# Amortization Schedule Calculator - PowerPoint

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```					   FINC3131

Chapter 5:
Time Value of Money –

1
Learning Objectives
1. Use a financial calculator to solve TVM
problems involving multiple periods and
multiple cash flows.
2. Solve TVM problems when the period of
compounding is less than a year.
3. Tell the difference between an ordinary annuity
and an annuity due.
4. Solve TVM problems involving an annuity due.
5. Prepare an amortization schedule
2
Preparing BAII Plus for use

1. Press „2nd‟ and [Format]. The screen will
display the number of decimal places that the
calculator will display. If it is not eight, press
„8‟ and then press „Enter‟.
2. Press „2nd‟ and then press [P/Y]. If the display
does not show one, press „1‟ and then „Enter‟.
3. Press „2nd‟ and [BGN]. If the display is not
END, that is, if it says BGN, press „2nd‟ and
then [SET], the display will read END.

3
The Formula for Future Value
Future Value         Number of periods

FV  PV  (1  r )   n

Rate of return or
Present Value
discount rate or
interest rate or
growth per period   4
The Formula for Present Value
From before, we know that

FV  PV  1  r 
n

Solving for PV, we get             Unless otherwise
stated, r stated on
FV
PV                     an annual basis.

(1  r ) n

Again, now we deal with PV problems where n > 2
5
Special keys used for TVM
problems
1. N: Number of periods (e.g., years)
2. I/Y: Interest rate/ discounting/
compounding rate per period
1. PV: Present value
2. PMT: The periodic fixed cash flow in an
annuity
3. FV: Future value
4. CPT: Compute
6
What is the future value (FV) of an initial
\$100 after 3 years, if I/YR = 10%?

1.   Finding the FV of a cash flow or series of cash
flows is called compounding.
2.   FV can be solved by using the step-by-step,

0                  1              2                3
10%

100                                            FV = ?       7
The step-by-step and formula methods

1. After 1 year:
FV1 = PV (1 + I) = \$100 (1.10)
= \$110.00
2. After 2 years:
FV2 = PV (1 + I)2 = \$100 (1.10)2
=\$121.00
3. After 3 years:
FV3 = PV (1 + I)3 = \$100 (1.10)3
=\$133.10
4. After N years (general case):
FVN = PV (1 + I)N
8
The calculator method

1. Solves the general FV equation.
2. Requires 4 inputs into calculator, and will
solve for the fifth.

INPUTS       3      10     -100     0
N      I/YR     PV     PMT      FV
OUTPUT                                    133.10

9
Multi-period, Find PV
 Find the present value of \$6,000 that
occurs at t = 6. The discount rate is 14
percent.

Use PV = FV/(1+r)6

FV=6000, N = 6, I/Y = 14, PMT = 0.
Press CPT and then PV

10
Multi-period, find FV
 Suppose you deposit \$150 in an account
today and the interest rate is 6 percent
p.a.. How much will you have in the
account at the end of 33 years?

Use FV = PV x (1+r)33

Press PV=-150, N=33,I/Y=6, PMT=0
Press CPT then FV

11
Multi-period, find r
 You deposited \$15,000 in an account 22 years
ago and now the account has \$50,000 in it.
What was the annual rate of return that you

Use r = (FV/PV)1/n – 1.

PV = - 15000, N = 22, PMT = 0, FV = 50000,
I/Y = ?
12
Multi-period, find n
 You currently have \$38,000 in an account that
has been paying 5.75 percent p.a.. You
remember that you had opened this account
quite some years ago with an initial deposit of
\$19,000. You forget when the initial deposit
was made. How many years (in fractions) ago
did you make the initial deposit?

PV = - 19000, PMT = 0, FV = 38000, I/Y = 5.75,
N=?

13
Perpetuity 1
 Perpetuity: a stream of equal cash flows
( C ) that occur at the end of each period
and go on forever.

C
PV of perpetuity =
r
C is the cash flow at the end of each period
r is the discount rate
14
Perpetuity 2
So what?
 We use the idea of a perpetuity to
determine the value of
 A preferred stock
 A perpetual debt

15
Perpetuity questions
 Suppose the value of a perpetuity is \$38,900
and the discount rate is 12 percent p.a.. What
must be the annual cash flow from this
perpetuity?
   Use C = PV x r. Verify that C = \$4,668.

 An asset that generates \$890 per year forever
is priced at \$6,000. What is the required rate
of return?
   Use r = C/PV. Verify that r = 14.833 percent

16
Annuity
 An annuity: a cash flow stream where a fixed
amount is received every period for a fixed
number of periods.
 Example: You rent out a property for \$12,000
per year for ten years.
 In many TVM problems, the cash flow stream is
 An annuity combined with a single cash flow (often at
the beginning or the end)
 A combination of two or more annuities.
17
Annuity, find PV
You are considering buying a rental property. The
yearly rent from this property is \$18,000. You
expect that the property will yield (i.e., generate)
this rent for the next twenty years after which you
will be able to sell it for \$250,000. If your required
rate of return is 12 percent p.a., what is the
maximum amount that you would pay for this
property?

PMT=18000, FV=250,000, I/Y=12, N=20, PV=?
18
Annuity, find FV
You open an account today with \$20,000
and at the end of each of the next 15 years,
you deposit \$2,500 in it. At the end of 15
years, what will be the balance in the
account if the interest rate is 7 percent p.a.?

PV=-20000, PMT=-2500, N=15, I/Y=7, FV=?

19
Annuity, find I/Y
You lend your friend \$100,000. He will
pay you \$12,000 per year for the ten
years and a balloon payment at t = 10 of
\$50,000. What is the interest rate that

PV=-100,000, FV=50,000, PMT=12,000,
N = 10, I/Y=?
20
Annuity, find PMT
Next year, you will start to make 35 deposits of
\$3,000 per year in your Individual Retirement
Account (so you will contribute from t=1 to t=35).
With the money accumulated at t=35, you will
then buy a retirement annuity of 20 years with
equal yearly payments from a life insurance
company (payments from t=36 to t=55).
If the annual rate of return over the entire period is
8%, what will be the annual payment of the
annuity?
21
Uneven Cash Flows

0           1      2          3
I%

-50         100     75     50

22
Uneven cash flows 1
 Your account pays interest at a rate of 5
percent p.a. You deposit \$8,000 in it
today. You must have exactly \$3,000 in
the account at the end of two years. How
much should you withdraw at the end of
the first year to ensure this?

23
What is the PV of this uneven cash
flow stream?

0         1   2     3      4
10%

100     300   300    -50

24
What is the PV of this uneven cash
flow stream?

0         1   2     3      4
10%

100     300   300    -50
90.91
247.93
225.39
-34.15
530.08 = PV
25
Solving for PV:
Uneven cash flow stream
1. Input cash flows in the calculator‟s “CF”
register: Press CF key
1.   CF0 = 0, ENTER,
2.   C01 = 100, ENTER, F01=1, ENTER,
3.   C02 = 300, ENTER,     F02=2, ENTER,
4.   C03= 50, +/- key, ENTER,
2. Press NPV key
3. I = 10, ENTER, press CPT key to get
NPV = 530.087. (Here NPV = PV.)
26
Uneven cash flows 2
 An asset promises to produce the following
series of cash flows. At the end of each of the
first three years, \$5,000. At the end of each of
the following four years, \$7,000. And, at the
end of each of following five years, \$9,000. If
your required rate of return is 10 percent, how
much is this asset worth to you?

 Find PV of this series of cash flows.
 PV = \$46,612.68
27
Uneven cash flows 3
   You will need to pay for your son‟s private school tuition (first
grade through 12th grade) a sum of \$8,000 per year for Years 1
through 5, \$10,000 per year for Years 6 through 8, and \$12,500
per year for Years 9 through 12. Assume that all payments are
made at the beginning of the year, that is, tuition for Year 1 is paid
now (i.e., at t = 0), tuition for Year 2 is paid one year from now,
and so on. In addition to the tuition payments you expect to incur
graduation expenses of \$2,500 at the end of Year 12. If a bank
account can provide a certain 10 percent p.a. rate of return, how
much money do you need to deposit today to be able to pay for
the above expenses?

28
Special topics
1.   Compounding period is less than 1 year
2.   Continuous compounding
3.   Annuity due
4.   Loan amortization

29
Compounding period is less than 1 year

Saying that compounding period is less than
1 year is equivalent to saying that
frequency of compounding is more than
once per year

30
Common examples
Compounding period   Compounding frequency

Six-months /              2
semiannual
Quarter                 4

Month                   12

Day                  365

31
Example (1)
Suppose that your bank “states” that the interest
on your account is eight percent p.a.. However,
interest is paid semi-annually, that is every six
months or twice a year.

The 8% is called the stated interest rate.
(also called the nominal interest rate)

But, the bank will pay you 4% interest every 6
months.
32
Example (2)
Ok, so we know how much interest is paid
every 6 months. Over a year, what is the
percentage interest I actually earn?

In other words,
I want to know the effective annual interest
rate
(or effective interest rate, or annual percentage
yield)
33
Example (3)
Suppose you deposit \$100 into the account
today.
Account balance at end of 6 months:
100 x 1.04 = 104
Account balance at end of 1 year:
104 x 1.04 =108.16
Effective interest rate
= (108.16 – 100)/100 = 0.0816 or 8.16%
34
When frequency of compounding is
more than once a year
FV                    „n‟ = number of
PV              m n
years
   r 
1  
„m‟ = frequency of
   m                  compounding per
year
m n
   r                „r‟ = stated interest
FV  PV  1                 rate
   m
m
   r 
Effective interest rate  1    1
   m
35
Can the effective rate ever be
equal to the nominal rate?

1. Yes, but only if annual compounding
is used, i.e., if m = 1.
2. If m > 1, effective rate will always be
greater than the nominal rate.

36
Effective rate example
You have decided to buy a car whose price is
\$45,000. The dealer offers to finance the entire
amount and requires 60 monthly payments of \$950
per month. What are the yearly stated and
effective interest rates for this financing?

stated = 9.723 % p.a.
effective = 10.168 % p.a.
37
 You are considering buying a new car. The
sticker price is \$15,000, and you have \$3000
for down payment. You obtain a 5-year car
loan at a nominal annual interest rate of 12%.
What is your monthly loan payment?
 Read question carefully when you work out the
size of the loan. What is PV?

38
Annuity with monthly compounding
 Compute the future value at the end of
year 25 of a \$100 deposited every month
for 10 years (with the first deposit made
one month from today) into an account
that pays 9 percent p.a.

39
Annuity with semiannual compounding
 You would like to accumulate \$16,500
over the next 8 years. How much must
you deposit every six months, starting six
months from now, given a 4 percent per
annum rate with semiannual
compounding?

40
Effective rate
Your bank‟s stated interest rate on a three
month certificate of deposit is 4.68 percent
p.a. and the interest is paid quarterly. What
is the effective interest rate?

41
Find period
 The stated interest rate for a bank
account is 7 percent and interest is paid
semi-annually. How many years will it
take you to double your money in this
account?

42
More frequent compounding,
more \$
All else constant, for a given nominal interest rate, an
increase in the number of compounding periods per year
will cause the future value of some current sum of money
to:
A.Increase
B.Decrease
C.Remain the same
D.May increase, decrease or remain the same depending
on the number of years until the money is to be received.
E.Will increase if compounding occurs more often than 12
times per year and will decrease if compounding occurs
less than 12 times per year.
43
Annuity Due 1
1. Up till now, we deal with ordinary
annuities.
2. For an ordinary annuity, payment
occurs at the end of each period.
3. For an annuity due, payment occurs at
the beginning of each period.
The difference becomes clear when we look
at time lines.
44
Consider an annuity that pays \$300 per
year for three years.
If ordinary annuity, time line is:
\$300            \$300   \$300

T=0           T=1              T=2    T=3

If annuity due, time line is:
\$300          \$300             \$300

T=0          T=1               T=2    T=3
45
Is there a relationship between
ordinary annuity and annuity due?
Yes !
PV of annuity due
= (PV of ordinary annuity) x (1 + r)

FV of annuity due
= (FV of ordinary annuity) x (1 + r)

„ordinary annuity‟ and „regular annuity‟ mean the same thing.

46
Example
   You have a rental property that you want to rent for 10
years. Prospective tenant A promises to pay you a rent
of \$12,000 per year with the payments made at the end
of each year. Prospective tenant B promises to pay
\$12,000 per year with payments made at the beginning
of each year. Which is a better deal for you if the
appropriate discount rate is 10 percent?

   Set PMT = 12,000, N = 10, I/Y = 10, FV=0
   To answer question, focus on dollar amount of each PV.

47
Another example
 What is the present value of an annuity of
\$1200 per year for 10 years (with the first
payment to be made today and the last
payment to be made 9 years from today) given
an interest rate of 5.5 percent p.a.?

48
Loan Amortization
Amortization is the process of separating a
payment into two parts:
 The interest payment
 The repayment of principal

Note:
 Interest payment decreases over time
 Principal repayment increases over time

49
Example of loan amortization 1
You have borrowed \$8,000 from a bank and have
promised to repay the loan in five equal yearly
payments. The first payment is at the end of the
first year. The interest rate is 10 percent. Draw up
the amortization schedule for this loan.

Amortization schedule is just a table that shows
how each payment is split into principal repayment
and interest payment.

50
Example of loan amortization 2
1) Compute periodic payment.
PV=8000, N=5, I/Y=10, FV=0, PMT=?
Verify that PMT = -2,110.38

Amortization for first year
Interest payment = 8000 x 0.1 = 800
Principal repayment
= 2,110.38 – 800 = 1310.38
Immediately after first payment, the principal
balance is = 8000 – 1310.38 = 6,689.62
51
Example of loan amortization 3
Amortization for second year
Interest payment = 6689.62 x 0.1 = 668.96
(using the new balance!)
Principal repayment
= 2,110.38 – 668.96 = 1441.42
Immediately after second payment, the principal
balance is = 6,689.62 – 1441.42 = 5,248.20

Verify the entire schedule (on following slide)

52
Verify the amortization schedule
Beg.                                 End.
Year Balance Payment Interest Principal   Balance
0                                        8,000.00
1   8,000.00 2,110.38 800.00 1,310.38 6,689.62
2   6689.62   2,110.38 668.96 1,441.42 5,248.20
3   5248.20   2,110.38 524.82 1,585.56 3,662.64
4   3662.64   2,110.38 366.26 1,744.12 1,918.53
5   1918.53   2,110.38 191.85 1,918.53     0.00

53
Using financial calculator to generate
amortization schedule 1
Very often, amortization problems involve long
periods of time, e.g., 30 year mortgage with
monthly payments => 360 periods.
To generate amortization schedule in such
problems, it‟s more efficient to use the financial
calculator.
Let‟s reuse the last problem (Problem 7.25). First,
find the monthly payment. Key in:
PV=8000, N=5, I/Y=10, FV=0, PMT=?
We already worked out that PMT = -2,110.38.

54
Using financial calculator to generate
amortization schedule 2
Suppose we want to work out the remaining
balance immediately after the 2nd
payment.
1. Press [2ND], [AMORT] to activate the
Amortization worksheet in BA II Plus.
2. Press P1=2, [ENTER], ,
3. Press P2=2, [ENTER], ,
4. You will see BAL=5,248.20
55
Using financial calculator to generate
amortization schedule 2
5. Press  again and you see the portion of
the year 2 payment going towards
repaying principal, i.e., PRN = -1,441.42
6. Press  again and you see the portion of
year 2 payment going towards interest,
i.e., INT = -668.96
To get out of the Amortization schedule,
press [2ND], Quit.
56
All together now (1)
    Which of the following statements is most correct?

A.   A 5-year \$100 annuity due will have a higher future value than a
5-year \$100 ordinary annuity.

B.   A 15-year mortgage will have smaller monthly payments than a
30-year mortgage of the same amount and same interest rate.

C.   All else being constant, for a given nominal interest rate, an
increase in the number of compounding periods per year will
cause the future value of some current sum of money to increase.

D.   Statements A and C are correct.

E.   All of the statements above are correct.

57
All together now (2)
Which of the following statements is most correct?

A.   An investment that compounds interest semiannually, and has a
nominal rate of 15 percent, will have an effective rate less than 15
percent.

B.   The present value of a three-year \$1000 annuity due is less than
the present value of a three-year \$1000 ordinary annuity.

C.   The portion of the payment of a fully amortized loan that goes
toward interest declines over time.

D.    Statements A and C are correct.

E.   None of the answers above is correct.

58
Summary
1. TVM problems with multiple periods and
multiple cash flows
2. Solving TVM problems using financial
calculator and time lines
3. Special topics
•   Compounding period < one year
•   Continuous compounding
•   Annuity due
•   Loan amortization
59
Assignment
Chapter2
Self-test ST-3 ST-4
Questions 5-2 5-3 5-4
Problems 5-1 5-2 5-3 5-4 5-5 5-6 5-7 5-8
5-9 5-10 5-12 5-13 5-14 5-15 5-16 5-17
5-18 5-19 5-21 5-22 5-23 5-24 5-25 5-27
5-33 5-34

60

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