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?

How were these answers obtained?

1 st deposit deposits/yr EAR Last deposit 5 yrs ago 2 8.00% 1,000.00 $ now 1 yr 1 st withdrawal in First 6 next 6 New rate compounded 2C 8.00% 2 times/yr Past eff period rate # deposits Equivalent amount in bank 3.9230% 11 5.5 yrs ago $ 8,796.95 $ 13,432.68 Amount in bank today Therefore the amount in the bank today is the PV of all future withdrawals Future withdrawals consist of two annuities, one standard and one deferred PV of standard = C/r* (1 - 1/(1 + r)) PV of deferred = 2C/r* (1 - 1/(1 + r)y (1 + r) Note the "2" Thus PV = C/r* (1 - 1/(1 + r)) + 2C/r* (1 - 1/(1 + r@y(1 + r) We now factorise out the "C" and get PV = C * [1/r * (1 - 1/(1 + r)) + 2/r * (1 - 1/(1 + ry(1 + r)1 Solving for C, we get C = PV/[1/r * (1 - 1/(1 + r)) + 2/r * (1 - 1/(1 + r)y(1 + r)] Future eff rate So C = So 20 = $ $ 8.16% 1,298.16 2,596.32 1 st deposit deposits/yr EAR Last deposit 5 yrs ago 2 8.00% 1,000.00 $ now 1 yr 1 st withdrawal in First 6 next 6 New rate compounded 2C 8.00% 2 times/yr Past eff period rate # deposits Equivalent amount in bank 3.9230% 11 5.5 yrs ago $ 8,796.95 $ 13,432.68 Amount in bank today Therefore the amount in the bank today is the PV of all future withdrawals Future withdrawals consist of two annuities, one standard and one deferred PV of standard = C/r* (1 - 1/(1 + r)) PV of deferred = 2C/r* (1 - 1/(1 + r)y (1 + r) Note the "2" Thus PV = C/r* (1 - 1/(1 + r)) + 2C/r* (1 - 1/(1 + r@y(1 + r) We now factorise out the "C" and get PV = C * [1/r * (1 - 1/(1 + r)) + 2/r * (1 - 1/(1 + ry(1 + r)1 Solving for C, we get C = PV/[1/r * (1 - 1/(1 + r)) + 2/r * (1 - 1/(1 + r)y(1 + r)] Future eff rate So C = So 20 = $ $ 8.16% 1,298.16 2,596.32