Learning Unit #8: Time Value of Money

1. Learning Unit #8Learning Unit #8
Time Value of MoneyTime Value of Money

2. Objectives of Learning Unit #8Objectives of Learning Unit #8
• Concept of time value of money
• Cash flows and timeline
• Present value and future value
• Value of bond and market price of bond
• Relationship between bond price and market
interest rate
Note: this learning unit covers the same materials as in Business
Finance course. If you have taken that course, you may simply refresh
your knowledge of Business Finance. However, unlike Business
Finance course, this Money & Banking course only deals with the
concept of cash flows and the present value and addresses simpler
cases. Although you do not need a business calculator in this course,
you must have at least a scientific calculator to do some calculation
problems.

3. Cash FlowsCash Flows
Every financial instrument involves cash flows
between saver/lenders and borrower/spenders.
• When funds are loaned, funds flow out from
saver/lenders and into borrower/spenders.
• When the funds are paid back, funds flow into
saver/lenders and from borrower/spenders.
Some financial instruments involve only two
cash flows, other involve more than two cash
flows.

4. Cash Flows and TimelineCash Flows and Timeline
• Timeline depicts amounts and timings of
cash flows of a particular financial
instrument in a diagram.
• Timeline is used to distinguish one financial
instrument with particular cash flows from
another financial instrument with different
cash flows.

5. Example of TimelineExample of Timeline
Ex. There are $20 cash flow out from a lender today
and $24 cash flow into the lender two years later.
$20
$24
0 1 2
• A horizontal line is a timeline.
• Numbers (0, 1, 2) indicate years where “0” means the beginning of
cash flows (today) and “2” means two years later. Note that the
beginning of the cash flows can be today, past, or future.
• Arrows indicate flows of funds.
• Dollar amounts at arrows indicate amounts of funds flowed at
particular points of time.
• Notice that there is no cash flow one year later.

6. Another Example of TimelineAnother Example of Timeline
Ex. There are $20 cash flow out from a lender today, then $12
cash flow into the lender one year later and another $12 cash
flow into the lender two years later.
$20
$12 $12
0 1 2
• A lender still receives total of $24 as the previous example,
but at different timings.
• These two timelines present two different cash flows
(financial instruments).
• Which one is better? You need to understand the concept of
time value of money first!

7. Four Types of Cash FlowsFour Types of Cash Flows
There are four basic types of cash flows.
• Simple Loan
• Fixed-Payment Bond
• Coupon Bond
• Discount (Zero-coupon) Bond
Of course, in real world you can arrange any pattern of
cash flows beside these four types.

8. Simple LoanSimple Loan
• Simple loan: The borrower receives from the lender an
amount of funds (principal) and agrees to repay the lender
the principal plus an additional amount (interest) at given
date (maturity).
• Ex. You borrow $1,000 from BankAmerica today, and pay
$1,200 back to the bank ($1,000 principal + $200 interest)
ten years later.
0 1 2 ..... 10
$1,000
$1,200

9. Fixed-Payment LoanFixed-Payment Loan
• Fixed payment loan: The borrower makes regular
periodic equal payments (part of principal and interest)
for a set number of years.
• Fixe payment loans are used when a borrower borrows
a large sum amount of funds and pays back a little bit
each time over an extended period of time.
• On a fixed payment loan, a borrower pays each year an
accrued interest and a portion of principal.
• Examples of fixed payment loans: Car loan, mortgage
loan, student loan

10. Example of Fixe Payment LoanExample of Fixe Payment Loan
• Ex. You borrows $1,000 student loan this year and
promises to pay back $120 every year for the next
10 years starting from the next year.
0 1 2 ..... 10
$1,000
$120 $120 ..... $120
• So, how much will you pay in total? A simple sum of all
payments is $1,200 (= $120 x 10), of which $1,000 is the
principal and $200 is an interest payment.

11. Coupon BondCoupon Bond
• Coupon bond: The borrower receives an amount of funds
(principal), make multiple payments of interest (coupon) at
regular intervals, and repays the face value at maturity.
• Each coupon bond has three basic information which determine
its cash flows: face value, coupon rate, and maturity.
− Face value is an amount that a borrower promises to pay back at
maturity.
− Maturity is the date when the financial instrument expires.
− Coupon rate determines an annual coupon payment that a borrower
promises to pay where
Annual coupon payment = Coupon rate x Face value.
Face value = $1,000
Coupon rate = 2%
Maturity = 10 year

12. Example of Coupon BondExample of Coupon Bond
Ex. U.S. Treasury Note has $1,000 face value, 2% coupon rate,
and 10 year of maturity.
• Annual coupon payment = 2% x $1,000 = $20
0 1 2 ..... 10
$20 $20 ..... $1,020
• $20 cash inflow in year 1 is a payment for holding the bond for
one year from year 0 to year 1.
• $1,020 cash inflow in year 10 is a sum of face value payment
($1,000) and the final coupon payment ($20) for holding the
bond for one year from year 9 to year 10.
• A sum of all payments is $1,200 (= $20 x 10 + $1,000), of
which $1,000 is the principal and $200 is an interest payment.

13. Discount (Zero-coupon) BondDiscount (Zero-coupon) Bond
• Zero-coupon bond: The borrower pays the lender
the amount of the loan or the face value of the
bond at maturity, but receives less than that
amount initially.
• Since a borrower promises to pay a set-amount of
funds in future and no (coupon) payment [so it is
called zero-coupon bond] between now and
maturity, a lender is willing to lend no more than
the face value today [so it is called discount bond].
• A difference between how much a lender lends
today and the face value constitutes the interest on
the zero-coupon bond.

14. Example of Discount BondExample of Discount Bond
Ex. A corporation issues $1,000 face value zero-coupon bond
maturing in 10 years.
• A lender is willing to pay only $800 to purchase the zero coupon
bond (lend $800 today and get $1,00 back 10 years later).
• $200, a difference between $1,000 face value and $800 that a
lender lends is an interest.
0 1 2 ..... 10
$800
$1,000
• Of $1,000 payment by a borrower, $800 is the principal and
$200 is an interest payment.

15. Difference between Simple Loan andDifference between Simple Loan and
Discount BondDiscount Bond
• Both simple loans and discount bonds have the
identical cash flows, in which there are only cash
flows, one cash flow today and another cash flow in
future (10 years later in our examples).
• A difference between them is a way that cash flows
are determined.
− On simple loans, a borrower first decides how much to
borrow today ($1,000), then a lender decides how much
he wants to be paid back later ($1,200).
− On discount bonds, a borrower first decides how much to
pay back later ($1,000), then a lender decides how much
to lend today ($800).

16. Which Cash flow is the Best?Which Cash flow is the Best?
• All four examples of cash flows provide
$200 interest. Which one is the best for a
lender? Which one is the best for a
borrower?
• To answer this question, first you need to
understand the concept of time value of
money, then how to calculate an interest rate
on each example.
• Then, a borrower should choose one with the
least interest rate, while a lender should
choose one with the highest interest rate.

17. Time Value of MoneyTime Value of Money
• Ex. Your boss tells you that you get a $1,000 bonus for your hard
work. Would you like to get the bonus today or next month?
Your answer may be “today.”
• Ex. You borrowed $2,000 from your mom to pay for the tuition
and fees and promised to pay back all one day. Would you like
to pay back today or next year? Your answer may be “next
year.”
• As these examples illustrate, people want one cash flow over
another. Why? The reason is as everyone knows
A dollar today worth more than a dollar tomorrow.
• This implies that the same $1 has different values for you today,
depending on when you receive it.

18. Reasons for Time Value of MoneyReasons for Time Value of Money
Three reasons for time value of money:
• Attitude: we want to spend it now than later to get
something that we need or want (i.e. satisfaction that you
get from something today is greater than satisfaction that
you will get from the same thing in future).
• Availability of opportunities: we have a good use of funds
now to earn greater income more than sufficient to pay back
in future.
• Inflation: as prices of goods and services increase over
time, the same dollar can buy less and less in future.

19. Example of Time Value of MoneyExample of Time Value of Money
• Two examples of time value of money are easy cases
since both involve the same amount of cash flows
today and future. How do we compare if they are
different. Here is a real world example that people
have to make such decision.
• Ex. Cash payment on winning lottery ticket
If you win $97 million on Powerball, will you like to
get the one time cash “lump sum” amount of
$48,478,863 (yes, not $97 million) or $3,233,333
each year for the next 30 years (totaling $97
million)?

20. Implication of Time Value of MoneyImplication of Time Value of Money
• Since a dollar today is different from a dollar
tomorrow, we cannot directly compare today’s cash
flow with past or future cash flows, nor simply add
one cash flow in one year and another cash flow in
another year.
• Use a different weight on each cash flow (i.e. giving
more weight on today’s one dollar that tomorrow’s
one dollar), so that they are comparable.
• This weighing scheme is called “present value
method.”

21. Present Value and Future ValuePresent Value and Future Value
• We will explain the concept of the present value
method and its applications with examples. Get
your calculator and work through those examples.
On each example, you must be able to draw a
diagram of cash flows and timeline!
• We start with a simple case of “future value,” then
move to “present value.”
• For simplicity, in our examples every cash flow
occurs exactly yearly interval (e.g. 1 year later, 2
years later, ... 10 years later) and no cash flow
between them (e.g. 9 months later, 1 year and 3
months later)

22. 0 1 2 ..... 10
$100
FV1
Future Value – Example #1Future Value – Example #1
• If you purchase $100 one-year CD today at 10% annual
interest rate, how much will you get back in total one year
later (assuming no payment between today and the
maturity)?
• This is a case of simple loan where today’s cash flow is
$100, but you do not know an amount of cash flow one year
later. The cash flows on timeline should look like
FV1 = $100 + $100x10% (i.e. principal plus interest)
= $100x(1+0.1) (i.e. 10% = 0.1)
= $100 + $10 (i.e. $10 interest)
= $110

23. 0 1 2 ..... 10
$100
FV2
Future Value – Example #2Future Value – Example #2
• If you purchase $100 two-year CD today at 10% annual
interest rate, how much will you get back in total two years
later (assuming no payment between today and the maturity)?
From the previous example, by next year your principal will grow to
FV1 = $100 + $100x10% = $110.
Then, you keep it one more year, so from year 1 to year 2 it will
grow to
FV2 = $110 + $110x10% (i.e. new principal $110 plus interest)
= $121
This is equivalent to
FV2 = $100x(1+0.1)x(1+0.1) = $100x(1+0.1)2
= $121

24. Compound InterestCompound Interest
• On the previous example, if you get an answer of
“total of $120 two years later” (i.e. $100 + $10 x2)
rather than $121, then the difference must come from
the concept of compound interest.
• Compound interest means that the interest accrued in
the first year will be added to the principal at the
beginning of the second year, so that in the second
year you will get an interest on the original principal
($100) as well as the first year interest ($10) – that is,
interest is compounded.
• In real business world, we always use “compound
interest,” so as in this course!

25. Future Value FormulaFuture Value Formula
You may use the following formula to find the
future value of cash flow:
Formula #1: FV = PV x (1+i)n
• FV (Future value): A future amount in $.
• PV (Present value): A present amount in $.
• i: Annual interest rate
• n: Number of years

26. Future Value – Example #3Future Value – Example #3
• If you purchase $100 ten-year CD today at 10% annual interest
rate, how much will you get back in total ten years later
(assuming no payment between today and the maturity)?
0 1 2 ..... 10
$100
FV10
Apply the future value formula,
FV10 = $100x(1+0.1)10
= $259.37

27. How to Compute Future Value on MyHow to Compute Future Value on My
Calculator?Calculator?
• The future value formula involves a “power”
(i.e. raised by n). You can use you scientific
calculator to perform this task very easily.
• Look for a key marked as “x^y” or “xy
”.
First, calculate 1 +0.1, that is 1.1. Next, push
this key and type 10. Then, push “=” or
“Enter” key. Presto! You got 2.5937...
Now, multiply by 100 to get the final answer,
259.37

28. Present Value – Example #1Present Value – Example #1
• You need $110 one year later. If you can purchase one-year CD at
10% annual interest rate, how much should you put in CD today?
• This is a case of simple loan where the next year cash flow is $110,
but you do not know an amount of cash flow today. The cash
flows on timeline should look like
According to the previous example,
FV = $110 = PV + PV x 10%
= PV x (1+0.1)
Solve for PV,
PV = $110/(1+0.1) = $100 (You know this)
0 1 2 ..... 10
PV
$110

29. 0 1 2 ..... 10
PV
$121
Present Value – Example #2Present Value – Example #2
• You need $121 two years later. If you can purchase two-
year CD at 10% annual interest rate, how much should you
put in CD today?
Applying the future value formula,
FV2 = $121 = PV x (1+0.1)2
Solve for PV,
PV = $121/(1+0.1)2
= $100 (You know this too )

30. Present Value FormulaPresent Value Formula
You may use the following formula to find
the present value of cash flow:
Formula #2: PV = FV / (1+i)n

31. Present Value – Example #3Present Value – Example #3
• You need $259.37 ten years later. If you can purchase ten-
year CD at 10% annual interest rate, how much should you
put in CD today?
0 1 2 ..... 10
PV
$259.37
Applying the present value formula,
PV = $259.37/(1+0.1)10
= $100 (You know this too )

32. Present Value and its DeterminantsPresent Value and its Determinants
The present value formula shows the
following relationships between PV and one
of three determinants:
PV = FV / (1+i)n
• PV ↓ as i↑ for given FV and n.
• PV ↑ as FV↑ for given i and n.
• PV ↓ as n↑ for given FV and i.
Note: ↓ means “decrease” and ↑ means “increase”.

33. Implications of Present Value FormulaImplications of Present Value Formula
The relationships between PV and one of three
determinants imply
• If you can get a higher interest rate (i↑), you need
to put less funds today (PV↓) to get the same
amount in future.
• If you need more in future (FV↑), you need to put
more funds today (PV↑).
• If you need funds in further distance future (n↑)
rather than tomorrow, you need to put less funds
today (PV↓) to get the same amount.

34. Implication to Your Financial LifeImplication to Your Financial Life
The relationships between PV and one of three
determinants give you the following advises for your
saving, in particular a retirement savings:
• You better start saving now than later. To get the same
retirement savings, you need to put much less funds today
(given the fact that you will retire 40 years later at an age of
60 years old). Earlier you save, more time your saving can
grow and bigger your retirement saving.
• If you can get a higher interest rate, your retirement saving
grow even bigger. Even one percentage point difference in
interest rates will make a big difference in future.
• Of course, if you want more retirement saving, you must save
more.

35. What “Present” and “Future” ValuesWhat “Present” and “Future” Values
Really Mean?Really Mean?
• In the present value method, the “present”
value does not necessarily means the
“today’s” value. It simply means the value at
the beginning of timeline (i.e. initial cash
flow).
• Accordingly, the “future” value does not
necessarily means the “next year or later”
value. It simply means the value at the later
year(s) in the timeline.

36. Example of “Present” and “Future”Example of “Present” and “Future”
ValuesValues
• You loaned $100 to your brother last year at 10% interest
rate. Now you need to get your loan back. How much
should your brother pay you back today in total.
• You know the answer, $110!
• How does the cash flows and timeline look on this example?
• As you see on the timeline, $100 cash flow last year is
“present value” (the initial cash flow) and $110 cash flow this
year is “future value” (cash flow in later date).
Last year Today
$100
$110

37. Present Value of Cash FlowsPresent Value of Cash Flows
• Previous examples only involve one future cash
flow (i.e. simple loan cases). How can you find a
present value if more than one future cash flows?
• First, convert each of future cash flows to a present
value. Once all future cash flows are converted to
present values, you can compare and add/subtract
each others (because all cash flows are valued at the
same unit).
• Then, the sum of present values of all future cash
flows is the present value of cash flows.

38. Formula of Present Value of CashFormula of Present Value of Cash
FlowsFlows
You may use the following formula to find the present
value of cash flows:
Formula #3:
n
n
2
21
n21
)i1(
FV
...
)i1(
FV
)i1(
FV
PV...PVPVPV
+
++
+
+
+
=
+++=
0 1 2 ..... 10
PV
FV1 FV2 .... FVn

39. Relationship between Formula andRelationship between Formula and
TimelineTimeline
• You should notice a close relationship
between the timeline diagram and the
formula of present value.
• Each term in the formula corresponds to a
present value of each future cash flows.
• Each future cash flow is converted to a present
value by dividing it by (1+i) powered by year.
• Once you can show cash flows on timeline, it
is straightforward to apply the present value
formula.

40. Present Value of Coupon BondPresent Value of Coupon Bond
Coupon bonds with maturity of more than one year
involve more than one future cash flows. We apply the
formula to find a present value of coupon bond.

41. Example of Present Value of CouponExample of Present Value of Coupon
BondBond
A coupon bond has 3-year maturity, 5% coupon rate,
and $1,000 face value.
An annual coupon payment = 5% x $1,000 = $50
Cash flows for next three years are
0 1 2 3 Year
PV
$50 $50 $1,050
• In year 3 (at maturity) a cash flow is $1,050, which is sum of
face value $1,000 and the last coupon payment $50.

42. Example of Present Value of CouponExample of Present Value of Coupon
BondBond
When the market interest rate is 10%, how much is a present
value of the coupon bond?
0 1 2 3 Year
PV
$50 $50 $1,050
65.875$
88.788$32.41$45.45$
)1.01(
050,1$
)1.01(
50$
)1.01(
50$
32
=
++=
+
+
+
+
+
=PV

43. Present Value of Bond, Interest Rate,Present Value of Bond, Interest Rate,
and Maturityand Maturity
The present value of coupon bond depends on its
maturity (n) and the market interest rate (i).
MaturityMarket Interest
Rate 1 year 2 year
5% $1,047.62 $1,092.97
10% $1,000.00 $1,000.00
15% $956.52 $918.71
Ex. A coupon bond with $1,000 face value and 10% coupon rate.
Can you compute a present value of this coupon bond if its
maturity is 2 year and a market interest rate is 15%? How about 1
year maturity and 5% market interest rate?

44. Present Value of Coupon Bond and PricePresent Value of Coupon Bond and Price
The present value of coupon bond is the price
of the bond in market today.
• If a holder of the bond knows that his bond is worth
$875.65, then he will offer it at least $875.65.
• If a buyer of the bond knows that the bond is worth
$875.65, then she will ask it at most $875.65.
• So, how much is a price bond that both the holder
and buyer of the bond agree on?
• $875.65, the present value of the bond!

45. Bond Price and Interest RateBond Price and Interest Rate
There is an inverse relationship between a price of
coupon bond and a market interest rate.
• As the market interest rate increases, the value of the
bond decreases.
This inverse relationship between a bond price and a
market interest rate depends on its maturity.
• The longer the maturity, the greater the changes in
price as the market interest rate changes.
Note. You verify these relationship by inspecting the table on “Present value
of Bond, Interest Rate, and maturity” earlier.

46. 0 1 2 3 4
PV
$275 $275 $275 $275
Example of Present Value of FixedExample of Present Value of Fixed
Payment LoanPayment Loan
Fixed payment loans also involve more than one future cash flows.
Ex. How much is a present value of fixed payment loan with 4
year maturity and $275 annual payment if the annual interest
rate is 5%?
13.975$
24.226$56.237$43.249$90.261$
)05.01(
275$
)05.01(
275$
)05.01(
275$
)05.01(
275$
PV 432
=
+++=
+
+
+
+
+
+
+
=

47. Present Value and Financial DecisionPresent Value and Financial Decision
• When you borrow a certain amount of funds and
have three different payment options, which one
should you choose?
• Assuming that you can afford any of three payment
options, then you should choose one with the lowest
present value.
− Notice that it is not the lowest sum of actual payments.
You must take into account of time value of money.
• Now, apply this to three examples given earlier this
learning unit; a simple loan, a fixed payment loan,
and a coupon bond if the market interest rate is 10%.

48. DisclaimerDisclaimer
Please do not copy, modify, or distribute this presentation
without author’s consent.
This presentation was created and owned by
Dr. Ryoichi Sakano
North Carolina A&T State University