The Time The Time Value of Value of MoneyMoney

The Time Value of MoneyThe Time Value of Money

Compounding and Compounding and Discounting Single Discounting Single

SumsSums

We know that receiving $1 today is We know that receiving $1 today is worth worth moremore than $1 in the future. than $1 in the future. This is dueThis is due toto opportunity costsopportunity costs..

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

Today Future

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

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

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

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

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

Today

?

Future

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

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

Translate $1 in the future into its equivalent Translate $1 in the future into its equivalent todaytoday (discounting)(discounting)..

Today

?

Future

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

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

Translate $1 in the future into its equivalent Translate $1 in the future into its equivalent todaytoday (discounting)(discounting)..

?

Today Future

Today

?

Future

Compound Compound Interest Interest and and

Future ValueFuture Value

Future Value - single sumsFuture Value - single sums

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

after 1 year?after 1 year?

Future Value - single sumsFuture Value - single sums

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

after 1 year?after 1 year?

0 1

PV =PV = FV = FV =

Future Value - single sumsFuture Value - single sums

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

after 1 year?after 1 year?

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 =

Future Value - single sumsFuture Value - single sums

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

after 1 year?after 1 year?

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 = 106106

Future Value - single sumsFuture Value - single sums

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

after 1 year?after 1 year?

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, ) (use FVIF table,

or)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

Future Value - single sumsFuture Value - single sums

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

after 5 years?after 5 years?

Future Value - single sumsFuture Value - single sums

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

after 5 years?after 5 years?

00 5 5

PV =PV = FV = FV =

Future Value - single sumsFuture Value - single sums

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

after 5 years?after 5 years?

Calculator Solution:Calculator Solution:

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 =

Future Value - single sumsFuture Value - single sums

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

after 5 years?after 5 years?

Calculator Solution:Calculator Solution:

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

Future Value - single sumsFuture Value - single sums

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

after 5 years?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, ) (use FVIF table, or)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% If you deposit $100 in an account earning 6%

with with quarterly compoundingquarterly compounding, how much , how much would you have in the account after 5 years?would you have in the account after 5 years?

0 ?

PV =PV = FV = FV =

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

with with quarterly compoundingquarterly compounding, how much , how much would you have in the account after 5 years?would you have in the account after 5 years?

Calculator Solution:Calculator Solution:

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 =

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

with with quarterly compoundingquarterly compounding, how much , how much would you have in the account after 5 years?would you have in the account after 5 years?

Calculator Solution:Calculator Solution:

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

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

with with quarterly compoundingquarterly compounding, how much , how much would you have in the account after 5 years?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 .015, 20.015, 20 ) ) (can’t use FVIF (can’t use FVIF table)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% If you deposit $100 in an account earning 6%

with with quarterly compoundingquarterly compounding, how much , how much would you have in the account after 5 years?would you have in the account after 5 years?

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

with with monthly compoundingmonthly compounding, how much would , how much would you have in the account after 5 years?you have in the account after 5 years?

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

with with monthly compoundingmonthly compounding, how much would , how much would you have in the account after 5 years?you have in the account after 5 years?

0 ?

PV =PV = FV = FV =

Calculator Solution:Calculator Solution:

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 =

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

with with monthly compoundingmonthly compounding, how much would , how much would you have in the account after 5 years?you have in the account after 5 years?

Calculator Solution:Calculator Solution:

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

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

with with monthly compoundingmonthly compounding, how much would , how much would you have in the account after 5 years?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 .005, 60.005, 60 ) ) (can’t use FVIF (can’t use FVIF table)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% If you deposit $100 in an account earning 6%

with with monthly compoundingmonthly compounding, how much would , how much would you have in the account after 5 years?you have in the account after 5 years?

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?

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?

0 ?

PV =PV = FV = FV =

Mathematical Solution:Mathematical Solution:

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 =

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 compounding, after 100 years?continuous compounding, after 100 years?

00 100 100

PV = -1000PV = -1000 FV = FV = $2.98m$2.98m

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 compounding, after 100 years?continuous compounding, after 100 years?

Mathematical Solution:Mathematical Solution:

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 receive $100 one year from now, what is If you receive $100 one year from now, what is the PV of that $100 if your opportunity cost is the PV of that $100 if your opportunity cost is

6%?6%?

0 ?

PV =PV = FV = FV =

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

6%?6%?

Calculator Solution:Calculator Solution:

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

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

6%?6%?

Calculator Solution:Calculator Solution:

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

N = 1 N = 1 FV = FV = 100100

PV = PV = -94.34-94.34

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

00 1 1

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

6%?6%?

Mathematical Solution:Mathematical Solution:

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

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

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

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

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

00 1 1

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

6%?6%?

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

is 6%?is 6%?

0 ?

PV =PV = FV = FV =

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

is 6%?is 6%?

Calculator Solution:Calculator Solution:

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

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

is 6%?is 6%?

Calculator Solution:Calculator Solution:

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

N = 5 N = 5 FV = FV = 100100

PV = PV = -74.73-74.73

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

is 6%?is 6%?

00 5 5

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

Mathematical Solution:Mathematical Solution:

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

PV = 100 (PVIF PV = 100 (PVIF .06, 5.06, 5 ) (use PVIF table, ) (use PVIF table, or)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 If you receive $100 five years from now, what is the PV of that $100 if your opportunity cost is the PV of that $100 if your opportunity cost

is 6%?is 6%?

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 What is the PV of $1,000 to be received 15 years from now if your opportunity cost is years from now if your opportunity cost is

7%?7%?

00 15 15

PV = PV = FV = FV =

Present Value - single sumsPresent Value - single sumsWhat is the PV of $1,000 to be received 15 What is the PV of $1,000 to be received 15 years from now if your opportunity cost is years from now if your opportunity cost is

7%?7%?

Calculator Solution:Calculator Solution:

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

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

PV = PV = -362.45-362.45

Present Value - single sumsPresent Value - single sumsWhat is the PV of $1,000 to be received 15 What is the PV of $1,000 to be received 15 years from now if your opportunity cost is years from now if your opportunity cost is

7%?7%?

00 15 15

PV = PV = FV = 1000 FV = 1000

Calculator Solution:Calculator Solution:

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

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

PV = PV = -362.45-362.45

Present Value - single sumsPresent Value - single sumsWhat is the PV of $1,000 to be received 15 What is the PV of $1,000 to be received 15 years from now if your opportunity cost is years from now if your opportunity cost is

7%?7%?

00 15 15

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

Mathematical Solution:Mathematical Solution:

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

PV = 100 (PVIF PV = 100 (PVIF .07, 15.07, 15 ) (use PVIF table, ) (use PVIF table, or)or)

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

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

Present Value - single sumsPresent Value - single sumsWhat is the PV of $1,000 to be received 15 What is the PV of $1,000 to be received 15 years from now if your opportunity cost is years from now if your opportunity cost is

7%?7%?

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 If you sold land for $11,933 that you bought 5 years ago for $5,000, what is your annual rate years ago for $5,000, what is your annual rate

of return?of return?

00 5 5

PV = PV = FV = FV =

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

of return?of return?

Calculator Solution:Calculator Solution:

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 = -5000PV = -5000 FV = 11,933 FV = 11,933

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

of return?of return?

Mathematical Solution:Mathematical Solution:

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 = i = .19.19

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

of return?of return?

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

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

00

PV = PV = FV = FV =

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

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

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

00 ? ?

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

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

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

Mathematical Solution:Mathematical Solution:

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

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

given three variables and you will given three variables and you will solve for the fourth variable.solve for the fourth variable.

Keeping this in mind makes solving Keeping this in mind makes solving time value problems much easier!time value problems much easier!

The Time Value of MoneyThe Time Value of Money

Compounding and Compounding and DiscountingDiscounting

Cash Flow StreamsCash Flow Streams

0 1 2 3 4

AnnuitiesAnnuities Annuity:Annuity: a sequence of a sequence of equalequal

cash flowscash flows, occurring at the , occurring at the endend of each period.of each period.

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

0 1 2 3 4

AnnuitiesAnnuities

Examples of Annuities:Examples of Annuities:

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

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

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

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

0 1 2 3

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

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

Calculator Solution:Calculator Solution:

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

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

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

0 1 2 3

10001000 10001000 1000 1000

Calculator Solution:Calculator Solution:

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

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

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

0 1 2 3

10001000 10001000 1000 1000

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

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

Mathematical Solution:Mathematical Solution:

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

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

Mathematical Solution:Mathematical Solution:

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

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

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

Mathematical Solution:Mathematical Solution:

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

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

or)or)

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

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

Mathematical Solution:Mathematical Solution:

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

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

or)or)

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

ii

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

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

Mathematical Solution:Mathematical Solution:

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

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

or)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 If you invest $1,000 each year at 8%, how

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

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

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

0 1 2 3

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

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

Calculator Solution:Calculator Solution:

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

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

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

Calculator Solution:Calculator Solution:

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

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

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

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

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

Mathematical Solution:Mathematical Solution:

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

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

Mathematical Solution:Mathematical Solution:

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

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

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

Mathematical Solution:Mathematical Solution:

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)

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

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

Mathematical Solution:Mathematical Solution:

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

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

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

Mathematical Solution:Mathematical Solution:

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 What is the PV of $1,000 at the end of each of

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

Other Cash Flow PatternsOther Cash Flow Patterns

0 1 2 3

The Time Value of Money

PerpetuitiesPerpetuities

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

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

Present Value of a Present Value of a PerpetuityPerpetuity

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

Present Value of a Present Value of a PerpetuityPerpetuity

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

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

Mathematically, Mathematically,

Mathematically, Mathematically,

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

Mathematically, Mathematically,

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

(1 + i)(1 + i)nn

ii

Mathematically, Mathematically,

(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,

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

1 -

1

(1 + i)n

i

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

this becomes this becomes zero.zero.1 -

1

(1 + i)n

i

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

this becomes this becomes zero.zero.

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

1 i

1 - 1

(1 + i)n

i

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

Present Value of a Perpetuity

PMT i

PV =

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

Present Value of a Perpetuity

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

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

PMT $10,000PMT $10,000 i .08 i .08

PV = =PV = =

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

PMT $10,000PMT $10,000 i .08 i .08

= $125,000= $125,000

PV = =PV = =

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

Begin Mode vs. End ModeBegin Mode vs. End Mode

1000 1000 10001000 1000 1000

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

Begin Mode vs. End ModeBegin Mode vs. End Mode

1000 1000 10001000 1000 1000

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

PVPVinin

ENDENDModeMode

Begin Mode vs. End ModeBegin Mode vs. End Mode

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

Begin Mode vs. End ModeBegin Mode vs. End Mode

1000 1000 10001000 1000 1000

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

PVPVinin

BEGINBEGINModeMode

Begin Mode vs. End ModeBegin Mode vs. End Mode

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:

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

0 1 2 3

10001000 10001000 1000 1000

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

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

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

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

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

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

Using an interest rate of 8%, we Using an interest rate of 8%, we find that:find 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, Same 3 $1000 cash flows,

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

you have at the end of year 3? you have at 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

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

you have at the end of year 3? you have at the end of year 3?

0 1 2 3

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

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

you have at the end of year 3? you have at 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

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

you have at the end of year 3? you have at the end of year 3?

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

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

you have at the end of year 3? you have at the end of year 3?

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

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

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

you have at the end of year 3? you have at the end of year 3?

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

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

you have at the end of year 3? you have at the end of year 3?

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

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

you have at the end of year 3? you have at the end of year 3?

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

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 What is the PV of $1,000 at the beginning of each of the next 3 years, if your opportunity each of the next 3 years, if your opportunity

cost is 8%? cost is 8%?

0 1 2 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

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

0 1 2 3

10001000 1000 1000 1000 1000

Present Value - annuity duePresent Value - annuity due What is the PV of $1,000 at the beginning of What is the PV of $1,000 at the beginning of each of the next 3 years, if your opportunity each of the next 3 years, if your opportunity

cost is 8%? cost is 8%?

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

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

0 1 2 3

10001000 1000 1000 1000 1000

Present Value - annuity duePresent Value - annuity due What is the PV of $1,000 at the beginning of What is the PV of $1,000 at the beginning of each of the next 3 years, if your opportunity each of the next 3 years, if your opportunity

cost is 8%? cost is 8%?

Present Value - annuity duePresent Value - annuity due

Mathematical Solution:Mathematical Solution:

Present Value - annuity duePresent Value - annuity due

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

Present Value - annuity duePresent Value - annuity due

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

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

Present Value - annuity duePresent Value - annuity due

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

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)

Present Value - annuity duePresent Value - annuity due

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

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)

Present Value - annuity duePresent Value - annuity due

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

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

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 How do we find the PV of a cash

flow stream when all of the cash flow stream when all of the cash flows are different? (Use a 10% flows are different? (Use a 10% discount rate.)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 Sorry! There’s no quickie for this one. We have to discount each one. We have to discount each cash flow back separately.cash flow 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

Uneven Cash FlowsUneven Cash Flows

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

Uneven Cash FlowsUneven Cash Flows

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

Uneven Cash FlowsUneven Cash Flows

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

Uneven Cash FlowsUneven Cash Flows

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

Uneven Cash FlowsUneven Cash Flows

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, or7.85%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 interest rates, because they don’t include the same number of compounding periods the same number of compounding periods per year!per year!

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

.0785.078544

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

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

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

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

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

This type of cash flow This type of cash flow sequence is often called a sequence is 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

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

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

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

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

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

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.

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 Then discount this single sum back to time 0.back to 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 After graduation, you plan to invest invest $400$400 per month per month in the in the stock market. If you earn stock market. If you earn 12%12% per per yearyear on your stocks, how much on your stocks, how much will you have accumulated when will you have accumulated when you retire in you retire in 3030 years years??

Retirement ExampleRetirement Example

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

for the next 30 years at 12%, how much would for the next 30 years at 12%, how much would you have at the end of year 30? you have at the end of year 30?

Retirement ExampleRetirement Example If you invest $400 at the end of each month If you invest $400 at the end of each month

for the next 30 years at 12%, how much would for the next 30 years at 12%, how much would you have at the end of year 30? you have at the end of year 30?

Mathematical Solution:Mathematical Solution:

Retirement ExampleRetirement Example If you invest $400 at the end of each month If you invest $400 at the end of each month

for the next 30 years at 12%, how much would for the next 30 years at 12%, how much would you have at the end of year 30? you have at the end of year 30?

Mathematical Solution:Mathematical Solution:

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

Retirement ExampleRetirement Example If you invest $400 at the end of each month If you invest $400 at the end of each month

for the next 30 years at 12%, how much would for the next 30 years at 12%, how much would you have at the end of year 30? you have at the end of year 30?

Mathematical Solution:Mathematical Solution:

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)

Retirement ExampleRetirement Example If you invest $400 at the end of each month If you invest $400 at the end of each month

for the next 30 years at 12%, how much would for the next 30 years at 12%, how much would you have at the end of year 30? you have at the end of year 30?

Mathematical Solution:Mathematical Solution:

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

Retirement ExampleRetirement Example If you invest $400 at the end of each month If you invest $400 at the end of each month

for the next 30 years at 12%, how much would for the next 30 years at 12%, how much would you have at the end of year 30? you have at the end of year 30?

Mathematical Solution:Mathematical Solution:

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 = 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 interest for fixed interest for 3030 years years in in

order to buy a house, what will order to buy a house, what will be your be your monthly house monthly house

paymentpayment??

House Payment ExampleHouse Payment Example

House Payment ExampleHouse Payment Example

If you borrow If you borrow $100,000$100,000 at at 7%7% fixed interest for fixed interest for 3030 years in years in

order to buy a house, what will order to buy a house, what will be your monthly house be your monthly house

payment?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 = PMT = -$665.30-$665.30

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

? ? ? ?? ? ? ?

House Payment ExampleHouse Payment Example

Mathematical Solution:Mathematical Solution:

House Payment ExampleHouse Payment Example

Mathematical Solution:Mathematical Solution:

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

House Payment ExampleHouse Payment Example

Mathematical Solution:Mathematical Solution:

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 (can’t use PVIFA

table)table)

House Payment ExampleHouse Payment Example

Mathematical Solution:Mathematical Solution:

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 (can’t use PVIFA

table)table)

11

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

ii

House Payment ExampleHouse Payment Example

Mathematical Solution:Mathematical Solution:

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 (can’t use PVIFA

table)table)

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 Upon retirement, your goal is to spend spend 55 years traveling around the years traveling around the world. To travel in style will require world. To travel in style will require $250,000$250,000 per year at the per 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 , what are the equal are the equal monthlymonthly payments payments necessary to achieve this goal? necessary to achieve this goal? The funds in your retirement The funds in your retirement account will compound at account will compound at 10%10% annually.annually.

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

Using your calculator,Using your calculator,

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 at the end of year 30?end 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

Using your calculator,Using your calculator,

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, in your retirement account, which earns 10% annually, at the which earns 10% annually, at the end of each of the next 360 months end of each of the next 360 months to finance the 5-year world tour.to finance the 5-year world tour.