fm10e ch05

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

2005, Pearson Prentice Hall

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 moremore than $1 in the future. This is due 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 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 futureTranslate $1 today into its equivalent in the future (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 futureTranslate $1 today into its equivalent in the future (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 futureTranslate $1 today into its equivalent in the future (compounding)(compounding)..

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

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

?

Today Future

Today

?

Future

Compound Interest Compound Interest and Future Valueand Future 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?

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 =

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?

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%, 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?

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%, 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?

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

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?

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?

00 5 5

PV =PV = FV = FV =

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?

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%, 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?

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%, 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

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 =

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?

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% 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?

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% 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?

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 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?

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?

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?

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% 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?

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% 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?

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 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?

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 compoundingcontinuous compounding, after 100 years?, 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 compoundingcontinuous compounding, after 100 years?, 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 the If you receive $100 one year 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%?

0 ?

PV =PV = FV = FV =

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

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 the If you receive $100 one year 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%?

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 the If you receive $100 one year 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%?

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, 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

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 the If you receive $100 one year 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%?

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%?

0 ?

PV =PV = FV = FV =

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%?

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 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%?

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 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

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, 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

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%?

00 15 15

PV = PV = FV = FV =

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%?

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 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%?

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 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%?

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, or)) (use PVIF table, 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 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%?

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?

00 5 5

PV = PV = FV = FV =

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?

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 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?

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 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?

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

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

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?

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 given When doing problems, you will be given

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

Keeping this in mind makes solving time Keeping this in mind makes solving 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:Annuity: a sequence of a sequence of equalequal cash cash

flowsflows, occurring at the , occurring at the endend of each of each period.period.

Annuity:Annuity: a sequence of a sequence of equalequal cash cash flows, occurring at the end of each flows, occurring at the end of each period.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 coupon receive equal semi-annual coupon interest payments over the life of interest payments over the life of the bond.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 coupon receive equal semi-annual coupon interest payments over the life of interest payments over the life of the bond.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 much If you invest $1,000 each year at 8%, how much

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

0 1 2 3

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?

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

would you have after 3 years?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 much If you invest $1,000 each year at 8%, how much

would you have after 3 years?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 much If you invest $1,000 each year at 8%, how much

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

Mathematical Solution:Mathematical Solution:

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?

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

would you have after 3 years?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, or)(use FVIFA table, or)

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?

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, or)(use FVIFA table, 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 much If you invest $1,000 each year at 8%, how much

would you have after 3 years?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, 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?

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

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%?

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 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%?

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 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%?

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%?

Mathematical Solution:Mathematical Solution:

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%?

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 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%?

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 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%?

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 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%?

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 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%?

Other Cash Flow PatternsOther Cash Flow Patterns

0 1 2 3

The Time Value of Money

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 Present Value of a PerpetuityPerpetuity

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:

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 following we think of the following relationship: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 zero.this becomes zero.1 -

1

(1 + i)n

i

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

this becomes zero.this becomes zero.

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

1 i

1 - 1

(1 + i)n

i

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

Present Value of a Perpetuity

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:

Present Value of a Perpetuity

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

PV = =PV = =

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 $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, 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

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

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

0 1 2 3

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

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

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

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 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?

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:

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?

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)

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?

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)

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?

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)

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?

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

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

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 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%?

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 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%?

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

Present Value - annuity duePresent Value - annuity due

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:

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

Uneven Cash FlowsUneven Cash Flows

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

Uneven Cash FlowsUneven Cash Flows

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

Uneven Cash FlowsUneven Cash Flows

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

Uneven Cash FlowsUneven Cash Flows

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

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, 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.

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

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 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

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

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?

Mathematical Solution:Mathematical Solution:

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?

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 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?

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 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?

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 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?

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 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

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

? ? ? ?

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 table)(can’t use PVIFA 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 table)(can’t use PVIFA 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 table)(can’t use PVIFA 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 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

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 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

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 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.