Ch 4:Introduction to Time

Value of Money

Dr. Yi

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Outlines of Ch 4 and 5: Time Value of Money (TVM)

Basics Simple present / future value problem Simple vs. Compound Interest Compounding Frequency Annuity / Annuity Due / Perpetuity Uneven cash flow Amortized loan Effective annual rate

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Time lines show timing of cash flows.

CF0 CF1 CF3CF2

0 1 2 3i%

Tick marks at ends of periods, so Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period 2.

Today End of second year

Start of third year

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The Time Value of Money and Interest Rates

Timelines– Constructing a Timeline– Identifying Dates on a Timeline

• Date 0 is today, the beginning of the first year

• Date 1 is the end of the first year

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The Time Value of Money and Interest Rates

The Time Value of Money– In general, a dollar today is worth more than a

dollar in one year • If you have $1 today and you can deposit it in a bank at

10%, you will have $1.10 at the end of one year

– The difference in value between money today and money in the future is the time value of money.

• $1.10 minus $1.00 = 10 cents

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The Time Value of Money and Interest Rates

The Interest Rate: Converting Cash Across Time– By depositing money, we convert money today into money

in the future– By borrowing money, we exchange money today for money

in the future– Interest Rate (i)

• The rate at which money can be borrowed or lent over a given period

– Interest Factor• It is the rate of exchange between dollars today and dollars in the

future• It has units of “$ in one year/$ today”• It is expressed as

(1 + i)

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The Time Value of Money and Interest Rates

The Interest Rate: Converting Cash Across Time– Discount Factors and Rates

• Money in the future is worth less today so its price reflects a discount

• Discount Rate ( i )– The appropriate rate to discount a cash flow to determine its value at an

earlier time

• Discount Factor– The value today of a dollar received in the future, expressed as:

$1 / (1 + i)

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The Rules of Time Travel

Rule 1: Comparing and Combining Values– It is only possible to compare or combine values at the same point in time

Rule 2: Compounding– To calculate a cash flow’s future value, you must compound it

Rule 3: Discounting– To calculate the value of a future cash flow at an earlier point in time, we

must discount it.

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Compounding vs. Discounting

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Four variables

PV FV Interest rate, i Period, time length, or investment horizon, n

FV

0 1 2 Year, ni%

PV 1n

nFV PV i

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FV and PV

Future ValueFuture Value is the value at some future time of a present amount of money, or a series of payments, evaluated at a given interest rate.

Present ValuePresent Value is the current value of a future amount of money, or a series of payments, evaluated at a given interest rate.

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What’s the FV of an $100 after 3 years if i = 10%?

FV = ?

0 1 2 310%

Finding FVs (moving to the righton a time line) is called compounding. Multiply by (1+i)N

100

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10%

What’s the PV of $100 due in 3 years if i = 10%?

Finding PVs is discounting, and it’s the reverse of compounding. Divide by (1+i)N

100

0 1 2 3

PV = ?

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Present Values

How much do I have to invest today to have some amount in the future?– FV = PV(1 + i)n

– Rearrange to solve for PV = FV / (1 + i)n

When we talk about discounting, we mean finding the present value of some future amount.

When we talk about the “value” of something, we are talking about the present value unless we specifically indicate that we want the future value.

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Financial calculators solve this equation:

There are 4 variables. If 3 are known, the calculator will solve for the 4th.

FV PV inn 1 .

Three Known Variables and One Unknown Variable

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Simple present / future value problem:Type I (FV Problem)

How much will I get tomorrow?

FV = ?

0 N=3i%=10

PV=100

FV = PV * (1 + i%)n

= 100 * (1+10%)3

= 100 * FVIF10%, 3

= 100 * 1.3310= $133.10

3 N

10 I

100 PV

Hit FV

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Simple present / future value problem:Type II (PV Problem) How much should I save today?

FV=100

0 N=3i%=10

PV = ?

PV = FV / (1 + i%)n

= 100 * (1/(1+10%)3)= 100 * PVIF10%, 3

= 100 * 0.7513= $75.13

3 N

10 I

100 FV

Hit PV

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Simple present / future value problem:Type III (# of Period Problem) How long it will take to reach my goal?

FV=133.10

0 N = ?i%=10

PV=-100

10 I

-100 PV

133.10 FV

Hit N

FV = PV * (1 + i%)n

FV / PV = (1 + i%)n

ln(FV/PV) = ln(1 + i%)n ln(FV/PV) = n*ln(1 + i%)ln(FV/PV) / ln(1+i%) = N

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Simple present / future value problem:Type IV (Interest Rate Problem)

At what interest rate should I save to reach my goal?

FV=133.10

0 N=3i% = ?

PV=-100

FV = PV * (1 + i%)n

FV / PV = (1 + i%)n

(FV / PV)1/n = 1 + i%(FV / PV)1/n - 1 = i%

3 N

-100 PV

133.10 FV

Hit I

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Using a Financial Calculator:Setting the keys

HP 10BII

– Gold → C All (Hold down [C] button)• Check P/YR

– # → Gold → P/YR• Sets Periods per Year to #

– Gold → DISP → #• Gold and [=] button

• Sets display to # decimal places

CGold

Gold# P/YR

Gold #DISP

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Examples

Example 1: Heather is considering the purchase of an apartment building. She estimates the building's current value at $400,000. At an annual increase of 8 percent, how much would the investment be worth at the end of 5 years?

Example 2: Chris is considering the purchase of an apartment building. He estimates the building's future value at $587,731 at 5 years from now. At an annual discount rate of 8%, how much would he pay now?

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Examples

Example 3: Suppose you want to buy a new house. You currently have $15,000 and you figure you need to have a 10% down payment plus an additional 5% in closing costs. If the type of house you want costs about $150,000 and you can earn 7.5% per year, how long will it be before you have enough money for the down payment and closing costs?

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Finding an interest rate, discount rate, or rate of return on the investment.

Example 4: The entrance cover charge at a local bar at the end of 2000 was $3 and the cover charge at the end of 2002 was $4. At what annual rate did cover charge grow?

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Another Example

Your daughter is currently 8 years old who will be going to college in ten years. You would like to have $100,000 at that time. You can deposit $60,000 now at 3% per year. Is this enough? How much will you be short of? How much extra dollars do you need to deposit today to achieve $100,000 goal?

FV of 60,000 = 80,634 Short of 19,365 PV of 19,365 = 14,409 Any short-cut available ? PV of 100,000 = 74,409

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Growth of a $100 Investment atVarious Compound Interest Rates

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Present Value of $100 at Various Discount Rates

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Simple vs. Compound Interest(Power of Compound) Simple Interest

– Simple $100 saving today at 10%– One year from today = $100 + 10 =$110.00– Two years from today = $100 + 10 + 10 = $120.00– Three years from today = $100 + 10 + 10 + 10 = $130.00– ….. 50 yrs later = $100 + 50($10) = $600

Compound Interest– Compound $100 saving today at 10%– One year from today = $100 + 10 =$110.00– Two years from today = $100 + 10 + (10 + 1) = $121.00– Three years from today = $100 + 10 + (10 + 1) + {10 + 1 +

1 + .1} = $133.10– ….. 50 yrs later = 100(1.1)50 = $11,739

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Mechanics

Invest $1 @ 10% per year for 1 yearFV = $1 x 1.1 = $1.1000“Get your money back ($1) plus interest (10 cents)”

Invest $1 @ 10% per year for 2 yearFV = $1 x 1.1 x 1.1 = $1 x 1.12 = $1.2100“Get your money back ($1) plus interests for two years (20 cents) plus interest on interest (1 cent) for the second year”

Invest $1 @ 10% per year for 1 yearFV = $1 x 1.1 x 1.1 x 1.1 = $1 x 1.13 = $1.3310……

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Example: Compound Interest

After 1st year, $1 ($1) + 10 cents = $1.1000

After 2nd year,$1 + 10 cents ($1 + 10 cents) + 10 cents + 1 cent= $1 + 20 cents + 1 cent= $1.2100

After 3rd year, $1 + 10 cents + 10 cents + 1 cent($1 + 10 cents + 10 cents + 1 cent) + 10 cents + 1 cent + 1 cent +.1 cent= $1 + 30 cents + 3 cents + .1 cent= $1.3310

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FV for $100 investing at 10%/Yr

FV at 1st year = $100 (1+10%) FV at 2nd year = $100(1+10%)(1+10%) FV at 3rd year = $100(1+10%)(1+10%)(1+10%)

…..

FV at 50th year

= $100(1+10%)(1+10%)……..(1+10%)50

$100 will be “compounded” 50 times. This is nothing new. However, the first interest ($10) will be compounded 49 times, and the second $10 will be compounded 48 times. And money will keep growing itself exponentially!!!

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0

5000

10000

15000

20000

1st Year 10thYear

20thYear

30thYear

Future Value of a Single $1,000 Deposit

10% SimpleInterest

10% CompoundInterest

Fu

ture

Va

lue

(U

.S. D

olla

rs)

Power of Compounding

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0 1 2 310%

0 1 2 3

5%

4 5 6

134.01

100 133.10

1 2 30

100

Annually: FV3 = $100(1.10)3 = $133.10.

Semiannually: FV6 = $100(1.05)6 = $134.01.

Compounding Frequency: Annual vs. Semiannual Compounding for FV of $100 after 3 years

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Future Values at Various Compounding

Annual FV=100(1+10%)1 =110

Semi-Annual FV=100(1+10%/2)2 =110.25

Quarterly FV=100(1+10%/4)4 =110.38

Monthly FV=100(1+10%/12)12 =110.47

Daily FV=100(1+10%/365)365 =110.52

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Present Values : Annual Vs. Monthly Comp

What is the PV of $100 one year from now with 12 percent interest?

Monthly comp:

PV0

= $100 1/(1 + .12/12)(1 12)

= $100 1/(1.126825)

= $100 (.887449)

= $ 88.74

Annual comp:

PV0

= $100 1/(1 + .12)1

= $100 1/(1.1200)

= $100 (.892857)

= $ 89.29

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A New Formula incorporating compounding frequencyTVM Formula Annual

Compoundingm compounded per year

FV of a lump sum FV = PV (1+i)N FV = PV (1+i/m)Nm

PV of a lump sum

The interest rate required for a PV to grow to a FV

The length of time required for a PV to grow to a FV

(1 )N

FVPV

i

(1 / )Nm

FVPV

i m

1

1NFV

iPV

1

1NmFV

i mPV

ln

ln(1 )

FVPV

Ni

ln

ln(1 / )

FVPV

Nm i m

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An Important Question

Will the FV of a lump sum be larger or smaller if we compound more often, holding the stated I% constant? Why?

LARGER! If compounding is morefrequent than once a year--for example, semiannually, quarterly,or daily--interest is earned on interestmore often.

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More Practice Problems

Example 5: Your family plan to go to Paris in three years from now. Your travel agency informs that you will need $4,000. Currently, your stock mutual fund yield 12 percent per year. How much must you invest today to achieve the goal?– with annual compounding

– with quarterly compounding

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More Practice Problems

Example 6: You are saving to buy a new $200,000 Ferrari. You have $100,000 today that can be invested at your bank. The bank pays 5 percent annual interest. How long will it be before you have enough to buy the car?– with annual compounding

– with quarterly compounding

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Credit Card Example: Daily Compounding

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General Rules

Rule 1– N could be days, months, quarters, semi-years, or years,

depending on various ways to compound interest.

– For instance, with one-year period,

• Enter N = 365, 12, 4, 2, or, 1. Rule 2

– You always input rates on an annual basis, but you must do this with appropriate pmt per year setting.

– For instance, with one-year period at 12% annual rate,

• Enter I = 12 always but with 365 P_YR, 12 P_YR, 4 P_YR, 2 P_YR, or 1 P_YR, depending on different compounding assumption.

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“Wordings” are confusing!!!

How much will you have per dollar invested in one year?– If you have an investment that will pay you 1.12

percent per month.

– If you have an investment that will pay you an interest rate of 13.44 percent compounded monthly.

– If you have an investment that will pay you an interest rate of 13.44 percent but interest accrues monthly.

– If you have an investment that will pay you an APR of 13.44 percent compounded monthly.

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More Practice Problems

P 1: Grandparents offer the following:– Option A: pay you $10,000 at the end of year 2, and

$20,000 at the end of year 4, – Option B: pay you $20,000 today– Which option would you prefer? Assume R = 10%

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P2: College Tuition

Your daughter is currently 8 years old who will be going to college in ten years. You will need to pay – $20,000 when she becomes 18 years old– $23,000 when she becomes sophomore– $25,000 when she becomes junior– $29,000 when she becomes senior

How much do you need to deposit today at 3% per year to be able to pay tuition as estimated above?

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At-the-end-of-chapter problems

See class syllabus

Practice! Practice! Practice!

Remember

“You will never have enough practices to master TVM

problems!”