Session2 Time Value Of Money (2)

The Time Value of MoneyThe Time Value of Money

Compounding and Compounding and Discounting Single SumsDiscounting Single Sums

We know that receiving $1 today is worth We know that receiving $1 today is worth more than $1 in the future. This is duemore than $1 in the future. This is due toto OPPORTUNITY COSTSOPPORTUNITY COSTS..

The opportunity cost of receiving $1 in The opportunity cost of receiving $1 in the future is thethe future is the interestinterest we could have we could have earned if we had received the $1 sooner.earned if we had received the $1 sooner.

Today Future

If we can MEASURE this If we can MEASURE this opportunity cost, we can:opportunity cost, we can:

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

Today Future

If we can MEASURE this If we can MEASURE this opportunity cost, we can:opportunity cost, we can:

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

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

Today Future

Future ValueFuture Value

Future Value - single sumsFuture Value - single sums

If you deposit $100 in an account earning 6%, how If you deposit $100 in an account earning 6%, how much would you have in the account after 1 year?much would you have in the account after 1 year?

PV =PV = FV = FV =

Calculator Solution:Calculator Solution:

P/Y = 1P/Y = 1 I = 6I = 6

N = 1 N = 1 PV = -100 PV = -100

FV = FV = $106$106

00 1 1

PV = -100PV = -100 FV = FV =

P/Y = 1P/Y = 1 I = 6I = 6

N = 1 N = 1 PV = -100 PV = -100

FV = FV = $106$106

00 1 1

PV = -100PV = -100 FV = FV = 106106

Mathematical Solution:Mathematical Solution:

FV = PV (FVIF FV = PV (FVIF i, ni, n ))

FV = 100 (FVIF FV = 100 (FVIF .06, 1.06, 1 ) (use FVIF table, or)) (use FVIF table, or)

FV = PV (1 + i)FV = PV (1 + i)nn

FV = 100 (1.06)FV = 100 (1.06)1 1 = = $106$106

00 1 1

PV = -100PV = -100 FV = FV = 106106

If you deposit $100 in an account earning 6%, how If you deposit $100 in an account earning 6%, how much would you have in the account after 5 years?much would you have in the account after 5 years?

00 5 5

PV =PV = FV = FV =

P/Y = 1P/Y = 1 I = 6I = 6

N = 5 N = 5 PV = -100 PV = -100

FV = FV = $133.82$133.82

00 5 5

PV = -100PV = -100 FV = FV =

P/Y = 1P/Y = 1 I = 6I = 6

N = 5 N = 5 PV = -100 PV = -100

FV = FV = $133.82$133.82

00 5 5

PV = -100PV = -100 FV = FV = 133.133.8282

FV = 100 (FVIF FV = 100 (FVIF .06, 5.06, 5 ) (use FVIF table, or)) (use FVIF table, or)

FV = PV (1 + i)FV = PV (1 + i)nn

FV = 100 (1.06)FV = 100 (1.06)5 5 = = $$133.82133.82

00 5 5

PV = -100PV = -100 FV = FV = 133.133.8282

Future Value - single sumsFuture Value - single sumsIf you deposit $100 in an account earning 6% with If you deposit $100 in an account earning 6% with quarterly compoundingquarterly compounding, how much would you have , how much would you have

in the account after 5 years?in the account after 5 years?

PV =PV = FV = FV =

P/Y = 4P/Y = 4 I = 6I = 6

N = 20 N = 20 PV = PV = -100-100

FV = FV = $134.68$134.68

00 20 20

PV = -100PV = -100 FV = FV =

P/Y = 4P/Y = 4 I = 6I = 6

N = 20 N = 20 PV = PV = -100-100

FV = FV = $134.68$134.68

00 20 20

PV = -100PV = -100 FV = FV = 134.134.6868

FV = 100 (FVIF FV = 100 (FVIF .015, 20.015, 20 ) ) (can’t use FVIF table)(can’t use FVIF table)

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

FV = 100 (1.015)FV = 100 (1.015)20 20 = = $134.68$134.68

00 20 20

PV = -100PV = -100 FV = FV = 134.134.6868

Future Value - single sumsFuture Value - single sumsIf you deposit $100 in an account earning 6% with If you deposit $100 in an account earning 6% with monthly compoundingmonthly compounding, how much would you have , how much would you have

PV =PV = FV = FV =

P/Y = 12P/Y = 12 I = 6I = 6

N = 60 N = 60 PV = PV = -100-100

FV = FV = $134.89$134.89

00 60 60

PV = -100PV = -100 FV = FV =

P/Y = 12P/Y = 12 I = 6I = 6

N = 60 N = 60 PV = PV = -100-100

FV = FV = $134.89$134.89

00 60 60

PV = -100PV = -100 FV = FV = 134.134.8989

FV = 100 (FVIF FV = 100 (FVIF .005, 60.005, 60 ) ) (can’t use FVIF table)(can’t use FVIF table)

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

FV = 100 (1.005)FV = 100 (1.005)60 60 = = $134.89$134.89

00 60 60

PV = -100PV = -100 FV = FV = 134.134.8989

Future Value - continuous compoundingFuture Value - continuous compoundingWhat is the FV of $1,000 earning 8% with What is the FV of $1,000 earning 8% with continuous compoundingcontinuous compounding, after 100 years?, after 100 years?

PV =PV = FV = FV =

FV = PV (e FV = PV (e inin))

FV = 1000 (e FV = 1000 (e .08x100.08x100) = 1000 (e ) = 1000 (e 88) )

FV = FV = $2,980,957.$2,980,957.9999

00 100 100

PV = -1000PV = -1000 FV = FV =

00 100 100

PV = -1000PV = -1000 FV = FV =

$2.98m$2.98m

FV = PV (e FV = PV (e inin))

FV = 1000 (e FV = 1000 (e .08x100.08x100) = 1000 (e ) = 1000 (e 88) )

FV = FV = $2,980,957.$2,980,957.9999

Present ValuePresent Value

Present Value - single sumsPresent Value - single sumsIf you will receive $100 one year from now, what is If you will receive $100 one year from now, what is the PV of that $100 if your opportunity cost is 6%?the PV of that $100 if your opportunity cost is 6%?

PV =PV = FV = FV =

P/Y = 1P/Y = 1 I = 6I = 6

N = 1 N = 1 FV = FV = 100100

PV = PV = -94.34-94.34

00 1 1

PV = PV = FV = 100 FV = 100

P/Y = 1P/Y = 1 I = 6I = 6

N = 1 N = 1 FV = FV = 100100

PV = PV = -94.34-94.34

00 1 1

PV = PV = -94.-94.3434 FV = 100 FV = 100

PV = FV (PVIF PV = FV (PVIF i, ni, n ))

PV = 100 (PVIF PV = 100 (PVIF .06, 1.06, 1 ) (use PVIF table, or)) (use PVIF table, or)

PV = FV / (1 + i)PV = FV / (1 + i)nn

PV = 100 / (1.06)PV = 100 / (1.06)1 1 = = $94.34$94.34

00 1 1

PV = PV = -94.-94.3434 FV = 100 FV = 100

Present Value - single sumsPresent Value - single sumsIf you will receive $100 5 years from now, what is If you will receive $100 5 years from now, what is the PV of that $100 if your opportunity cost is 6%?the PV of that $100 if your opportunity cost is 6%?

PV =PV = FV = FV =

P/Y = 1P/Y = 1 I = 6I = 6

N = 5 N = 5 FV = FV = 100100

PV = PV = -74.73-74.73

00 5 5

PV = PV = FV = 100 FV = 100

00 5 5

PV = PV = FV = 100 FV = 100

-74.-74.7373

P/Y = 1P/Y = 1 I = 6I = 6

N = 5 N = 5 FV = FV = 100100

PV = PV = -74.73-74.73

PV = FV / (1 + i)PV = FV / (1 + i)nn

PV = 100 / (1.06)PV = 100 / (1.06)5 5 = = $74.73$74.73

00 5 5

PV = PV = -74.-74.7373 FV = 100 FV = 100

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

from now if your opportunity cost is 7%?from now if your opportunity cost is 7%?

PV =PV = FV = FV =

P/Y = 1P/Y = 1 I = 7I = 7

N = 15 N = 15 FV = FV = 1,0001,000

PV = PV = -362.45-362.45

00 15 15

PV = PV = FV = 1000 FV = 1000

P/Y = 1P/Y = 1 I = 7I = 7

N = 15 N = 15 FV = FV = 1,0001,000

PV = PV = -362.45-362.45

00 15 15

PV = PV = -362.-362.4545 FV = 1000 FV = 1000

PV = FV / (1 + i)PV = FV / (1 + i)nn

PV = 100 / (1.07)PV = 100 / (1.07)15 15 = = $362.45$362.45

00 15 15

PV = PV = -362.-362.4545 FV = 1000 FV = 1000

Present Value - single sumsPresent Value - single sumsIf you sold land for $11,933 that you bought 5 years If you sold land for $11,933 that you bought 5 years ago for $5,000, what is your annual rate of return?ago for $5,000, what is your annual rate of return?

PV =PV = FV = FV =

P/Y = 1P/Y = 1 N = 5N = 5

PV = -5,000 PV = -5,000 FV = 11,933FV = 11,933

I = I = 19%19%

00 5 5

PV = -5,000PV = -5,000 FV = 11,933 FV = 11,933

PV = FV (PVIF PV = FV (PVIF i, ni, n ) )

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

PV = FV / (1 + i)PV = FV / (1 + i)nn

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

.419 = ((1/ (1+i).419 = ((1/ (1+i)55))

2.3866 = (1+i)2.3866 = (1+i)55

(2.3866)(2.3866)1/51/5 = (1+i) = (1+i) i = .19i = .19

Present Value - single sumsPresent Value - single sumsSuppose you placed $100 in an account that pays Suppose you placed $100 in an account that pays 9.6% interest, compounded monthly. How long 9.6% interest, compounded monthly. How long

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

PV = PV = FV = FV =

Calculator Solution:Calculator Solution: P/Y = 12P/Y = 12 FV = 500FV = 500 I = 9.6I = 9.6 PV = -100PV = -100 N = N = 202 months202 months

00 ? ?

PV = -100PV = -100 FV = 500 FV = 500

PV = FV / (1 + i)PV = FV / (1 + i)nn

100 = 500 / (1+ .008)100 = 500 / (1+ .008)NN

5 = (1.008)5 = (1.008)NN

ln 5 = ln (1.008)ln 5 = ln (1.008)NN

ln 5 = N ln (1.008)ln 5 = N ln (1.008)

1.60944 = .007968 N1.60944 = .007968 N N = 202 monthsN = 202 months

Hint for single sum problems:Hint for single sum problems:

In every single sum future value In every single sum future value and present value problem, there and present value problem, there are 4 variables: are 4 variables:

FVFV, , PVPV, , ii, and , and nn When doing problems, you will be When doing problems, you will be

given 3 of these variables and given 3 of these variables and asked to solve for the 4th variable.asked to solve for the 4th variable.

Keeping this in mind makes “time Keeping this in mind makes “time value” problems much easier!value” problems much easier!

The Time Value of MoneyThe Time Value of Money

Compounding and DiscountingCompounding and Discounting

Cash Flow StreamsCash Flow Streams

0 1 2 3 4

AnnuitiesAnnuities

Annuity: a sequence of Annuity: a sequence of equal cash equal cash flowsflows, occurring at the , occurring at the endend of each of each period.period.

AnnuitiesAnnuities

Annuity: a sequence of equal cash Annuity: a sequence of equal cash flows, occurring at the end of each flows, occurring at the end of each period.period.

0 1 2 3 4

Examples of Annuities:Examples of Annuities:

If you buy a bond, you will If you buy a bond, you will receive equal coupon interest receive equal coupon interest payments over the life of the payments over the life of the bond.bond.

If you borrow money to buy a If you borrow money to buy a house or a car, you will pay a house or a car, you will pay a stream of equal payments.stream of equal payments.

Examples of Annuities:Examples of Annuities:

If you buy a bond, you will If you buy a bond, you will receive equal coupon interest receive equal coupon interest payments over the life of the payments over the life of the bond.bond.

If you borrow money to buy a If you borrow money to buy a house or a car, you will pay a house or a car, you will pay a stream of equal payments.stream of equal payments.

Future Value - annuityFuture Value - annuityIf you invest $1,000 at the end of the next 3 years, If you invest $1,000 at the end of the next 3 years, at 8%, how much would you have after 3 years?at 8%, how much would you have after 3 years?

0 1 2 3

P/Y = 1P/Y = 1 I = 8I = 8 N = 3N = 3

PMT = -1,000 PMT = -1,000

FV = FV = $3,246.40$3,246.40

0 1 2 3

10001000 10001000 1000 1000

P/Y = 1P/Y = 1 I = 8I = 8 N = 3N = 3

PMT = -1,000 PMT = -1,000

FV = FV = $3,246.40$3,246.40

0 1 2 3

10001000 10001000 1000 1000

FV = PMT (FVIFA FV = PMT (FVIFA i, ni, n ))

FV = 1,000 (FVIFA FV = 1,000 (FVIFA .08, 3.08, 3 ) ) (use FVIFA table, or)(use FVIFA table, or)

FV = PMT (1 + i)FV = PMT (1 + i)nn - 1 - 1

FV = 1,000 (1.08)FV = 1,000 (1.08)33 - 1 = - 1 = $3246.40$3246.40

.08 .08

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

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

0 1 2 3

P/Y = 1P/Y = 1 I = 8I = 8 N = 3N = 3

PMT = -1,000 PMT = -1,000

PV = PV = $2,577.10$2,577.10

0 1 2 3

10001000 10001000 1000 1000

P/Y = 1P/Y = 1 I = 8I = 8 N = 3N = 3

PMT = -1,000 PMT = -1,000

PV = PV = $2,577.10$2,577.10

0 1 2 3

10001000 10001000 1000 1000

PV = PMT (PVIFA PV = PMT (PVIFA i, ni, n ))

PV = 1,000 (PVIFA PV = 1,000 (PVIFA .08, 3.08, 3 ) (use PVIFA table, or)) (use PVIFA table, or)

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

PV = 1000 1 - (1.08 )PV = 1000 1 - (1.08 )33 = = $2,577.10$2,577.10

.08.08

Other Cash Flow PatternsOther Cash Flow Patterns

0 1 2 3

PerpetuitiesPerpetuities

Suppose you will receive a fixed Suppose you will receive a fixed payment every period (month, year, payment every period (month, year, etc.) forever. This is an example of etc.) forever. This is an example of a perpetuity.a perpetuity.

You can think of a perpetuity as an You can think of a perpetuity as an annuityannuity that goes on that goes on foreverforever..

Present Value of a PerpetuityPresent Value of a Perpetuity

When we find the PV of an annuity, When we find the PV of an annuity, we think of the following we think of the following relationship:relationship:

When we find the PV of an When we find the PV of an annuityannuity, , we think of the following we think of the following relationship:relationship:

PV = PMT (PVIFA PV = PMT (PVIFA i, ni, n ) )

Mathematically, Mathematically,

(PVIFA i, n ) = (PVIFA i, n ) =

(PVIFA i, n ) = (PVIFA i, n ) = 1 - 1 - 11

(1 + i)(1 + i)nn

(PVIFA i, n ) = (PVIFA i, n ) =

We said that a perpetuity is an We said that a perpetuity is an annuity where n = infinity. What annuity where n = infinity. What happens to this formula when happens to this formula when nn gets very, very large? gets very, very large?

1 - 1 - 11

(1 + i)(1 + i)nn

When n gets very large,When n gets very large,

this becomes zero.this becomes zero.1 -

(1 + i)n

1 1 i i

When n gets very large,When n gets very large,

this becomes zero.this becomes zero.

So we’re left with PVIFA =So we’re left with PVIFA =

So, the PV of a perpetuity is very So, the PV of a perpetuity is very simple to find:simple to find:

What should you be willing to pay in What should you be willing to pay in order to receive order to receive $10,000$10,000 annually annually forever, if you require forever, if you require 8%8% per year per year on the investment?on the investment?

PMT PMT

iiPV =PV = ==

$10,000 $10,000

.08.08

= = $125,000$125,000

Ordinary AnnuityOrdinary Annuity vs. vs.

Annuity Due Annuity Due

$1000 $1000 $1000

4 5 6 7 8

Begin Mode vs. End ModeBegin Mode vs. End Mode

1000 1000 10001000 1000 1000

4 5 6 7 8 4 5 6 7 8

1000 1000 10001000 1000 1000

4 5 6 7 8 4 5 6 7 8 year year year 5 6 7

1000 1000 10001000 1000 1000

4 5 6 7 8 4 5 6 7 8 year year year 5 6 7

PVPVinin

ENDENDModeMode

1000 1000 10001000 1000 1000

4 5 6 7 8 4 5 6 7 8 year year year 5 6 7

PVPVinin

ENDENDModeMode

FVFVinin

ENDENDModeMode

1000 1000 10001000 1000 1000

4 5 6 7 8 4 5 6 7 8 year year year 6 7 8

1000 1000 10001000 1000 1000

4 5 6 7 8 4 5 6 7 8 year year year 6 7 8

PVPVinin

BEGINBEGINModeMode

1000 1000 10001000 1000 1000

4 5 6 7 8 4 5 6 7 8 year year year 6 7 8

PVPVinin

BEGINBEGINModeMode

FVFVinin

BEGINBEGINModeMode

Earlier, we examined this Earlier, we examined this “ordinary” annuity:“ordinary” annuity:

Using an interest rate of 8%, we find Using an interest rate of 8%, we find that:that:

The The Future ValueFuture Value (at 3) is (at 3) is $3,246.40.$3,246.40.

The The Present ValuePresent Value (at 0) is (at 0) is $2,577.10.$2,577.10.

0 1 2 3

10001000 10001000 1000 1000

What about this annuity?What about this annuity?

Same 3-year time line,Same 3-year time line, Same 3 $1000 cash flows, butSame 3 $1000 cash flows, but The cash flows occur at the The cash flows occur at the

beginningbeginning of each year, rather of each year, rather than at the than at the endend of each year. of each year.

This is an This is an “annuity due.”“annuity due.”

0 1 2 3

10001000 1000 1000 1000 1000

Future Value - annuity dueFuture Value - annuity due If you invest $1,000 at the beginning of each of the If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at next 3 years at 8%, how much would you have at

the end of year 3? the end of year 3?

0 1 2 3

Mode = BEGIN P/Y = 1Mode = BEGIN P/Y = 1 I = 8I = 8

N = 3N = 3 PMT = -1,000 PMT = -1,000

FV = FV = $3,506.11$3,506.11

0 1 2 3

-1000-1000 -1000 -1000 -1000 -1000

0 1 2 3

-1000-1000 -1000 -1000 -1000 -1000

N = 3N = 3 PMT = -1,000 PMT = -1,000

FV = FV = $3,506.11$3,506.11

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

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

FV = 1,000 (FVIFA FV = 1,000 (FVIFA .08, 3.08, 3 ) (1.08) ) (1.08) (use FVIFA table, or)(use FVIFA table, or)

FV = PMT (1 + i)FV = PMT (1 + i)nn - 1 - 1

ii(1 + i)(1 + i)

FV = PMT (1 + i)FV = PMT (1 + i)nn - 1 - 1

FV = 1,000 (1.08)FV = 1,000 (1.08)33 - 1 = - 1 = $3,506.11$3,506.11

.08 .08

(1 + i)(1 + i)

(1.08)(1.08)

Present Value - annuity duePresent Value - annuity due What is the PV of $1,000 at the beginning of each What is the PV of $1,000 at the beginning of each

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

0 1 2 3

N = 3N = 3 PMT = 1,000 PMT = 1,000

PV = PV = $2,783.26$2,783.26

0 1 2 3

10001000 1000 1000 1000 1000

N = 3N = 3 PMT = 1,000 PMT = 1,000

PV = PV = $2,783.26$2,783.26

0 1 2 3

10001000 1000 1000 1000 1000

Present Value - annuity duePresent Value - annuity due

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

PV = 1,000 (PVIFA PV = 1,000 (PVIFA .08, 3.08, 3 ) (1.08) ) (1.08) (use PVIFA table, or)(use PVIFA table, or)

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

ii(1 + i)(1 + i)

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

PV = 1000 1 - (1.08 )PV = 1000 1 - (1.08 )33 = = $2,783.26$2,783.26

.08.08

(1 + i)(1 + i)

(1.08)(1.08)

Is this an annuity?Is this an annuity? How do we find the PV of a cash flow How do we find the PV of a cash flow

stream when all of the cash flows are stream when all of the cash flows are different? (Use a 10% discount rate).different? (Use a 10% discount rate).

Uneven Cash FlowsUneven Cash Flows

00 11 22 33 44

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

Sorry! There’s no quickie for this one. Sorry! There’s no quickie for this one. We have to discount each cash flow We have to discount each cash flow back separately.back separately.

00 11 22 33 44

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

00 11 22 33 44

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

00 11 22 33 44

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

00 11 22 33 44

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

00 11 22 33 44

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

periodperiod CF CF PV (CF)PV (CF)

00 -10,000 -10,000 -10,000.00-10,000.00

11 2,000 2,000 1,818.181,818.18

22 4,000 4,000 3,305.793,305.79

33 6,000 6,000 4,507.894,507.89

44 7,000 7,000 4,781.094,781.09

PV of Cash Flow Stream: $ 4,412.95PV of Cash Flow Stream: $ 4,412.95

00 11 22 33 44

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

ExampleExample Cash flows from an investment are Cash flows from an investment are

expected to be expected to be $40,000$40,000 per year at the per year at the end of years 4, 5, 6, 7, and 8. If you end of years 4, 5, 6, 7, and 8. If you require a require a 20%20% rate of return, what is rate of return, what is the PV of these cash flows?the PV of these cash flows?

ExampleExample

00 11 22 33 44 55 66 77 88

0 0 0 0 40 40 40 40 40 0 0 0 0 40 40 40 40 40

Cash flows from an investment are Cash flows from an investment are expected to be expected to be $40,000$40,000 per year at the per year at the end of years 4, 5, 6, 7, and 8. If you end of years 4, 5, 6, 7, and 8. If you require a require a 20%20% rate of return, what is rate of return, what is the PV of these cash flows?the PV of these cash flows?

This type of cash flow sequence is This type of cash flow sequence is often called a often called a “deferred annuity.”“deferred annuity.”

00 11 22 33 44 55 66 77 88

0 0 0 0 40 40 40 40 40 0 0 0 0 40 40 40 40 40

How to solve:How to solve:

1) 1) Discount each cash flow back to time Discount each cash flow back to time 0 separately.0 separately.

Or,Or,

00 11 22 33 44 55 66 77 88

0 0 0 0 40 40 40 40 40 0 0 0 0 40 40 40 40 40

2) 2) Find the PV of the annuity:Find the PV of the annuity:

PVPV3:3: End mode; P/YR = 1; I = 20; End mode; P/YR = 1; I = 20; PMT = 40,000; N = 5 PMT = 40,000; N = 5

PVPV33= = $119,624$119,624

00 11 22 33 44 55 66 77 88

0 0 0 0 40 40 40 40 40 0 0 0 0 40 40 40 40 40

119,624119,624

00 11 22 33 44 55 66 77 88

0 0 0 0 40 40 40 40 40 0 0 0 0 40 40 40 40 40

Then discount this single sum back to Then discount this single sum back to time 0.time 0.

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

N = 3; FV = 119,624; N = 3; FV = 119,624;

Solve: PV = $69,226Solve: PV = $69,226

119,624119,624

00 11 22 33 44 55 66 77 88

0 0 0 0 40 40 40 40 40 0 0 0 0 40 40 40 40 40

119,624119,62469,22669,226

00 11 22 33 44 55 66 77 88

0 0 0 0 40 40 40 40 40 0 0 0 0 40 40 40 40 40

119,624119,62469,22669,226

The PV of the cash flow The PV of the cash flow stream is $69,226.stream is $69,226.

00 11 22 33 44 55 66 77 88

0 0 0 0 40 40 40 40 40 0 0 0 0 40 40 40 40 40

ExampleExample

After graduation, you plan to invest After graduation, you plan to invest $400$400 per month per month in the stock market. in the stock market. If you earn If you earn 12%12% per year per year on your on your stocks, how much will you have stocks, how much will you have accumulated when you retire in accumulated when you retire in 30 30 yearsyears??

Retirement ExampleRetirement Example

After graduation, you plan to invest After graduation, you plan to invest $400$400 per month in the stock market. per month in the stock market. If you earn If you earn 12%12% per year on your per year on your stocks, how much will you have stocks, how much will you have accumulated when you retire in 30 accumulated when you retire in 30 years?years?

00 11 22 33 . . . 360. . . 360

400 400 400 400400 400 400 400

00 11 22 33 . . . 360. . . 360

400 400 400 400400 400 400 400

Using your calculator,Using your calculator,

P/YR = 12P/YR = 12

N = 360 N = 360

PMT = -400PMT = -400

I%YR = 12I%YR = 12

FV = $1,397,985.65FV = $1,397,985.65

00 11 22 33 . . . 360. . . 360

400 400 400 400400 400 400 400

Retirement ExampleRetirement Example If you invest $400 at the end of each month for the If you invest $400 at the end of each month for the next 30 years at 12%, how much would you have at next 30 years at 12%, how much would you have at

FV = PMT (FVIFA FV = PMT (FVIFA i, ni, n ) )

FV = 400 (FVIFA FV = 400 (FVIFA .01, 360.01, 360 ) ) (can’t use FVIFA table)(can’t use FVIFA table)

FV = PMT (1 + i)FV = PMT (1 + i)nn - 1 - 1

FV = 400 (1.01)FV = 400 (1.01)360360 - 1 = - 1 = $1,397,985.65$1,397,985.65

.01 .01

If you borrow If you borrow $100,000 at 7%$100,000 at 7% fixed fixed interest for interest for 30 years30 years in order to in order to buy a house, what will be your buy a house, what will be your

monthly house paymentmonthly house payment??

House Payment ExampleHouse Payment Example

If you borrow $100,000 at 7% fixed If you borrow $100,000 at 7% fixed interest for 30 years in order to interest for 30 years in order to buy a house, what will be your buy a house, what will be your

monthly house payment?monthly house payment?

0 1 2 3 . . . 360

? ? ? ?

P/YR = 12P/YR = 12

N = 360N = 360

I%YR = 7I%YR = 7

PV = $100,000PV = $100,000

PMT = -$665.30PMT = -$665.30

00 11 22 33 . . . 360. . . 360

? ? ? ?? ? ? ?

100,000 = PMT (PVIFA 100,000 = PMT (PVIFA .07, 360.07, 360 ) ) (can’t use PVIFA table)(can’t use PVIFA table)

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

100,000 = PMT 1 - (1.005833 )100,000 = PMT 1 - (1.005833 )360360 PMT=$665.30PMT=$665.30

.005833.005833

Team AssignmentTeam Assignment

Upon retirement, your goal is to spend Upon retirement, your goal is to spend 55 years traveling around the world. To years traveling around the world. To travel in style will require travel in style will require $250,000$250,000 per per year at the year at the beginningbeginning of each year. of each year.

If you plan to retire in If you plan to retire in 30 years30 years, what are , what are the equal the equal monthlymonthly payments necessary payments necessary to achieve this goal? to achieve this goal?

The funds in your retirement account will The funds in your retirement account will compound at compound at 10%10% annually. annually.

How much do we need to have by How much do we need to have by the end of year 30 to finance the the end of year 30 to finance the trip?trip?

PVPV3030 = PMT (PVIFA = PMT (PVIFA .10, 5.10, 5) (1.10) =) (1.10) =

= 250,000 (3.7908) (1.10) == 250,000 (3.7908) (1.10) =

= = $1,042,470$1,042,470

2727 2828 2929 3030 3131 3232 3333 3434 3535

250 250 250 250 250 250 250 250 250 250

Mode = BEGINMode = BEGIN

PMT = -$250,000PMT = -$250,000

N = 5N = 5

I%YR = 10I%YR = 10

P/YR = 1P/YR = 1

PV = PV = $1,042,466$1,042,466

2727 2828 2929 3030 3131 3232 3333 3434 3535

250 250 250 250 250 250 250 250 250 250

Now, assuming 10% annual Now, assuming 10% annual compounding, what monthly compounding, what monthly payments will be required for payments will be required for you to have you to have $1,042,466$1,042,466 at the end at the end of year 30?of year 30?

2727 2828 2929 3030 3131 3232 3333 3434 3535

250 250 250 250 250 250 250 250 250 250

1,042,4661,042,466

Mode = ENDMode = END

N = 360N = 360

I%YR = 10I%YR = 10

P/YR = 12P/YR = 12

FV = $1,042,466FV = $1,042,466

PMT = PMT = -$461.17-$461.17

2727 2828 2929 3030 3131 3232 3333 3434 3535

250 250 250 250 250 250 250 250 250 250

1,042,4661,042,466

So, you would have to place So, you would have to place $461.17$461.17 in your retirement account, which in your retirement account, which earns 10% annually, at the end of earns 10% annually, at the end of each of the next 360 months to each of the next 360 months to finance the 5-year world tour.finance the 5-year world tour.

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