8.Time Value of Money 2

Chapter 5 - The Time Chapter 5 - The Time Value of MoneyValue of Money

2005, Pearson Prentice Hall

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 5 years?much would you have in the account after 5 years?

Mathematical Solution:Mathematical Solution:

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

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



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 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?



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 - 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

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

Present ValuePresent Value


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

PV = 100 (PVIF PV = 100 (PVIF .06, 5.06, 5 ) (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)5 5 = = $74.73$74.73

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

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

00 5 5

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

The Time Value of MoneyThe Time Value of Money

Compounding and DiscountingCompounding and Discounting

Cash Flow StreamsCash Flow Streams

0 1 2 3 4


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

ii

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

.08 .08

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

would you have after 3 years?would you have after 3 years?


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)

11PV = PMT 1 - (1 + i)PV = PMT 1 - (1 + i)nn

ii

11PV = 1000 1 - (1.08 )PV = 1000 1 - (1.08 )33 = = $2,577.10$2,577.10

.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%?

Ordinary AnnuityOrdinary Annuity vs. vs.

Annuity Due Annuity Due

$1000 $1000 $1000$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:


0 1 2 3

10001000 10001000 1000 1000


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

0 1 2 3

10001000 10001000 1000 1000



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

0 1 2 3

10001000 10001000 1000 1000



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

0 1 2 3

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?

Calculator Solution:Calculator Solution:

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

ii

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)





11

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

ii(1 + i)(1 + i)





11

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

ii

11

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 Is this an annuityannuity?? 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 1 1 2 2 3 3 4 4

-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 1 1 2 2 3 3 4 4

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



00 1 1 2 2 3 3 4 4

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



00 1 1 2 2 3 3 4 4

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



00 1 1 2 2 3 3 4 4

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



00 1 1 2 2 3 3 4 4

-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 1 1 2 2 3 3 4 4

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

Annual Percentage Yield (APY)Annual Percentage Yield (APY)

Which is the better loan:Which is the better loan: 8%8% compounded compounded annuallyannually, or, or 7.85%7.85% compounded compounded quarterlyquarterly?? We can’t compare these nominal (quoted) We can’t compare these nominal (quoted)

interest rates, because they don’t include the interest rates, because they don’t include the same number of compounding periods per same number of compounding periods per year!year!

We need to calculate the APY.We need to calculate the APY.



APY = APY = (( 1 + 1 + ) ) m m - 1- 1quoted ratequoted ratemm


Find the APY for the quarterly loan:Find the APY for the quarterly loan:





APY = APY = (( 1 + 1 + ) ) 4 4 - 1- 1.0785.078544




APY = APY = (( 1 + 1 + ) ) 4 4 - 1- 1

APY = .0808, or 8.08%APY = .0808, or 8.08%

.0785.078544



The quarterly loan is more expensive than The quarterly loan is more expensive than the 8% loan with annual compounding!the 8% loan with annual compounding!


APY = APY = (( 1 + 1 + ) ) 4 4 - 1- 1

APY = .0808, or 8.08%APY = .0808, or 8.08%

.0785.078544

Practice ProblemsPractice Problems

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 0 0 0 0 4040 4040 4040 4040 4040

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 annuitydeferred annuity.”.”

00 11 22 33 44 55 66 77 88

$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040

How to solve:How to solve:

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

00 11 22 33 44 55 66 77 88

$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040



00 11 22 33 44 55 66 77 88

$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040



00 11 22 33 44 55 66 77 88

$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040



00 11 22 33 44 55 66 77 88

$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040



00 11 22 33 44 55 66 77 88

$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040



00 11 22 33 44 55 66 77 88

$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040



Or,Or,

00 11 22 33 44 55 66 77 88

$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040

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

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

PV = PV = $119,624$119,624

00 11 22 33 44 55 66 77 88

$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040

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 0 0 0 0 4040 4040 4040 4040 4040

119,624119,624

00 11 22 33 44 55 66 77 88

$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040

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

119,624119,624

00 11 22 33 44 55 66 77 88

$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040

69,22669,226

00 11 22 33 44 55 66 77 88

$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040

119,624119,624

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

69,22669,226

00 11 22 33 44 55 66 77 88

$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040

119,624119,624

Retirement ExampleRetirement Example

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 3030 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 = FV = $1,397,985.65$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 = 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

ii






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

ii

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$100,000 at at 7%7% fixed fixed interest for interest for 3030 years 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 If you borrow $100,000$100,000 at at 7%7% fixed fixed interest for interest for 3030 years in order to 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 = PMT = -$665.30-$665.30

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

? ? ? ?? ? ? ?





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



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

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





11

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

ii





11

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

ii

11

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 30 yearsyears, what are , what are the equal the equal monthlymonthly payments necessary payments necessary to achieve this goal? The funds in your to achieve this goal? The funds in your retirement account will compound at retirement account will 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 you payments will be required for you to have to have $1,042,466$1,042,466 at the end of at the end of year 30?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 in your retirement account, which earns your retirement account, which earns 10% annually, at the end of each of the 10% annually, at the end of each of the next 360 months to finance the 5-year next 360 months to finance the 5-year world tour.world tour.

Practice Practice

A Client has $202.971,39 in an account that A Client has $202.971,39 in an account that earns 8% per year, compounded monthly. The earns 8% per year, compounded monthly. The client’s 35client’s 35thth birthday was yesterday and she will birthday was yesterday and she will retire when the account value is $1 million.retire when the account value is $1 million.

A.A.At what age can she retire if she puts no more At what age can she retire if she puts no more money in the account?money in the account?

B.B.At what age can she retire if she puts $250 per At what age can she retire if she puts $250 per month into the account every month, beginning month into the account every month, beginning one month from now?one month from now?

AnswerAnswer

A.A. PV = $ 202.971,39 PV = $ 202.971,39

I/Y = 8%/12 = 0.6667%. A = 0. FV = $1.000.000,- I/Y = 8%/12 = 0.6667%. A = 0. FV = $1.000.000,-

N = 240 months = 20 years.N = 240 months = 20 years.

She will be 55 years old.She will be 55 years old.

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

$ 1.000.000,- = $202.971,39 (1+0.6667%)$ 1.000.000,- = $202.971,39 (1+0.6667%)nn

(1.000.000,- / 202.971,39) = (1.006667)(1.000.000,- / 202.971,39) = (1.006667) n n

ln 4.9268 = n x ln (1.006667)ln 4.9268 = n x ln (1.006667)

1.5947 = n x 0.006645 1.5947 = n x 0.006645 n = 240 bulan n = 240 bulan

B. A = $250/monthB. A = $250/month

N = 18.335 yearsN = 18.335 years

She will be 53 years old.She will be 53 years old.

Documents

8.Time Value of Money 2