Q1. Sally plans to send her daughter to DCU in 18 years’ time. She has decided to invest a
sum of money each year on which will accumulate to €100,000 when her daughter is
entering college. If Sally can obtain a return of 13% per annum, how much must she save
each year?
Q2. Suppose that you take out a $250,000 house mortgage from your local savings bank.
The bank requires you to repay the mortgage in equal annual installments over the next 30
years. How much will you have to pay per annum if the interest rate is 12% a year?
Chapter 4
The Time Value
of Money
(Part 2)
Learning Objectives
1. Compute the future value of multiple
cash flows.
2. Determine the future value of an
annuity.
3. Determine the present value of an
annuity.
Copyright ©2016 Pearson Education, Ltd. All rights reserved.
4-2
4.1 Future Value of Multiple Payment
Streams
• With unequal periodic cash flows, treat each
of the cash flows as a lump sum and
calculate its future value over the relevant
number of periods.
• Sum up the individual future values to get
the future value of the multiple payment
streams.
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4-3
Figure 4.1 The time line of a
nest egg
Copyright ©2016 Pearson Education, Ltd. All rights reserved.
4-4
4.1 Future Value of Multiple
Payment Streams (continued)
Example 1: Future Value of an Uneven Cash
Flow Stream:
Jim deposits $3,000 today into an account that
pays 10% per year, and follows it up with 3 more
deposits at the end of each of the next three years.
Each subsequent deposit is $2,000 higher than the
previous one. How much money will Jim have
accumulated in his account by the end of three
years?
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4-5
4.1 Future Value of Multiple Payment
Streams (Example 1 Answer)
FV = PV x (1+r)n
FV
FV
FV
FV
of
of
of
of
Cash
Cash
Cash
Cash
Flow
Flow
Flow
Flow
at
at
at
at
T0
T1
T2
T3
=
=
=
=
$3,000
$5,000
$7,000
$9,000
Copyright ©2016 Pearson Education, Ltd. All rights reserved.
x
x
x
x
(1.10)3
(1.10)2
(1.10)1
(1.10)0
=
=
=
=
$3,000
$5,000
$7,000
$9,000
x
x
x
x
1.331 = $3,993.00
1.210 = $6,050.00
1.100 = $7,700.00
1.000 = $9,000.00
Total = $26,743.00
4-6
4.1 Future Value of Multiple Payment
Streams (Example 1 Answer)
ALTERNATIVE METHOD:
Using the Cash Flow (CF) key of the calculator, enter the
respective cash flows.
CF0=-$3000;CF1=-$5000;CF2=-$7000;
CF3=-$9000;
Next calculate the NPV using I=10%; NPV=$20,092.41;
Finally, using PV=-$20,092.41; n=3; i=10%;PMT=0;
CPT FV=$26,743.00
Copyright ©2016 Pearson Education, Ltd. All rights reserved.
4-7
4.2 Future Value of an Annuity
Stream
• Annuities are equal, periodic outflows/inflows., e.g. rent,
lease, mortgage, car loan, and retirement annuity payments.
• An annuity stream can begin at the start of each period
(annuity due) as is true of rent and insurance payments or at
the end of each period, (ordinary annuity) as in the case of
mortgage and loan payments.
• The formula for calculating the future value of an annuity
stream is as follows:
FV = PMT * (1+r)n -1
r
• where PMT is the term used for the equal periodic cash flow, r
is the rate of interest, and n is the number of periods
involved.
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4-8
4.2 Future Value of an Annuity
Stream (continued)
Example 2: Future Value of an Ordinary
Annuity Stream
Jill has been faithfully depositing $2,000 at the end
of each year since the past 10 years into an
account that pays 8% per year. How much money
will she have accumulated in the account?
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4-9
4.2 Future Value of an Annuity
Stream (continued)
Example 2
Answer
Future Value of Payment One = $2,000 x 1.089 =
$3,998.01
Future Value of Payment Two = $2,000 x 1.088 =
$3,701.86
Future Value of Payment Three = $2,000 x 1.087 = $3,427.65
Future Value of Payment Four = $2,000 x 1.086 =
$3,173.75
Future Value of Payment Five = $2,000 x 1.085 =
$2,938.66
Future Value of Payment Six = $2,000 x 1.084 =
$2,720.98
Future Value of Payment Seven = $2,000 x 1.083 = $2,519.42
Future Value of Payment Eight = $2,000 x 1.082 =
$2,332.80
Future Value of Payment Nine = $2,000 x 1.081 =
$2,160.00
Future Value of Payment Ten = $2,000 x 1.080 =
$2,000.00
Total Value of Account at the end of 10 years
$28,973.13
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4-10
4.2 Future Value of an Annuity
Stream (continued)
Example 2 (Answer)
FORMULA METHOD
FV = PMT * (1+r)n -1
r
where, PMT = $2,000; r = 8%; and n=10.
FVIFA ➔[((1.08)10 - 1)/.08] = 14.486562,
FV = $2000*14.486562 ➔ $28,973.13
USING A FINANCIAL CALCULATOR
N= 10; PMT = -2,000; I = 8; PV=0; CPT FV = 28,973.13
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4-11
4.2 Future Value of an Annuity
Stream (continued)
USING AN EXCEL SPREADSHEET
Enter =FV(8%, 10, -2000, 0, 0); Output = $28,973.13
Rate, Nper, Pmt, PV,Type
Type is 0 for ordinary annuities and 1 for annuities
due
USING FVIFA TABLE (A-3)
Find the FVIFA in the 8% column and the 10 period
row; FVIFA = 14.486
FV = 2000*14.4865 = $28.973.13
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4-12
FIGURE 4.3 Interest and principal
growth with different interest rates
for $100-annual payments.
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4-13
4.3 Present Value of an Annuity
To calculate the value of a series of equal
periodic cash flows at the current point in time,
we can use the following simplified formula:
1
1 −
n
(
)
1
+
r
PV = PMT
r
The last portion of the equation, is the
Present Value Interest Factor of an Annuity (PVIFA).
Practical applications include figuring out the nest egg needed
prior to retirement or lump sum needed for college expenses.
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4-14
FIGURE 4.4 Time line of present
value of annuity stream.
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4-15
4.3 Present Value of an Annuity
(continued)
Example 3: Present Value of an
Annuity.
John wants to make sure that he has saved up
enough money prior to the year in which his
daughter begins college. Based on current
estimates, he figures that college expenses will
amount to $40,000 per year for 4 years
(ignoring any inflation or tuition increases
during the 4 years of college). How much
money will John need to have accumulated in
an account that earns 7% per year, just prior to
the year that his daughter starts college?
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4-16
4.3 Present Value of an Annuity
(continued)
Example 3 Answer
Using the following equation:
1
1−
n
(1+ r )
PV = PMT
r
1. Calculate the PVIFA value for n=4 and r=7%➔3.387211.
2. Then, multiply the annuity payment by this factor to get
the PV,
PV = $40,000 x 3.387211 = $135,488.45
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4-17
4.3 Present Value of an Annuity
(continued)
Example 3 Answer—continued
FINANCIAL CALCULATOR METHOD:
Set the calculator for an ordinary annuity (END mode) and
then enter:
N= 4; PMT = 40,000; I = 7; FV=0; CPT PV = 135,488.45
SPREADSHEET METHOD:
Enter =PV(7%, 4, 40,000, 0, 0); Output = $135,488.45
Rate, Nper, Pmt, FV, Type
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4-18
4.3 Present Value of an Annuity
(continued)
Example 3 Answer—continued
PVIFA TABLE (APPENDIX A-4) METHOD
For r =7% and n = 4; PVIFA =3.3872
PVA = PMT*PVIFA = 40,000*3.3872
= $135,488 (Notice the slight rounding error!)
Copyright ©2016 Pearson Education, Ltd. All rights reserved.
4-19
Chapter 3
The Time Value
of Money
(Part 1)
Learning Objectives
1. Calculate future values and understand
compounding.
2. Calculate present values and understand
discounting.
3. Calculate implied interest rates and waiting time
from the time value of money equation.
4. Apply the time value of money equation using
formula, and spreadsheet.
Copyright ©2016 Pearson Education, Ltd. All rights reserved.
3-2
3.1 Future Value and
Compounding Interest
• The value of money at the end of the stated
period is called the future or compound
value of that sum of money.
– Determine the attractiveness of alternative
investments
– Figure out the effect of inflation on the future
cost of assets, such as a car or a house.
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3-3
3.1 (A) The Single-Period
Scenario
FV = PV + PV x interest rate, or
FV = PV(1+interest rate)
(in decimals)
Example 1: Let’s say John deposits $200 for a
year in an account that pays 6% per year. At
the end of the year, he will have:
FV = $200 + ($200 x .06) = $212
= $200(1.06)
= $212
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3-4
3.1 (B) The Multiple-Period
Scenario
FV = PV x (1+r)n
Example 2: If John closes out his account after 3 years, how
much money will he have accumulated? How much of that is
the interest-on-interest component? What about after 10
years?
FV3 = $200(1.06)3 = $200*1.191016 = $238.20,
where, 6% interest per year for 3 years = $200 x.06 x 3=$36
Interest on interest = $238.20 - $200 - $36 =$2.20
FV10 = $200(1.06)10 = $200 x 1.790847 = $358.17
where, 6% interest per year for 10 years = $200 x .06 x 10 = $120
Interest on interest = $358.17 - $200 - $120 = $38.17
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3-5
3.1 (C) Methods of Solving
Future Value Problems
• Method 1: The formula method
– Time-consuming, tedious
• Method 2: The financial calculator approach
– Quick and easy
• Method 3: The spreadsheet method
– Most versatile
• Method 4: The use of Time Value tables:
– Easy and convenient but most limiting in scope
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3-6
3.1 (C) Methods of Solving Future
Value Problems (continued)
Example 3: Compounding of Interest
Let’s say you want to know how much money you
will have accumulated in your bank account after 4
years, if you deposit all $5,000 of your high-school
graduation gifts into an account that pays a fixed
interest rate of 5% per year. You leave the money
untouched for all four of your college years.
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3-7
3.1 (C) Methods of Solving Future
Value Problems (continued)
Example 3: Answer
Formula Method:
FV = PV x (1+r)n➔$5,000(1.05)4=$6,077.53
Calculator method:
PV =-5,000; N=4; I/Y=5; PMT=0; CPT FV=$6077.53
Spreadsheet method:
Rate = .05; Nper = 4; Pmt=0; PV=-5,000; Type =0;
FV=6077.53
Time value table method:
FV = PV(FVIF, 5%, 4) = 5000*(1.215506)=6077.53,
where (FVIF, 5%,4) = Future value interest factor listed
under the 5% column and in the 4-year row of the Future
Value of $1 table.
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3-8
3.1 (C) Methods of Solving Future
Value Problems (continued)
Example 4: Future Cost due to Inflation
Let’s say that you have seen your dream
house, which is currently listed at $300,000,
but unfortunately, you are not in a position to
buy it right away and will have to wait at least
another 5 years before you will be able to
afford it. If house values are appreciating at
the average annual rate of inflation of 5%,
how much will a similar house cost after 5
years?
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3-9
3.1 (C) Methods of Solving Future
Value Problems (continued)
Example 4 (Answer)
PV = current cost of the house = $300,000;
n = 5 years;
r = average annual inflation rate = 5%.
Solving for FV, we have
FV = $300,000*(1.05)(1.05)(1.05)(1.05)(1.05)
= $300,000*(1.276282)
= $382,884.5
So the house will cost $382,884.5 after 5 years
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3-10
3.1 (C) Methods of Solving Future
Value Problems (continued)
Calculator method:
PV =-300,000; N=5; I/Y=5; PMT=0; CPT
FV=$382,884.5
Spreadsheet method:
Rate = .05; Nper = 5; Pmt=0; PV=-$300,000;
Type =0; FV=$382,884.5
Time value table method:
FV = PV(FVIF, 5%, 5) =
300,000*(1.27628)=$382,884.5;
where (FVIF, 5%,5) = Future value interest factor
listed under the 5% column and in the 5-year row
of the future value of $1 table=1.276
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3-11
3.2 Present Value and
Discounting
• Involves discounting the interest that would have
been earned over a given period at a given rate of
interest.
• It is therefore the exact opposite or inverse of
calculating the future value of a sum of money.
• Such calculations are useful for determining today’s
price or the value today of an asset or cash flow
that will be received in the future.
• The formula used for determining PV is as follows:
PV = FV x 1/ (1+r)n
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3-12
3.2 (A) The Single-Period
Scenario
When calculating the present or discounted
value of a future lump sum to be received
one period from today, we are basically
deducting the interest that would have been
earned on a sum of money from its future
value at the given rate of interest.
i.e. PV = FV/(1+r)➔ since n = 1
So, if FV = 100; r = 10%; and n =1;
➔PV = 100/1.1=90.91
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3-13
3.2 (B) The Multiple-Period
Scenario
When multiple periods are involved…
The formula used for determining PV is as
follows:
PV = FV x 1/(1+r)n
where the term in brackets is the present
value interest factor for the relevant rate of
interest and number of periods involved,
and is the reciprocal of the future value
interest factor (FVIF)
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3-14
3.2 Present Value and
Discounting (continued)
Example 5: Discounting Interest
Let’s say you just won a jackpot of $50,000
at the casino and would like to save a
portion of it so as to have $40,000 to put
down on a house after 5 years. Your bank
pays a 6% rate of interest. How much
money will you have to set aside from the
jackpot winnings?
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3-15
3.2 Present Value and
Discounting (continued)
Example 5 (Answer)
FV = amount needed = $40,000
N = 5 years; Interest rate = 6%;
• PV = FV x 1/ (1+r)n
• PV = $40,000 x 1/(1.06)5
• PV = $40,000 x 0.747258
• PV = $29,890.33➔ Amount needed to set
aside today
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3-16
3.2 Present Value and
Discounting (continued)
Calculator method:
FV 40,000; N=5; I/Y =6%; PMT=0; CPT PV=-$29,890.33
Spreadsheet method:
Rate = .06; Nper = 5; Pmt=0; Fv=$40,000; Type =0;
Pv=-$29,890.33
Time value table method:
PV = FV(PVIF, 6%, 5) = 40,000*(0.7473)=$29,892
where (PVIF, 6%,5) = Present value interest factor listed
under the 6% column and in the 5-year row of the Present
Value of $1 table=0.7473
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3-17
3.2 (C) Using Time Lines
• When solving time value of money
problems, especially the ones involving
multiple periods and complex combinations
(which will be discussed later) it is always a
good idea to draw a time line and label the
cash flows, interest rates and number of
periods involved.
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3-18
3.2 (C) Using Time Lines
(continued)
FIGURE 3.1 Time lines of growth rates (top)
and discount rates (bottom) illustrate
present value and future value.
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3-19
3.4 Applications of the Time
Value of Money Equation
• Calculating the amount of saving required
for retirement
• Determining future value of an asset
• Calculating the cost of a loan
• Calculating growth rates of cash flows
• Calculating number of periods required to
reach a financial goal.
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3-20
Example 3.3 Saving for
retirement
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3-21
Example 3.3 Saving for
retirement (continued)
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3-22
Example 3.3 Saving for
retirement (continued)
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3-23
Example 3.4 Let’s make a deal
(future value)
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3-24
Example 3.4 Let’s make a deal
(continued)
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3-25
Example 3.4 Let’s make a deal
(continued)
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3-26
Example 3.5 What’s the cost of
that loan?
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3-27
Example 3.5 What’s the cost of
that loan? (continued)
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3-28
Example 3.5 What’s the cost of
that loan? (continued)
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3-29
Example 3.6 Boomtown, USA
(growth rate)
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3-30
Example 3.6 Boomtown, USA
(continued)
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3-31
Example 3.6 Boomtown, USA
(continued)
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3-32
Example 3.7 When will I be rich?
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3-33
Example 3.7 When will I be rich?
(continued)
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3-34
Example 3.7 When will I be rich?
(continued)
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3-35
Additional Problems with Answers
Problem 1
Joanna’s Dad is looking to deposit a sum of money
immediately into an account that pays an annual
interest rate of 9% so that her first-year college
tuition costs are provided for. Currently, the
average college tuition cost is $15,000 and is
expected to increase by 4% (the average annual
inflation rate). Joanna just turned 5, and is
expected to start college when she turns 18. How
much money will Joanna’s Dad have to deposit into
the account?
Copyright ©2016 Pearson Education, Ltd. All rights reserved.
3-36
Additional Problems with Answers
Problem 1 (Answer)
Step 1. Calculate the average annual college tuition cost
when Joanna turns 18, i.e., the future compounded
value of the current tuition cost at an annual increase
of 4%.
PV = -15,000; n= 13; i=4%; PMT=0; CPT FV=$24,
976.10
OR
FV= $15,000 x (1.04)13 = $15,000 x 1.66507 =
$24,976.10
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3-37
Additional Problems with Answers
Problem 1 (Answer) (continued)
Step 2. Calculate the present value of the annual tuition
cost using an interest rate of 9% per year.
FV = 24,976.10; n=13; i=9%; PMT = 0; CPT PV = $8,146.67
(rounded to 2 decimals)
OR
PV = $24,976.10 x (1/(1+0.09)13=$24,976.10 x 0.32618 =
$8,146.67
So, Joanna’s Dad will have to deposit $8,146.67 into the
account today so that she will have her first-year tuition
costs provided for when she starts college at the age of
18.
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3-38
Additional Problems with Answers
Problem 2
Bank A offers to pay you a lump sum of $20,000
after 5 years if you deposit $9,500 with them
today. Bank B, on the other hand, says that they
will pay you a lump sum of $22,000 after 5 years if
you deposit $10,700 with them today. Which offer
should you accept, and why?
Copyright ©2016 Pearson Education, Ltd. All rights reserved.
3-39
Additional Problems with Answers
Problem 2 (Answer)
To answer this question, you have to calculate the
rate of return that will be earned on each investment
and accept the one that has the higher rate of return.
Bank A’s Offer:
PV = -$9,500; n=5; FV =$20,000; PMT = 0; CPT I =
16.054%
OR
Rate = (FV/PV)1/n - 1 = ($20,000/$9,500)1/5 – ...

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