Question

Bella Inc. wishes to accumulate funds to provide a retirement annuity for its Vice President of Research, Edward Cullen. Mr. Cullen by contract will retire at the end of exactly 20 years. On retirement, he is entitled to receive an annual end-of-year payment of $35,000 for exactly 30 years. If he dies prior to the end of the 30-year period, the annual payments will pass to his heirs. During the 20-year ‘accumulation period’, Bella Inc. wishes to fund the annuity by making equal annual end-of-year deposits into an account earning 7 percent interest compounded quarterly. Once the 30-year ‘distribution period’ begins, Bella Inc. plans to move the accumulated monies into an account earning a guaranteed 12 percent per year compounded annually. At the end of the distribution period the account balance will equal zero. Note that the first deposit will be made at the end of year 1 and the first distribution payment will be received at the end of year 21.

Required:

a) How large must Bella Inc.’s equal annual end-of-year deposits into the account be over the 20-year accumulation period to fund fully Mr. Cullen’s retirement annuity?

b) How much would Bella Inc. have to deposit annually during the accumulation period if it could earn 8 per cent rather than 7 percent?

c) How much would Bella Inc. have to deposit annually during the accumulation period if Mr. Cullen’s retirement annuity was perpetuity and all other terms were the same as initially described?

Answer 1

(a)

Retirement Annuity: Annual Year-End Withdrawals = $ 35000, Interest Rate = 12 % compounded annually, Withdrawal Tenure = 30 years

Therefore, PV of Annual Withdrawals (at retirement) = 35000 x (1/0.12) x [1-{1/(1.12)^(30)}] = 281931.44

Annual Deposits: Interest Rate = 7%, Compounding Frequency: Quarterly, Applicable Equivalent Annual Rate = [1+(0.07/4)]^(4) - 1 = 0.071859 or 7.1859 %

Let the required year end deposits be $ p and Deposit Tenure = 20 years

Therefore, p x (1.071859)^(19) +..............+ p = 281931.44

p x [{(1.0718591)^(20) - 1} / {1.0718591-1}] = 281931.44

p = $ 6738.7488 ~ $ 6738.75

(b) Interest Rate = 8%, Compounding Frequency: Quarterly, Applicable Interest Rate = [1+(0.08/4)]^(4) - 1 = 0.08243 or 8.243 %

Let the required annual deposits be $m

Therefore, m x (1.08243)^(19) +.............+ m = 281931.44

m x [{(1.08243)^(20)-1}/{1.08243-1}] = 281931.44

m = $ 5996.79

(c) If the retirement withdrawals are a perpetuity, then PV (at retirement) of the Perpetuity = 35000 / 0.12 = $ 291666.67

Let the required annual deposits in such a case be $n

Therefore, 291666.67 = n x (1.0718591)^(19) +..............+ n

291666.67 = [{(1.0718591)^(20)-1} / {(1.0718591)-1}] x n

n = $ 6971.44