FINC3131
Business Finance
Chapter 5:
Time Value of Money –
Advanced Topics
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 )
Present Value
n
Rate of return or
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
FV
PV 
n
(1  r )
Unless otherwise
stated, r stated on
an annual basis.
Again, now we deal with PV problems where n > 2
5
Special keys for TVM problems
1. N: Number of periods (e.g., years)
2. I/Y: Interest rate/ discounting rate per
period
3. PV: Present value
4. PMT: Periodic fixed cash flow
5. FV: Future value
6. CPT: Compute
6
What is the future value of an initial $100
after 3 years, if I/YR = 10%?
1.
2.
Finding the FV of a cash flow or series of cash
flows is called compounding.
FV can be solved by using the step-by-step,
financial calculator, and spreadsheet methods.
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
OUTPUT
3
10
-100
0
CPT
N
I/YR
PV
PMT
FV
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 after
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 on this investment?
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 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
 Annuity (ordinary annuity): a cash flow stream
where a fixed amount is received at the end of
every period for a fixed number of periods.
 Example: You borrow a loan for $12,000 and
pay 5% interest at the end of every year.
 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 to be paid
at year-end. You expect that the property will
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
Time Line Analysis
PV
250000
18000
0
1.
2.
3.
1
18000
2
18000
19
18000
20
18,000 lasts for 20 years.
250,000 occurs one minute right after the last 18,000.
PV actually is the price your are willing to pay.
19
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 15th
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=?
20
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
you are charging your friend?
PV=-100,000, FV=50,000, PMT=12,000,
N = 10, I/Y=?
21
Annuity, find PMT
Next year, you will start to make deposits of
$3,000 per year for 35 years in your Individual
Retirement Account. 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?
22
continued
-3000 -3000
0
1
2
-3000 -3000
35
34
PV
35
•
•
•
FV
PMT=?
36
55
There are two annuity. One is from t=0 to 35, the other is from 35 to 55.
To find the PMT in the second annuity, we need to input PV=, N=20,
FV=0, I/Y=8. However, PV (the price) is not known. But the PV in the
second annuity equals the FV in the first annuity because you use all the
money accumulated up to t=35 to buy the second annuity.
To find FV in the first annuity, you need to input PV=0, N=35, PMT=3000, I/Y=8.
23
Uneven Cash Flows
0
1
2
3
100
75
50
I%
-50
24
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 after two years. How much
should you withdraw at the end of the
first year to ensure this?
25
What is the PV of this uneven cash
flow stream?
0
1
2
3
4
100
300
300
-50
10%
26
What is the PV of this uneven cash
flow stream?
0
1
2
3
4
100
300
300
-50
10%
90.91
247.93
225.39
-34.15
530.08 = PV
27
Uneven cash flow stream
1. Input cash flows in the calculator’s “CF”
register: Press CF key. Clear the chip
1.
2.
3.
4.
CF0 = 0, ENTER,
C01 = 100, ENTER, F01=1, ENTER,
C02 = 300, ENTER,
F02=2, ENTER,
C03= 50, +/- key, ENTER, F03=1,
ENTER,
2. Press NPV key
3. I = 10, ENTER, press CPT key to get
NPV = 530.087. (Here NPV = PV.)
28
Use calculator
1.
Remember to clear the chip: 2 steps
2nd, CLR TVM; 2nd, CLR WORK
2. Remember to push ENTER and
keys after each step.
3. ENTER is not = key.
4.
F01, F02,… refers to the frequency of the cash flows.
Since only one 100, so F01=1, but F02=2 because
there are two consecutive 300.
29
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
30
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?
31
Special topics
1. Compounding period is less than 1 year
2. Annuity due
3. Loan amortization
32
Compounding period is less than 1 year
Saying that compounding period is less than
1 year is equivalent to saying frequency of
compounding is more than once per year
33
Common examples
Compounding period
Compounding frequency
Six-months /
semiannual
Quarter
2
Month
12
Day
365
4
34
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.
35
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?
I want to know the effective annual interest rate
36
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%
37
When frequency of compounding is
more than once a year
PV 
‘n’ = number of
years
FV
r 

1  
m

mn
r 

FV  PV  1  
m

mn
‘m’ = frequency of
compounding per
year
‘r’ = stated interest
rate
m
r 

Effective interest rate  1    1
m

38
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.
39
Can’t remember those formula?
1. Formula are hard to remember.
2. The examples in the following slides are
easy to remember.
3.
r
m
is the rate for each period, that is,
periodic rate.
40
Examples
1. If you deposit $100 today for 3 years.
The stated annual interest rate is 12% a
year. How much can you withdraw after
3 years? What is the effective rate?
* The answer depends on how often the interest is
compounded or paid. There are 4 possible answers.
41
Possible answers
1. The interest is paid once 1 year:
r
0.12 13
FV  PV (1  )1n  100(1 
)  140.49
1
1
0.12 1
Effective rate  (1 
)  1  0.12
1
r
2
2 n
2. The interest is paid every 6 months: FV  PV (1  )  100(1 
Effective rate  (1 
0.12 23
)  141.86
2
0.12 2
)  1  0.1236
2
3. The interest is paid every 3 months: FV  PV (1  r ) 4n  100(1  0.12 ) 43  142.58
4
Effective rate  (1 
4. The interest is paid every month:
4
0.12 4
)  1  0.1255
4
r 12n
0.12 123
)
 100(1 
)  143.08
12
12
0.12 12
Effective rate  (1 
)  1  0.1268
12
FV  PV (1 
42
Effective rate example
You have decided to buy a car priced at $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?
Answer:
stated = 9.723 % p.a.
effective = 10.168 % p.a.
43
Car buying with down payment
 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?
44
Annuity with monthly compounding
 Compute the future value at the end of year 25
of a $100 deposited every month (with the first
deposit made one month from today) into an
account that pays 9 percent p.a.
45
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 annual rate with
semiannual compounding?
46
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?
47
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?
48
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.
49
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.
50
Consider an annuity that pays $300 per
year for three years.
If ordinary annuity, time line is:
$300
T=0
T=1
$300
T=2
$300
T=3
If annuity due, time line is:
$300
T=0
$300
T=1
$300
T=2
T=3
51
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.
52
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.
53
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.?
54
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
55
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.
56
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
57
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)
58
Verify the amortization schedule
Beg.
Year Balance Payment Interest Principal
End.
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
59
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.
60
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
61
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.
62
What are P1 and P2?
1. P1 is the first payment of the period your are
interested in, while p2 is the ending payment of
the period you are interested in.
2. In the example, you are interested in the
second payment only. So P1=P2=2.
3. If you want to find the total interest paid from
the 3rd, 4th, and 5th payments, then P1=3 and
P2=5.
63
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.
64
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.
65
One more example
You borrowed $300,000 mortgage loan to buy a house at
the first day of 2009. The loan is for 15 years and the
stated rate is 12% p.a. All mortgage loan computes interest
monthly. Compute the total mortgage interest paid in 2012.
66
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
67
Practice Assignment
Chapter 5
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
68