Question

a. What is the amount of the annuity purchase required if you wish to receive a fixed payment of $200,000 for 20 years? Assume that the annuity will earn 12 percent per year.

b. Calculate the annual cash flows (annuity payments) from a fixed-payment annuity if the present value of the 20-year annuity is $1.1 million and the annuity earns a guaranteed annual return of 12 percent. The payments are to begin at the end of the current year.

c. Calculate the annual cash flows (annuity payments) from a fixed-payment annuity if the present value of the 20-year annuity is $1.1 million and the annuity earns a guaranteed annual return of 12 percent. The payments are to begin at the end of five years.

Answer 1

a.

Formula for PV of annuity can be used to compute present value of cash flows as:

PV = P x [1 – (1+r)-n]/r

PV = Present value of fund

P = Periodic cash withdrawal = $ 200,000

r = Rate per period = 0.12 p.a.

n = Number of periods = 20

PV = $ 200,000 x [1 – (1+0.12)-20]/0.12

   = $ 200,000 x [1 – (1.12)-20]/0.12

   = $ 200,000 x [(1 – 0.103666765080688)/0.12]

= $ 200,000 x (0.896333234919312/0.12)

   = $ 200,000 x 7.4694436243276

   = $ 1,493,888.72486552 or $ 1,493,888.72

Current purchase price of annuity is $ 1,493,888.72

b.

Formula for PV of annuity can be used to compute periodic cash withdrawal as:

PV = P x [1 – (1+r)-n]/r

P = PV/ [1 – (1+r)-n]/r

PV = Present value of fund = $ 1,100,000

P = Periodic cash withdrawal

r = Rate per period = 0.12 p.a.

n = Number of periods = 20

P = $ 1,100,000 / [1 – (1+0.12)-20]/0.12

   = $ 1,100,000 / [1 – (1.12)-20]/0.12

   = $ 1,100,000 / [(1 – 0.103666765080688)/0.12]

   = $ 1,100,000 / (0.896333234919312/0.12)

   = $ 1,100,000 / 7.4694436243276

   = $ 147,266.658043627 or $ 147,266.66

Annual cash flow will be $ 147,266.66

c)

Value of fund at the end of year 4 can be computed as:

FV = PV x (1+r) n

PV = $ 1,100,000; r = 0.12; n = 4

FV = $ 1,100,000 x (1+0.12) 4

      = $ 1,100,000 x (1.12) 4

      = $ 1,100,000 x 1.57351936

      = $ 1,730,871.296

Fund of $ 1,730,871.296 will facilitate cash flow for 20 years, annual annuity can be computed as:

P = $ 1,730,871.296 / [1 – (1+0.12)-20]/0.12

   = $ 1,730,871.296 / [1 – (1.12)-20]/0.12

   = $ 1,730,871.296 / [(1 – 0.103666765080688)/0.12]

   = $ 1,730,871.296 / (0.896333234919312/0.12)

   = $ 1,730,871.296 / 7.4694436243276

   = $ 2,31,726.937514146 or $ 2,31,726.94

Annual cash flow will be $ 2,31,726.94 starting at the end of 5 years from now.