15]
= $ 2,600 x [1-(1.15)-8/0.15]
= $ 2,600 x [(1-0.326901773846168)/0.15]
= $ 2,600 x (0.673098226153833/0.15)
= $ 2,600 x 4.48732150769222
= $ 11,667.03592 or $ 11,667.04
Formula for PV of annuity due:
PV = P x [1-(1+r)-n/r] x (1+r)
= PV of ordinary annuity x (1+r)
= $ 11,667.03592 x 1.15 = $ 13,417.091308 or $ 13,417.09
Present value of ordinary annuity and annuity due are $ 11,667.04 and $ 13,417.09 respectively.
g.
Formula for FV of annuity is:
FV = P x [(1+r) n – 1/r]
P = Periodic cash flow = $ 2,600
r = Rate per period = 15 % or 0.15 p.a.
n = Numbers of periods = 8
FV = $ 2,600 x [(1+0.15)8 – 1/0.15]
= $ 2,600 x [(1.15)8 – 1/0.15]
= $ 2,600 x [(3.059022862539060) – 1/0.15]
= $ 2,600 x (2.059022862539060/0.15)
= $ 2,600 x 13.7268190835937
= $ 35,689.7296173437 or $ 35,689.73
Formula for FV of annuity due:
FV = P x [(1+r) n – 1/r] x (1+r)
= FV of ordinary annuity x (1+r)
= $ 35,689.7296173437 x 1.15 = $ 41,043.1890599453 or $ 41,043.19
Future value of ordinary annuity and annuity due are $ 35,689.73 and $ 41,043.19 respectively.
h.
Formula for Equal Annual payment:
Annual payment = (r x P)/ [1-(1+r)-n]
P = Principal = $ 230,000
r = rate of interest = 14 %
n = No. of periods = 15
Annual payment = (0.14 x $ 230,000)/ [1-(1+0.14)-15]
= (0.14 x $ 230,000)/ [1-(1.14)-15]
= (0.14 x $ 230,000)/ [1- 0.140096482069159]
= $ 32,200/ 0.859903517930841
= $ 37,446.0614808064 or $ 37,446.06
Annual payment size will be $ 37,446.06
i.
PV of perpetuity = Periodic cash flow/Rate of interest
= $ 1,900/0.18 = $ 10,555.5555555556 or $ 10,555.56
Present value of perpetuity is $ 10,555.56
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