7 - 1 copyright © 2002 by harcourt, inc.all rights reserved. future value present value rates of...

7 - 1

Copyright © 2002 by Harcourt, Inc. All rights reserved.

Future value

Present value

Rates of return

Amortization

CHAPTER 7Time Value of Money

7 - 2


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.

7 - 3


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

100

0 1 2 Yeari%

7 - 4


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

100 100100

0 1 2 3i%

7 - 5


Time line for uneven CFs -$50 at t = 0 and $100, $75, and $50 at the end of

Years 1 through 3.

100 50 75

0 1 2 3i%

-50

7 - 6


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

FV = ?

0 1 2 310%

100

Finding FVs is compounding.

7 - 7


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

After 2 years:

FV2 = PV(1 + i)2

= $100(1.10)2

= $121.00.

After 1 year:

7 - 8


After 3 years:

FV3 = PV(1 + i)3

= $100(1.10)3

= $133.10.

In general,

FVn = PV(1 + i)n.

7 - 9


Four Ways to Find FVs

Solve the equation with a regular calculator.

Use tables.

Use a financial calculator.

Use a spreadsheet.

7 - 10


FVn = PV(1 + i)n.

Algebraic Solution

FV3 = 100(1 + .10) 3

FV3 = 100(1.331) = 133.10

7 - 11


FVn = PV(FVIF i,n).

Solution Using Tables

FV3 = 100(FVIF 10%, 3)

Use FVIF table from pages A-6 & 7, Table A3

FV3 = 100(1.331) = 133.10

7 - 12


Financial calculators solve this equation:

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

FVn = PV(1 + i)n.

Financial Calculator Solution

7 - 13


Here’s the setup to find FV:

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

Set: P/YR = 1, END

INPUTS

OUTPUT

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

133.10

7 - 14


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 = ?

7 - 15


Solve FVn = PV(1 + i )n for PV:

PV = = FVn .FVn

(1 + i)n 11 + i

PV = $100 = $100(PVIFi,n) Table A1

= $100(0.7513) = $75.13.

So the (PVIF 10%,3) = .7513

11.10

3

n

7 - 16



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

7 - 17


If sales grow at 20% per year, how long before sales double?

Solve for n:

FVn = $1(1 + i)n; $2 = $1(1.20)n

ln 2 = ln 1.2n.693147=.18232n3.801 = n

Use calculator to solve, see next slide.

7 - 18


If sales grow at 20% per year, how long before sales double?

Solve for n:

FVn = PV(FVIF i,n); $2 = $1(FVIF 20,n) 2.00 = (FVIF 20,n) n between 3 and 4 years 1.728 and 2.0736

Use calculator to solve, see next slide.

7 - 19


20 -1 0 2N I/YR PV PMT FV

3.8

Graphical Illustration:

01 2 3 4

1

2

FV

3.8

Year

INPUTS

OUTPUT

7 - 20


Compound GrowthHow do you find the compound growth rate for your company to analyze sales

growth ?

1062021 (1 + i )9 = 5284371 (1 + i )9 = 5284371/1062021(1 + i)9 = 4.976(1 + i) = 4.976 .111

(1 + i) = 1.195i = .195 or 19.5%

Can use either PV or FV formula, use FV

7 - 21


Tabular Solution

5284371 (PVIF i,9) = 1062021

PVIF i,9 = 1062021/5284371PVIF i,9 = .20097Use table A-3, for 9 Periods, find .20097i is between 18% and 20%

Use PV formula and table A-1

7 - 22


Calculator Solution, Compound Growth

9 -1062021 0 5284371

INPUTS

OUTPUT

N I/YR PV PMT FV

19.51

7 - 23


Ordinary Annuity

PMT PMTPMT

0 1 2 3i%

PMT PMT

0 1 2 3i%

PMT

Annuity Due

What’s the difference between an ordinary annuity and an annuity due?

7 - 24


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

100 100100

0 1 2 310%

110 121FV = 331

7 - 25


FVA =( PMT)* ( 1 +i)n – 1 I

FVA =( 100)* ( 1 + .1)3 – 1 .1

FVA =( 100)* 1.331 – 1 = 100 * 3.31 = .1

FVA = 331.00

Algebraic Solution

7 - 26


FVA i,n =( PMT) * (FVIFA i,n )Use Table A-4 on pages A-8 & 9

FVA 10%,3 =( 100) * (FVIFA 10%,3)

FVA 10%,3 =( 100)* 3.31 =

FVA 10%,3 = 331.00

Tabular Solution

7 - 27


3 10 0 -100

331.00


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

INPUTS

OUTPUTI/YRN PMT FVPV

7 - 28


What’s the PV of this ordinary annuity?

100 100100

0 1 2 310%

90.91

82.64

75.13248.68 = PV

7 - 29


1 . PVA =( PMT)* 1 – (1 + i)n

i 1 . PVA =( 100) * 1 – (1 + .1)3

.1

PVA =( 100)* 1 - .7513 = 100 * 2.48685 = .1

PVA = 248.69

Algebraic Solution

7 - 30


PVA i,n =( PMT) * (PVIFA i,n )Use Table A-2 on pages A-4 & 5

PVA 10%,3 =( 100) * (PVIFA 10%,3)

PVA 10%,3 =( 100)* 2.4869 =

PVA 10%,3 = 248.69

Tabular Solution

7 - 31


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

3 10 100 0

-248.69

INPUTS

OUTPUTN I/YR PV PMT FV

7 - 32


Find the FV and PV if theannuity were an annuity due.

100 100

0 1 2 3

10%

100

Easiest way, multiply results by (1 + i).

7 - 33


1 . PVAD i,n =( PMT)* 1 – (1 + i)n * (1 + i)

i 1 . PVAD 10%,3 =( 100) * 1 – (1 + .1)3 * (1 + .1)

.1

PVAD 10%,3 =( 100)* 1 - .7513 * (1.1) = .1 100 * 2.48685 * (1.1)=

PVAD 10%,3 = 273.55

Algebraic Solution

7 - 34


PVAD i,n =( PMT) * (PVIFA i,n )* (1 + i)Use Table A-2 on pages A-4 & 5

PVAD10%,3 =( 100) * (PVIFA 10%,3) * (1 + i)

PVAD 10%,3 =( 100)* 2.4869 * 1.1 =

PVAD 10%,3 = 273.55

Tabular Solution

7 - 35


3 10 100 0

-273.55

Switch from “End” to “Begin.”Then enter variables to find PVA3 = $273.55.

Then enter PV = 0 and press FV to findFV = $364.10.

INPUTS

OUTPUTN I/YR PV PMT FV

7 - 36


What is the PV of this uneven cashflow stream?

0

100

1

300

2

300

310%

-50

4

90.91247.93225.39 -34.15530.08 = PV

7 - 37


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.)

7 - 38


A 20-year old student wants to start saving for retirement. She plans to save $3 a day. Every day, she puts $3 in her drawer. At the end of the year, she invests the accumulated savings ($1,095) in an online stock account. The stock account has an expected annual return of 12%.

The Power of Compound Interest

7 - 39


How much money by the age of 65?

45 12 0 -1095

1,487,261.89

INPUTS

OUTPUT

N I/YR PV PMT FV

If she begins saving today, and sticks to her plan, she will have $1,487,261.89 by the age of 65.

7 - 40


How much would a 40-year old investor accumulate by this method?

25 12 0 -1095

146,000.59

INPUTS

OUTPUT

N I/YR PV PMT FV

Waiting until 40, the investor will only have $146,000.59, which is over $1.3 million less than if saving began at 20. So it pays to get started early.

7 - 41


How much would the 40-year old investor need to save to accumulate as

much as the 20-year old?

25 12 0 1487261.89

-11,154.42

INPUTS

OUTPUT

N I/YR PV PMT FV

The 40-year old investor would have to save $11,154.42 every year, or $30.56 per day to have as much as the investor beginning at the age of 20.

7 - 42


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.

7 - 43


Rules for Non-annual Compounding

95% of the time, the method for adjusting for non-annual compounding is:

Divide i by m, m being the # of compounding periods in a year.

Multiply n by m

7 - 44


FV of $100 after 3 years under 10% semiannual compounding? Quarterly?

= $100(1.05)6 = $134.01.FV3Q = $100(1.025)12 = $134.49.

FV = PV 1 .+ imnNom

m*n

FV = $100 1 + 0.10

23S

2*3

7 - 45


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.

7 - 46


Exam Question (Example)

Your uncle has given you a choice between receiving $20,000 today on your 18th birthday, or waiting until your 25th birthday and receiving $40,000. If you would invest in a junk bond fund if you took the $20,000, expecting to average 10% per year, compounded semiannually, which would you prefer?

7 - 47



See board for timeline. Algebraic solution:

FV = PV(1 + i/m)n*m

FV = 20,000 (1 + .1/2)7 * 2

FV = 20,000 ( 1.9799) = 39,598.63

Prefer the $40,000 in 7 years.

7 - 48



Tabular solution:

PV = FV(PVIF i/2,n*2)

PV = 40,000 (PVIF 10/2,7*2) Table A-1

PV = 40,000 ( .505) = 20,202

Prefer the $40,000 in 7 years (same conclusion.

7 - 49



14 10 ? 0 40,000

-20,202.72

INPUTS

OUTPUT

N I/YR PV PMT FV

Could have solved for FV inputting PV calcuation: P/Y set to 2

7 - 50


We will deal with 3 different rates:

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

iPer = periodic rate.

EAR= EFF% = .effective annual

rate

7 - 51


iNom is stated in contracts. Periods per year (m) must also be given.

Examples:

8%; Quarterly

8%, Daily interest (365 days)

7 - 52


Periodic rate = iPer = iNom/m, where m is number of compounding periods per year. m = 4 for quarterly, 12 for monthly, and 360 or 365 for daily compounding.

Examples:

8% quarterly: iPer = 8%/4 = 2%.

8% daily (365): iPer = 8%/365 = 0.021918%.

7 - 53


Effective Annual Rate (EAR = EFF%):The annual rate that causes PV to grow to the same FV as under multi-period compounding.Example: EFF% for 10%, semiannual:

FV = (1 + iNom/m)m

= (1.05)2 = 1.1025.

EFF% = 10.25% because (1.1025)1 = 1.1025.

Any PV would grow to same FV at 10.25% annually or 10% semiannually.

7 - 54


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.

7 - 55


How do we find EFF% for a nominal rate of 10%, compounded

semiannually?

EFF = – 1

Or use a financial calculator.

= – 1.0

= (1.05)2 – 1.0

= 0.1025 = 10.25%.

1 + iNom

m

1 + 0.102

2

m

7 - 56


EAR = EFF% of 10%

EARAnnual = 10%.

EARQ = (1 + 0.10/4)4 – 1 = 10.38%.

EARM = (1 + 0.10/12)12 – 1 = 10.47%.

EARD(365) = (1 + 0.10/365)365 – 1 = 10.52%.

7 - 57


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.

7 - 58


When is each rate used?

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

7 - 59


iPer: Used in calculations, shown on time lines.

If iNom has annual compounding,then iPer = iNom/1 = iNom.

7 - 60


(Used for calculations if and only ifdealing with annuities where payments don’t match interest compounding periods.)

EAR = EFF%: Used to compare returns on investments with different payments per year.

7 - 61


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

quoted interest rate is 10%, compounded semiannually?

0 1

100

2 35%

4 5 6-mos. periods

100 100

6

7 - 62


Payments occur annually, but compounding occurs each 6 months.

So we can’t use normal annuity valuation techniques.

7 - 63


1st Method: Compound Each CF

0 1

100

2 35%

4 5 6

100 100.00110.25121.55331.80

FVA3 = $100(1.05)4 + $100(1.05)2 + $100= $331.80.

7 - 64


Could you find FV with afinancial calculator?

Yes, by following these steps:

a. Find the EAR for the quoted rate:

2nd Method: Treat as an Annuity

EAR = (1 + ) – 1 = 10.25%. 0.10

22

7 - 65


Or, to find EAR with a calculator:

NOM% = 10.

P/YR = 2.

EFF% = 10.25.

7 - 66


EFF% = 10.25P/YR = 1NOM% = 10.25

3 10.25 0 -100 INPUTS

OUTPUT

N I/YR PV FVPMT

331.80

b. The cash flow stream is an annual annuity. Find kNom (annual) whose EFF% = 10.25%. In calculator,

c.

7 - 67


What’s the PV of this stream?

0

100

15%

2 3

100 100

90.7082.27

74.62247.59

7 - 1 copyright © 2002 by harcourt, inc.all rights reserved. future value present value rates of...

Documents

tables fv

year i

pvif i

pvfvif i

niyr pv pmt fv

algebraic solution fv

table a3 fv

time value of money