Slide 1 The Time Value of Money Time Value of Money Concept Future and Present Values of single...

Slide 1

The Time Value of Money Time Value of Money Concept Future and Present Values of single payments Future and Present values of periodic payments

(Annuities) Present value of perpetuity Future and Present values of annuity due Annual Percentage Yield (APY)

Slide 2

The Time Value of Money Concept We know that receiving $1 today is worth more

than $1 in the future. This is due to opportunity costs

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

Slide 3

The Future Value Future Value equation:

nn iPVFV

nformulatio general Value Future

iiiPVFV

periods three for Value Future

iiPVFV

periods two for Value Future

iPVFV

period one for Value Future

1

111

11

1

0

03

02

101

Slide 4

Future Value – Single Sums If you deposit $100 in an account earning 6%,

how much would you have in the account after 1 year?

Mathematical Solution:

FVn = $1 x (1 + i)n

FVn = $100 x (1 + 0.06)1

FVn = $106$106

Slide 5

Future Value – Single Sums (Continued)

N I/Y P/Y PV PMT FV MODE

1 6 1 -100 0 106

Calculator Solution (TI BA II PLUS)

N Number of periods

I/Y Interest per Year

P/Y Payment per Year

C/Y Compounding per Year

PV Present Value

PMT PayMenT (Periodic and Fixed)

FV Future Value

MODE END for ending and BGN for beginning

Slide 6

Future Value – Single Sums (Continued) If you deposit $100 in an account earning 6%,

how much would you have in the account after 5 years?

Mathematical Solution:

FVn = $1 x (1 + i)n

FVn = $100 x (1 + 0.06)5

FVn = $133.82$133.82

N I/Y P/Y PV PMT FV MODE

5 6 1 -100 0 133.82

Slide 7

Future Value – Single Sums (Continued) If you deposit $100 in an account earning 6% with

quarterly compounding, how much would you have in the account after 5 year?

Mathematical Solution:

FVn = $1 x (1 + i)n

FVn = $100 x (1 + 0.06/4)5x4

FVn = $134.69$134.69

N I/Y P/Y PV PMT FV MODE

20 6 4 -100 0 134.69

Slide 8

Future Value – Single Sums (Continued) If you deposit $100 in an account earning 6% with

monthly compounding, how much would you have in the account after 5 year?

Mathematical Solution:

FVn = $1 x (1 + i)n

FVn = $100 x (1 + 0.06/12)5x12

FVn = $134.89$134.89

N I/Y P/Y PV PMT FV MODE

60 6 12 -100 0 134.89

Slide 9

Future Value – Single Sums (Continued) If you deposit $1,000 in an account earning 8%

with daily compounding, how much would you have in the account after 100 year?

Mathematical Solution:

FVn = $1 x (1 + i)n

FVn = $1,000 x (1 + 0.08/365)100x365

FVn = $2,978,346.07$2,978,346.07

N I/Y P/Y PV PMT FV MODE

36,500 8 365 -1000 0 2,978,346.07

Slide 10

The Present Value Present Value equation:

together. them add

canyou future the in made payments

of value present the finding after

is that additive are Values Present

i

FVPV

:(payment) flow cash futureany of (today) Value Present

nn

10

Slide 11

Present Value – Single Sums (Continued) If you receive $100 one year from now, what is

the PV of that $100 if your opportunity cost is 6%?

Mathematical Solution:

PV0 = $1 / (1 + i)n

PV0 = $100 / (1 + 0.06)1

PV0 = -$94.34-$94.34

N I/Y P/Y PV PMT FV MODE

1 6 1 -94.37 0 100

Slide 12

Present Value – Single Sums (Continued) If you receive $100 five year from now, what is

the PV of that $100 if your opportunity cost is 6%?

Mathematical Solution:

PV0 = $1 / (1 + i)n

PV0 = $100 / (1 + 0.06)5

PV0 = -$74.73-$74.73

N I/Y P/Y PV PMT FV MODE

5 6 1 -74.73 0 100

Slide 13

Present Value – Single Sums (Continued) If you sold land for $11,933 that you bought 5

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

Mathematical Solution:

N I/Y P/Y PV PMT FV MODE

5 19 1 -5,000 0 11,933

1

1

n

PV

FVi %191

000,5

933,11 5

1

i

Slide 14

Present Value – Single Sums (Continued) Suppose you placed $100 in an account that pays

9.6% interest, compounded monthly. How long will it take for your account to grow to $500?

Mathematical Solution:

N I/Y P/Y PV PMT FV MODE

202 9.6 12 -100 0 500

frequency gcompoundin:m

1ln

ln

mi

PVFV

n months. 202

12096.0

1ln

100500

ln

n

Slide 15

Hint for Single Sum Problems In every single sum future value and present 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!

Slide 16

Compounding and Discounting Cash Flow Streams Annuity: a sequence of equal cash flows,

occurring at the end of each period If you buy a bond, you will receive 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

Slide 17

Future Value – Annuity If you invest $1,000 each year at 8%, how much

would you have after 3 years?

Mathematical Solution:

N I/Y P/Y PV PMT FV MODE

3 8 1 0 -1000 3,246.40

i

iPMTFV

n

n

11

40.246,3$

08.0

108.01000,1$

3

3

3

FV

FV

Slide 18

Present Value – Annuity What is the PV of $1,000 at the end of each of the

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

Mathematical Solution:

N I/Y P/Y PV PMT FV MODE

3 8 1 2,577.10 -1000 0

ii

PMTPVn1

11

0

10.577,2$

08.008.01

11

000,1$

0

3

0

PV

PV

Slide 19

Perpetuities Suppose you will receive a fixed payment every

period (month, year, etc.) forever. This is an example of a perpetuity

You can think of a perpetuity as an annuity that goes on

Slide 20

Perpetuities (Continued)

toreducesequation PV theTherefore

zero. approaches 1

1 term then thelarge getsn If

1

11

:as PVfor equation theknow We

0

n

n

i)(

ii

PMTPV

i

PMTPV

iPMTPV

00 or 1

Slide 21

Perpetuities (Continued) What should you be willing to pay in order to

receive $10,000 annually forever, if you require 8% per year on the investment?

PV = $10,000 / 0.08 = $125,000

Slide 22

Future Value – Annuity DueAnnuity Due: The cash flows occur at the beginning of each

year, rather than at the end of each year If you invest $1,000 at the beginning of each of

the next 3 years at 8%, how much would you have at the end of year 3?

Mathematical Solution:

N I/Y P/Y PV PMT FV MODE

3 8 1 0 -1000 3,506.11 BEGIN

)1(

11i

i

iPMTFV

n

n

11.506,3$

)08.01(08.0

108.01000,1$

3

3

3

FV

FV

Slide 23

Present Value – Annuity DueAnnuity Due: The cash flows occur at the beginning of each

year, rather than at the end of each year What is the PV of $1,000 at the beginning of each

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

Mathematical Solution:

N I/Y P/Y PV PMT FV MODE

3 8 1 2,783.26 -1000 0 BEGIN

)1(

1

11

0 ii

iPMTPV

n

26.783,2$

)08.1(08.0

08.01

11

000,1$

0

3

0

PV

PV

Slide 24

Uneven Cash FlowsHow do we find the PV of a cash flow stream when all of the cash flows are different? (Use a 10% discount rate)

Period CF PVCF

0 -10,000 -10,000.00

1 2,000 1,818.15

2 4,000 3,305.79

3 6,000 4,507.89

4 7,000 4,781.09

Total 4,412.95

Slide 25

CF

CF0 -10000 ENTER

C01 2000 ENTER F01 1.00 ENTER

C02 4000 ENTER F02 1.00 ENTER

C03 6000 ENTER F03 1.00 ENTER

C04 7000 ENTER F04 1.00 ENTER

NPV 10 ENTER CPT 4,412.95

Uneven Cash Flows

Slide 26

Annual Percentage Yield (APY) or Effective Annual Rate (EAR)

Which is the better loan: 8.00% compounded annually, or 7.85% compounded quarterly?

We can’t compare these nominal (quoted) interest rates, because they don’t include the same number of compounding periods per year!

We need to calculate the APY

Slide 27

Annual Percentage Yield (APY) or Effective Annual Rate (EAR) (Continued)

Find the APY for the quarterly (m = 4) loan:

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

frequency gcompoundin theis m

1rate quoted

1

m

mAPY

0808014

078501

4

..

APY

Slide 28

Annual Percentage Yield (APY) or Effective Annual Rate (EAR) (Continued)

2nd ICONV NOM 7.85 ENTER (EFF) C/Y 4 ENTER (EFF) CPT 8.08