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The Time Value of Money The Time Value of Money Compounding and Compounding and Discounting Single Sums Discounting Single Sums

Session2 Time Value Of Money (2)

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Page 1: Session2  Time Value Of Money (2)

The Time Value of MoneyThe Time Value of Money

Compounding and Compounding and Discounting Single SumsDiscounting Single Sums

Page 2: Session2  Time Value Of Money (2)

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

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

Today Future

Page 3: Session2  Time Value Of Money (2)

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

?

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

Today Future

Page 4: Session2  Time Value Of Money (2)

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

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

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

?

?

Today Future

Today Future

Page 5: Session2  Time Value Of Money (2)

Future ValueFuture Value

Page 6: Session2  Time Value Of Money (2)

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 =

Page 7: Session2  Time Value Of Money (2)

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 =

Page 8: Session2  Time Value Of Money (2)

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

Page 9: Session2  Time Value Of Money (2)

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

Page 10: Session2  Time Value Of Money (2)

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 =

Page 11: Session2  Time Value Of Money (2)

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 =

Page 12: Session2  Time Value Of Money (2)

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

Page 13: Session2  Time Value Of Money (2)

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

Page 14: Session2  Time Value Of Money (2)

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 =

Page 15: Session2  Time Value Of Money (2)

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?

Page 16: Session2  Time Value Of Money (2)

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?

Page 17: Session2  Time Value Of Money (2)

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?

Page 18: Session2  Time Value Of Money (2)

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 =

Page 19: Session2  Time Value Of Money (2)

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?

Page 20: Session2  Time Value Of Money (2)

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?

Page 21: Session2  Time Value Of Money (2)

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?

Page 22: Session2  Time Value Of Money (2)

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 =

Page 23: Session2  Time Value Of Money (2)

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?

Page 24: Session2  Time Value Of Money (2)

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?

$2.98m$2.98m

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

Page 25: Session2  Time Value Of Money (2)

Present ValuePresent Value

Page 26: Session2  Time Value Of Money (2)

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

0 ?

PV =PV = FV = FV =

Page 27: Session2  Time Value Of Money (2)

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

Page 28: Session2  Time Value Of Money (2)

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

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

Page 29: Session2  Time Value Of Money (2)

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

00 1 1

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

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

Page 30: Session2  Time Value Of Money (2)

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

0 ?

PV =PV = FV = FV =

Page 31: Session2  Time Value Of Money (2)

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

Page 32: Session2  Time Value Of Money (2)

00 5 5

PV = PV = FV = 100 FV = 100

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

-74.-74.7373

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

Page 33: Session2  Time Value Of Money (2)

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

00 5 5

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

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

Page 34: Session2  Time Value Of Money (2)

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

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

0 ?

PV =PV = FV = FV =

Page 35: Session2  Time Value Of Money (2)

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

00 15 15

PV = PV = FV = 1000 FV = 1000

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%?

Page 36: Session2  Time Value Of Money (2)

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

00 15 15

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

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%?

Page 37: Session2  Time Value Of Money (2)

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

00 15 15

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

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%?

Page 38: Session2  Time Value Of Money (2)

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

0 ?

PV =PV = FV = FV =

Page 39: Session2  Time Value Of Money (2)

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 = -5,000PV = -5,000 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?

Page 40: Session2  Time Value Of Money (2)

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 = .19i = .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?

Page 41: Session2  Time Value Of Money (2)

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 =

Page 42: Session2  Time Value Of Money (2)

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

Page 43: Session2  Time Value Of Money (2)

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

Page 44: Session2  Time Value Of Money (2)

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

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

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

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

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

Page 45: Session2  Time Value Of Money (2)

The Time Value of MoneyThe Time Value of Money

Compounding and DiscountingCompounding and Discounting

Cash Flow StreamsCash Flow Streams

0 1 2 3 4

Page 46: Session2  Time Value Of Money (2)

AnnuitiesAnnuities

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

Page 47: Session2  Time Value Of Money (2)

AnnuitiesAnnuities

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

0 1 2 3 4

Page 48: Session2  Time Value Of Money (2)

Examples of Annuities:Examples of Annuities:

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

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

Page 49: Session2  Time Value Of Money (2)

Examples of Annuities:Examples of Annuities:

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

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

Page 50: Session2  Time Value Of Money (2)

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

0 1 2 3

Page 51: Session2  Time Value Of Money (2)

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

0 1 2 3

10001000 10001000 1000 1000

Page 52: Session2  Time Value Of Money (2)

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

0 1 2 3

10001000 10001000 1000 1000

Page 53: Session2  Time Value Of Money (2)

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

Page 54: Session2  Time Value Of Money (2)

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

Page 55: Session2  Time Value Of Money (2)

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

Page 56: Session2  Time Value Of Money (2)

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

Page 57: Session2  Time Value Of Money (2)

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%?

Page 58: Session2  Time Value Of Money (2)

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%?

Page 59: Session2  Time Value Of Money (2)

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%?

Page 60: Session2  Time Value Of Money (2)

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%?

Page 61: Session2  Time Value Of Money (2)

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

11

PV = 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%?

Page 62: Session2  Time Value Of Money (2)

Other Cash Flow PatternsOther Cash Flow Patterns

0 1 2 3

Page 63: Session2  Time Value Of Money (2)

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..

Page 64: Session2  Time Value Of Money (2)

Present Value of a PerpetuityPresent Value of a Perpetuity

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

Page 65: Session2  Time Value Of Money (2)

Present Value of a PerpetuityPresent Value of a Perpetuity

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 ) )

Page 66: Session2  Time Value Of Money (2)

Mathematically, Mathematically,

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

Page 67: Session2  Time Value Of Money (2)

Mathematically, Mathematically,

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

(1 + i)(1 + i)nn

ii

Page 68: Session2  Time Value Of Money (2)

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

Page 69: Session2  Time Value Of Money (2)

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

Page 70: Session2  Time Value Of Money (2)

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

this becomes zero.this becomes zero.1 -

1

(1 + i)n

i

Page 71: Session2  Time Value Of Money (2)

1 - 1

(1 + i)n

i

1 1 i i

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

this becomes zero.this becomes zero.

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

Page 72: Session2  Time Value Of Money (2)

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 PerpetuityPresent Value of a Perpetuity

Page 73: Session2  Time Value Of Money (2)

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 PerpetuityPresent Value of a Perpetuity

Page 74: Session2  Time Value Of Money (2)

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?

Page 75: Session2  Time Value Of Money (2)

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

PMT PMT

iiPV =PV = ==

$10,000 $10,000

.08.08

= = $125,000$125,000

Page 76: Session2  Time Value Of Money (2)

Ordinary AnnuityOrdinary Annuity vs. vs.

Annuity Due Annuity Due

$1000 $1000 $1000

4 5 6 7 8

Page 77: Session2  Time Value Of Money (2)

Begin Mode vs. End ModeBegin Mode vs. End Mode

1000 1000 10001000 1000 1000

4 5 6 7 8 4 5 6 7 8

Page 78: Session2  Time Value Of Money (2)

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

Page 79: Session2  Time Value Of Money (2)

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

Page 80: Session2  Time Value Of Money (2)

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

Page 81: Session2  Time Value Of Money (2)

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

Page 82: Session2  Time Value Of Money (2)

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

Page 83: Session2  Time Value Of Money (2)

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

Page 84: Session2  Time Value Of Money (2)

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

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

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

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

0 1 2 3

10001000 10001000 1000 1000

Page 85: Session2  Time Value Of Money (2)

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

Page 86: Session2  Time Value Of Money (2)

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

Page 87: Session2  Time Value Of Money (2)

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?

Page 88: Session2  Time Value Of Money (2)

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

Page 89: Session2  Time Value Of Money (2)

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)

Page 90: Session2  Time Value Of Money (2)

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)

Page 91: Session2  Time Value Of Money (2)

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)

Page 92: Session2  Time Value Of Money (2)

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

Page 93: Session2  Time Value Of Money (2)

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%?

Page 94: Session2  Time Value Of Money (2)

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%?

Page 95: Session2  Time Value Of Money (2)

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)

Page 96: Session2  Time Value Of Money (2)

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)

Page 97: Session2  Time Value Of Money (2)

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)

Page 98: Session2  Time Value Of Money (2)

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

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

Uneven Cash FlowsUneven Cash Flows

00 11 22 33 44

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

Page 99: Session2  Time Value Of Money (2)

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.

Uneven Cash FlowsUneven Cash Flows

00 11 22 33 44

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

Page 100: Session2  Time Value Of Money (2)

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 11 22 33 44

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

Page 101: Session2  Time Value Of Money (2)

Uneven Cash FlowsUneven Cash Flows

00 11 22 33 44

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

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

Page 102: Session2  Time Value Of Money (2)

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 11 22 33 44

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

Page 103: Session2  Time Value Of Money (2)

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 11 22 33 44

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

Page 104: Session2  Time Value Of Money (2)

periodperiod CF CF PV (CF)PV (CF)

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

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

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

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

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

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

00 11 22 33 44

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

Page 105: Session2  Time Value Of Money (2)

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?

Page 106: Session2  Time Value Of Money (2)

ExampleExample

00 11 22 33 44 55 66 77 88

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

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

Page 107: Session2  Time Value Of Money (2)

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

00 11 22 33 44 55 66 77 88

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

Page 108: Session2  Time Value Of Money (2)

How to solve:How to solve:

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

Or,Or,

00 11 22 33 44 55 66 77 88

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

Page 109: Session2  Time Value Of Money (2)

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

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

PVPV33= = $119,624$119,624

00 11 22 33 44 55 66 77 88

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

Page 110: Session2  Time Value Of Money (2)

119,624119,624

00 11 22 33 44 55 66 77 88

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

Page 111: Session2  Time Value Of Money (2)

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

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

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

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

119,624119,624

00 11 22 33 44 55 66 77 88

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

Page 112: Session2  Time Value Of Money (2)

119,624119,62469,22669,226

00 11 22 33 44 55 66 77 88

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

Page 113: Session2  Time Value Of Money (2)

119,624119,62469,22669,226

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

00 11 22 33 44 55 66 77 88

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

Page 114: Session2  Time Value Of Money (2)

ExampleExample

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

Page 115: Session2  Time Value Of Money (2)

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

Page 116: Session2  Time Value Of Money (2)

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

400 400 400 400400 400 400 400

Page 117: Session2  Time Value Of Money (2)

Using your calculator,Using your calculator,

P/YR = 12P/YR = 12

N = 360 N = 360

PMT = -400PMT = -400

I%YR = 12I%YR = 12

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

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

400 400 400 400400 400 400 400

Page 118: Session2  Time Value Of Money (2)

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)

Page 119: Session2  Time Value Of Money (2)

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

Page 120: Session2  Time Value Of Money (2)

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

Page 121: Session2  Time Value Of Money (2)

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

monthly house paymentmonthly house payment??

House Payment ExampleHouse Payment Example

Page 122: Session2  Time Value Of Money (2)

House Payment ExampleHouse Payment Example

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

monthly house payment?monthly house payment?

Page 123: Session2  Time Value Of Money (2)

0 1 2 3 . . . 360

? ? ? ?

Page 124: Session2  Time Value Of Money (2)

Using your calculator,Using your calculator,

P/YR = 12P/YR = 12

N = 360N = 360

I%YR = 7I%YR = 7

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

PMT = -$665.30PMT = -$665.30

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

? ? ? ?? ? ? ?

Page 125: Session2  Time Value Of Money (2)

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)

Page 126: Session2  Time Value Of Money (2)

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

Page 127: Session2  Time Value Of Money (2)

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

Page 128: Session2  Time Value Of Money (2)

Team AssignmentTeam Assignment

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

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

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

Page 129: Session2  Time Value Of Money (2)

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

Page 130: Session2  Time Value Of Money (2)

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

Page 131: Session2  Time Value Of Money (2)

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

2727 2828 2929 3030 3131 3232 3333 3434 3535

250 250 250 250 250 250 250 250 250 250

1,042,4661,042,466

Page 132: Session2  Time Value Of Money (2)

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

Page 133: Session2  Time Value Of Money (2)

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