Download Brooks PP Ch 4 RTP

The Time Value of Money
(Part Two)
LEARNING OBJECTIVES
1. Explain and illustrate an annuity.
2. Determine the future value of an annuity.
3. Determine the present value of an annuity.
4. Adjust the annuity equation for present value and
future value for an annuity due.
5. Distinguish between the different types of loan
repayments.
6. Build and analyze amortization schedules.
7. Calculate waiting time and interest rates for an
annuity.
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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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Fig. 4.1 The time-line of a nest egg
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4.2 Future Value of an Annuity Stream
• Annuities are equal, periodic outflows/inflows at regular intervals,
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 ordinary annuity
stream is as follows:
FV = PMT x (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 payments, one at the end of
each period (ordinary annuity).
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4.2 Future Value of an Annuity Stream
Example: Future Value of an Ordinary Annuity
Stream
Jill has been faithfully depositing $2,000 at the end of
each year over the past 10 years into an account that
pays a guaranteed 8% per year. How much money has
she have accumulated in the account?
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4.2 Future Value of an Annuity Stream
Example Answer (via the long way)
Future Value of Payment One = $2,000 x 1.089 =
Future Value of Payment Two = $2,000 x 1.088 =
Future Value of Payment Three = $2,000 x 1.087 =
Future Value of Payment Four = $2,000 x 1.086 =
Future Value of Payment Five = $2,000 x 1.085 =
Future Value of Payment Six = $2,000 x 1.084 =
Future Value of Payment Seven = $2,000 x 1.083 =
Future Value of Payment Eight = $2,000 x 1.082 =
Future Value of Payment Nine = $2,000 x 1.081 =
Future Value of Payment Ten = $2,000 x 1.080 =
Total Value of Account at the end of 10 years
$3,998.01
$3,701.86
$3,427.65
$3,173.75
$2,938.66
$2,720.98
$2,519.42
$2,332.80
$2,160.00
$2,000.00
$28,973.13
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4.2 Future Value of an Annuity Stream
Example Answer (short way)
FORMULA METHOD
FV = PMT x (1+r)n -1
r
where, PMT = $2,000; r = 8%; and n=10.
FVIFA =[((1.08)10 - 1)/.08] = 14.486562,
FV = $2000 x 14.486562 = $28,973.13
USING A FINANCIAL CALCULATOR
N= 10; PMT = -2,000; I/Y = 8; PV=0; CPT FV = $28,973.13
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4.2 Future Value of an Annuity Stream
USING AN EXCEL SPREADSHEET
Enter =FV(8%, 10, -2000, 0, 0); Output = $28,973.13
Rate = 0.08, Nper = 10, Pmt = -2,000, PV =0, and
Type is 0, for ordinary annuities
USING FVIFA TABLE (A-3), page 575
Find the FVIFA in the 8% column and the 10 period
row; FVIFA = 14.4866
FV = 2000 x 14.4866 = $28.973.20 (off by 7 cents)
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FIGURE 4.3 Interest and principal growth with
different interest rates for $100 annual payments.
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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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FIGURE 4.4 Time line of present value of
annuity stream.
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4.3 Present Value of an Annuity
 Example problem for the four solution methods
 You are now holding the winning lottery ticket that will
pay the holder of the ticket $10,000 per year for the next
20 years. A friend has offered to buy the winning ticket
from you. What should you sell the ticket for assuming
you have a discount rate of 6% on future dollars (this is
your opportunity rate for the future cash flow)?
 Four ways to solve




Formula
Calculator
Spreadsheet
Table
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4.3 Present Value of an Annuity
 Formula
 Inputs? N = 20, r = 0.06, PMT = $10,000 and
 Compute PV,
 PV = $10,000 x [1 – 1/(1.06)20] / 0.06 = $114,699.21
 Calculator
 Inputs? N = 20, I/Y = 6.0, PMT = 10,000, FV = 0
 Compute PV
 PV = -$114,699.21
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4.3 Present Value of an Annuity
 Spreadsheet, use PV function
 Inputs? Rate = 0.06, Nper = 20, Pmt = 10,000, Fv = 0
 PV = -$114,699.21
 Table
 First find the PVIFA with n = 20 and r = 6.0% on page
576, PVIFA = 11.4699
 Calculate PV = $10,000 x 11.4699 = $114,699.00 (off by 21
cents)
4-#
4.4 Annuity Due and Perpetuity
A cash flow stream such as rent, lease, and insurance
payments, which involves equal periodic cash flows
that begin right away or at the beginning of each time
interval is known as an annuity due.
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4.4 Annuity Due and Perpetuity
Formula Adjustment
PV annuity due = PV ordinary annuity x (1+r)
FV annuity due = FV ordinary annuity x (1+r)
PV annuity due > PV ordinary annuity
FV annuity due > FV ordinary annuity
Can you see why?
Financial calculator
Mode set to BGN for annuity due
Mode set to END for an ordinary annuity
Spreadsheet
Type = 0 or omitted for an ordinary annuity
Type = 1 for an annuity due.
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4.4 Annuity Due and Perpetuity
Example: Annuity Due versus Ordinary Annuity
Let’s say that you are saving up for retirement and
decide to deposit $3,000 each year for the next 20 years
into an account which pays a rate of interest of 8% per
year. By how much will your accumulated nest egg
vary if you make each of the 20 deposits at the
beginning of the year, starting right away, rather than
at the end of each of the next twenty years?
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4.4 Annuity Due and Perpetuity
Example Answer
Given information: PMT = -$3,000; n=20; i= 8%; PV=0;
FV ordinary annuity = $3,000 x [((1.08)20 - 1)/.08]
= $3,000 x 45.76196
= $137,285.89
FV of annuity due = FV of ordinary annuity x (1+r)
FV of annuity due = $137,285.89 x (1.08) = $148,268.76
Difference is $10,982.87
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4.4 Annuity Due and Perpetuity
Perpetuity
A Perpetuity is an equal periodic cash flow stream that
will never cease.
The PV of a perpetuity is calculated by using the
following equation:
PMT
PV 
r
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4.4 Annuity Due and Perpetuity
Example: PV of a perpetuity
If you are considering the purchase of a consol that pays
$60 per year forever, and the rate of interest you want to
earn is 10% per year, how much money should you pay for
the consol?
Answer:
r=10%, PMT = $60; and PV = ($60/0.10) = $600
$600 is the most you should pay for the consol.
You can think of it this way, if you put $600 in the bank
earning 10% you can withdraw the annual interest of
$60 every year forever without touching the principal.
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4.5 Three Payment Methods
Loan payments can be structured in one of 3 ways:
Discount loan
1)
•
Principal and interest is paid in lump sum at end
2) Interest-only loan
•
Periodic interest-only payments, principal due at
end.
3) Amortized loan
•
Equal periodic payments of principal and interest
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4.5 Three Payment Methods
Example: Discount versus Interest-only versus Amortized loans
Roseanne wants to borrow $40,000 for a period of 5 years.
The lenders offers her a choice of three payment structures:
1)
Pay all of the interest (10% per year) and principal in one lump
sum at the end of 5 years;
2)
Pay interest at the rate of 10% per year for 4 years and then a
final payment of interest and principal at the end of the 5th
year;
3)
Pay 5 equal payments at the end of each year inclusive of
interest and part of the principal.
Under which of the three options will Roseanne pay the least interest
and why? Calculate the total amount of the payments and the amount
of interest paid under each alternative.
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4.5 Three Payment Methods
Method 1: Discount Loan.
Since all the interest and the principal is paid at the
end of 5 years we can use the FV of a lump sum
equation to calculate the payment required, i.e.
FV
= PV x (1 + r)n
FV5 = $40,000 x (1+0.10)5
= $40,000 x 1.61051
= $64, 420.40
Interest paid = Total payment - Loan amount
Interest paid = $64,420.40 - $40,000 = $24,420.40
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4.5 Three Payment Methods
Method 2: Interest-Only Loan.
Annual Interest Payment (Years 1-4)
= $40,000 x 0.10 = $4,000 each year ($16,000)
Year 5 payment
= Annual interest payment + Principal payment
= $4,000 + $40,000 = $44,000
Total payment = $16,000 + $44,000 = $60,000
Interest paid = $60,000 - $40,000 = $20,000
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4.5 Three Payment Methods
Method 3: Amortized Loan.
n = 5; I/Y = 10.0; PV=$40,000; FV = 0;
CPT PMT= $10,551.89923
Total payments = 5 x $10,551.89923 = $52,759.50
Interest paid = Total Payments - Loan Amount
Interest paid = $52,759.50 - $40,000
Interest paid = $12,759.50
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4.5 Three Payment Methods
Loan Type
Discount Loan
Interest-only Loan
Amortized Loan
Total Payment
$64,420.40
$60,000.00
$52,759.31
Interest Paid
$24,420.40
$20,000.00
$12,759.31
Why does the equal annual payments of principal and
interest each period have the lowest total interest?
4-#
4.6 Amortization Schedules
Tabular listing of the allocation of each loan payment towards
interest and principal reduction
Helps borrowers and lenders figure out the payoff balance on an
outstanding loan.
Procedure:
1) Compute the amount of each equal periodic
payment (PMT) using the ordinary annuity formula.
2) Calculate interest on unpaid balance at the end of
each period, minus it from the PMT, reduce the loan
balance by the remaining amount,
3) Continue the process for each payment period, until we
get a zero loan balance.
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4.6 Amortization Schedules
Example: Loan amortization schedule.
Prepare a loan amortization schedule for the
amortized loan option given in the previous Example
with the five annual payments for the $40,000 at 10%
annual interest rate. What is the loan payoff amount
at the end of 2 years?
Step One, determine the annual payment:
PV = $40,000; n=5; I/Y=10.0; FV=0;
CPT PMT = $10,551.90 (rounded to nearest whole cent)
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Amortization Table
Year
Beg. Bal
Payment
Interest
Prin. Red
End. Bal
1
40,000.00 10,551.90 4,000.00
6,551.90
33,448.10
2
33,448.10 10,551.90 3,344.81
7,207.09
26,241.01
3
26,241.01 10,551.90 2,264.10
7,927.80
18,313.21
4
18,313.21 10,551.90 1,831.32
8,720.58
9,592.64
9,592.64
0.00
5
9,592.64 10,551.90
959.26
The loan payoff amount at the end of 2 years is
$26,241.01
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4.7 Waiting Time and Interest Rates for
Annuities
Problems involving annuities typically have 4 variables, i.e.
PV or FV, PMT, r, n
If any 3 of the 4 variables are given, we can easily solve for
the fourth one.
This section deals with the procedure of solving problems
where either n or r is not given.
For example:
–
–
Finding out how many deposits (n) it would take to reach a
retirement or investment goal;
Figuring out the rate of return (r) required to reach a
retirement goal given fixed monthly deposits,
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4.7 Waiting Time and Interest Rates for
Annuities
Example: Solving for the number of annuities
involved
Martha wants to save up $100,000 as soon as possible
so that she can use it as a down payment on her dream
house. She figures that she can easily set aside $8,000
per year and earn 8% annually on her deposits. How
many years will Martha have to wait before she can buy
that dream house?
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4.7 Waiting Time for Annuities
Method 2: Using a financial calculator
INPUT
?
8.0
0
-8000
100000
TVM KEYS
N
I/Y
PV
PMT
FV
Compute 9.006467
Method 3: Using an Excel spreadsheet
Using the “NPER” function we enter the following:
Rate = 8%; Pmt = -8000; PV = 0; FV = 100000;
Type = 0 or omitted;
display in excel = NPER(8%,-8000,0,100000,0)
The cell displays 9.006467.
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4.7 Waiting Time for Annuities
Method 1: Formula (uses natural logs)
N = ln ([FV x r]/PMT + 1) / ln (1+r)
N = ln ([100,000 x 0.08]/8,000 + 1) / ln 1.08
N = ln 2 / ln 1.08 = 0.693147181 / 0.07691041 = 9.006467
Method 4: Tables
You need to interpret from the tables…
Take FV / PMT to find the FVIFA, 12.50
Look for 12.50 under the 8% column, find its close to n
= 9 (its between 9 and 10 but very close to 9)
4-#
4.8 Finding the interest rate
Solving a Lottery Problem
In the case of lottery winnings, 2 choices
1) Annual lottery payment for fixed
number of years, OR
2) Lump sum payout.
How do we make an informed judgment?
Need to figure out the implied rate of return of both
options using TVM functions.
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4.8 Finding the interest rate
Example: Calculating an implied rate of return
given an annuity
Let’s say that you have just won the state lottery. The
authorities have given you a choice of either taking a
lump sum of $26,000,000 or a 30-year annuity of
$1,500,000. Both payments are assumed to be aftertax. What will you do?
The missing variable is the implied interest rate on the
two payment choices.
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4.8 Finding the interest rate
Using the TVM keys of a financial calculator, enter:
PV=26,000,000; FV=0; N=30; PMT = -1,500,000;
CPT I/Y = 3.98%
3.98% = rate of interest used to determine the 30-year
annuity of $1,500,000 versus the $26,000,000 lump
sum pay out.
Choice: If you can earn an annual after-tax rate of
return higher than 4.0% over the next 30 years, go
with the lump sum.
Otherwise, take the annuity option.
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4.8 Finding the interest rate
 We could use the Spreadsheet functions (Rate
function) to find the 3.98%.
 We could use the Tables to estimate the interest rate by
looking at the PVIFA at 30 years with the PVIFA
calculated as PV / PMT but again we will need to
estimate between to interest rates although in this case
it will be very close to 4.0% (PVIFA is 17.3333)
 We can not use the formula to solve for interest rate, it
is an iterative process (trial and error)
4-#
4.9 Ten Important Points about the TVM
Equation
1.
2.
3.
4.
5.
Amounts of money can be added or subtracted only
if they are at the same point in time.
The timing and the amount of the cash flow are what
matters.
It is very helpful to lay out the timing and amount of
the cash flow with a timeline.
Present value calculations discount all future cash
flow back to current time.
Future value calculations value cash flows at a single
point in time in the future
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4.9 Ten Important Points about the TVM
Equation
6. An annuity is a series of equal cash payments at regular
intervals across time.
7. The time value of money equation has four variables but
only one basic equation, and so you must know three of
the four variables before you can solve for the missing or
unknown variable.
8. There are three basic methods to solve for an unknown
time value of money variable:
(1) Using equations and calculating the answer;
(2) Using the TVM keys on a calculator;
(3) Using financial functions from a spreadsheet.
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4.9 Ten Important Points about the TVM
Equation
9. There are 3 basic ways to repay a loan:
(1)
(2)
(3)
Discount loans,
Interest-only loans, and
Amortized loans.
10. Despite the seemingly accurate answers from the time
value of money equation, in many situations not all the
important data can be classified into the variables of
present value, i.e., time, interest rate, payment, or
future value.
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