Future and present values Lump sums Annuities Uneven cash flow streams Solving for I and N

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Copyright © 1999 by the Foundation of the American College of Healthcare Executives

Future and present valuesLump sumsAnnuitiesUneven cash flow streams

Solving for I and NTypes of interest ratesAmortization

Time Value of Money Analysis

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The cost of time -- opportunity cost of money

Relationship between asset value and future cash flows

Examples - real estate and GE stock

Time Value -- What is it??

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Relationship between future cash flows and asset value changes

The role of time value analysisTypes of analysis - future value

and present value analysis

Time Value -- What is it??

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Present valueFuture value Interest rateDiscount rateCash flow

Terminology of Time Value Analysis

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Time LineLump sum AnnuityUneven cash flows

Terminology of Time Value Analysis

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Time Lines

CF0 CF1 CF3CF2

0 1 2 3I%

Tick marks designate ends of periods. Time 0 is today (the beginning of Period 1); Time 1 is the end of Period 1 (the beginning of Period 2); and so on.

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Time Line Illustration 1 (Lump sum)

$100

0 1 2 5%

What does this time line show?

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Time Line Illustration 2 (annuity)

$100 $100$100

0 1 2 310%


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Time Line Illustration 3 (uneven cash flows)

100 50 75

0 1 2 36%

-50


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What is the FV after 3 years ofa $100 lump sum invested at 10%?

FV = ?

0 1 2 310%

-$100

Finding future values (moving to the right along the time line) is called compounding.

For now, assume interest is paid annually.

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After 1 year:

FV1 = PV + INT1 = PV + (PV x I)= PV x (1 + I)= $100 x 1.10 = $110.00.

After 2 years:

FV2 = FV1 + INT2

= FV1 + (FV1 x I) = FV1 x (1 + I)

= PV x (1 + I) x (1 + I) = PV x (1 + I)2

= $100 x (1.10)2 = $121.00.

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After 3 years:

FV3 = FV2 + I3

= PV x (1 + I)3

= 100 x (1.10)3

= $133.10.

In general,

FVN = PV x (1 + I)N .

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Three Primary Methods to Find FVs

Solve the FV equation using a regular (non-financial) calculator.

Use a financial calculator; that is, one with financial functions.

Use a computer with a spreadsheet program such as Excel, Lotus 1-2-3, or Quattro Pro.

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Non-Financial Calculator Solution

$133.10

0 1 2 310%

-$100 $110.00 $121.00

$100 x 1.10 x 1.10 x 1.10 = $133.10.

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Financial Calculator Solution

Financial calculators are pre-programmed to solve the FV equation:

FVN = PV x (1 + I)N.

There are four variables in the equation: FV, PV, I and N. If any three are known, the calculator can solve for the fourth (unknown).

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3 10 -100 0N I/YR PV PMT FV

133.10

Using a calculator to find FV (lump sum):

(1) For lump sums, the PMT key is not used. Either clear before the calculation or enter PMT = 0.

(2) Set your calculator on P/YR = 1, END.

INPUTS

OUTPUT

Notes:

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Ordinary Annuity

PMT PMTPMT

0 1 2 3I%

PMT PMT

0 1 2 3I%

PMT

Annuity Due

Types of Annuities

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What is the FV of a 3-year ordinary annuity of $100 invested at 10%?

$100 $100$100

0 1 2 310%

110 121 FV = $331

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3 10 0 -100

331.00


Have payments but no lump sum, so enter 0 for present value.

INPUTS

OUTPUTI/YRN PMT FVPV

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What is the FV if the annuity were an annuity due?

Do ordinary annuity calculation as described before

Multiply result by (1 + interest rate)

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10%

What is the PV of $100 duein 3 years if I = 10%? (lump sum)

$100

0 1 2 3

PV = ?

Finding present values (moving to the left along the time line) is called discounting.

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Solve FVN = PV x (1 + I )N for PV:

PV = $100 / (1.10)3

= $100(0.7513) = $75.13.

PV = FVN / (1 + I )N.

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3 10 0 100

-75.13

Either PV or FV must be negative on most calculators. Here, PV = -75.13. Put in $75.13 today, take out $100 after 3 years.

INPUTS

OUTPUTN I/YR PV PMT FV

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Opportunity Costs

On the last illustration we needed to apply a discount rate. Where did it come from?The discount rate is the opportunity cost rate.It is the rate that could be earned on alternative

investments of similar risk.It does not depend on the source of the

investment funds.

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What is the PV of this ordinary annuity?

$100 $100$100

0 1 2 310%

$90.91

82.64

75.13$248.68 = PV

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This problem has payments but no lump sum, so enter 0 for future value.

3 10 100 0

-248.69

INPUTS


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What is the PV if the annuity were an annuity due?

$100 $100

0 1 2 310%

$100 ??

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What is the PV if the annuity were an annuity due?

Calculate present value of ordinary annuity as described above

Multiply result by (1 + interest rate)

Annuities due not as common as ordinary annuities

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3 10 100 0

-273.55

Switch from End to Begin mode on a financial calculator. Repeat the annuity calculations. PV = $273.55.

INPUTS


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Perpetuities

A perpetuity is an annuity that lasts forever.

What is the present value of a perpetuity?

PV (Perpetuity) = .PMTI

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Uneven Cash Flow Streams: Setup

0

$100

1

$300

2

$300

310%

-$50

4

$ 90.91247.93225.39-34.15

$530.08 = PV

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Input into “CF” registers:CF0 = 0CF1 = 100CF2 = 300CF3 = 300CF4 = -50

Enter I = 10%, then press NPV button to get NPV (PV) = $530.09.

Uneven Cash Flow Streams:Financial Calculator Solution

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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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0 1 2 310%

0 1 2 35%

4 5 6

134.01

-100 133.10

1 2 30

-100

Annual: FV3 = 100 x (1.10)3 = 133.10.

Semiannual: FV6 = 100 x (1.05)6 = 134.01.

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We will deal with 3 different rates:

IStated = stated, or nominal, or quoted, rate per year.

IPeriod = periodic rate.

EAR= effective annual rate.

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IStated is the rate given in contracts.Often an annual rate.Compounding periods (M) may be given:8% compounded quarterly.12% compounded monthly.

IPeriod is the rate per period.IPeriod = IStated / M.For 8% compounded quarterly:

periodic rate = 2%.For 12% compounded monthly:

periodic rate = 1%.

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EAR is the annual rate which causes any PV to grow to the same FV as under intra-year compounding.

What is the EAR for 10%, semiannual compounding?Consider the FV of $1 invested for one year. FV = $1 x (1.05)2 = $1.1025.EAR = 10.25%, because this rate would produce the same ending amount ($1.1025) under annual compounding.

Effective Annual Rate (EAR)

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The EAR Formula

Or use the EFF% key on a financial calculator.

EAR = 1 + - 1.0

IStated

M

M

= 1 + - 1.0

0.10

2

2

= (1.05)2 - 1.0 = 0.1025 = 10.25%.

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EAR of 10% at Various Compounding

EARAnnual = 10%.

EARQ = (1 + 0.10/4)4 - 1.0 = 10.38%.

EARM = (1 + 0.10/12)12 - 1.0 = 10.47%.

EARD(360) = (1 + 0.10/360)360 - 1.0 = 10.52%.

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What’s the value at the end of Year 3of the following CF stream if the stated

interest rate is 10%, compounded semiannually?

0 1

$100

2 35%

4 5 6 6-month periods

$100 $100 Note that payments occur annually, but

compounding occurs semiannually, so we can not use normal annuity valuation techniques.

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First Method: Compound Each CF

0 1

$100

2 35%

4 5 6

$100 $100.00110.25121.55

$331.80

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Second Method: Treat as an Annuity

EAR = (1 + ) - 1 = 10.25%. 0.10

22

Find the EAR for the stated rate:

Then use standard annuity techniques:

3 10.25 0 -100

INPUTS

OUTPUT

N I/YR PV FVPMT

331.80

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Amortization

Construct an amortization schedulefor a $1,000, 10% annual rate loanwith 3 equal payments.

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Step 1: Find the required payments.

PMT PMTPMT

0 1 2 310%

-$1,000

3 10 -1000 0

402.11

INPUTS

OUTPUT

N I/YR PV FVPMT

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Step 2: Find interest charge for Year 1.

INTt = Beginning balance x I.INT1 = $1,000 x 0.10 = $100.

Step 3: Find repayment of principal in Year 1.

Repmt = PMT - INT = $402.11 - $100 = $302.11.

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Step 4: Find ending balance at end of Year 1.

End bal = Beg balance - Repayment= $1,000 - $302.11 = $697.89.

Repeat these steps for Years 2 and 3to complete the amortization table.

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Note that annual interest declines over time while the principal payment increases.

BEG PRIN ENDYR BAL PMT INT PMT BAL

1 $1,000 $402 $100 $302 $698

2 698 402 70 332 366

3 366 402 37 366 0

TOT $1,206.34 $206.34 $1,000

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$

0 1 2 3

402.11Interest

302.11

Level payments. Interest declines because outstanding balance declines. Lender earns10% on loan outstanding, which is falling.

Principal Payments

Future and present values Lump sums Annuities Uneven cash flow streams Solving for I and N

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