Chapter 5: Extensions and
Applications of Time Value
of Money: Exchange Rates,
Inflation, Taxes, and the Life
Cycle
Objective
1
Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley
Financial decisions in an
uncertain world; Human
capital, permanent income
decisions over
life cycle
Chapter 5 Contents
– 5.1 A Life-Cycle Model of Savings
– 5.2 Taking Account of Social Security
– 5.3 Deferring Taxes through Voluntary
Retirement Plans
– 5.4 Should you Invest in a Professional
Degree?
– 5.5 Should you Buy or Rent?
2
Objectives
– How much to save for retirement
– Whether to defer taxes or pay them now
– Whether to get a professional degree
– Whether to buy or rent an apartment
3
5.1 A Life-Cycle Model of
Saving
• Assume that you are currently 35 years
old, expect to retire in 30 years at 65,
and then live for 15 more years until 80
• Your real labor income is $30,000/year
until age 65
• Interest rates exceed inflation by 3%/
year
4
How Much Should I Save and
Consume?
• Consider two approaches:
– Target replacement rate of pre-retirement
income
– Maintain the same level of consumption
spending
5
Target replacement rate of
pre-retirement income
• First compute the retirement income.
Many experts recommend a rate of 75%
of the pre-retirement income.
– $30,000*0.75 = $22,500/year
– using your calculator compute the present
value of the retirement funds as an regular
annuity
n=15, i = 3, FV=0, PMT=-22,500 -> PV=268,604
6
Target replacement rate of
pre-retirement income (Cont.)
• Next compute the retirement income
• Next compute how much you need to
save each year
n=15, i = 3, PV=0, FV= -268,604 -> PMT=5,646
•
To obtain a real $22,500 you need to
save $5,646 per year
7
Target replacement rate
Conclusion
• You will have noticed that your preretirement consumption is $30,000 $5,646 = 24,354; but the real retirement
income is only $22,500
• The next method equates consumption
8
Maintain the same level of
consumption spending
– Assume that your level of real consumption
is C
– The present value of consumption over the
next 45 years must equal the present value
of earnings over the next 30 years
n = 30, i = 3, FV = 0, PMT = 3,000, CPT PV, n = 45
CPT PMT gives $23,982
• The savings are then $30,000 - $23,982 = $6,018
9
Human Capital and
Permanent Income
• Human capital
– The present value of one’s future labor
income
• Permanent income
– The constant level of (real) consumption
spending that has a present value equal to
one’s human capital
10
Labor Income and Consumption
35000
30000
25000
Real $
20000
lab_inc
consump
15000
10000
5000
0
35
40
45
50
55
60
65
-5000
Age
11
70
75
80
Human Capital and Wealth
700000
600000
fund
HumanCap
Capital
500000
Real $
400000
300000
200000
100000
0
35
45
55
65
-100000
Age
12
75
The Inter-temporal Budget
Constraint
T
R
Ct
Yt
B

 W0  

t
T
t
1  i 
t 1 1  i 
t 1 1  i 
i = real interest rate
R = number of years to retirement
T = number of years of remaining life
W0 = initial wealth
B = bequest
13
5.2 Taking Account of Social
Security
• In many countries the government
obliges citizens to participate in a
mandatory retirement income system
called social security
• Contributors pay a tax during their
working years, and in return qualify for a
lifetime annuity in their old age
14
Social Security as Investment
Substitute
– If social security pays a return equal to 3% in the last
example, then just reduce the savings by the social
security tax
– The analysis becomes progressively more complex as we
make the assumptions more realistic.
• What if you don’t know your date of death., returns are risky,
et cetera?
15
5.3 Deferring Taxes Through
Voluntary Retirement Plans
– Many countries encourage voluntary savings for retirement
through provisions of the tax code.
– In the US employees are permitted to set up Individual
Retirement Accounts (IRA) that defer payment of taxes
until retirement
– The rules are a little complex, but an IRA may be used by
an investor to save money for retirement. Payments into
the plan are tax-deductible, but the flows from the plan
after retirement are taxed
– It is usual for marginal tax rates to be lower after
retirement, but this is not the key benefit
16
IRA Benefits
– The major benefits are more subtle. Assume:
• You can reserve $2,000 of pre-taxed income
for investment, starting next year, for the next
40-years. This will grow at the rate of
inflation of 3%
• That the investment will return 10%/year
• That you plan to remain retired for 20-years,
and will require income that is indexed to
inflation
• The tax rate on all taxable income streams is
30%, both now and after retirement
17
Sheltered and Unsheltered
Cases
• Sheltered case
– The full (real) $2,000 enters the plan.
Accumulations are not taxed, dispersions are
taxed. Result: 1st year after tax retirement
benefits = $82,785 ($24,639 in real terms)
• Unsheltered case
– Only a (real) $2,000 (1-0.30), enters the
plan. Earnings and realized capital gains are
taxable, dispersions are not taxed. Result:
1st year after tax retirement benefits
18
$31,671 ($9,426 in real terms)
IRA Conclusion
• Not taking into account the advantages
of differential taxation, the investor will
be 2.61 times better off using the
sheltered plan
19
5.4 Should You Invest in a
Professional Degree?
• Education may be viewed as an
investment in human capital
– One purpose of additional schooling is to
increase one’s earning power
• Example: Getting a Graduate Degree
20
• The Data:
– You've decided to obtain practical experience for 10 years,
and then get a Ph.D. In three years
– You want the same standard of income over the next 13
years
– Assume that all cash flows occur at year-end
– Your starting salary is $50,000. Because you are smart,
this will increase by 15% / year. You have agreed to be
paid this at the end of the first year, and yearly thereafter
– Ph.D. Fees are currently $15,000 per year, and increase by
3%/year with general inflation. Fees are paid at the end
of each year, so the fees for the period from 10 to 11 are
paid at year 11
21
Personal Planning Application
• Data (continued)
– Taxes are 30%, and are assumed to be constant. Assume
that lending and borrowing rates have been adjusted for
tax
– A fund with acceptable risk yields 10% / year
– You may also borrow at 10%
– Lending rate = borrowing rate! The real reason for this is
to simplify the math, but the fund could be moderately
aggressive, and the debt be consumer loans
22
Data Extraction
– Let the expenditure required for your
standard of living be X at the end of year 0
(beginning of year 1), X*1.03 in year 2,
X*1.032 in year 3, et cetera
– The fees start at $15,000*1.0310 (year 11),
and continue to grow at a rate of 3%
– Your net salary starts at $35,000 in year 1,
and grows at 15% for 10 years
– Everything is discounted at 10%
23
Salary cash flow (rate issue)
– The after-tax nominal cash flow in the first
year is $35,000, grows at a nominal 15% for
10 years
– We treat the 15% as the combined effect of
inflation and real growth
– Inflation is the interest rate
– The real rate is (0.10-0.15)/1.15= - 4.35%
24
Salary cash flow ($ issue)
– Now, remember, we have assumed that the
cash flows occur at the end of each year
– The first net income occurs at time 1, and so
must be discounted to year 0
– The real salary is not $35000 but
$35000/1.15 = $30,434.78
25
Salary Cash Flow Computation
– Using your financial calculators
• 10 - > n; 4.3478261 “+/-” -> I; PV = ?;
$30,434.78261 -> PMT; 0 -> FV
• Result PV = $391,816.3459 (in)
26
Fee cash flow ($ issues)
– The fees are already expressed in real terms,
but the first cash flow occurs in year 11, not
10 (the evaluation point) The year 11 real
cash flow must be adjusted to year 10
– 15000/1.03 = $14563.1068
27
Solution by Real Conversion
– The real interest rate is (0.10-0.03)/1.03 =
6.7961165% or about 6.80%
28
Fee Cash Flow Computation
– The present value of the fees at year 10 may
be obtained using your financial calculator:
• 3 -> n; 6.7961165 -> I; PV = ?; $14563.1068
-> PMT; 0 -> FV
• Result PV10 = -$38,361.00678
• 10 -> n; 6.7961165 -> I; PV = ?; PMT -> 0;
FV10 = 38,361.00678 (= PV10)
• Result PV0 = $19,876.2931 (out)
29
Living Expenditure CF $ issues
– Cash flows are assumed to occur at the end
of each year
– Let us compute the real amount today.
Denote this nominal amount in terms of the
unknown amount X in year 1
X/1.03 = 0.970873786 x
30
Living Expenditure CF Rate
Issues
– This case is easy. The real rate has been
computed to be 6.7961165% in the fee
section
31
Living Expenditure CF
Computation
– We do not know both the PV nor the PNT.
Set the payment to $1 for now, and multiply
by X later
– Using your financial calculators
• 13 -> n; 6.7961165 -> I; PV = ?;
$0.970873786 -> PMT; 0 -> FV
• Result PV = $8.208829899 * X (out)
32
Solution by Real Conversion
We are almost done. All that remains is to assemble
the parts, and solve the resulting equation
PV  391816.3459
 8.208829897 * X
 19876.29308
 391816.3459  19876.29308
X
 $45,309.75
 8.208829897
33
Conclusion
– This amount is the actual amount that will be expended
for the first year, paid at the end of that year
– This is a simple, but not a trivial, example, but it is loaded
with traps of even the most experienced. It requires
multistage logic
– The use of two distinct interest rates will bother some of
you
– Some thinkers believe that it is better to avoid quantities
that can not be observed directly
• While we certainly feel the influence of real cash flows
and real rates, observation is through the inflation rate
• We live in the world of the nominal, and another
approach is to recognize this in our system of financial
analysis
34
Solution by Growing Annuity
• Notation
– A is the starting year of a cash flow
– B is the ending year of a cash flow
– R is the nominal discount rate from 0 to b
– G is the geometric growth rate in nominal
cash flows
– Xa is the starting cash flow in year a
35
Solution by Growing Annuity
• Equation
 1 g 
Xa
1  
PV 

a 1 
r  g 1  r    1  r 
36
b  a 1




Solution method
– Just apply the equation three times
– To avoid error, you may wish to summarize
the data in a table before using it
37
Solution by Growing Annuity
j
aj bj
gj
Xaj
PVj
1 1
10 10% 35,000
?
2 1
13 3%
?
3 11 13 3%
X1
10
15,000*1.03
38
?
Algebra
  1.15 10 
  1.03 13 
35000
x
1  
1  
PV  0  
  
 
11 
11 
0.1  0.151.1   1.1   0.1  0.031.1   1.1  
10
  1.03 13111 
150001.03
1  



111 

0.1  0.031.1   1.1 


0  391816.3459  8.208829897x  19876.29308

x  45309.75
39
Interpretation
• The plan projects your going into debt in
the early years. You are probably
collaterallizing your existing human
capital, or future earnings potential
40
Year
NetSalary Fees
0
1
2
3
4
5
6
7
8
9
10
11
12
13
StartExp
35000
40250
46288
53231
61215
70398
80957
93101
107066
123126
-20159
-20764
-21386
Expend
-45310
-46669
-48069
-49511
-50997
-52526
-54102
-55725
-57397
-59119
-60893
-62719
-64601
45309.75
41
Sum
-10310
-6419
-1782
3719
10219
17871
26855
37375
49669
64007
-81051
-83483
-85987
Account
-10310
-17760
-21317
-19730
-11484
5239
32618
73255
130249
207281
146957
78170
0
How Much Must I Save?
• The real method is already set up. The
key strokes are
– N -> 8; I -> 8.7378641; PV -> 0; PMT -> ?;
FV -> 20000 “=/-”;
– Result: save at a real $1,830.79/year. That
is, starting in year1, save these nominal
amounts:
• $1885.71, $1942.28, $2000.55, ...
42
How Much Must I Save? The
Nominal Approach
n
n


X 0 1  g    1  g  
1 g 

 P
1 
PV 


r  g    1  r    1  r 
200000.12  0.03
Pr  g 

X0 
n
  1.12 8 

 1 r 

1  g  1.03 

1



1


  1.03 

  1  g 




X 0  $1,830.79
43
Concluding Remark
• The real (= nominal) value, xo, is not a
flow. It is just a base from which the
nominal flows may be computed
44
5.5 Should You Buy or Rent
• Interest paid on a mortgage is taxdeductible, and this provides a
substantial tax break if you are in a
higher tax bracket
• This may make buying a house attractive
to you when compared to renting
• Remember that this tax shield decays
45