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

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 rateNote: 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.

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.

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.

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.

Another Example of TimelineAnother Example of TimelineEx. 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!

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.

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

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

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.

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 = $1,000Coupon rate = 2%Maturity = 10 year

Example of Coupon BondExample of Coupon BondEx. 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.

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.

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.

Difference between Simple Loan and Difference 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).

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.

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.

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.

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)?

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.”

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)

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

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) = $121This is equivalent to FV2 = $100x(1+0.1)x(1+0.1) = $100x(1+0.1)2 = $121

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!

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

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

How to Compute Future Value on My How 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

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

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 )

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

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 )

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”.

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.

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.

What “Present” and “Future” Values What “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.

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

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.

Formula of Present Value of Cash Formula of Present Value of Cash FlowsFlowsYou may use the following formula to find the present value of cash flows:

Formula #3:

nn

221

n21

)i1(FV...

)i1(FV

)i1(FV

PV...PVPVPV

0 1 2 ..... 10 PV

FV1 FV2 .... FVn

Relationship between Formula and Relationship 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.

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.

Example of Present Value of Coupon Example 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 = $50Cash 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.

Example of Present Value of Coupon Example of Present Value of Coupon BondBondWhen the market interest rate is 10%, how much is a present value of the coupon bond?

0 1 2 3 YearPV

$50 $50 $1,050

65.875$88.788$32.41$45.45$

)1.01(050,1$

)1.01(50$

)1.01(50$

32

PV

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 InterestRate 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?

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!

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.

0 1 2 3 4 PV

$275 $275 $275 $275

Example of Present Value of Fixed Example 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

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%.

DisclaimerDisclaimer

Please do not copy, modify, or distribute this presentation without author’s consent.

This presentation was created and owned byDr. Ryoichi Sakano

North Carolina A&T State University