Question

Advanced Time Value of Money Problems

Question (Retirement planning)

You have just graduated Hofstra University at age 22. You hard work has paid off as you already have a job as an investment banker at Goldman Sachs waiting for you. You plan to work continuously until age 65 and retire exactly on that day. You expect to live until exactly 90 and enjoy your golden years and leave you heirs NOTHING. Assume your investments earn 8% per year.

You plan to contribute $10,000 to your retirement fund every year on your birthday starting at age 23. Your last deposit will be at exactly age 65 and your first withdrawal will be at age 66. Your last withdrawal will be at the moment you die at age 90.

Ignore all tax considerations for this problem.
(i) How much you will be able to spend each year in retirement?

(ii)

FV (deposits) = PV (withdrawals)
NOTE: This could be at any time period but t=65 is particularly convenient

How much will you be able to spend each year in retirement if you begin deposits at age 30?

(iii) How much larger do your deposits have to be if deposits start at age 30 to equal your answer in part (i)?

Now let’s consider the effect of inflation.
The values calculated above are nominal values. However, what is more important is real, i.e. inflation-adjusted, values.
Assume inflation averages 4% per year.

4. (iv) Re-calculate parts (i), (ii) and (iii) above.

5. (v) Comment on above.

Answer 1

	Rate	Rate of investment Earning	8%
	Nper	Number of Years of savings	43	(65-22)
	Pmt	Annual Savings	$10,000
	FV	Future value of savings at retirement	$3,295,830	(Using FV function of excel with Rate=8%, Nper=43, Pmt=-10000)
	Pv	Amount available at retirement	$3,295,830
	Rate	Rate of investment Earning	8%
	Nper	Number of annual withdrawals	25	(90-65)
	PMT	Amount can be spent in each year	$308,749	(Using PMT function of excel with Rate=8%, Nper=25, Pv=-3295830)
	(ii)	If You Start Deposit at the age of 30
	Rate	Rate of investment Earning	8%
	Nper	Number of Years of savings	35	(65-30)
	Pmt	Annual Savings	$10,000
	FV	Future value of savings at retirement	$1,723,168	(Using FV function of excel with Rate=8%, Nper=35, Pmt=-10000)
	Pv	Amount available at retirement	$1,723,168
	Rate	Rate of investment Earning	8%
	Nper	Number of annual withdrawals	25	(90-65)
	PMT	Amount can be spent in each year	$161,424	(Using PMT function of excel with Rate=8%, Nper=25, Pv=-1723168)
	(iii)	Deposit Required if Savings start at age 30
	Rate	Rate of investment Earning	8%
	Nper	Number of Years of savings	35	(65-30)
	Fv	Future Value required at retirement	$3,295,830
	PMT	Amount of annual deposit required	$19,127	(Using PMT function of excel with Rate=8%, Nper=35, Fv=-3295830)
		Inflation Adjusted Return on Investment=r
		Nominal return =8%=0.08
		Inflation rate =4%=0.04
		(1+r)*(1+0.04)=(1+0.08)
		1+r=(1.08/1.04)=	1.038461538
		Inflation Adjusted Return on Investment=r	0.0385	3.85%
	Rate	Rate of investment Earning	8.00%
	Nper	Number of Years of savings	43	(65-22)
	Pmt	Annual Savings	$10,000
	FV	Future value of savings at retirement	$3,295,830	(Using FV function of excel with Rate=8%, Nper=43, Pmt=-10000)
	Pv	Amount available at retirement	$3,295,830
	Rate	Rate of investment Earning	3.85%
	Nper	Number of annual withdrawals	25	(90-65)
	PMT	Amount can be spent in each year	$207,556	(Using PMT function of excel with Rate=3.85%, Nper=25, Pv=-3295830)
		Amount at todays dollar =207556/(1.04^43)	$38,432.74
	(ii)	If You Start Deposit at the age of 30
	Rate	Rate of investment Earning	8.00%
	Nper	Number of Years of savings	35	(65-30)
	Pmt	Annual Savings	$10,000
	FV	Future value of savings at retirement	$1,723,168	(Using FV function of excel with Rate=8%, Nper=35, Pmt=-10000)
	Pv	Amount available at retirement	$1,723,168
	Rate	Rate of investment Earning	3.85%
	Nper	Number of annual withdrawals	25	(90-65)
	PMT	Amount can be spent in each year	$108,517	(Using PMT function of excel with Rate=3.85%, Nper=25, Pv=-1723168)
		Amount at price at age 30 =108517/(1.04^35)	$40,707
	(iii)	Deposit Required if Savings start at age 30
	Rate	Rate of investment Earning	8%
	Nper	Number of Years of savings	35	(65-30)
	Fv	Future Value required at retirement	$3,295,830
	PMT	Amount of annual deposit required	$19,127	(Using PMT function of excel with Rate=8%, Nper=35, Fv=-3295830)