Compounding and Discounting Single Sums 03, 2011 · Compounding and Discounting Single Sums 2 We know that receiving $1 today is worth ... PV = 100 (PVIF .06, 5) (use PVIF table,

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The Time Value of Money

Compounding and

Discounting Single Sums

http://www.deden08m.com

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We know that receiving $1 today is worth

more than $1 in the future. This is due

to OPPORTUNITY COSTS.

The opportunity cost of receiving $1 in

the future is the interest we could have

earned if we had received the $1 sooner.

Today Future


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If we can MEASURE this

opportunity cost, we can:

Translate $1 today into its equivalent in

the future (COMPOUNDING).

Translate $1 in the future into its

equivalent today (DISCOUNTING).

?

?

Today Future

Today Future


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Future Value


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Future Value - single sums

If you deposit $100 in an account earning 6%, how

much would you have in the account after 1 year?

Solution:

FV = PV (FVIF i, n )

FV = 100 (FVIF .06, 1 ) (use FVIF table, or)

FV = PV (1 + i)n

FV = 100 (1.06)1 = $106

0 1

PV = -100 FV = 106


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Future Value - single sums

If you deposit $100 in an account earning 6%, how

much would you have in the account after 5 years?

Solution:


FV = 100 (FVIF .06, 5 ) (use FVIF table, or)

FV = PV (1 + i)n

FV = 100 (1.06)5 = $133.82

0 5

PV = -100 FV = 133.82


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Mathematical Solution:


FV = 100 (FVIF .015, 20 ) (cant use FVIF table)

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

FV = 100 (1.015)20 = $134.68

0 20

PV = -100 FV = 134.68

Future Value - single sumsIf you deposit $100 in an account earning 6% with

quarterly compounding, how much would you have

in the account after 5 years?


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FV = 100 (FVIF .005, 60 ) (cant use FVIF table)

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

FV = 100 (1.005)60 = $134.89

0 60

PV = -100 FV = 134.89

Future Value - single sumsIf you deposit $100 in an account earning 6% with

monthly compounding, how much would you have

in the account after 5 years?


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FV = PV (e in)

FV = 1000 (e .08x100) = 1000 (e 8)

FV = $2,980,957.99

0 100

PV = -1000 FV =

Future Value - continuous compoundingWhat is the FV of $1,000 earning 8% with

continuous compounding, after 100 years?


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0 100

PV = -1000 FV =

Future Value - continuous compoundingWhat is the FV of $1,000 earning 8% with

continuous compounding, after 100 years?

$2.98m


FV = PV (e in)

FV = 1000 (e .08x100) = 1000 (e 8)

FV = $2,980,957.99http://www.deden08m.com

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Present Value


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PV = FV (PVIF i, n )

PV = 100 (PVIF .06, 1 ) (use PVIF table, or)

PV = FV / (1 + i)n

PV = 100 / (1.06)1 = $94.34

0 1

PV = -94.34 FV = 100

Present Value - single sumsIf you will receive $100 one year from now, what is

the PV of that $100 if your opportunity cost is 6%?


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PV = FV / (1 + i)n

PV = 100 / (1.06)5 = $74.73

0 5

PV = -74.73 FV = 100

Present Value - single sumsIf you will receive $100 5 years from now, what is

the PV of that $100 if your opportunity cost is 6%?


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PV = FV / (1 + i)n

PV = 100 / (1.07)15 = $362.45

0 15

PV = -362.45 FV = 1000

Present Value - single sumsWhat is the PV of $1,000 to be received 15 years

from now if your opportunity cost is 7%?


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5,000 = 11,933 (PVIF ?, 5 )

PV = FV / (1 + i)n

5,000 = 11,933 / (1+ i)5

.419 = ((1/ (1+i)5)

2.3866 = (1+i)5

(2.3866)1/5 = (1+i) i = .19

Present Value - single sumsIf you sold land for $11,933 that you bought 5 years

ago for $5,000, what is your annual rate of return?


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Present Value - single sumsSuppose you placed $100 in an account that pays

9.6% interest, compounded monthly. How long

will it take for your account to grow to $500?


PV = FV / (1 + i)n

100 = 500 / (1+ .008)N

5 = (1.008)N

ln 5 = ln (1.008)N

ln 5 = N ln (1.008)

1.60944 = .007968 N N = 202 monthshttp://www.deden08m.com

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Hint for single sum problems:

In every single sum future value

and present value problem, there

are 4 variables:

FV, PV, i, and n

When doing problems, you will be

given 3 of these variables and

asked to solve for the 4th variable.

Keeping this in mind makes time

value problems much easier!http://www.deden08m.com

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The Time Value of Money

Compounding and Discounting

Cash Flow Streams

0 1 2 3 4


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Annuities

Annuity: a sequence of equal cash

flows, occurring at the end of each

period.

0 1 2 3 4


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Examples of Annuities:

If you buy a bond, you will

receive equal coupon interest

payments over the life of the

bond.

If you borrow money to buy a

house or a car, you will pay a

stream of equal payments.


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Examples of Annuities:

If you buy a bond, you will

receive equal coupon interest

payments over the life of the

bond.

If you borrow money to buy a

house or a car, you will pay a

stream of equal payments.


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Calculator Solution:

P/Y = 1 I = 8 N = 3

PMT = -1,000

FV = $3,246.40

Future Value - annuityIf you invest $1,000 at the end of the next 3 years,

at 8%, how much would you have after 3 years?

0 1 2 3

1000 1000 1000


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P/Y = 1 I = 8 N = 3

PMT = -1,000

FV = $3,246.40



0 1 2 3

1000 1000 1000


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FV = PMT (FVIFA i, n )

FV = 1,000 (FVIFA .08, 3 ) (use FVIFA table, or)

FV = PMT (1 + i)n - 1

i

FV = 1,000 (1.08)3 - 1 = $3246.40

.08




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P/Y = 1 I = 8 N = 3

PMT = -1,000

PV = $2,577.10

0 1 2 3

1000 1000 1000

Present Value - annuityWhat is the PV of $1,000 at the end of each of the

next 3 years, if the opportunity cost is 8%?


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P/Y = 1 I = 8 N = 3

PMT = -1,000

PV = $2,577.10

0 1 2 3

1000 1000 1000




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PV = PMT (PVIFA i, n )

PV = 1,000 (PVIFA .08, 3 ) (use PVIFA table, or)

1

PV = PMT 1 - (1 + i)n

i

1

PV = 1000 1 - (1.08 )3 = $2,577.10

.08




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Other Cash Flow Patterns

0 1 2 3


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Perpetuities

Suppose you will receive a fixed

payment every period (month, year,

etc.) forever. This is an example of

a perpetuity.

You can think of a perpetuity as an

annuity that goes on forever.


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Present Value of a Perpetuity

When we find the PV of an annuity,

we think of the following

relationship:


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When we find the PV of an annuity,

we think of the following

relationship:



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Mathematically,

(PVIFA i, n ) =

We said that a perpetuity is an

annuity where n = infinity. What

happens to this formula when n

gets very, very large?

1 -1

(1 + i)n

i


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1 -1

(1 + i)n

i

1

i

When n gets very large,

this becomes zero.

So were left with PVIFA =


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PMT

iPV =

So, the PV of a perpetuity is very

simple to find:



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What should you be willing to pay in

order to receive $10,000 annually

forever, if you require 8% per year

on the investment?

PMT

iPV = =

$10,000

.08

= $125,000


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Ordinary Annuity

vs.

Annuity Due

$1000 $1000 $1000

4 5 6 7 8


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Begin Mode vs. End Mode

1000 1000 1000

4 5 6 7 8 year year year

5 6 7

PVin

END

Mode

FVin

END

Modehttp://www.deden08m.com

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Begin Mode vs. End Mode

1000 1000 1000

4 5 6 7 8 year year year

6 7 8

PVin

BEGIN

Mode

FVin

BEGIN

Modehttp://www.deden08m.com

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Earlier, we examined this

ordinary annuity:

Using an interest rate of 8%, we

find that:

The Future Value (at 3) is

$3,246.40.

The Present Value (at 0) is

$2,577.10.

0 1 2 3

1000 1000 1000


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What about this annuity?

Same 3-year time line,

Same 3 $1000 cash flows, but

The cash flows occur at the

beginning of each year, rather

than at the end of each year.

This is an annuity due.

0 1 2 3

1000 1000 1000


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0 1 2 3

-1000 -1000 -1000

Future Value - annuity dueIf you invest $1,000 at the beginning of each of the

next 3 years at 8%, how much would you have at

the end of year 3?


Mode = BEGIN P/Y = 1 I = 8

N = 3 PMT = -1,000

FV = $3,506.11http://www.deden08m.com

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Future Value - annuity dueIf you invest $1,000 at the beginning of each of the


the end of year 3?

Mathematical Solution: Simply compound the FV of the ordinary annuity one more period:

FV = PMT (FVIFA i, n ) (1 + i)

FV = 1,000 (FVIFA .08, 3 ) (1.08) (use FVIFA table, or)

FV = PMT (1 + i)n - 1

i

FV = 1,000 (1.08)3 - 1 = $3,506.11

.08

(1 + i)

(1.08)http://www.deden08m.com

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Mode = BEGIN P/Y = 1 I = 8

N = 3 PMT = 1,000

PV = $2,783.26

0 1 2 3

1000 1000 1000

Present Value - annuity dueWhat is the PV of $1,000 at the beginning of each

of the next 3 years, if your opportunity cost is 8%?


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Present Value - annuity due

Mathematical Solution: Simply compound the FV of the ordinary annuity one more period:

PV = PMT (PVIFA i, n ) (1 + i)

PV = 1,000 (PVIFA .08, 3 ) (1.08) (use PVIFA table, or)

1

PV = PMT 1 - (1 + i)n

i

1

PV = 1000 1 - (1.08 )3 = $2,783.26

.08

(1 + i)

(1.08)


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Is this an annuity?

How do we find the PV of a cash flow

stream when all of the cash flows are

different? (Use a 10% discount rate).

Uneven Cash Flows

0 1 2 3 4

-10,000 2,000 4,000 6,000 7,000


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Sorry! Theres no quickie for this one.

We have to discount each cash flow

back separately.

Uneven Cash Flows

0 1 2 3 4

-10,000 2,000 4,000 6,000 7,000


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Uneven Cash Flows



back separately.

0 1 2 3 4

-10,000 2,000 4,000 6,000 7,000


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Uneven Cash Flows

0 1 2 3 4

-10,000 2,000 4,000 6,000 7,000



back separately.http://www.deden08m.com

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Uneven Cash Flows



back separately.

0 1 2 3 4

-10,000 2,000 4,000 6,000 7,000


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Uneven Cash Flows



back separately.

0 1 2 3 4

-10,000 2,000 4,000 6,000 7,000


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period CF PV (CF)

0 -10,000 -10,000.00

1 2,000 1,818.18

2 4,000 3,305.79

3 6,000 4,507.89

4 7,000 4,781.09

PV of Cash Flow Stream: $ 4,412.95

0 1 2 3 4

-10,000 2,000 4,000 6,000 7,000


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Example

Cash flows from an investment are

expected to be $40,000 per year at the

end of years 4, 5, 6, 7, and 8. If you

require a 20% rate of return, what is

the PV of these cash flows?


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Example

0 1 2 3 4 5 6 7 8

0 0 0 0 40 40 40 40 40

Cash flows from an investment are

expected to be $40,000 per year at the

end of years 4, 5, 6, 7, and 8. If you

require a 20% rate of return, what is

the PV of these cash flows?


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This type of cash flow sequence is

often called a deferred annuity.

0 1 2 3 4 5 6 7 8

0 0 0 0 40 40 40 40 40


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How to solve:

1) Discount each cash flow back to

time 0 separately.

Or,

0 1 2 3 4 5 6 7 8

0 0 0 0 40 40 40 40 40


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2) Find the PV of the annuity:

PV3: End mode; P/YR = 1; I = 20;

PMT = 40,000; N = 5

PV3= $119,624

0 1 2 3 4 5 6 7 8

0 0 0 0 40 40 40 40 40


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119,624

0 1 2 3 4 5 6 7 8

0 0 0 0 40 40 40 40 40


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Then discount this single sum back to

time 0.

PV: End mode; P/YR = 1; I = 20;

N = 3; FV = 119,624;

Solve: PV = $69,226

119,624

0 1 2 3 4 5 6 7 8

0 0 0 0 40 40 40 40 40


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119,62469,226

0 1 2 3 4 5 6 7 8

0 0 0 0 40 40 40 40 40


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119,62469,226

The PV of the cash flow

stream is $69,226.

0 1 2 3 4 5 6 7 8

0 0 0 0 40 40 40 40 40


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Example

After graduation, you plan to invest

$400 per month in the stock market.

If you earn 12% per year on your

stocks, how much will you have

accumulated when you retire in 30

years?


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Retirement Example

After graduation, you plan to invest

$400 per month in the stock market.

If you earn 12% per year on your

stocks, how much will you have

accumulated when you retire in 30

years?

0 1 2 3 . . . 360

400 400 400 400


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Using your calculator,

P/YR = 12

N = 360

PMT = -400

I%YR = 12

FV = $1,397,985.65

0 1 2 3 . . . 360

400 400 400 400


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Retirement Example

If you invest $400 at the end of each month for the


the end of year 30?


FV = PMT (FVIFA i, n )

FV = 400 (FVIFA .01, 360 ) (cant use FVIFA table)

FV = PMT (1 + i)n - 1

i

FV = 400 (1.01)360 - 1 = $1,397,985.65

.01 http://www.deden08m.com

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House Payment Example

If you borrow $100,000 at 7% fixed

interest for 30 years in order to

buy a house, what will be your

monthly house payment?


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P/YR = 12

N = 360

I%YR = 7

PV = $100,000

PMT = -$665.30

0 1 2 3 . . . 360

? ? ? ?


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House Payment Example



100,000 = PMT (PVIFA .07, 360 ) (cant use PVIFA table)

1

PV = PMT 1 - (1 + i)n

i

1

100,000 = PMT 1 - (1.005833 )360 PMT=$665.30

.005833http://www.deden08m.com

68

Team Assignment

Upon retirement, your goal is to spend 5

years traveling around the world. To

travel in style will require $250,000 per

year at the beginning of each year.

If you plan to retire in 30 years, what are

the equal monthly payments necessary

to achieve this goal?

The funds in your retirement account will

compound at 10% annually.http://www.deden08m.com

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How much do we need to have by

the end of year 30 to finance the

trip?

PV30 = PMT (PVIFA .10, 5) (1.10) =

= 250,000 (3.7908) (1.10) =

= $1,042,470

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250 250 250 250 250


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Mode = BEGIN

PMT = -$250,000

N = 5

I%YR = 10

P/YR = 1

PV = $1,042,466

27 28 29 30 31 32 33 34 35

250 250 250 250 250


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Now, assuming 10% annual

compounding, what monthly

payments will be required for

you to have $1,042,466 at the end

of year 30?

27 28 29 30 31 32 33 34 35

250 250 250 250 250

1,042,466


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Mode = END

N = 360

I%YR = 10

P/YR = 12

FV = $1,042,466

PMT = -$461.17

27 28 29 30 31 32 33 34 35

250 250 250 250 250

1,042,466


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So, you would have to place

$461.17 in your retirement account,

which earns 10% annually, at the

end of each of the next 360 months

to finance the 5-year world tour.http://www.deden08m.com

Documents

Compounding and Discounting Single Sums 03, 2011 · Compounding and Discounting Single Sums 2 We know that receiving $1 today is worth ... PV = 100 (PVIF .06, 5) (use PVIF table,