Question

Exactly 5 years ago, Y made the first of several semi-annual deposits in the bank earning interest at the rate of 8% p.a. effective. Each of these deposits was $1,000. The last deposit occurred a few minutes ago. This bank account will fund a series of withdrawals. These will occur annually with the first in exactly 1 year. There will be a total of 12 withdrawals. Each of the first 6 will be for the same amount. Each of the remaining 6 will be exactly twice the amount of each of the first 6. (E.g. if the first 6 are for $1,000 each then each of the remaining 6 will be $2,000). If interest rates are now changing to 8% p.a. compounded semi-annually, what are the magnitudes of the withdrawals?

Answer 1

1)

Future value of Deposit (means At today's date)

P = $1000 (Semi - annually)

Rate of interest (r) = 1.08^0.5 = 1.03923 or 3.923%

(Since Rate is Effective per annum, means compounded. so this will be the method of finding the rate. where, 0.5 = 6/12 months )

No. of deposit (n) = 5*2 + 1 = 11 deposits

(Since question is saying that the first deposit was exactly 5 years ago and thereafter he made semi annual deposit till date. It means there will be total 11 deposit.)

Now, We are required to calculate the future value (at today's date) of first 10 deposits because the last (11th deposit) is already at today's date.

So, following will be our formula for finding FV of first 10 deposit.

FV of Annuity = 1000*12.43265 = 12,432.65

So, Total Deposit value till date = 12432.65 + 1000 (11th deposit)

= $13,432.65

2)

Present value of all the withdrawl = $13,432.65

Interest (r) = 8*0.5 = 4% = 0.04

(Since it is already Semi annualy rate.)

n = 12 ( 6 +6 )

FIrst of all, in the case, we will calculate the Present value annuity factor (PVFA)

a)   PVFA_1-6 = [1- (1+r)^-n]/r =  [1- (1+0.04)^-6]/0.04 = 5.2421

b) PVFA_1-12 = [1- (1+r)^-n]/r =  [1- (1+0.04)^-12]/0.04 = 9.3850

c) PVFA_7-12 = 9.3850-5.2421 = 4.1429

Now, Lets Consder

Amount of Withdrawl for year 1-6 = P

then, Amount of Wirthdrawl for year 7-12 = 2*P = 2P  

So, our Equation will be lile as followed :

13,432.65 = P*5.2421 + 2P*4.1429

13,432.65 = P*5.2421 + P*8.2858

13,432.65 = P*13.5279

P = $ 992.96

So, Mr. Y will withdrawl $992.96 for 1-6 years and $1,985.92 (992.96*2) for Next 6 years.