The Time Value of Money Compounding and Discounting Single Sums

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

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?

0 1

PV =PV = FV = FV =



Calculator Solution:Calculator Solution:

I = 6I = 6

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

FV = FV = $106$106

00 1 1

PV = 100PV = 100 FV = FV =




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 =




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 =




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?

0 ?

PV =PV = FV = FV =


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

N = 20 N = 20 PV = PV = 100100

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 = 100100

FV = FV = $134.68$134.68

00 20 20

PV = 100PV = 100 FV = FV = 134.134.6868





FV = 100 (FVIF FV = 100 (FVIF .06, 20.06, 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


0 ?

PV =PV = FV = FV =


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

N = 60 N = 60 PV = PV = 100100

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 = 100100

FV = FV = $134.89$134.89

00 60 60

PV = 100PV = 100 FV = FV = 134.134.8989





FV = 100 (FVIF FV = 100 (FVIF .06, 60.06, 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



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%?

0 ?

PV =PV = FV = FV =


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

N = 1 N = 1 FV = FV = 100100

PV = PV = 94.3494.34

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 = 94.3494.34

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%?

0 ?

PV =PV = FV = FV =


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

N = 5 N = 5 FV = FV = 100100

PV = PV = 74.7374.73

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 = 74.7374.73




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%?

0 ?

PV =PV = FV = FV =


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

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

PV = PV = 362.45362.45

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 = 362.45362.45

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?

0 ?

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?

00

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

ii





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


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)






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

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

ii


(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

ii

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


this becomes zero.this becomes zero.1 -

1

(1 + i)n

i

1 - 1

(1 + i)n

i

1 1 i i


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:


PMT i

PV =

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?

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

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).

Cash FlowsCash 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

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


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

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

the end of year 30? the end of year 30?


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 = 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 = 400 (FVIFA FV = 400 (FVIFA .01, 360.01, 360 ) )

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 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

? ? ? ?

Using your calculator,Using your calculator,

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 ) )





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

Documents

The Time Value of Money Compounding and Discounting Single Sums