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Ch. 5 - The Time Value

of Money

2002, Prentice Hall, Inc.

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What is r ?

Discount Rate

Compound rate

Interest rate

Rate of return

Required rate of return Expected rate of return

Opportunity cost of funds

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Cont.

Weighted average cost of capital (WACC)

Internal rate of return (IRR)

Yield to maturity (YTM)

Accounting rate of return

Economic rate of return Social rate of return

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The Time Value of Money

Compounding and

Discounting Single Sums

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We know that receiving $1 today is worth

morethan $1 in the future. This is due

toopportunity costs.The opportunity cost of receiving $1 in

the future is theinterestwe could have

earned if we had received the $1 sooner.Today Future

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I f we can measure this opportunity cost,

we can:

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I f we can measure this opportunity cost,

we can:

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

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I f we can measure this opportunity cost,

we can:

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

Today

?

Future

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I f we can measure this opportunity cost,

we can:

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

Translate $1 in the future into its equivalent today

(discounting).

Today

?

Future

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I f we can measure this opportunity cost,

we can:

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

Translate $1 in the future into its equivalent today

(discounting).

?

Today Future

Today

?

Future

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Future Value

F t V l i l

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Future Value - single sums

I f you deposit $100 in an account earning 6%, how

much would you have in the account after 1 year?

F t V l i l

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Future Value - single sums

I f you deposit $100 in an account earning 6%, how

much would you have in the account after 1 year?

0 1

PV = FV =

F t V l i l

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Future Value - single sums

I f you deposit $100 in an account earning 6%, how

much would you have in the account after 1 year?

Calculator Solution:

P/Y = 1 I = 6

N = 1 PV = -100

FV = $106

0 1

PV = -100 FV =

F t V l i l

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Future Value - single sums

I f you deposit $100 in an account earning 6%, how

much would you have in the account after 1 year?

Calculator Solution:

P/Y = 1 I = 6

N = 1 PV = -100

FV = $106

0 1

PV = -100 FV = 106

F t V l i l

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Future Value - single sums

I f you deposit $100 in an account earning 6%, how

much would you have in the account after 1 year?

Mathematical Solution:

FV = PV (FVIF i, n)

FV = 100 (FVIF .06, 1) (use FVIF table, or)

FV = PV (1 + i)n

FV = 100 (1.06)1

= $106

0 1

PV = -100 FV = 106

F t V l i l

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Future Value - single sums

I f you deposit $100 in an account earning 6%, how

much would you have in the account after 5 years?

F t V l i l

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Future Value - single sums

I f you deposit $100 in an account earning 6%, how

much would you have in the account after 5 years?

0 5

PV = FV =

F t re Val e single s ms

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Future Value - single sums

I f you deposit $100 in an account earning 6%, how

much would you have in the account after 5 years?

Calculator Solution:

P/Y = 1 I = 6

N = 5 PV = -100

FV = $133.82

0 5

PV = -100 FV =

Future Value single sums

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Future Value - single sums

I f you deposit $100 in an account earning 6%, how

much would you have in the account after 5 years?

Calculator Solution:

P/Y = 1 I = 6

N = 5 PV = -100

FV = $133.82

0 5

PV = -100 FV = 133.82

Future Value single sums

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Future Value - single sums

I f you deposit $100 in an account earning 6%, how

much would you have in the account after 5 years?

Mathematical Solution:

FV = PV (FVIF i, n)

FV = 100 (FVIF .06, 5) (use FVIF table, or)

FV = PV (1 + i)n

FV = 100 (1.06)5

= $133.82

0 5

PV = -100 FV = 133.82

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Future Value - single sumsI f you deposit $100 in an account earning 6% with

quarterly compounding, how much would you have

in the account after 5 years?

F t V l i l

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0 ?

PV = FV =

Future Value - single sumsI f you deposit $100 in an account earning 6% with

quarterly compounding, how much would you have

in the account after 5 years?

F t V l i l

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Calculator Solution:

P/Y = 4 I = 6

N = 20 PV = -100

FV = $134.68

0 20

PV = -100 FV =

Future Value - single sumsI f you deposit $100 in an account earning 6% with

quarterly compounding, how much would you have

in the account after 5 years?

F t V l i l

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Calculator Solution:

P/Y = 4 I = 6

N = 20 PV = -100

FV = $134.68

0 20

PV = -100 FV = 134.68

Future Value - single sumsI f you deposit $100 in an account earning 6% with

quarterly compounding, how much would you have

in the account after 5 years?

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Future Value single sums

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Future Value - single sumsI f you deposit $100 in an account earning 6% with

monthly compounding, how much would you have

in the account after 5 years?

Future Value single sums

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Future Value - single sumsI f you deposit $100 in an account earning 6% with

monthly compounding, how much would you have

in the account after 5 years?

0 ?

PV = FV =

Future Value single sums

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Calculator Solution:

P/Y = 12 I = 6

N = 60 PV = -100

FV = $134.89

0 60

PV = -100 FV =

Future Value - single sumsI f you deposit $100 in an account earning 6% with

monthly compounding, how much would you have

in the account after 5 years?

Future Value single sums

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Calculator Solution:

P/Y = 12 I = 6

N = 60 PV = -100

FV = $134.89

0 60

PV = -100 FV = 134.89

Future Value - single sumsI f you deposit $100 in an account earning 6% with

monthly compounding, how much would you have

in the account after 5 years?

Future Value single sums

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Mathematical Solution:

FV = PV (FVIF i, n)FV = 100 (FVIF .005, 60) (cant use FVIF table)

FV = PV (1 + i/m) m x n

FV = 100 (1.005)60 = $134.89

0 60

PV = -100 FV = 134.89

Future Value - single sumsI f you deposit $100 in an account earning 6% with

monthly compounding, how much would you have

in the account after 5 years?

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Future Value - continuous compoundingWhat is the FV of $1,000 earning 8% with

continuous compounding, after 100 years?

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Future Value - continuous compoundingWhat is the FV of $1,000 earning 8% with

continuous compounding, after 100 years?

0 ?

PV = FV =

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Mathematical Solution:

FV = PV (e in)

FV = 1000 (e .08x100) = 1000 (e 8)

FV = $2,980,957.99

0 100

PV = -1000 FV =

Future Value - continuous compoundingWhat is the FV of $1,000 earning 8% with

continuous compounding, after 100 years?

F V l i di

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0 100

PV = -1000 FV = $2.98m

Future Value - continuous compoundingWhat is the FV of $1,000 earning 8% with

continuous compounding, after 100 years?

Mathematical Solution:

FV = PV (e in)

FV = 1000 (e .08x100) = 1000 (e 8)

FV = $2,980,957.99

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Present Value

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Present Value - single sumsI f you receive $100 one year from now, what is the

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

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0 ?

PV = FV =

Present Value - single sumsI f you receive $100 one year from now, what is the

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

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Calculator Solution:

P/Y = 1 I = 6

N = 1 FV = 100

PV = -94.34

0 1

PV = FV = 100

Present Value - single sumsI f you receive $100 one year from now, what is the

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

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Calculator Solution:

P/Y = 1 I = 6

N = 1 FV = 100

PV = -94.34

PV = -94.34 FV = 100

0 1

Present Value - single sumsI f you receive $100 one year from now, what is the

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

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Mathematical Solution:

PV = FV (PVIFi, n

)

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

PV = FV / (1 + i)n

PV = 100 / (1.06)1

= $94.34

PV = -94.34 FV = 100

0 1

Present Value - single sumsI f you receive $100 one year from now, what is the

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

P t V l i l

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Present Value - single sumsI f you receive $100 five years from now, what is the

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

P t V l i l

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0 ?

PV = FV =

Present Value - single sumsI f you receive $100 five years from now, what is the

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

Present Value single sums

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Calculator Solution:

P/Y = 1 I = 6

N = 5 FV = 100

PV = -74.73

0 5

PV = FV = 100

Present Value - single sumsI f you receive $100 five years from now, what is the

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

Present Value single sums

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Calculator Solution:

P/Y = 1 I = 6

N = 5 FV = 100

PV = -74.73

Present Value - single sumsI f you receive $100 five years from now, what is the

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

0 5

PV = -74.73 FV = 100

Present Value single sums

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Mathematical Solution:

PV = FV (PVIF i, n

)PV = 100 (PVIF .06, 5) (use PVIF table, or)

PV = FV / (1 + i)n

PV = 100 / (1.06)5 = $74.73

Present Value - single sumsI f you receive $100 five years from now, what is the

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

0 5

PV = -74.73 FV = 100

Present Value single sums

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

from now if your opportunity cost is 7%?

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Present Value single sums

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Calculator Solution:

P/Y = 1 I = 7

N = 15 FV = 1,000

PV = -362.45

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

from now if your opportunity cost is 7%?

0 15

PV = FV = 1000

Present Value single sums

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Calculator Solution:

P/Y = 1 I = 7

N = 15 FV = 1,000

PV = -362.45

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

from now if your opportunity cost is 7%?

0 15

PV = -362.45 FV = 1000

Present Value single sums

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Mathematical Solution:

PV = FV (PVIFi, n

)

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

PV = FV / (1 + i)n

PV = 100 / (1.07)15

= $362.45

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

from now if your opportunity cost is 7%?

0 15

PV = -362.45 FV = 1000

Present Value single sums

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Present Value - single sumsI f you sold land for $11,933 that you bought 5 years

ago for $5,000, what is your annual rate of return?

Present Value single sums

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0 5

PV = FV =

Present Value - single sumsI f you sold land for $11,933 that you bought 5 years

ago for $5,000, what is your annual rate of return?

Present Value - single sums

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Calculator Solution:

P/Y = 1 N = 5

PV = -5,000 FV = 11,933

I = 19%

0 5

PV = -5000 FV = 11,933

Present Value - single sumsI f you sold land for $11,933 that you bought 5 years

ago for $5,000, what is your annual rate of return?

Present Value - single sums

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Mathematical Solution:

PV = FV (PVIF i, n)

5,000 = 11,933 (PVIF ?, 5)PV = FV / (1 + i)n

5,000 = 11,933 / (1+ i)5

.419 = ((1/ (1+i)5)

2.3866 = (1+i)5

(2.3866)1/5 = (1+i) i = .19

Present Value - single sumsI f you sold land for $11,933 that you bought 5 years

ago for $5,000, what is your annual rate of return?

Present Value - single sums

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gSuppose you placed $100 in an account that pays

9.6% interest, compounded monthly. How long wil l

i t take for your account to grow to $500?

0

PV = FV =

Present Value - single sums

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Calculator Solution:

P/Y = 12 FV = 500

I = 9.6 PV = -100

N = 202 months

gSuppose you placed $100 in an account that pays

9.6% interest, compounded monthly. How long wil l

i t take for your account to grow to $500?

0 ?

PV = -100 FV = 500

Present Value - single sums

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gSuppose you placed $100 in an account that pays

9.6% interest, compounded monthly. How long wil l

i t take for your account to grow to $500?

Mathematical Solution:

PV = FV / (1 + i)n

100 = 500 / (1+ .008)N

5 = (1.008)N

ln 5 = ln (1.008)N

ln 5 = N ln (1.008)

1.60944 = .007968 N N = 202 months

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H int for single sum problems:

In every single sum future value andpresent value problem, there are 4

variables:

FV, PV, i, and n When doing problems, you will be

given 3 of these variables and asked to

solve for the 4th variable. Keeping this in mind makes time

value problems much easier!

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The Time Value of Money

Compounding and Discounting

Cash Flow Streams

0 1 2 3 4

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Annuities

Annuity: a sequence of equalcash

flows, occurring at the endof each

period.

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Annuity: a sequence of equalcash

flows, occurring at the end of each

period.

0 1 2 3 4

Annuities

E l f A iti

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Examples of Annuities:

If you buy a bond, you willreceive equal semi-annual coupon

interest payments over the life of

the bond. If you borrow money to buy a

house or a car, you will pay a

stream of equal payments.

E l f A iti

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If you buy a bond, you willreceive equal semi-annual coupon

interest payments over the life of

the bond. If you borrow money to buy a

house or a car, you will pay a

stream of equal payments.

Examples of Annuities:

Future Value - annui ty

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

would you have after 3 years?

Future Value - annui ty

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0 1 2 3

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

would you have after 3 years?

Future Value - annui ty

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Calculator Solution:

P/Y = 1 I = 8 N = 3

PMT = -1,000

FV = $3,246.40

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

would you have after 3 years?

0 1 2 3

1000 1000 1000

Future Value - annui ty

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Calculator Solution:

P/Y = 1 I = 8 N = 3

PMT = -1,000

FV = $3,246.40

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

would you have after 3 years?

0 1 2 3

1000 1000 1000

Future Value - annui ty

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

would you have after 3 years?

Future Value - annui ty

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Mathematical Solution:

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

would you have after 3 years?

Future Value - annui ty

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Mathematical Solution:

FV = PMT (FVIFA i, n)

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

would you have after 3 years?

Future Value - annui ty

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Mathematical Solution:

FV = PMT (FVIFA i, n)

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

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

would you have after 3 years?

Future Value - annui ty

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Mathematical Solution:

FV = PMT (FVIFA i, n)

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

FV = PMT (1 + i)n- 1

i

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

would you have after 3 years?

Future Value - annui ty

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Mathematical Solution:

FV = PMT (FVIFA i, n)

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

FV = PMT (1 + i)n- 1

iFV = 1,000 (1.08)3 - 1 = $3246.40

.08

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

would you have after 3 years?

Present Value - annuity

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

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

Present Value - annuity

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0 1 2 3

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

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

Present Value - annuity

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Calculator Solution:

P/Y = 1 I = 8 N = 3

PMT = -1,000

PV = $2,577.10

0 1 2 3

1000 1000 1000

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

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

Present Value - annuity

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Calculator Solution:

P/Y = 1 I = 8 N = 3

PMT = -1,000

PV = $2,577.10

0 1 2 3

1000 1000 1000

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

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

Present Value - annuity

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

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

Present Value - annuity

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Mathematical Solution:

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

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

Present Value - annuity

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Mathematical Solution:

PV = PMT (PVIFA i, n)

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

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

Present Value - annuity

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Mathematical Solution:

PV = PMT (PVIFA i, n)

PV = 1,000 (PVIFA .08, 3) (use PVIFA table, or)

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

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

Present Value - annuity

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Mathematical Solution:

PV = PMT (PVIFA i, n)

PV = 1,000 (PVIFA .08, 3) (use PVIFA table, or)

1

PV = PMT 1 - (1 + i)n

i

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

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

Present Value - annuity

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Mathematical Solution:

PV = PMT (PVIFA i, n)

PV = 1,000 (PVIFA .08, 3) (use PVIFA table, or)

1

PV = PMT 1 - (1 + i)n

i

1

PV = 1000 1 - (1.08 )3 = $2,577.10

.08

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

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

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Other Cash F low Patterns

0 1 2 3

The Time Value of Money

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Perpetuities

Suppose you will receive a fixed

payment every period (month, year,

etc.) forever. This is an example ofa perpetuity.

You can think of a perpetuity as an

annuitythat goes on forever.

Present Value of a

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Present Value of a

Perpetuity

When we find the PV of an annuity,

we think of the following

relationship:

Present Value of a

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Present Value of a

Perpetuity

When we find the PV of an annuity,

we think of the following

relationship:

PV = PMT (PVIFA i, n)

Mathematically,

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Mathematically,

Mathematically,

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at e at ca y,

(PVIFA i, n ) =

Mathematically,

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y,

(PVIFA i, n ) = 1 -

1

(1 + i)n

i

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When n gets very large,

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g y g

When n gets very large,

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g y g

1 -1

(1 + i)n

i

When n gets very large,

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this becomes zero.1 -

1(1 + i)

n

i

When n gets very large,

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this becomes zero.

So were left with PVIFA =

1

i

1 -1

(1 + i)n

i

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Cont.

FV = PV + PV (r)

x+100 = x + x (r)

x (r) = 100

x =

PV =

100

r

C

r

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What should you be willing to pay in

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What should you be willing to pay in

order to receive $10,000annually

forever, if you require 8%per year

on the investment?

PMT $10,000i .08

PV = =

What should you be willing to pay in

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What should you be willing to pay in

order to receive $10,000annually

forever, if you require 8%per year

on the investment?

PMT $10,000i .08

= $125,000

PV = =

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Ordinary Annui ty

vs.

Annui ty Due

$1000 $1000 $1000

4 5 6 7 8

Begin Mode vs. End Mode

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1000 1000 1000

4 5 6 7 8

Begin Mode vs. End Mode

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1000 1000 1000

4 5 6 7 8year year year

5 6 7

Begin Mode vs. End Mode

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1000 1000 1000

4 5 6 7 8year year year

5 6 7

PV

inEND

Mode

Begin Mode vs. End Mode

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1000 1000 1000

4 5 6 7 8year year year

5 6 7

PV

inEND

Mode

FV

inEND

Mode

Begin Mode vs. End Mode

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1000 1000 1000

4 5 6 7 8year year year

6 7 8

Begin Mode vs. End Mode

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1000 1000 1000

4 5 6 7 8year year year

6 7 8

PV

inBEGIN

Mode

Begin Mode vs. End Mode

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1000 1000 1000

4 5 6 7 8year year year

6 7 8

PV

inBEGIN

Mode

FV

inBEGIN

Mode

Earlier, we examined this

di i

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ordinary annuity:

Earlier, we examined this

di i

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ordinary annuity:

0 1 2 3

1000 1000 1000

Earlier, we examined this

di i

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ordinary annuity:

Using an interest rate of 8%, wefind that:

0 1 2 3

1000 1000 1000

Earlier, we examined this

di i

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ordinary annuity:

Using an interest rate of 8%, wefind that:

The Future Value(at 3) is

$3,246.40.

0 1 2 3

1000 1000 1000

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What about this annuity?

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Same 3-year time line,

Same 3 $1000 cash flows, but

The cash flows occur at the

beginningof each year, ratherthan at the endof each year.

This is an annuity due.

0 1 2 3

1000 1000 1000

Future Value - annui ty dueI f you invest $1,000 at the beginning of each of the

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0 1 2 3

y g g

next 3 years at 8%, how much would you have at the

end of year 3?

Future Value - annui ty dueI f you invest $1,000 at the beginning of each of the

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Calculator Solution:

Mode = BEGIN P/Y = 1 I = 8

N = 3 PMT = -1,000

FV = $3,506.11

0 1 2 3

-1000 -1000 -1000

y g g

next 3 years at 8%, how much would you have at the

end of year 3?

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Future Value - annui ty dueI f you invest $1,000 at the beginning of each of the

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next 3 years at 8%, how much would you have at the

end of year 3?Mathematical Solution: Simply compound the FV of the

ordinary annuity one more period:

Future Value - annui ty dueI f you invest $1,000 at the beginning of each of the

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next 3 years at 8%, how much would you have at the

end of year 3?Mathematical Solution: Simply compound the FV of the

ordinary annuity one more period:

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

Future Value - annui ty dueI f you invest $1,000 at the beginning of each of the

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next 3 years at 8%, how much would you have at the

end of year 3?Mathematical Solution: Simply compound the FV of the

ordinary annuity one more period:

FV = PMT (FVIFA i, n) (1 + i)FV = 1,000 (FVIFA .08, 3) (1.08) (use FVIFA table, or)

Future Value - annui ty dueI f you invest $1,000 at the beginning of each of the

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next 3 years at 8%, how much would you have at the

end of year 3?Mathematical Solution: Simply compound the FV of the

ordinary annuity one more period:

FV = PMT (FVIFA i, n) (1 + i)FV = 1,000 (FVIFA .08, 3) (1.08) (use FVIFA table, or)

FV = PMT (1 + i)n- 1

i (1 + i)

Future Value - annui ty dueI f you invest $1,000 at the beginning of each of the

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next 3 years at 8%, how much would you have at the

end of year 3?Mathematical Solution: Simply compound the FV of the

ordinary annuity one more period:

FV = PMT (FVIFA i, n) (1 + i)FV = 1,000 (FVIFA .08, 3) (1.08) (use FVIFA table, or)

FV = PMT (1 + i)n- 1

i

FV = 1,000 (1.08)3 - 1 = $3,506.11

08

(1 + i)

(1.08)

Present Value - annuity dueWhat is the PV of $1,000 at the beginning of each of

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$ , g g

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

0 1 2 3

Present Value - annuity dueWhat is the PV of $1,000 at the beginning of each of

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Calculator Solution:

Mode = BEGIN P/Y = 1 I = 8N = 3 PMT = 1,000

PV = $2,783.26

0 1 2 3

1000 1000 1000

, g g

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

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Present Value - annuity due

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Mathematical Solution:

Present Value - annuity due

M h i l S l i

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

Present Value - annuity due

M th ti l S l ti

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

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

Present Value - annuity due

M th ti l S l ti

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

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

PV = 1,000 (PVIFA .08, 3) (1.08) (use PVIFA table, or)

Present Value - annuity due

M th ti l S l ti

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

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

PV = 1,000 (PVIFA .08, 3) (1.08) (use PVIFA table, or)

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

i(1 + i)

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Uneven Cash F lows

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Is this an annuity?

How do we find the PV of a cash flow

stream when all of the cash flows are

different? (Use a 10% discount rate).

0 1 2 3 4

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

Uneven Cash F lows

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Sorry! Theres no quickie for this one.

We have to discount each cash flow

back separately.

0 1 2 3 4

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

Uneven Cash F lows

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Sorry! Theres no quickie for this one.

We have to discount each cash flow

back separately.

0 1 2 3 4

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

Uneven Cash F lows

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Sorry! Theres no quickie for this one.

We have to discount each cash flow

back separately.

0 1 2 3 4

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

Uneven Cash F lows

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Sorry! Theres no quickie for this one.

We have to discount each cash flow

back separately.

0 1 2 3 4

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

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

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period CF PV (CF)

0 -10,000 -10,000.00

1 2,000 1,818.18

2 4,000 3,305.79

3 6,000 4,507.894 7,000 4,781.09

PV of Cash Flow Stream: $ 4,412.95

0 1 2 3 4

Annual Percentage Yield (APY)

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ua e ce age e d ( )

Which is the better loan:

8%compounded annually, or

7.85%compounded quarterly?

We cant compare these nominal (quoted)

interest rates, because they dont include the

same number of compounding periods peryear!

We need to calculate the APY.

Annual Percentage Yield (APY)

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Annual Percentage Yield (APY)

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APY = (1 + ) m - 1quoted rate

m

Annual Percentage Yield (APY)

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Find the APY for the quarterly loan:

APY= (1 + ) m - 1quoted rate

m

Annual Percentage Yield (APY)

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Find the APY for the quarterly loan:

APY= (1 + ) m - 1quoted rate

m

APY = (1 + ) 4 - 1.07854

Annual Percentage Yield (APY)

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Find the APY for the quarterly loan:

APY= (1 + ) m - 1quoted rate

m

APY = (1 + ) 4 - 1

APY = .0808, or 8.08%

.0785

4

Annual Percentage Yield (APY)

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Find the APY for the quarterly loan:

The quarterly loan is more expensive than

the 8% loan with annual compounding!

APY= (1 + ) m - 1quoted rate

m

APY = (1 + ) 4 - 1

APY = .0808, or 8.08%

.0785

4

Practice Problems

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Practice Problems

Example

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Cash flows from an investment are

expected to be $40,000per year at the

end of years 4, 5, 6, 7, and 8. If you

require a 20%rate of return, what is

the PV of these cash flows?

Example

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0 1 2 3 4 5 6 7 8

$0 0 0 0 40 40 40 40 40

Cash flows from an investment are

expected to be $40,000per year at the

end of years 4, 5, 6, 7, and 8. If you

require a 20%rate of return, what is

the PV of these cash flows?

$0 0 0 0 40 40 40 40 40

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This type of cash flow sequence is

often called a deferred annuity.

0 1 2 3 4 5 6 7 8

$0 0 0 0 40 40 40 40 40

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How to solve:

1) Discount each cash flow back to

time 0 separately.

0 1 2 3 4 5 6 7 8

$0 0 0 0 40 40 40 40 40

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How to solve:

1) Discount each cash flow back to

time 0 separately.

0 1 2 3 4 5 6 7 8

$0 0 0 0 40 40 40 40 40

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How to solve:

1) Discount each cash flow back to

time 0 separately.

0 1 2 3 4 5 6 7 8

$0 0 0 0 40 40 40 40 40

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How to solve:

1) Discount each cash flow back to

time 0 separately.

0 1 2 3 4 5 6 7 8

$0 0 0 0 40 40 40 40 40

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How to solve:

1) Discount each cash flow back to

time 0 separately.

0 1 2 3 4 5 6 7 8

$0 0 0 0 40 40 40 40 40

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How to solve:

1) Discount each cash flow back to

time 0 separately.

0 1 2 3 4 5 6 7 8

$0 0 0 0 40 40 40 40 40

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How to solve:

1) Discount each cash flow back to

time 0 separately.

Or,

0 1 2 3 4 5 6 7 8

$0 0 0 0 40 40 40 40 40

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2) Find the PV of the annuity:

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

PV = $119,624

0 1 2 3 4 5 6 7 8

$0 0 0 0 40 40 40 40 40

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2) Find the PV of the annuity:

PV3: End mode; P/YR = 1; I = 20;PMT = 40,000; N = 5

PV3= $119,624

0 1 2 3 4 5 6 7 8

$0 0 0 0 40 40 40 40 40

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119,624

0 1 2 3 4 5 6 7 8

$0 0 0 0 40 40 40 40 40

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Then discount this single sum back to

time 0.

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

N = 3; FV = 119,624;

Solve: PV = $69 226

119,624

0 1 2 3 4 5 6 7 8

$0 0 0 0 40 40 40 40 40

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69,226

0 1 2 3 4 5 6 7 8

119,624

$0 0 0 0 40 40 40 40 40

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The PV of the cash flow

stream is $69,226.

69,226

0 1 2 3 4 5 6 7 8

119,624

Retirement Example

After graduation, you plan to invest

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After graduation, you plan to invest

$400per monthin the stock market.If you earn 12%per yearon your

stocks, how much will you have

accumulated when you retire in 30years?

Retirement Example

After graduation, you plan to invest

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After graduation, you plan to invest

$400per month in the stock market.If you earn 12%per year on your

stocks, how much will you have

accumulated when you retire in 30years?

0 1 2 3 . . . 360

400 400 400 400

400 400 400 400

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0 1 2 3 . . . 360

0 1 360

400 400 400 400

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Using your calculator,

P/YR = 12

N = 360

PMT = -400I%YR = 12

FV = $1,397,985.65

0 1 2 3 . . . 360

Retirement ExampleI f you invest $400 at the end of each month for the

next 30 years at 12%, how much would you have at

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y y

the end of year 30?

Retirement ExampleI f you invest $400 at the end of each month for the

next 30 years at 12%, how much would you have at

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y y

the end of year 30?Mathematical Solution:

Retirement ExampleI f you invest $400 at the end of each month for the

next 30 years at 12%, how much would you have at

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y y

the end of year 30?Mathematical Solution:

FV = PMT (FVIFA i, n)

Retirement ExampleI f you invest $400 at the end of each month for the

next 30 years at 12%, how much would you have at

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the end of year 30?

Mathematical Solution:

FV = PMT (FVIFA i, n)FV = 400 (FVIFA .01, 360) (cant use FVIFA table)

Retirement ExampleI f you invest $400 at the end of each month for the

next 30 years at 12%, how much would you have at

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the end of year 30?

Mathematical Solution:

FV = PMT (FVIFA i, n)FV = 400 (FVIFA .01, 360) (cant use FVIFA table)

FV = PMT (1 + i)n- 1

i

Retirement ExampleI f you invest $400 at the end of each month for the

next 30 years at 12%, how much would you have at

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the end of year 30?

Mathematical Solution:

FV = PMT (FVIFA i, n)FV = 400 (FVIFA .01, 360) (cant use FVIFA table)

FV = PMT (1 + i)n- 1

i

FV = 400 (1.01)360- 1 = $1,397,985.65

01

If you borrow $100,000at 7%fixed

House Payment Example

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y ,

interest for 30yearsin order tobuy a house, what will be your

monthly house payment?

House Payment Example

If you borrow $100,000at 7%fixed

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y

interest for 30years in order tobuy a house, what will be your

monthly house payment?

0 1 2 3 360

? ? ? ?

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0 1 2 3 . . . 360

0 1 2 3 360

? ? ? ?

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Using your calculator,

P/YR = 12

N = 360

I%YR = 7PV = $100,000

PMT = -$665.30

0 1 2 3 . . . 360

House Payment Example

Mathematical Solution:

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House Payment Example

Mathematical Solution:

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PV = PMT (PVIFA i, n)

House Payment Example

Mathematical Solution:

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PV = PMT (PVIFA i, n)

100,000 = PMT (PVIFA .07, 360) (cant use PVIFA table)

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House Payment Example

Mathematical Solution:

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PV = PMT (PVIFA i, n)

100,000 = PMT (PVIFA .07, 360) (cant use PVIFA table)

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

i

1

100,000 = PMT 1 - (1.005833 )360 PMT=$665.30

.005833

Team Assignment

Upon retirement your goal is to spend 5

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Upon retirement, your goal is to spend 5

years traveling around the world. To

travel in style will require $250,000per

year at the beginningof each year.

If you plan to retire in 30 years, what are

the equal monthlypayments necessary

to achieve this goal? The funds in yourretirement account will compound at

10%annually.

27 28 29 30 31 32 33 34 35

250 250 250 250 250

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How much do we need to have by

the end of year 30 to finance the

trip?

PV30 = PMT (PVIFA .10, 5) (1.10) =

= 250,000 (3.7908) (1.10) =

= $1,042,470

27 28 29 30 31 32 33 34 35

250 250 250 250 250

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Using your calculator,

Mode = BEGIN

PMT = -$250,000

N = 5

I%YR = 10

P/YR = 1

PV = $1,042,466

27 28 29 30 31 32 33 34 35

250 250 250 250 250

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Now, assuming 10% annualcompounding, what monthly

payments will be required for

you to have $1,042,466at the endof year 30?

1,042,466

27 28 29 30 31 32 33 34 35

250 250 250 250 250

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Using your calculator,

Mode = ENDN = 360

I%YR = 10

P/YR = 12

FV = $1,042,466

PMT = -$461.17

1,042,466

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So, you would have to place $461.17in

your retirement account, which earns