L4 Financial Mathematic

7/30/2019 L4 Financial Mathematic

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Chapter 5 - The TimeValue of Money

2005, Pearson Prentice Hall


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The Time Value of Money

Compounding and

Discounting Single Sums


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We know that receiving $1 today is worth

more than $1 in the future. This is due

toopportunity costs.The opportunity cost of receiving $1 in

the future is theinterestwe could have

earned if we had received the $1 sooner.Today Future


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If we can measure this opportunitycost, we can:


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Translate $1 today into its equivalent in the future(compounding).


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Today

?

Future


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Translate $1 in the future into its equivalent today

(discounting).

Today

?

Future


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Translate $1 in the future into its equivalent today

(discounting).

?

Today Future

Today

?

Future


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Compound Interest

and 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?


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0 1

PV = FV =


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Mathematical Solution:

FV = PV (FVIFi, 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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much would you have in the account after 5 years?

i


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0 5

PV = FV =

F V l i l


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FV = PV (FVIF i, n)

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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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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0 ?

PV = FV =





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FV = PV (FVIF i, n)

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





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monthly compounding, how much would you have


F t V l i l


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Future Value - single sumsIf you deposit $100 in an account earning 6% withmonthly compounding, how much would you have


0 ?

PV = FV =

F t V l i l


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FV = PV (FVIFi, n

)

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% withmonthly compounding, how much would you have



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

continuous compounding, after 100 years?


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0 ?

PV = FV =


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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 =




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0 100

PV = -1000 FV = $2.98m




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

FV = $2,980,957.99


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Present Value


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

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


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0 ?

PV = FV =




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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

PV = -94.34 FV = 100

0 1




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Present Value - single sumsIf you receive $100 five years from now, what is the



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0 ?

PV = FV =



i


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PV = FV (PVIFi, n

)


PV = FV / (1 + i)n

PV = 100 / (1.06)5 = $74.73



0 5

PV = -74.73 FV = 100

P V l i l


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

from now if your opportunity cost is 7%?

P V l i l


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0 15

PV = FV =



P t V l i l


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PV = FV (PVIF i, n)


PV = FV / (1 + i)n

PV = 100 / (1.07)15 = $362.45



0 15

PV = -362.45 FV = 1000

P t V l i l


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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?

P t V l i l


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0 5

PV = FV =



P t V l i l


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PV = FV (PVIF i, n )

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



P t V l i l


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Present Value - single sumsSuppose you placed $100 in an account that pays9.6% interest, compounded monthly. How long

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

0

PV = FV =

P t V l i l


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Present Value - single sumsSuppose you placed $100 in an account that pays9.6% 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)Nln 5 = N ln (1.008)

1.60944 = .007968 N N = 202 months


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Hint for single sum problems:

In every single sum present value andfuture value problem, there are fourvariables:

FV, PV, i and n. When doing problems, you will be given

three variables and you will solve for the

fourth variable. Keeping this in mind makes solving time

value problems much easier!


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Compounding and Discounting

Cash Flow Streams

0 1 2 3 4


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Annuities

Annuity: a sequence ofequal cash

flows, occurring at the end of each

period.


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Annuity: a sequence ofequal cash

flows, occurring at the end of each

period.

0 1 2 3 4

Annuities


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If you buy a bond, you willreceive equal semi-annual 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.

Examples of Annuities:

Future Value annuity


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Future Value - annuityIf you invest $1,000 each year at 8%, how much

would you have after 3 years?

Future Value annuity


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0 1 2 3



Future Value - annuity


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FV = PMT (FVIFA i, n)





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





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FV = PMT (1 + i)n - 1

i





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FV = PMT (1 + i)n - 1

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

.08



Present Value - annuity


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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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0 1 2 3

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




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PV = PMT (PVIFA i, n)





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





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1

PV = PMT 1 - (1 + i)n

i





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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 ofa perpetuity.

You can think of a perpetuity as anannuity 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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Present Value of a

Perpetuity

When we find the PV of an annuity,

we think of the following

relationship:


M th ti ll


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Mathematically,

Mathematicall


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Mathematically,

(PVIFA i, n ) =

Mathematically


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Mathematically,

(PVIFA i, n ) = 1 -

1

(1 + i)n

i

Mathematically


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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 ngets very, very large?

1 -

1

(1 + i)n

i

When n gets very large


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When n gets very large,



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1 -1

(1 + i)n

i



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this becomes zero.1 -

1(1 + i)n

i



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this becomes zero.

So were left with PVIFA =

1

i

1 -1

(1 + i)n

i


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So, the PV of a perpetuity is verysimple to find:

Present Value of a Perpetuity


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PMT

iPV =

So, the PV of a perpetuity is verysimple to find:

Present Value of a Perpetuity


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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?


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on the investment?

PMT $10,000i .08

PV = =


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on the investment?

PMT $10,000i .08

= $125,000

PV = =


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Ordinary Annuity

vs.

Annuity Due

$1000 $1000 $1000

4 5 6 7 8

Begin Mode vs. End Mode


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1000 1000 1000

4 5 6 7 8



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1000 1000 1000

4 5 6 7 8year year year

5 6 7



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1000 1000 1000


5 6 7

PV

inEND

Mode



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g

1000 1000 1000


5 6 7

PV

inEND

Mode

FV

inEND

Mode



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g

1000 1000 1000


6 7 8



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g

1000 1000 1000


6 7 8

PV

inBEGIN

Mode



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g

1000 1000 1000


6 7 8

PV

inBEGIN

Mode

FV

inBEGIN

Mode

Earlier, we examined this


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,

ordinary annuity:



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,

ordinary annuity:

0 1 2 3

1000 1000 1000



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,

ordinary annuity:

Using an interest rate of 8%, wefind that:

0 1 2 3

1000 1000 1000



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,

ordinary annuity:


The Future Value (at 3) is

$3,246.40.

0 1 2 3

1000 1000 1000



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,

ordinary annuity:


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

What about this annuity?


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Same 3-year time line,

Same 3 $1000 cash flows, but

The cash flows occur at the

beginning of each year, ratherthan at the end of each year.

This is an annuity due.

0 1 2 31000 1000 1000

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


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0 1 2 3

If you invest $1,000 at the beginning of each of thenext 3 years at 8%, how much would you have at

the end of year 3?



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Calculator Solution:

Mode = BEGIN P/Y = 1 I = 8N = 3 PMT = -1,000

FV = $3,506.11

0 1 2 3

-1000 -1000 -1000


the end of year 3?



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0 1 2 3

-1000 -1000 -1000


the end of year 3?


Mode = BEGIN P/Y = 1 I = 8N = 3 PMT = -1,000

FV = $3,506.11



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the end of year 3?Mathematical Solution: Simply compound the FV of the

ordinary annuity one more period:



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



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FV = PMT (FVIFA i, n) (1 + i)FV = 1,000 (FVIFA .08, 3) (1.08) (use FVIFA table, or)



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FV = PMT (1 + i)n - 1

i (1 + i)



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FV = PMT (1 + i)n - 1

i

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

.08

(1 + i)

(1.08)

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


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What is the PV of $1,000 at the beginning of eachof the next 3 years, if your opportunity cost is 8%?

0 1 2 3



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Mode = BEGIN P/Y = 1 I = 8N = 3 PMT = 1,000

PV = $2,783.26

0 1 2 3

1000 1000 1000




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Mode = BEGIN P/Y = 1 I = 8N = 3 PMT = 1,000

PV = $2,783.26

0 1 2 3

1000 1000 1000


Present Value - annuity due


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



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



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



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1PV = PMT 1 - (1 + i)n

i(1 + i)



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1PV = PMT 1 - (1 + i)n

i

1

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

.08

(1 + i)

(1.08)

Uneven Cash Flows


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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.)

0 1 2 3 4

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

Uneven Cash Flows


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Sorry! Theres no quickie for this one.

We have to discount each cash flow

back separately.

0 1 2 3 4

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

Uneven Cash Flows


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back separately.

0 1 2 3 4

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

Uneven Cash Flows


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back separately.

0 1 2 3 4

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

Uneven Cash Flows


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back separately.

0 1 2 3 4

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

Uneven Cash Flows


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back separately.

0 1 2 3 4

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

-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 4,507.894 7,000 4,781.09

PV of Cash Flow Stream: $ 4,412.95

0 1 2 3 4

Annual Percentage Yield (APY)


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Which is the better loan: 8% compounded annually, or

7.85% compounded quarterly?

We cant compare these nominal (quoted)

interest rates, because they dont include the

same number of compounding periods per

year!

We need to calculate the APY.



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APY = ( 1 + ) m - 1quoted ratem



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Find the APY for the quarterly loan:




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APY = ( 1 + ) 4 - 1.07854



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APY = ( 1 + ) 4 - 1

APY = .0808, or 8.08%

.0785

4



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The quarterly loan is more expensive than

the 8% loan with annual compounding!


APY = ( 1 + ) 4 - 1

APY = .0808, or 8.08%

.0785

4

The Effective Annual Rate


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122

EAR can be calculated for any compounding

period using the formula

Effect of more frequent compounding is greater

at higher interest rates

mnom

m

kEAR

1

Practice Problems


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Practice Problems

Example


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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?

Example


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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?

$0 0 0 0 40 40 40 40 40


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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.

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:


time 0 separately.

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:


time 0 separately.

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:


time 0 separately.

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:


time 0 separately.

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:


time 0 separately.

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:


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:

PV: End mode; P/YR = 1; I = 20;PMT = 40,000; N = 5

PV = $119,624

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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69,226

0 1 2 3 4 5 6 7 8

119,624

$0 0 0 0 40 40 40 40 40


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The PV of the cash flow

stream is $69,226.

69,226

0 1 2 3 4 5 6 7 8

119,624

Retirement Example

Aft d ti l t i t


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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 30years?

Retirement Example

Aft d ti l t i t


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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 30years?

0 1 2 3 . . . 360

400 400 400 400

400 400 400 400


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0 1 2 3 . . . 360

Retirement ExampleIf you invest $400 at the end of each month for the

next 30 years at 12% how much would you have at


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next 30 years at 12%, how much would you have at

the end of year 30?Mathematical Solution:




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




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FV = PMT (1 + i)n - 1

i




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FV = PMT (1 + i)n - 1

i

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

01

House Payment Example

If you borrow $100 000 at 7% fixed


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If you borrow $100,000 at 7% fixed

interest for 30 years in order tobuy a house, what will be your

monthly house payment?

? ? ? ?


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0 1 2 3 . . . 360




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




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1PV = PMT 1 - (1 + i)n

i




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1PV = PMT 1 - (1 + i)n

i

1

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

.005833

Exercise


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1) If you deposit $10,000 in a bank accountthat pay 10% interest annually, how much will

it be in your account after 5 years?

Compounded annually Compounded monthly

Compounded quarterly

Compounded semi-annually

Exercise


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What is the value of security that will pay$5,000 in 20 years if securities of equal risk

pay 7% annually?

Compounded annually Compounded monthly

Compounded quarterly

Compounded semi-annually

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L4 Financial Mathematic