Financial Management

Session -#

Time Value of Money

Time Value of Money

Would you prefer to have $1 million now or $1 million 10 years from now?

Of course, we would all prefer the money now! This illustrates that there is an inherent monetary

value attached to time.

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Time Value of Money A dollar received today is worth more than a

dollar received tomorrow The time value of money is a recognition that

money received today is worth more than an equal amount of money received months or years in the future: Why TIME is such an important element in decision? This is because a dollar received today can be invested

to earn interest Cash flows for the future are uncertain The amount of interest earned depends on the rate of

return that can be earned on the investment

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Time Value of Money

Compounding Technique Translating a current value into its equivalent

future value is referred to as compounding. Used to estimate the future value of money FV is higher than the PV

Discounting Technique Translating a future cash flow or value into its equivalent

value in a prior period is referred to as discounting. Used to estimate the present value of money PV is less than the future value

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Time Value of Money

Time Value of Money, or TVM, is a concept that is used in all aspects of finance including: Bond valuation Stock valuation Accept/reject decisions for project management Financial analysis of firms And many others!

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PV & FV

If you were to invest $10,000 at 5-percent interest for one year, your investment would grow to $10,500.

In the one-period case, the formula for FV can be written as:

FV = C0(1 + r)

Where C0 is cash flow today (time zero), and r is the appropriate interest rate.

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PV & FVContd

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In the one-period case, the formula for PV can be written as:

rCPV 1

1

WhereC1 iscashflowatdate1,and

ristheappropriateinterestrate.

FV for Multi-Period

The general formula for the future value of an investment over many periods can be written as:

FV = C0(1 + r)TWhere

C0 is cash flow at date 0,

r is the appropriate interest rate, and

T is the number of periods over which the cash is invested.

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Exercise: FV

Suppose a stock currently pays a dividend of $1.10, which is expected to grow at 40% per year for the next five years.

What will the dividend be in five years?

FV = C0(1 + r)T

$5.92 = $1.10(1.40)5

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Exercise: FV

Assume Joe has the same cash flow stream from his investment but wants to know what it will be worth at the end of the fourth year

1. Draw a timeline:

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0 1 2 3 4

$100 $300 $500 $1000

i = 10%$1000

??

?

Exercise: PV

What is the PV of this uneven cash flow stream? We can always treat each CF as a separate lump sum,

discount each CF to PV separately, and sum up the PVs

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0

$100

1

$300

2

$300

310%-$50

4

90.91247.93225.39-34.15$530.08 = PV

The

Compounding an investment m times a year for T years provides for future value of wealth:

If you invest $50 for 3 years at 12% compounded semi-annually, what amount will you expect at the end of 3 years?

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Tm

mrCFV

10

93.70$)06.1(50$212.150$ 6

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FV

Continuous Compounding The general formula for the future value of an

investment compounded continuously over many periods can be written as:

FV = C0erTWhere

C0 is cash flow at date 0,

r is the stated annual interest rate,

T is the number of years, and

e is a transcendental number approximately equal to 2.718.

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The Power of Compound Interest A 20-year-old student wants to save $3 a day for her

retirement. Every day she places $3 in a drawer. At the end of the year, she invests the accumulated savings ($1,095) in a brokerage account with an expected annual return of 12%.

How much money will she have when she is 65 years old? If she sticks to her plan, she will have $1,487,261.89 when

she is 65. What if she sticks to the plan but returns compounded

on daily basis?

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The Power of Compound InterestContd

If you dont start saving until you are 40 years old, how much will you have at 65?

If a 40-year-old investor begins saving today, and sticks to the plan, he or she will have $146,000.59 at age 65. This is $1.3 million less than if starting at age 20.

Lesson: It pays to start saving early.

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The Power of Compound InterestContd

How much must the 40-year old deposit annually to catch the 20-year old?

To find the required annual contribution, enter the number of years until retirement and the final goal of $1,487,261.89, and solve for PMT.

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The Power of Compound InterestContd

If you invest $1000 for a year @10% rate of annual interest, what will be the value after the end of the year, if interest is compounded yearly, half-yearly, Quarterly.

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Frequency No. Future valueYearly 1 1100Half-yly 2 1102.5Qtly 4 1103.81Monthly 12 1104.71Weekly 52 1105.06Daily 365 1105.16Continuous 1105.17

Simplifications

Perpetuity A constant stream of cash flows that lasts forever

Growing perpetuity A stream of cash flows that grows at a constant rate forever

Annuity A stream of constant cash flows that lasts for a fixed

number of periods

Growing annuity A stream of cash flows that grows at a constant rate for a

fixed number of periods

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Perpetuity

A constant stream of cash flows that lasts forever

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0

1

C

2

C

3

C

32 )1()1()1( rC

rC

rCPV

rCPV

Exercise: Perpetuity

What is the value of a bond that promises to pay 15 every year for ever? The interest rate is 10-percent.

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0

1

15

2

15

3

15

15010.

15 PV

Growing Perpetuity

A growing stream of cash flows that lasts forever

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0

1

C

2

C(1+g)

3

C (1+g)2

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2 )1()1(

)1()1(

)1( rgC

rgC

rCPV

grCPV

Exercise: Growing Perpetuity The expected dividend next year is $1.30, and

dividends are expected to grow at 5% forever. If the discount rate is 10%, what is the value of this promised dividend stream?

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0

1

$1.30

2

$1.30(1.05)

3

$1.30 (1.05)2

00.26$05.10.

30.1$ PV

Annuity

A constant stream of cash flows with a fixed maturity

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0 1

C

2

C

3

C

TrC

rC

rC

rCPV

)1()1()1()1( 32

Trr

CPV)1(

11

T

C

Exercise: Annuity If you can afford a $400 EMI for car loan, how much

car loan can you afford if interest rates are 7% for 36-month?

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0 1

$400

2

$400

3

$400

59.954,12$)1207.1(

1112/07.

400$36

PV

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$400

Exercise: AnnuityContd

What is the present value of a four-year annuity of $100 per year that makes its first payment two years from today if the discount rate is 9%?

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97.323$)09.1(

100$)09.1(

100$)09.1(

100$)09.1(

100$)09.1(

100$4321

4

11

ttPV

22.297$09.1

97.327$0 PV

$100 $100 $100 $100$323.97$297.22

Exercise: AnnuityContd Whats the FV of a 3-year $100 annuity, if the quoted

interest rate is 10%, compounded semiannually? Compound each cash flow

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Payments occur annually, but compounding occurs every 6 months.

Cannot use normal annuity valuation techniques.

Exercise: AnnuityContd

Whats the FV of a 3-year $100 annuity, if the quoted interest rate is 10%, compounded semiannually?

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110.25121.55331.80

0 1

100

2 35%

4 5

100

6

100

FV3 = $100(1.05)4 + $100(1.05)2 + $100FV3 = $331.80

Growing Annuity

A growing stream of cash flows with a fixed maturity

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0 1

C

T

T

rgC

rgC

rCPV

)1()1(

)1()1(

)1(

1

2

T

rg

grCPV

)1(11

2

C(1+g)

3

C (1+g)2

T

C(1+g)T-1

Exercise: Growing Annuity A defined-benefit retirement plan offers to pay

$20,000 per year for 40 years and increase the annual payment by 3% each year. What is the present value at retirement if the discount rate is 10%?

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0 1

$20,000

57.121,265$10.103.11

03.10.000,20$ 40

PV

2

$20,000(1.03)

40

$20,000(1.03)39

Exercise: Growing Annuity You are evaluating an income generating property. Net rent is

received at the end of each year. The first year's rent is expected to be $8,500, and rent is expected to increase 7% each year. What is the present value of the estimated income stream over the first 5 years if the discount rate is 12%?

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0 1 2 3 4 5

500,8$ )07.1(500,8$

2)07.1(500,8$095,9$ 65.731,9$

3)07.1(500,8$87.412,10$

4)07.1(500,8$

77.141,11$

$34,706.26

What Is a Firm Worth?

Conceptually, a firm should be worth the present value of the firms cash flows.

The tricky part is determining the size, timing, and risk of those cash flows.

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Thank You!

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