Question

Discussion Question 2 – (CLOs covered: 1, 3)

Retirement Planning.

(a) By the end of this year you would be 35 years old and you want to plan for your retirement. You wish to retire at the age of 65 and you expect to live 20 years (I hope more) after retirement. Upon retirement you wish to have an annual sum of $50,000 to supplement your social security benefits. A conservative bond fund has been returning 7% annually and you decide to invest your retirement money in this fund. Assuming that the fund continues to return at least 7% during your planning horizon before and after retirement, how much should you invest in the fund starting from now, annually, in order to be able to withdraw $50,000 per year during your retirement?

Now let’s extend the problem so that you protect yourself against inflation.

(b) Suppose you think if you were to retire right now you would have needed $50,000 each year to supplement your social security and maintain your desired life style.

But because there is on average 3% annual inflation, when you retire in 30 years from now you need more than $50,000 per year to maintain the life style you like. (Hint: first calculate what future amount in 30 years is equivalent to $50,000 of now and then solve the rest of the problem).

Provide your explanations and definitions in detail and be precise. Comment on your findings. Provide references for content when necessary. Provide your work in detail

Answer 1

Part (a)

Assumption: All the cash flows whether deposit or withdrawal take place at the end of the period.

Kitty size required at the age of 65 years should be good enough to allow withdrawal of $ 50,000 every year for next 20 years. Interest rate , R = 7%

Kitty size required at the age of 65 years = Sum of PV of annuity, A of $ 50,000 over time , N of 20 years at interest rate, R of 7% = A / R x [1 - (1 + R)^-N] = 50,000 / 7% x [1 - (1 + 7%)^-20] = $  529,701

This has to be funded by an annual investment D over next 65 - 30 = 35 years.

Sum of FV of annuity of D over T = 35 years at interest rate, R of 7% will be = D / R x [(1 + R)^T - 1] = D / 7% x [(1 + 7%)³⁰ - 1] = $ 94.46078632 x D

Hence, FV of annual investments = = $ 94.46078632 x D = kitty size required at the time of retirement = $ 529,701

Hence, D = $ 529,701 / 94.46078632 = $  5,607.63

Hence, annual investment = D = $ 5,607.63 (please round it off as per your requirement)

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(b) Annual amount required per year post retirement = A = 50,000 x (1 + inflation) ^{Time to retire} = 50,000 x (1 + 3%)³⁰ = $  121,363

Kitty size required at the age of 65 years should be good enough to allow withdrawal of $ 121,363 every year for next 20 years. Interest rate , R = 7%

Kitty size required at the age of 65 years = Sum of PV of annuity, A of $ 121,363 over time , N of 20 years at interest rate, R of 7% = A / R x [1 - (1 + R)^-N] = 121,363 / 7% x [1 - (1 + 7%)^-20] = $ 1,285,723

This has to be funded by an annual investment D over next 65 - 30 = 35 years.

Sum of FV of annuity of D over T = 35 years at interest rate, R of 7% will be = D / R x [(1 + R)^T - 1] = D / 7% x [(1 + 7%)³⁰ - 1] = $ 94.46078632 x D

Hence, FV of annual investments = = $ 94.46078632 x D = kitty size required at the time of retirement = $ 1,285,723

Hence, D = $ 1,285,723 / 94.46078632 = $ 13,611.18

Hence, annual investment = D = $  13,611.18 (please round it off as per your requirement)