OBJECTIVES
- Analyze how much to save for retirement.
- Determine whether to defer taxes or pay them now.
- Determine whether to get a professional degree.
- Determine whether to buy or rent an apartment.
CONTENTS
5.1 A Life-Cycle Model of Saving
5.2 Taking Account of Social Security
5.3 Deferring Taxes Through Voluntary Retirement Plans
5.4 Shoule You Invest in a Professional Degree?
5.5 Should You Buy or Rent?
In chapter 4 we explained how to use discounted cash flow analysis in making financial decisions. In this chapter
we apply theos discounted cash flow concepts to the major financial decisions we all must make at different stages of
our lives.Starting with the decision about how much to save for retirement, we develop a lifecycle model for
comprehensive financial planning.We then analyze whether you should defer taxes or pay them immediately,whether
you should invest in a professional education,and whether you should buy or rent a house.
5.1 A LIFE-CYCLE MODEL OF SAVING
Consider the following example. You are currently 35 years old, expect to retire in 30 years at age 65, and then to live
for 15 more years until age 80. Your current labor income is $30,OOO per year, and you have not yet accumulated any
assets.
We simp1ify the example by ignoring taxes. Also let us assume that your real labor income adjusted for inflation
remains at $30,OOO per year until age 65. In other words, we assume that your income will keep pace with inflation,
but not beat it.
How much should you spend for consumption now and how much should you save for retirement?
Every dollar you save will earn interest until you take it out.Of course, the cost of living will be going up, too.
We will assume that the interest rate you earn will exceed the rate of inflation by 3% per year.In other words, the real
rate of interest is 3% per year.
There are two approaches that you could take to computing how much you should save for your retirement: (1)
Aim for a target replacement rate of preretirement income, and (2) aim for maining the same level of consumption
spending before and after retirement. We will examine both approaches next.
5.1.1 Approach 1: Target Replacement Rate of Preretirement Income
Many experts recommend that in making a savings plan,you should aim for a replacement rate equal to 75% of your
preretirement income.Let us apply this r to our situation.With a real income before retirement of $ 30,000,the target
level of preretirement income is 0.75 × $30,000,or $22,500 per year.
The method for computing how much saving is needed to reach the desired target consists of two steps:
First compute the amount you need to have accumuiated in your personal retirement account when you reach
retirement age.
Then compute the annual amount of saving needed to reach that future value.
So, first we compute the amount that you have to have in your retirement fund at age 65 to be able to withdraw
$22,500 per year for 15 years:
n
15
i
3
PV
?
FV
0
PMT
-22,550
Result
PV=$268,604
Next we compute how much you need to save each year to have $268,604 accumulated 30 years from now:
n
i
PV
FV
PMT
Result
30
3
0 $268.604
?
PMT=$5,646
So the conclusion from this procedure is that in order to be able to take out a retirement benefit of $22,500 per year
for 15 years, you would need to save $5,646 per year in each of the next 30 years.
Now, let us consider a problem that arises when you use approach 1. Note that approach does not necessarily
result in your having the same consumptin level after retirement as you did during your working years. In the
preceding example, your consumption spending during the working years when you are saving $5,646 of your
$30,OOO annual income will be $24,354 per year,but then in retiremnt you will have only $22,500 to spend each
year.
One way to deal with this problem is to redo the calculations using a higher replacement rate than 75%.If that
replacement rate turns out to be too high, then try a lower one.You could continue applying this trial-and-error
procedure until you find a replacement rate that results in the same consumption spending before and after retirement.
Approach 2 addresses the problem directly without resortmg to a trial-and-error search.
5.1.2 Approach 2: Maintain the Same level of Consumption Spending
Let us now consider how much you need to save if your goal is to spend the same amount on consumption before and
after retirement.This implies a constant stream of the same amount in each of the next 45 years, denoted by C. The
amount saved each year from age 35 to 65 is $30,OOO minus C. At age 65,the total accumulation will be $47.58 ×
($30,000-C).The amount withdrawn from the retirement account each year after age 65 will be C.Its present value at
age 65 is $11.94C.
To find C we set the two amounts equal to each other:
47.58(30,000-C)=11.94C
C=$23,982
So consumption spending is $23,982 per year.Annual savings during the working years must,therefore,be
$ 6,018 per year (i.e.,$30,000 - $23,982).The total accumulation at age 65 will be $286,298.
Columns 1 through 4 in Table 5.1 and Figure 5.1 show the time profiles of income,consumption,and saving
derived in this example.They demonstrate that income is $30,OOO until age 65, and then it drops off to zero.
Consumption stays level at $23,982 per year from age 35 until age 80.
The equation that we have solved in order to find C can be written in a slightly different and more general way:
45
30
Yt
C



t
t
i 1 (1  r )
t 1 (1  r )
where r is the interest rate and Yt is labor income in year t.
1The
computation is:
n
i
PV
30
3
0
2The computation is:
n
i
PV
15
3
?
FV
?
PMT
1
Result
FV=$47.58
FV
0
PMT
1
Result
PV=$11.94
Equation 5.1 says that the present value of consumption spending over the next 45 years equals the present value
of labor income over the next 30 years.Economists call the present value of one's future labor income human capital,
and they call the constant level of consumption spending that has a present value equa to one’s human capital
permanent income.(See Box Mabout the economists who were awarded Nobel Prizes for their contributions to the
theory of human capital and consumption spending.)
In our example, with labor income of $30,OOO per year for 30 years,your human capital is $588,013 at age
35,and your permanent income is $23,982 per year. As you get older, the PV of your remaining labor income declines,
so your human capital falls steadily until it reaches zero at age 65.
Figure 5.2 and columns 5 and 6 in Table 5.1 show the time profiles of human capital and the accumulated
amount in the retirement fund implied by the pattern of income and saving in Figure 5.1 and columns 2 and 4 of Table
5.1 .The retirement fund starts out an zero at age 35, and it gradually grows to a high of $285,309 at age 65.It then
declines to zero at age 80.The individual’s wealth,defined as human capital plus retirement assets,declines continuously
between ages 35 and 80.
Let us consider what effect a different interest rate would have on both permanent income and human
capital.Table 5.2 shows that the higher the interest rate,the lower the value of human capital,but the higher the level of
permanent income. Because you save throughout your work years, you are better off with a higher real interest rate,
even though the value of your human capital is lower.
Now suppose that instead of starting out at age 35 with no accumulated assets, you have $10,000 in a savings
account.How does that affect the amount you can consume over your lifetime?The answer is that it enables you to
increase consumption spending in each of the next 45years by $407.85,assuming the interest rate is 3% per year.
On the other hand, suppose you wanted to leave a bequest of $10,OOO to your children after you die at age
80.With unchanged lifetime income,how does the intended bequest affect your lifetime consumption stream?Answer:It
would reduce your consumption by $107.85in each of the next 45 years.
Assumptions:You are currently 35 years old,expect to retire in 30 years at age 65, and then to live for 15 more years until age 80.Your real
salary is $30,OOO per year, and you have not yet accumulated any assets.
The generalformula that expresses the lifetime consumption possibilities open to you as a function of your
income, initial wealth,and bequests is:
T
Ct
 (1  r )
t 1
t

R
Bt
Yt

W


0
t
t
(1  r )
t 1 (1  r )
where:
Ct = consumption spending in year t
Yt =labor income in year t
r = zInterest rate
R = number of years until retirement
T = number of years of iife
Wo = va111e of initial wealth
B = bequest
Equation 5.2 says that the present value of your lifetime consumption spending and bequests equals the present
value of your lifetime resources-initial wealth and future labor income.This is the intertemporal budget constraint
that you face in deciding on a lifetime consumption spending plan.
Note that any lifetime consumption spending plan that satisfies your budget constraint (i.e.,equation 52) is a
feasible plan.There are many possible feasible plans.To choose among them you must specify a criterion for
quantitatively assessing the welfare or satisfaction (economists use the term utility)that you receive from each feasible
plan.A quantitative model that enables you to choose the best among all feasible plans is called an optimization
model.Developing optimization models for lifetime financial planning is beyond the scope of this text.
Now let us consider the effect of changes in real income over the life cycle.For example, Dr. Omar Ben Holim
has just graduated from medical school at age 3O and has started training to be a surgeon at Mount Heaven
Hospital.Omar's real salary for the next five years wili be $25,OOO per year.After completing his residency,however,
Omar’s expects to earn $300,OOO per year in real terms until he retires at age 65. Given his future expectations, he
decides to start enjoying a high standard of living immediately.If he wants to maintain the same level of real
consumption spending for the rest of his life and his life expectancy is 85 years, how much should he plan to save now
and in the future? Assume that the real interest rate is 3% per year, and that Omar can either borrow or lend at that
same rate.
Table 5.3 and Figure 5.3 show Omar's expected pattern of salary and planned consumption spending and saving
on the assumption that he wants to have the same real consumption every year.His human capital initially is
$5,186,747,and his permanent income is $193,720.In order to spend $193,720 per year during the five years of his
residency,he will have to borrow(“dissave”)$169,720 each year to suplement his $25,000 salary.His total indebtedness
will grow to a maximum of $895,758 at age 35,and then decline thereafter as a reault of his saving $106,280 every year
from age 36 until he retires at age 65.Note that he will not have paid off his debt until age 45.One is never too young to
start learning finance(see Box 5.2).
5.2 TAKING ACCOUNT OF SOCIAL SECURITY
In many countries the government obliges ciqzens to participate in a Mandatory retirement income system called social
security.Under such systems, people pay a tax during their working years and in return qualify for a lifetime annuity in
their old age.Such a system of mandatory saving shouid influence the amount of voluntary saving that we do for
retirement.Let us use our life-cycle planning model to examine the right way to take account of social security.
To address this question in the context of our life-cycle discounted cash now planning model, we first recognize
that social security alters the profile of our life-time net cash inflows. We return to our first example in which you are
35 years old, and your salary will be $30,OOO per year for the next 30 years.Your human capital is $588,013--the PV
of labor income at an interest rate of 3% per year. Suppose the optimal level of consumption spending is a constant,
$23,982 per year (equal to your permanent income).Annual savings in the preretirement years, must, therefore be
$6,018($30,000 - $23,982) per year. The total accumulation at age 65 will be $286, 309, which is enough to support a
retirement income of $23,982 per year for 15 years.
Suppose that social security benefits are equal to what you would have if you had saved each year an amount
equal to the amount you pay in social security taxes and earned a red interest rate of 3% per year. Thus if you pay
$2,OOO per year in social security taxes for 30 years,you will receive in benefits $7,970 per year for 15 years starting
at age 66.What impact will social security have on your savings and your welfare under these circumstances?
The answer is that you will simply reduce your personal voluntary savings by the amount of social security
taxes.So your savings will fall from $6,018 per year to $4,018.The difference of $2,000 is the amount you will pay in
social security taxes.Thus,you reduce your private saving by an amount equal to the "savings"imposed on you by the
social security system.Your pool of private savings will suffice to provide a life annuity of $16,012,which when added
to your social security benefit of $7,970 will give you a total retirement income of $23,982 per year.
Thus,if social secunty pays you the same rate of return you could earn on your private saving,your life time
comumption plan will not be affected by the existence of social security.There wiil only be a substitution of forced
saving for voluntary private saving.
But what happens if social security pays an implied real interest rate that differs from 3%? If it pays a rate
higher than 3%,you willbe able to afford a higher lifetime consumption stream than $23,982;if it pays less than 3%
per year,your consumption stream will be lower.
In many countries, the social security system offers a higher rate of return to people in the lower end of the
income distribution than to those in the upper end.But the fact that benefits are paid in the form of a lifetime annuity
implies that no matter how rich or poor you are, the longer you live, the higher your actual rate of return. The
effective rate of return earned in the social security system is an important issz (see Box 5.3).
5.3 DEFERRING TAXES THROUGH VOLUNTARY RETIREMENT PLANS
In many countries governments encourage voluntary saving for retirement through provisions of the tax code.In the
United States,people are permitted to establish tax-advantaged accounts,known as individual retirement accounts
(IRAs),to which contributions are deductible from current income for tax purposes,and interest on these contributions is
not taxed until the money is withdrawn.These plans are called tax deferred rather than tax exempt because any amounts
withdrawn from the plan are taxed at the time of withdrawal.
6The
n
PMT
30
calculation 15:
i
3
PV
0
?
FV=$95,151
FV
2,000
PMT
15
n
3
i
95,151
0
PMT=$7,970
PV
FV
?
Some people believe that there is an advantage to such tax deferral only if you will be in a lower tax bracket
when you withdraw the money.But that is not correct. Tax deferral is quite advantageous even for people who remain
in the same tax bracket after retirement.
To see why,consider the following example,summarized in Figure 5.4.Suppose that you face a tax rate of 20%
both before and after retirement.The interest rate is 8% per year .You are 30 years before your retirement date and
contribute $1,000 to the plan.Your total before-tax amount accumulated at retirement will be $1,000 × 1.0830
=$10,062.65.You will have to pay taxes at the rate of 20% on the entire amount,if you choose to withdraw it at that
time.Thus,your taxes will be 02 × $10,062.65 = $2,012.53,and you will be left with $8,050.12 after taxes.
If,instead,you choose not to participate in the retirement plan and invest in an ordinary savings plan,you have to
pay 20% of the $1,000 or $200 immediately in additional taxes.The remaining $800 will go into the ordinary savings
plan,and interest earnings on the $800 will be taxed each year.The after-tax interest rate earned is,therefore,(1-0.2) ×
8% or 6.4%.The amount accumulated at retirement from this ordinary savings plan is $800 ×
l .06430=$5,144.45.Because you have paid the taxes on the original contribution and on the interest along the way, the
amount accumulated is not subject to further tax.
Clearly, the tax-deferred savings plan provides a larger after-tax benefit because $8,050.12 is greater than
$5,144.45. Thus, even though you remain in the same 20% tax bracket both before and after retirement, the amount you
have to spend in the future is almost twice as much under the tax-deferred savings plan.
When your tax rate remains unchanged,the benefit of deferral can be summarized in the rule:deferral earns you
the pretax rate of return atfer tax.That is ,if you paid the initial tax and ivevsted %800 at the pretax rate of 8%,you
would have 800$×1.0830=$8,050.12
5.4 SHOULD YOU INVEST IN A PROFESSIONAL DEGREE?
Education and training can be viewed as an investment in human capital.Although there may be many reasons for
acquiring additional schooling,one purpose is to increase people's earning power,that is,increase their human capital.
Let us consider the costs and benefits of additional education.The economic costs consist of explicit costs such
as tuition and the implicit cost of forgone earnings during the time spent in school.The economic benefits consist of the
value of the increased stream of earnings attributable to the additional years of education.Like other investment
decisions,the investment is worthwhile if the present value of the expected incremental benefits exceeds the present
value of the expected incremental costs.
For example,consider Joe Grad who has just graduated from college and is deciding whether to go on for his
master's degree.Joe figures that if he takes a job immediately,he can earn $30,OOO per year in real terms for the
remainder of his working years.If he goes on for two more years of graduate study, however,he can increase his
earnings to $35,OOO per year.The cost of tuition is $15,OOO per year in real terms.Is this a worthwhile investment if
the real interest rate is 3% per year?
Ignoring uncertainty, Joe must give up $45,000(tuition plus forgone earnings) in each of the next two years in
order to increase his earnings by $5,OOO per year over his remaining career.
Suppose Joe is now 20 years old and expects to retire at age 65.The relevant cash flows for this investment are
incremental outflows of $45,000 in each of the next two years and then incremental inflows of $5,OOO in each of the
succeeding 43 years.The present value of the outflows is $86,106;the present value of the inflows is $113,026.The net
present value of the investment in human capital is,therefore, $26,920,and it is worthwhile.
5.5 SHOULD YOU BUY OR RENT?
You are currently renting a house for $10,OOO per year and have an option to buy it for $200,000.Property taxes are
deductible for income tax purposes,and your tax rate is 30%.The maintenance and property taxes are estimated to be:
Mainterlance
$1,200
Property Taxes
$2,400
Total
$3,600
These costs are currently included in the rent.
Let us assume that your objective is to provide yourself with housing at the lowest present value of cost.Should
you buy or continue to rent?
The present value of cost equals the discounted value of the after-tax outflows discounted at the after-tax rate of
interest.Because property taxes can be deducted from income for federal income tax purposes,the after-tax outflow for
property taxes each year is 0.7 × $2,400,or $1,680.Because no date for eventually sel1ing the house has been
specified, we will assume for simplicity an infinite horizon.
7First,compute
the present value of the outflows
n
i
PV
FV
2
3
?
0
Next compute the PV of the inflows:
n
i
PV
FV
PMT
45,000
PMT
Resrlt
PV =$86,106
Resrlt
43
2
3
3
?
0
0
119,910
5,000
0
PV =$119,910
PV =$113,026
If you buy the house, then you will have to pay $200,OOO immediately,and the expected after-tax cash outflow
will consist of the maintenance expenses and property taxes net of the income tax savings from deductibi1ity of the
property taxes:
Cash Outflow in Year t = $1,200 + $1,680 =$2,880
Because we are assuming that the maintenance costs and property taxes are fixed in real terms, i should be a real
interest rate.Let us assume no inflation so that the real and nominal before-tax discount rate is 3% per year. Therefore,
you would be better off buying the house.
The buy-or-rent decision is really an investment decision.In effect you are laying out $200,OOO today in order
to receive future cash benefits equal to the after-tax savings in rental costs.In present value terms,you save
$139,047(i.e., $476,190 一 $337,143).This is the NPV of the investment in the house.
Of course, the relation between PV Cost of Renting and PV Cost of Owning depends on the rent charged.At
what rent would you be indifferent between buying and renting?
This break-even rent (i.e., the annual rental costs at which you would be indifferent between owning or renting)
is found by setting PV Cost of Owning equal to PV Cost of Renting and solving for X:
X = 0.021 × $200,000 + $2,880
X = $4,200 + $2,880 = $7,080
Summary





In making lifetime saving/consumption decisions:(1)Do the analysis in real terms (constant dollars)to simplify
the calculations and to avoid having to forecast inflation.(2)Start by computing the present value of your
iifetime resources.The present value of your lifetime spending cannot exceed this amount.
Social security or any other forced saving program will offset voluntary saving. It may have a positive or a
negative effect on the present value of your total lifetime resources.
Tax-deferred retirement accounts are advantageous because they allow you to earn a before-tax rate of return
until money is withdrawn from the account. They are advantageous if you are in the same tax bracket before
and after you retire, and even more so if your tax bracket is lower after you retire.
Getting a professional degree or other training can be evaluated as an investment in human capital.As such, it
should be undertaken if the present value of the benefits (such as increase in your earnings) exceeds the present
value of the costs (such as tuition and forgone salary.)
In deciding whether to buy or rent an apartment or a consumer durable, choose the alternative with the lower
present value of costs.