Question

3. You plan to retire in 40 years. Upon retirement, you plan to spend US$40,000.00 per year

for 30 years.

a. At a discount rate of 7.00% per year, how much do you need to have saved up in 40

years if you are to meet up with your retirement objectives?

b. Using the answer to part (a) and a compound rate of 8.00%, how much would you

have to save yearly for the next 40 years if you are to meet up with your retirement

objective?

c. Suppose you seek to spend US$40,000.00 per year for 30 years during retirement in

real purchasing power terms, as opposed to nominal purchasing power. Suppose

inflation equals 3% per year; how do your answers to parts (a) and (b) alter - number

answers to follow, parts (c) and (d).

Answer 1

a)

Annual spend at retirement = $40,000

Years in Retirement = 30

Discount rate, r = 7% per year

Amount needed at retirement can be calculated using PV function in spreadsheet

PV(rate, number of periods, payment amount, future value, when-due)

Where, rate = Discount rate = 7%

number of periods = years in retirement = 30

payment amount = Annual payment at retirement = $40,000

future value = 0

when-due = when is the withdrawal made each year = beginning = 1

Amount needed at retirement = PV(7%, 30, 40000, 0, 1) = $531,106.96 --------------(a)

b)

Years to retirement = 40

If you invest $1 each year, the future value of these investments after 40 years can be calculated using the FV function in spreadsheet

FV(rate, number of periods, payment amount, present value, when-due)

Where, rate = compounding rate = 8%

number of periods = years to retirement = 40

payment amount = yearly investment = $1

present value = present value of investments = 0

when-due = when is the investment made each year = end = 0

Future value of $1 investments at retirement = FV(8%, 40, 1, 0, 0) = $259.06

This is the value of $1 invested every year for 40 years, at retirement. If instead of $1, amount A is invested

Future Value of A invested for 40 years, at retirement = 259.06A -----------------(1)

Equating (a) and (1)

259.06A =  531,106.96

A =  531,106.96/259.06 = $2,050.16 -----------------(b)

You would need to invest $2,050.16 at the end of each year for 40 years, to cover for your retirement needs

c)

inflation = 3%

years to retirement = 40 years

Annual spend at retirement in today's terms = $40,000

Inflation adjusted Annual spend at retirement = Annual spend in today's terms * (1+inflation rate)^{years to retirement}

= 40000*(1+3%)⁴⁰ = 40000*3.262037792 = $130,481.51

Years in Retirement = 30

Discount rate, r = 7% per year

Inflation adjusted discount rate = [(1+discount rate)/(1+inflation rate)]-1 = [(1+7%)/(1+3%)]-1 = 3.8835%

Amount needed at retirement can be calculated using PV function in spreadsheet

PV(rate, number of periods, payment amount, future value, when-due)

Where, rate = Inflation adjusted discount rate = 3.8835%

number of periods = years in retirement = 30

payment amount = Inflation adjusted Annual spend at retirement = $130,481.51

future value = 0

when-due = when is the withdrawal made each year = beginning = 1

Amount needed at retirement = PV(3.8835%, 30, 130481.51, 0, 1) = $2,377,429.10 -----------------(c)

d)

Years to retirement = 40

If you invest $1 each year, the future value of these investments after 40 years can be calculated using the FV function in spreadsheet

FV(rate, number of periods, payment amount, present value, when-due)

Where, rate = compounding rate = 8%

number of periods = years to retirement = 40

payment amount = yearly investment = $1

present value = present value of investments = 0

when-due = when is the investment made each year = end = 0

Future value of $1 investments at retirement = FV(8%, 40, 1, 0, 0) = $259.06

This is the value of $1 invested every year for 40 years, at retirement. If instead of $1, amount A is invested

Future Value of A invested for 40 years, at retirement = 259.06A -----------------(2)

Equating (c) and (2)

259.06A =  2,377,429.10

A =  2,377,429.10 /259.06 = $9,177.26 -----------------(d)

You would need to invest $9,177.26 at the end of each year for 40 years, to cover for your retirement needs