ch04 ppt brighamfm1ce

PowerPoint Presentationprepared by

Traven ReedCanadore College

chapter 4Time Value of Money

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Copyright © 2011 by Nelson Education Ltd. All rights reserved. 4-3

Corporate Valuation and the Time Value of Money

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Time Value Topics

• Time lines• Future value• Present value• Effective rates of return• Amortization

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Time lines show timing of cash flows

CF0 CF1 CF3CF2

0 1 2 3I%

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

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Time line for a $100 lump sum due at the end of Year 2

100

0 1 2 YearI%

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Time line for an ordinary annuity of $100 for 3 years

100 100100

0 1 2 3I%

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Time line for uneven CFs

100 50 75

0 1 2 3I%

-50

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FV of an initial $100 after3 years (i = 5%)

FV = ?

0 1 2 35%

Finding FVs (moving to the righton a time line) is called compounding.

100

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After 1 year (Formula Approach)

FV1 = PV + INT1 = PV + PV (I)= PV(1 + I)= $100(1.05)= $105.00

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After 2 years (Formula Approach)

FV2 = FV1(1+I) = PV(1+I)(1+I)= PV(1+I)2

= $100(1.05)2

= $110.25

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After 3 years(Formula Approach)

FV3 = FV2(1+I)=PV(1 + I)2(1+I)= PV(1+I)3

= $100(1.05)3

= $115.76

In general,FVN = PV(1 + I)N

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Growth of $1

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Three Ways to Find FVs

• Solve the equation with a regular calculator.

• Use a financial calculator.• Use a spreadsheet.

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(Reference Only)Financial calculator: HP10BII

• Adjust display brightness: hold down ON and push + or -.

• Set number of decimal places to display: Orange Shift key, then DISP key (in orange), then desired decimal places (e.g., 3).

• To temporarily show all digits, hit Orange Shift key, then DISP, then =

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(Reference Only)HP10BII (cont’d)

• To permanently show all digits, hit ORANGE shift, then DISP, then . (period key)

• Set decimal mode: Hit ORANGE shift, then ./, key.

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(Reference Only) HP10BII: Set Time Value Parameters

• To set END (for cash flows occurring at the end of the year), hit ORANGE shift key, then BEG/END.

• To set 1 payment per period, hit 1, then ORANGE shift key, then P/YR

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Financial calculators solve this equation:

FVN + PV (1+I)N = 0

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

Financial Calculator Solution

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3 5 -100 0N I/YR PV PMT FV

115.76

Clearing automatically sets everything to 0, but for safety enter PMT = 0

Set: P/YR = 1, END

INPUTS

OUTPUT

Here’s the setup to find FV

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Spreadsheet Solution

• Use the FV function:

• = FV(I, N, PMT, PV)

• = FV(0.05, 3, 0, -100) = 115.76

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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.

100

0 1 2 3

PV = ?

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1.10

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

PV =

FVN

(1+I)N= FVN

1

1 + I

N

PV = $1001

= $100(0.7513) = $75.13

3

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3 10 0 100N I/YR PV PMT FV

-75.13

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

INPUTS

OUTPUT


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• Use the PV function:

• = PV(I, N, PMT, FV)

• = PV(0.10, 3, 0, 100) = -75.13

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?%

150

0 1 2 10

-100 FV = PV(1 + I)N

$150 = $100(1 + I)10

(1.5)(1/10) = (1 + I) 1.0414= (1 + I) I = 0.0414 = 4.14%

Finding the Interest Rate

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10 -100 0 150N I/YR PV PMT FV

4.14

INPUTS

OUTPUT

Financial Calculator

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• Use the RATE function:

• = RATE(N, PMT, PV, FV)

• = RATE(10, 0, -100, 150) = 0.0414

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Find the Number of Years, N

• Suppose we now have $100 and the interest rate is 20%. How long will it take to grow to $200?


20%1 2 ?

-100 200

0

FV = PV (1 + I)N

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Finding the Time

• Follow the formula approach:• $200 = $100 ( 1 + 0.20)N

• $200/$100 = 2 = (1.2)N

• Ln (2) = N × Ln (1.2)• N = Ln(2) / Ln(1.2)• N = 0.693 / 0.182 = 3.8 years


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• 20 -100 0 200

• N I/YR PV PMTFV

• 3.8

INPUTS

OUTPUTS

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• Use the NPER function:• = NPER(I, PMT, PV, FV)• = NPER (0.2, 0, -100, 200) = 3.8


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Ordinary Annuity

PMT PMTPMT

0 1 2 3I%

PMT PMT

0 1 2 3I%

PMT

Annuity Due

Ordinary Annuity vs. Annuity Due

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What’s the FV of a 3-year ordinary annuity of $100 at 5%?

100 100.00100

0 1 2 35%

105.00 110.25FV = 315.25

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FV Annuity Formula

• The future value of an annuity with N periods and an interest rate of I can be found with the following formula:

= PMT

(1+I)N-1

I

= 100

(1+0.05)3-1

0.05= 315.25

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Financial Calculator Formula for Annuities

• Financial calculators solve this equation:

FVN + PV(1+I)N + PMT

(1+I)N-1

I= 0

There are 5 variables. If 4 are known, the calculator will solve for the 5th.

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3 5 0 -100

315.25N I/YR PV PMT FV

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

INPUTS

OUTPUT


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• Use the FV function: see spreadsheet.

• = FV(I, N, PMT, PV)• = FV(0.05, 3, -100, 0) = 315.25

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Future Value of an Annuity Due

• When payments occur at the beginning of the period rather than at the end, those cash flows have more time to earn extra interest.

• FVAdue = FVAordinary (1 + I)

• FVAdue = ($315.25)(1.05) = $331.01

• Set the calculator to Begin Model • In Excel, FV(I, N, PMT, PV, Type) =

FV(0.05, 3, -100, 0, 1)Copyright © 2011 by Nelson Education Ltd. All rights reserved. 4-38

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What’s the PV of this ordinary annuity?

100 100100

0 1 2 35%

95.24

90.70

86.38272.32 = PV

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PV Annuity Formula

• The present value of an annuity with N periods and an interest rate of I can be found with the following formula:

= PMT

1

I

1−

I (1+I)N

= 100 1

0.05

1−

0.05(1+0.05)3

= $272.32

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

3 5 100 0N I/YR PV PMT FV

-272.32

INPUTS

OUTPUT


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• Use the PV function: see spreadsheet.

• = PV(I, N, PMT, FV)• = PV(0.05, 3, 100, 0) = -272.32

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Find the PV if theannuity were an annuity due

100 100

0 1 2 3

10%

100

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PV of Annuity Due

• Since each payment for an annuity due occurs one period earlier, the payments will all be discounted for one less period.

• Therefore, PVAdue > PVA

• PV of annuity due (PVAdue) :

• = (PV of ordinary annuity) (1+I) • = (272.32) (1+ 0.05) = $285.94

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3 5 100 0

-285.94 N I/YR PV PMT FV

INPUTS

OUTPUT

PV of Annuity Due: Switch from “End” to “Begin

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Excel Function for Annuities Due

• Change the formula to:• =PV(10%,3,-100,0,1)

• The fourth term, 0, tells the function there are no other cash flows. The fifth term (i.e. type) tells the function that it is an annuity due. A similar function gives the future value of an annuity due. Refer to the previous slide 4-38.


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5 6 0 10,000

-1,773.96

N I/YR PV PMT FVINPUTS

OUTPUT

Find Annuity Payment for Ordinary Annuity (End Mode) & Annuity Due (Begin Mode)

5 6 0 10,000

-1,673.55

INPUTS

INPUTS

OUTPUT

OUTPUT

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6 0 -1,200 10,000

6.96

N I/YR PV PMT FVINPUTS

OUTPUT

Find Period Number and Interestfor Ordinary Annuity (End Mode)

5 0 -1,200 10,000

25.78

INPUTS

INPUTS

OUTPUT

OUTPUT

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Perpetuities

• A perpetuity is simply an annuity with an extended life.

• Since the payments go on forever, you cannot apply the step-by-step approach.

• PV of a perpetuity = PMT/I• Example: A British consol with a face

value of $1,000 that pays $50 per year forever. What is its value today? The going interest rate is 2.5%.

• PV(P) = $50/0.025 = $2,000

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What is the PV of this uneven cash flow stream?

0

100

1

300

2

300

310%

-50

4

90.91247.93225.39-34.15530.08 = PV

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(Reference Only)Financial calculator: HP10BII

• Clear all: Orange Shift key, then C All key (in orange).

• Enter number, then hit the CFj key.• Repeat for all cash flows, in order.• To find NPV: Enter interest rate

(I/YR). Then Orange Shift key, then NPV key (in orange).

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(Reference Only)Financial calculator: HP10BII (cont’d)

• To see current cash flow in list, hit RCL CFj CFj

• To see previous CF, hit RCL CFj –• To see subsequent CF, hit RCL CFj

+• To see CF 0-9, hit RCL CFj 1 (to

see CF 1). To see CF 10-14, hit RCL CFj . (period) 1 (to see CF 11).

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• Input in “CFLO” register:– CF0 = 0– CF1 = 100– CF2 = 300– CF3 = 300– CF4 = -50

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

(Reference Only)Financial calculator: HP10BII (cont’d)

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Excel Formula in cell A3: =NPV(10%,B2:E2)

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Nominal or Quoted Rate (INOM)

• Stated in contracts, and quoted by banks and brokers.

• Not used in calculations or shown on time lines

• Compounding periods per year (M) must be given. Examples:– 8%; Quarterly– 8%, Daily interest (365 days)

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Periodic rate (IPER )

• IPER = INOM/M, where M is number of

compounding periods per year. M = 4 for

quarterly, 12 for monthly, and 365 for daily

compounding.

• Used in calculations, shown on time lines.

• Examples:

– 8% quarterly: IPER = 8%/4 = 2%.

– 8% daily (365): IPER = 8%/365 = 0.021918%.

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The Impact of Compounding

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

• Why?

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The Impact of Compounding (Answer)

• LARGER!

• If compounding is more frequent than once a year--for example, semiannually, quarterly, or daily--interest is earned on interest more often.

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FV Formula with Different Compounding Periods

INOMFVN = PV 1

+ M

(M)(N)

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$100 at a 12% nominal rate with quarterly compounding for 2 years

= $100(1.03)8 = $126.68

INOMFVN = PV 1 + M

M N

0.12FV5S = $100 1 + 4

4x2

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FV of $100 at a 12% nominal rate for 5 years with different compounding

FV(Ann.) = $100(1.12)5 = $176.23

FV(Semi.) = $100(1.06)10 = $179.08

FV(Quar.) = $100(1.03)20 = $180.61

FV(Mon.) = $100(1.01)60 = $181.67

FV(Daily) = $100(1+(0.12/365))(5x365) = $182.19

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Effective Annual Rate (EAR = EFF%)

• The EAR is the annual rate that produces the same result as if we had compounded at a given periodic rate M times per year.

• The effective percentage (EFF%) is the interest rate expressed as if it were compounded once per year.

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Effective Annual Rate Example

• Example: Invest $1 for one year at 12%, semiannual:

FV = PV(1 + INOM/M)M

FV = $1× (1.06)2 = 1.1236• EFF% = 12.36%, because $1 invested

for one year at 12% semiannual compounding would grow to the same value as $1 invested for one year at 12.36% annual compounding.

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Comparing Rates

• An investment with monthly payments is different from one with quarterly payments. Must put on EFF% basis to compare rates of return. Use EFF% only for comparisons.

• Banks say “interest paid daily.” Same as compounded daily.

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EFF% = 1 + − 1

INOM

M

M

EFF% for a nominal rate of 12%, compounded monthly

= 1 + − 1

0.12

12

12

= (1.01)12 - 1.0= 0.126825 = 12.6825%

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(Reference)Finding EFF with HP10BII

• Type in nominal rate, then Orange Shift key, then NOM% key (in orange).

• Type in number of periods, then Orange Shift key, then P/YR key (in orange).

• To find effective rate, hit Orange Shift key, then EFF% key (in orange).

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EAR (or EFF%) for a Nominal Rate of 12%

EARAnnual = 12.00%

EARQuarterly = (1 + 0.12/4)4 - 1 = 12.55%

EARMonthly = (1 + 0.12/12)12 - 1= 12.68%

EARDaily(365) = (1 + 0.12/365)365 - 1= 12.75%

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Can the effective rate ever be equal to the nominal rate?

• Yes, but only if annual compounding is used, i.e., if M = 1

• If M > 1, EFF% will always be greater than the nominal rate.

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When is each rate used?

INOM: Written into contracts, quoted by banks and brokers. Not used in actual calculations or shownon time lines.

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IPER: Used in actual calculations, shown on time lines.

If INOM has annual compounding,then IPER = INOM/1 = INOM.

Otherwise, adjust with the number of periods involved.

When is each rate used? (cont’d)

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When is each rate used? (cont’d)

• EAR (or EFF%): Used to compare returns on investments with different payments per year.

• Used for calculations if and only if dealing with annuities where payments don’t match interest compounding periods.

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Fractional Time Periods

• On January 1 you deposit $100 in an account that pays a nominal interest rate of 11.33463%, with daily compounding (365 days).

• How much will you have on October 1, or after 9 months (273 days)? (Days given.)


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Convert interest to daily rate


IPER = 11.33463%/365

= 0.031054% per day.

0 1 2 273

-100 FV=?

0.031054%

FV 273 = $100 (1.00031054)273

= $100 (1.08846) = $108.85

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Calculator Solution

• IPER = INOM/M = 11.33463%/365

=0.031054% per day• With inputs: N = 273, I/Y =

0.031054, PV = -100, PMT = 0• Find FV = 108.85


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Amortized Loans

• Construct an amortization schedule for a $1,000, 10% annual rate loan with three equal payments.

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Step 1: Find the required payments

PMT PMTPMT

0 1 2 310%

-1,000

3 10 -1000 0

INPUTS

OUTPUT

N I/YR PV FVPMT

402.11

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Step 2: Find interest charge for Year 1

INTt = Beginning balancet (I)

INT1 = $1,000(0.10) = $100

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

Step 3: Find repayment of principal in Year 1

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Step 4: Find ending balance after Year 1

Ending balance = Beginning bal - Repmt= $1,000 - $302.11 = $697.89

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

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Amortization Table

YEARBEG BAL PMT INT

PRIN PMT

END 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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Interest declines because outstanding balance declines

$0$50

$100$150$200$250$300$350$400$450

PMT 1 PMT 2 PMT 3

Interest

Principal

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• Amortization tables are widely used--for home mortgages, auto loans, business loans, retirement plans, and more. They are very important!

• Financial calculators (and spreadsheets) are great for setting up amortization tables.

Interest declines because outstanding balance declines (cont’d)

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Growing Annuities and Growing Perpetuities

• Cash flows are likely to grow over time, due to either to real growth or to inflation.

• Growing annuity is a series of finite cash flows that grow at a fixed rate.

• Growing perpetuity is a constant stream of cash flows without end that is expected to rise indefinitely

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Real Rate

• Rr = [(1 + INOM)/(1 + inflation)] – 1

• Example: If the nominal annual interest rate is 10% and the expected inflation rate is 5% per annum, what is the expected real rate of return?

• Rr = [1.10/1.05] – 1 = 0.0476 = 4.76%


ch04 ppt brighamfm1ce

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