lecture time value of money

8/3/2019 Lecture Time Value of Money

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Lecture

Time Value of Money


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Time Value of Money

Used for valuing:

Evaluate investment alternatives

Plan for retirement Estimate estate needs

Make credit decisions

Planning Insurance purchases


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TIME VALUE OF MONEY

TIME VALUE OF MONEY

BASIC PROBLEM FACED BY FINANCIAL

MANAGER IS

HOW TO VALUE FUTURE CASH FLOWS?

For example:I HAVE TO SPENDMONEY TODAY

TO BUILD A PLANT WHICH WILL GENERATECASH FLOWS IN THE FUTURE


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Time Value of Money An

Important Concept Even if we did not have inflation, a dollarreceived in the future is worth less than a

dollar receiv

ed today.

An obligation to pay a dollar in the future isless costly than paying a dollar today.


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TIME allows one the opportunitytopostpone consumption and earn

INTE

RE

ST.

NOThaving the opportunity to earninterest on money is called

OPPORTUNITY COST.

WhyTIME?WhyTIME?


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WHAT DETERMINES TRADE - OFFBETWEEN CURRENT DOLLARS

AND FUTUREDOLLARS?

WHAT DETERMINES TRADE - OFFBETWEEN CURRENT DOLLARS

AND FUTUREDOLLARS?

HOW MUCH I CAN EARN ON THEMONEYDURING THEYEAR

The opportunity cost of capital (k)


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How can one compare amounts

in different time periods?

How can one compare amounts

in different time periods?

One can adjust values from different timeperiods using the opportunity cost of capital(k).

Remember, one CANNOT comparenumbers in different time periods withoutfirst adjusting them using the opportunity

cost of capital (k).


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Measure the time value of money

Take also into consideration the risk of theinvestment decision alternatives

Basic Formula:

K = Rf + premium risk

Rf = nominal risk-free rate


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Depend on the type of investmentalternatives:

Bank deposits interest rate;

A company stocks company or activity

sector returns and so forth.


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FUTURE VALUE

COMPOUNDPRINCIPALAMOUNT

FORWARD

INTO THEFUTURE

PRESENT VALUE

DISCOUNTA FUTURE VALUEBACK

TO THE

PRE

SE

NT

Taking into consideration the timev

alue of money.


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The amount to which a cash flow or series ofcash flows will grow over a period of time

when compounded at a given opportunitycost .

Future Value


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Compound InterestCompound Interest

When interest is paid on not only the principal amountinvested, but also on any previous interest earned, this iscalled compound interest.

FV = Principal + (Principal x Interest)

= PV (1 + k)

= 2000 (1 + k)

= 2000 + (2000 x .06)

Note: PV refers to Present Value or Principal


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If you invested $2,000 today in an account that pays 6%

interest, with interest compounded annually, how much willbe in the account at the end of two years if there are no

withdrawals?

Future Value

(Graphic)

Future Value

(Graphic)

0 1 2

$2,000$2,000

FVFV

6%


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FV1 = PV (1+k)n = $2,000 (1.06)2

= $2,247.20

Future Value

(F

ormula)

Future Value

(F

ormula)

FV = future value, a value at some future point in time

PV = present value, a value today which is usually designated as time 0

k = rate of interest per compounding period

n = number of compounding periods


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0

2

4

6

8

10

12

14

16

18

0 5 10 15 20 25

Year

k = 5%

k = 10%

k = 15%

FUTURE VALUE

Year

1

2

5

10

20

5%

1.050

1.103

1.276

1.629

2.653

10%

1.100

1.210

1.331

2.594

6.727

15%

1.150

1.323

2.011

4.046

16.37

0 2 4 6 8 10 12 14 16 18 20

20

15

10

5

0

FUTURE VALUE OF $1

YEARS


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General Formula:

FVn = PV0(1 + [k/m])mn

n: Number ofYears

m: Compounding Periods perYear

k: Annual Interest Rate

FVn,m: FV at the end ofYear n

PV0: PV of the Cash Flow today

Frequency of

Compounding

Frequency of

Compounding


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Frequency of Compounding

Example Suppose you deposit $1,000 in an account that pays

12% interest, compounded quarterly. How much willbe in the account after eight years if there are no

withdrawals?

PV = $1,000

k = 12%/4 = 3% per quarter

n = 8 x 4 = 32 quartersAnswer:

FV= PV (1 + k)n = 1,000(1.03)32 = 2,575.10


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0 1 2 310%

100133.10

0 1 2 35% 4 5 6

134.01

1 2 30

100

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

Semi-annually: FV6/2 = 100(1.05)6 = 134.01.

Compounding

Annually vs. Semi-Annually


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kSIMPLE = Simple (Quoted) RateSimple (Quoted) Rate

used to compute the interest paid per period

EAR = Effective Annual RateEffective Annual Ratethe annual rate of interest actually being

earned

APR =Annual Percentage RateAnnual Percentage Rate = kSIMPLEperiodic rate X the number of periods per year

Distinguishing BetweenD

ifferent Interest Rates


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1-m

k+1=EAR

mSIMPLE

10.25%=0.1025=1.0-1.05=

1.0-2

0.10+1=

2

2

How do we find EAR for a simple rate of

10%, compounded semi-annually?


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nmSIMPLE

nm

k+1PV=FV

v

$134.0110)$100(1.3402

0.10

+1$100=FV

32

23 !!

v

v

FV of $100 after 3 years if interest is 10%compounded semi-annual? Quarterly?

$134.4989)$100(1.3444

0.10+1$100=FV

34

43 !!

v

v


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Future Value of an Annuity

Annuity:A series of payments of equalamounts at fixed intervals for a specifiednumber of periods.

Ordinary (deferred) Annuity:An annuitywhose payments occur at the end of eachperiod.

Annuity Due:An annuity whose paymentsoccur at the beginning of each period.


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PMT PMTPMT

0 1 2 3k%

PMT PMT

0 1 2 3k%

PMT

Ordinary AnnuityVersus

Annuity DueOrdinary Annuity

Annuity Due


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100 100100

0 1 2 310%

110

121

FV = 331

Whats the FV of a 3-year Ordinary

Annuity of $100 at 10%?


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Numerical Solution:

-

!

-

!

! k1k)(1PMTk)(1PMTFVA

n1n

0t

tn

$331.0000)$100(3.310

0.10

1(1.10)$100FVA

3

3

!!

-

!


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Present Value

Present Value is the current value of a

future amount of money, or a series ofpayments, evaluated at a givenopportunity cost.


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Present Values

How much do I have to invest today to have someamount in the future?

FV = PV(1 + k)t

Rearrange to solv

e for PV = FV / (1 + k)

t

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

When we talk about the value of something, weare talking about the present value unless wespecifically indicate that we want the future value.


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Assume that you need to have exactly $4,000 saved 10 yearsfrom now. How much must you deposit today in an accountthat pays 6% interest, compounded annually, so that youreach your goal of $4,000?

0 55 10

$4,000$4,000

6%

PVPV00

Present Value

(Graphic)

Present Value

(Graphic)


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PV0 = FV / (1+k)10 = $4,000 / (1.06)10

= $2,233.58

Present Value

(F

ormula)

Present Value

(F

ormula)

0 55 10

$4,000$4,000

6%

PVPV00


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PRESENT VALUE OF $1

0

0,2

0,4

0,6

0,8

1

1,2

0 2 4 6 8 10 12 14 16 18 20

k = 5%

k = 10%k = 15%

PRESENT VALUE

Year 5% 10% 15%1 .952 .909 .870

2 .907 .826 .756

5 .784 .621 .497

10 .614 .386 .247

20 .377 .149 .061

YEARS


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Present Value of an Annuity

PVAn = the present value of an annuitywith n payments.

Each payment is discounted, and the sumof the discounted payments is the presentvalue of the annuity.


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248.69 = PV

100 100100

0 1 2 310%

90.91

82.64

75.13

What is the PV of this Ordinary

Annuity?


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Numerical Solution

-

!

-

!

! k

-1PMT

k)(1

1PMTPVA

nk)(1

1n

1t tn

$248.6985)$100(2.486

0.10

-1$100PVA

3(1.10)

1

3

!!

-

!


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Short Cuts

Sometimes there are shortcuts that makeit very easy to calculate the present valueof an asset that pays off in differentperiods. These tolls allow us to cutthrough the calculations quickly.


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SHORTCUTS FORSHORTCUTS FOR

1. PERPETUITIES

2. GROWING PERPETUITIES

3. ANNUITIES


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1. PERPETUITIES PMT

k

2. GROWING PERPETUITIES

3. ANNUITIES

PVAn =n2 )(1.......)(1)(1 k

PMT

k

PMT

k

PMT n2

1

PVAn=

gk

PMTPVA

n

!1

!k

kPMT

n)(111

nPVA


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Quick Quiz (I)

All other things being equal, I'd rather have $1,000 todaythan to receive $1,000 in 10 years.A.TrueB.False

Comparing the values of undiscounted cash flows isanalogous to comparing apples to oranges.A.TrueB.False

Compound interest pays interest for each time period onthe original investment only.A.TrueB.False


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Quick Quiz (II)

Finding the present value is simply the reverse ofcompounding.A.TrueB.False

For a given amount, the greater the discount rate, theless the present value.A.TrueB.False

If you would like to double your money in 8 years, theapproximate compound annual return you need is 9percentA.TrueB.False


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Quick Quiz (III)

A How much must you deposit today in a bank accountpaying interest compounded quarterly:

If you wish to have $6,000 at the end of 12 months, if

the bank pays 9.0% APR? Answer: $5,489

A How much must you deposit today in a bank account

paying interest compounded monthly: If you wish to have: 6,000 at the end of 6 months, if the

bank pays 9.0% APR ?

Answer: 5,737


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Quick Quiz (IV)

Suppose you make an investment of $1,000. This firstyear the investment returns 12%, the second year itreturns 6%, and the third year in returns 8%. How muchwould this investment be worth, assuming no

withdrawals are made? Answer: 1000*(1.12) x (1.06) x (1.08) = $1,282

How much would you need to deposit every month in anaccount paying 6% a year to accumulate by $1,000,000by age 65 beginning at age 20?

Answer: PMT = $362.85


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Quick Quiz (V)

As a winner of a local competition, you canchoose one of the following prizes:

(a) $100,000 now

(b) $180,000 at the end of 4 years

(c) $11,400 a year forever

(d) $19,000 for each of 10 years

(e) $6,500 next year and increasingthereafter by 5% a year forever

If the interest rate is 12%, which is the most valuable prize?


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Quick Quiz (VI)

Your firm has a retirement plan that matchesall contributions on a one to two basis. That is, if youcontribute $1,000 per year, the company will add$500 to make it $1,500. The firm guarantees 8%

return on the funds. Alternatively, you can do it yourself; you think

you can earn 11% on your money by doing ityourself. The first contribution will be made one yearfrom today. At that time, and every year thereafter,

you will put $1,000 into the retirement account. If you want to retire in 25 years, which way are

you better off?


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Quick Quiz (VII)

A typical mortgage problem. You borrow $80,000 to berepaid in equal monthly installments for 30 years. TheAPR is 9%. What is the monthly payment?

Answer: PMT = $643.70

You will receive $100,000 dollars when you retire, fortyyears from today. If inflation averages 3% per year forthe next forty years, how much would that amount beworth measured in today's dollars? (Note, this is not atime value of money problem, but it solved with a similar

calculation. Such adjustments are necessary toovercome money illusion] Answer: $100,000 (1.03)^40

=100,000 3.26204 = $ 30,655


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Quick Quiz (VIII)

You will receive $100,000 dollars when you retire, fortyyears from today. If inflation averages 3% per year forthe next forty years, how much would that amount beworth measured in today's dollars? (Note, this is not a

time value of money problem, but it solved with asimilar calculation. Such adjustments are necessary toovercome money illusion]

Answer:$100,000 (1.03)40 =100,000 3.26204 = $ 30,655

lecture time value of money

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