Completed

Download Report

Transcript Completed

Discounted Cash Flow Valuation

BASIC PRINCIPAL

 Would you rather have $1,000 today or $1,000 in 30 years?



Why?

Can invest the $1,000 today let it grow This is a fundamental building block of finance

Present and Future Value

 Present Value: value of a future payment today  Future Value: value that an investment will grow to in the future  We find these by discounting or compounding at the discount rate  Also know as the hurdle rate or the opportunity cost of capital or the interest rate 3

One Period Discounting

 PV = Future Value / (1+ Discount Rate)  V 0 = C 1 / (1+r)  Alternatively  PV = Future Value * Discount Factor  V 0 = C 1 * (1/ (1+r))  Discount factor is 1/ (1+r) 4

PV Example

 What is the value today of $100 in one year, if r = 15%?



PV = 100 / 1.15 = 86.96

FV Example

 What is the value in one year of $100, invested today at 15%?



FV = 100 * (1.15) 1 = $115

Discount Rate Example

 Your stock costs $100 today, pays $5 in dividends at the end of the period, and then sells for $98. What is your rate of return?



PV =



FV =

Discount Rate Example

 Your stock costs $100 today, pays $5 in dividends at the end of the period, and then sells for $98. What is your rate of return?



PV = $100



FV =

Discount Rate Example

 Your stock costs $100 today, pays $5 in dividends at the end of the period, and then sells for $98. What is your rate of return?



PV = $100



FV = $103 = $98 + $5



($98 + $5)/$100 – 1 = 3%

NPV

 NPV = PV of all expected cash flows  Represents the value generated by the project  To compute we need: expected cash flows & the discount rate  Positive NPV investments generate value  Negative NPV investments destroy value 10

Net Present Value (NPV)

 NPV = PV (Costs) + PV (Benefit)  Costs: are negative cash flows  Benefits: are positive cash flows  One period example  NPV = C 0 + C 1 / (1+r)  For

Investments

positive

C 0 will be

negative

, and C 1 will  For

Loans negative

C 0 will be

positive

, and C 1 will be 11

Net Present Value Example

 Suppose you can buy an investment that promises to pay $10,000 in one year for $9,500. Should you invest?



We don’t know



We cannot simply compare cash flows that occur at different times

Net Present Value

 Since we cannot compare cash flow we need to calculate the NPV of the investment  If the discount rate is 5%, then NPV is?



NPV = -9,500 + 10,000/1.05



NPV = -9,500 + 9,523.81



NPV = 23.81

 At what price are we indifferent?

Net Present Value

 Since we cannot compare cash flow we need to calculate the NPV of the investment  If the discount rate is 5%, then NPV is?



NPV = -9,500 + 10,000/1.05



NPV = -9,500 + 9,523.81



NPV = 23.81

 At what price are we indifferent?

$9,523.81



NPV would be 0

Coffee Shop Example

 If you build a coffee shop on campus, you can sell it to Starbucks in one year for $300,000  Costs of building a coffee shop is $275,000  Should you build the coffee shop?

Step 1: Draw out the cash flows

Today Year 1

-$275,000 $300,000

Step 2: Find the Discount Rate

 Assume that the Starbucks offer is guaranteed  US T-Bills are risk-free and currently pay 7% interest  This is known as r f  Thus, the appropriate discount rate is 7%  Why?

Step 3: Find NPV

 The NPV of the project is?



– 275,000 + (300,000/1.07)



– 275,000 + 280,373.83



NPV = $5,373.83



Positive NPV → Build the coffee shop

If we are unsure about future?

 What is the appropriate discount rate if we are unsure about the Starbucks offer  r d = r f  r d > r f  r d < r f 19

If we are unsure about future?

 What is the appropriate discount rate if we are unsure about the Starbucks offer  r d = r f 

r d > r f

 r d < r f 20

The Discount Rate

 Should take account of two things: 1.

Time value of money Riskiness of cash flow  The appropriate discount rate is the opportunity cost of capital  This is the return that is offer on comparable investments opportunities 21

Risky Coffee Shop

 Assume that the risk of the coffee shop is equivalent to an investment in the stock market which is currently paying 12%  Should we still build the coffee shop?

Calculations

 Need to recalculate the NPV 

NPV = – 275,000 + (300,000/1.12)



NPV = – 275,000 + 267,857.14



NPV = -7,142.86



Negative NPV → Do NOT build the coffee shop

Future Cash Flows

 Since future cash flows are not certain, we need to form an expectation (best guess)  Need to identify the factors that affect cash flows (ex. Weather, Business Cycle, etc).

 Determine the various scenarios for this factor (ex. rainy or sunny; boom or recession)  Estimate cash flows under the various scenarios (sensitivity analysis)  Assign probabilities to each scenario 24

Expectation Calculation

 The expected value is the weighted average of X’s possible values, where the probability of any outcome is p  E(X) = p 1 X 1 + p 2 X 2 + …. p s X s  E(X) – Expected Value of X  X i  Outcome of X in state i  p i – Probability of state i  s – Number of possible states  Note that = p 1 + p 2 +….+ p s = 1 25

Risky Coffee Shop 2

 Now the Starbucks offer depends on the state of the economy Value Probability Recession 300,000 0.25

Normal 400,000 0.5

Boom 700,000 0.25

Calculations

 Discount Rate = 12%  Expected Future Cash Flow = 

(0.25*300) + (0.50*400) + (0.25*700) = 450,000

 NPV = 

-275,000 + 450,000/1.12



-275,000 + 401,786 = 126,790

 Do we still build the coffee shop?



Build the coffee shop, Positive NPV

Valuing a Project Summary

 Step 1: Forecast cash flows  Step 2: Draw out the cash flows  Step 3: Determine the opportunity cost of capital  Step 4: Discount future cash flows  Step 5: Apply the NPV rule 28

Reminder

 Important to set up problem correctly  Keep track of •

Magnitude

and

timing

of the cash flows •

TIMELINES

 You cannot compare cash flows @ t=3 and @ t=2 if they are not in present value terms!!

General Formula

PV 0 = FV N /(1 + r) N

FV N = PV o *(1 + r) N  Given any three, you can solve for the fourth  Present value (PV)  Future value (FV)  Time period  Discount rate 30

Four Related Questions

How much must you deposit today to have $1 million in 25 years? (r=12%) If a $58,823.31 investment yields $1 million in 25 years, what is the rate of interest?

How many years will it take $58,823.31 to grow to $1 million if r=12%?

What will $58,823.31 grow to after 25 years if r=12%?

FV Example

 Suppose a stock is currently worth $10, and is expected to grow at 40% per year for the next five years.

 What is the stock worth in five years?

$53.78

$10

14 19.6

27.44

38.42

0 1 2



$53.78 = $10×(1.40)

3 4 5

PV Example

 How much would an investor have to set aside today in order to have $20,000 five years from now if the current rate is 15%?

$20,000

0 1 2 3 4 5

PV Example

 How much would an investor have to set aside today in order to have $20,000 five years from now if the current rate is 15%?

9,943.53

$20,000

0 1 2 3



20,000/(1+0.15) 5 = 9,943.53

4 5

Historical Example

 From Fibonacci’s

Liber Abaci

, written in the year 1202: “A certain man gave 1 denari at interest so that in 5 years he must receive double the denari, and in another 5, he must have double 2 of the denari and thus forever. How many denari from this 1denaro must he have in 100 years?”  What is rate of return? Hint: what does the investor earn every 5 years 35

Historical Example

 From Fibonacci’s

Liber Abaci

100%



1 * (1+1) 20 = 1,048,576 denari.

Simple vs. Compound Interest

 Simple Interest: Interest accumulates only on the principal  Compound Interest: Interest accumulated on the principal as well as the interest already earned  What will $100 grow to after 5 periods at 35%?

• Simple interest •  FV 2 = (PV 0 * (r) + PV 0 *(r)) + PV 0 Compounded interest  FV 2 = PV 0 (1+r) (1+r)= PV 0 (1+r) 2 = PV 0 = (1 + 2r) = 37

Simple vs. Compound Interest

• Simple interest •  FV 2 = (PV 0 * (r) + PV 0 *(r)) + PV 0 Compounded interest  FV 2 = PV 0 (1+r) (1+r)= PV 0 (1+r) 2 = PV 0 = (1 + 2r) =

$275

Simple vs. Compound Interest

• • Simple interest  Compounded interest  FV 5 FV 5 = (PV 0 *(r) + PV 0 *(r)+…) + PV 0 = PV 0 (1+r) (1+r) * …= PV 0 = PV (1+r) 5 = 0 (1 + 5r) =

$448.40

$275

Compounding Periods

 We have been assuming that compounding and discounting occurs annually, this does not need to be the case 40

Non-Annual Compounding

 Cash flows are usually compounded over periods shorter than a year  The relationship between PV & FV when interest is not compounded annually 

FV N = PV * ( 1+ r / M) M*N



PV = FV N / ( 1+ r / M) M*N



M is number of compounding periods per year



N is the number of years

Compounding Examples

 What is the FV of $500 in 5 years, if the discount rate is 12%, compounded monthly? 

FV = 500 * ( 1+ 0.12 / 12) 12*5 = 908.35

 What is the PV of $500 received in 5 years, if the discount rate is 12% compounded monthly?



PV = 500 / ( 1+ 0.12 / 12) 12*5 = 275.22

Another Example

 An investment for $50,000 earns a rate of return of 1% each month for a year. How much money will you have at the end of the year?



$50,000 * 1.01

12 = $56,341

Interest Rates

 The 12% is the

Stated Annual Interest Rate

(also known as the

Annual Percentage Rate

)  This is the rate that people generally talk about  Ex. Car Loans, Mortgages, Credit Cards  However, this is not the rate people earn or pay  The

Effective Annual Rate

is what people actually earn or pay over the year  The more frequent the compounding the higher the

Effective Annual Rate

Compounding Example 2

 If you invest $50 for 3 years at 12% compounded semi-annually, your investment will grow to:

$70.93

FV = 50 * (1+(0.12/2)) 2*3 = $70.93

Compounding Example 2: Alt.

 If you invest $50 for 3 years at 12% compounded semi-annually, your investment will grow to:

$70.93

 Calculate the EAR: EAR = (1 + R/m) m – 1 

EAR = (1 + 0.12 / 2) 2 – 1 = 12.36%



FV = 50 * (1+0.1236) 3 = $70.93

 So, investing at

12.36%

compounded annually is the same as investing at 12% compounded semi-annually 46

EAR Example

 Find the Effective Annual Rate (EAR) of an 18% loan that is compounded weekly.



EAR = (1 + 0.18 / 52) 52 – 1 = 19.68%

Credit Card

 A bank quotes you a credit card with an interest rate of 14%, compounded daily. If you charge $15,000 at the beginning of the year, how much will you have to repay at the end of the year?

 EAR = 48

Credit Card

  A bank quotes you a credit card with an interest rate of 14%, compounded daily. If you charge $15,000 at the beginning of the year, how much will you have to repay at the end of the year?

EAR =

is (1+0.14/365) 365 – 1 = 15%



$15,000 * 1.15 = $17,250

Present Value Of a Cash Flow Stream

 ( 1

 1

1 )  ( 1

2 

2 ) 2  ( 1

3 

3 ) 3 =

t N

  1 ( 1

C t



r t

)

( 1 

C N r N

)

 Discount each cash flow back to the present using the appropriate discount rate and then sum the present values.

Insight Example

2 3 r = 10% Year 1 Project A 100 400 300 Project B 300 400 100 PV Which project is more valuable? Why?

Insight Example

2 3 r = 10% Year 1 Project A 100

90.91

400 300

330.58

225.39

Project B 300

272.73

400 100

330.58

75.13

646.88

Which project is more valuable? Why?

B, gets the cash faster 678.44

Various Cash Flows

 A project has cash flows of $15,000, $10,000, and $5,000 in 1, 2, and 3 years, respectively. If the interest rate is 15%, would you buy the project if it costs $25,000?



PV = 15,000/1.15+$10,000/1.15

2 +$5,000/1.15



PV = $23,892.50



NPV = –$25,000+$23,892.50 –$1,107.50

Example

(Given)

 Consider an investment that pays $200 one year from now, with cash flows increasing by $200 per year through year 4. If the interest rate is 12%, what is the present value of this stream of cash flows?

 If the issuer offers this investment for $1,500, should you purchase it?

Multiple Cash Flows

(Given)

0 1 2 3 4

178.57

318.88

427.07

508.41

1,432.93

200 400

Don’t buy

600 800 55

Various Cash Flow

(Given)

  A project has the following cash flows in periods 1 through 4: –$200, +$200, –$200, +$200. If the prevailing interest rate is 3%, would you accept this project if you were offered an up-front payment of $10 to do so?

PV = –$200/1.03 + $200/1.03

2 – $200/1.03

3 + $200/1.03



PV = –$10.99.



NPV = $10 – $10.99 = –$0.99.



You would not take this project

Common Cash Flows Streams

 Perpetuity, Growing Perpetuity  A stream of cash flows that lasts forever  Annuity, Growing Annuity  A stream of cash flows that lasts for a fixed number of periods 

NOTE:

All of the following formulas assume the first payment is next year, and payments occur annually 57

Perpetuity

 A stream of cash flows that lasts forever

C C C 0 1 2 3 PV



( 1 

)  ( 1 

C r

) 2  ( 1 

C r

) 3   …  PV: = C/r  What is PV if C=$100 and r=10%:

100/0.1 = $1,000

Perpetuity Example

 What is the PV of a perpetuity paying $30 each month, if the annual interest rate is a constant effective 12.68% per year?



Monthly rate: 1.1268

(1/2) – 1 = 1%



PV = $30/0.01 = $3,000.

Perpetuity Example 2

 What is the prevailing interest rate if a perpetual bond were to pay $100,000 per year

beginning next year

and costs $1,000,000 today?



r = C/PV = $100,000/$1,000,000 = 10%

Growing Perpetuities

 Annual payments grow at a constant rate, g 0 C 1 1 C 2 (1+g) 2 C 3 (1+g) 2

…

3  PV= C 1 /(1+r) + C 1 (1+g)/(1+r) 2 PV = C 1 /(r-g) + C 1 (1+g) 2 (1+r) 3 +…  What is PV if C 1 =$100, r=10%, and g=2%?



PV = 100 / (0.10 – 0.02) =1,250

Growing Perpetuity Example

 What is the interest rate on a perpetual bond that pays $100,000 per year with payments that grow with the inflation rate (2%) per year, assuming the bond costs $1,000,000 today?



r = C/PV+g = $100,000/$1,000,000+0.02 = 12%

Growing Perpetuity: Example

(Given)

 The expected dividend next year is $1.30, and dividends are expected to grow at 5% forever.  If the discount rate is 10%, what is the value of this promised dividend stream?

$1.30

$1.30×(1.05) = $1.37

$1.30 ×(1.05) 2 = $1.43

…

0 1 2 3

PV = 1.30 / (0.10 – 0.05) = $26

Example

An investment in a

growing perpetuity

costs $5,000 and is expected to pay $200 next year.

If the interest is 10%, what is the growth rate of the annual payment?

5,000 = 200/ (0.10 – g) 5,000 * (0.10 – g) = 200 0.10 – g = 200 / 5,000 0.10 – (200 / 5,000) = g = 0.06 = 6%

Annuity

A constant stream of cash flows with a fixed maturity

C C C C



0 1 2 3 T PV



( 1 

)  ( 1 

C r

) 2  ( 1 

C r

) 3   ( 1 

C r

)

T PV



C r

   1  ( 1  1

)

   65

Annuity Formula

PV C



C r

 ( 1 

r r

)

T C C C C C C C 0 1 2 3 T T+1 T+2 T+3

 Simply subtracting off the PV of the rest of the perpetuity’s cash flows 66

Annuity Example 1

 Compute the present value of a 3 year ordinary annuity with payments of $100 at r=10%  Answer: Or PVA 3 = 100 0.1

  1 1 (1.1

) 3   = $248.69

PVA 3 = 1 100 1.1

1 + 100 1.1

2 1 + 100 1.1

3 = $248.69

Alternative: Use a Financial Calculator

 Texas Instruments BA-II Plus, basic  N = number of periods  I/Y = periodic interest rate  P/Y must equal 1 for the I/Y to be the periodic rate  Interest is entered as a percent, not a decimal  PV = present value  PMT = payments received periodically  FV = future value  Remember to clear the registers (CLR TVM) after each problem  Other calculators are similar in format 68

Annuity Example 2



N I/Y

You agree to lease a car for 4 years at $300 per month. You are not required to pay any money up front or at the end of your agreement. If your opportunity cost of capital is 0.5% per month, what is the cost of the lease? Work through on your financial calculators

= 4 * 12 = 48 = 0.5

PV = ????

PMT = 300 FV = 0 Solve = 12,774.10

Annuity Example 3

 What is the value today of a 10-year annuity that pays $600 every other year? Assume that the stated annual discount rate is 10%.

 What do the payments look like?

 What is the discount rate?

Annuity Example 3

 What is the value today of a 10-year annuity that pays $600 every other year? Assume that the stated annual discount rate is 10%.

 What do the payments look like?

PV $600 $600 $600 $600 $600

0 2 4 6



We receive 5 payments of $600

8 10

Annuity Example 3

 What is the value today of a 10-year annuity that pays $600 every other year? Assume that the stated annual discount rate is 10%.

 What is the discount rate?



The discount rate is 10% each year, so over 2 years the discount rate is going to be

Annuity Example 3

 What is the value today of a 10-year annuity that pays $600 every other year? Assume that the stated annual discount rate is 10%.

 What is the discount rate?



The discount rate is 10% each year, so the two year stated rate SB A R is 20%, and the effective rate is



EB A R = (1 + SB A R/m) m -1



1.1

2 – 1 = 0.21 = 21%

Annuity Example 3

 What is the value today of a 10-year annuity that pays $600 every other year? Assume that the stated annual discount rate is 10%.

N = 5 I/Y we receive 5 payment over 10 years = 21 PV PMT FV = ????

= 600 = 0 Solve = 1,755.59

Annuity Example 4

 What is the present value of a four payment annuity of $100 per year that makes its first payment two years from today if the discount rate is 9%?

 What do the payments look like?

0 1 2 3 4 5

Annuity Example 4

 What is the present value of a four-payment annuity of $100 per year that makes its first payment two years from today if the discount rate is 9%?

100 100 100 100

1 2 3 4 5

Annuity Example 4

 What is the present value of a four-payment annuity of $100 per year that makes its first payment two years from today if the discount rate is 9%?

323.97

100 100 100 100 N I/Y PV

= 4 = 9 = ????

2 3 4 5

PMT = 100 FV = 0 PV = 323.97

But the $323.97 is a year 1 cash flow and we want to know the

year 0 value

Annuity Example 4

 What is the present value of a four-year annuity of $100 per year that makes its first payment two years from today if the discount rate is 9%?

297.22

323.97

100 100 100 100



1 2 3 4 5

To get PV today we need to discount the $323.97 back one more year 

323.97 / 1.09 = 297.22

Annuity Example 5

 What is the value today of a 10-pymt annuity that pays $300 a year if the annuity’s first cash flow is at the end of year 6. The interest rate is 15% for years 1-5 and 10% thereafter?

$300 $300 $300 $300 $300 $300 $300 $300 $300 $300 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 79

Annuity Example 5

  What is the value today of a 10-pymt annuity that pays $300 a year (at year-end) if the annuity’s first cash flow is at the end of year 6. The interest rate is 15% for years 1-5 and 10% thereafter?

Steps: Get value of annuity at t= 5 (year end)

= 10 I/Y PV PMT = 10 = ????

= 300 = 1,843.37

= 0

Bring value in step 1 to t=0

1,843.37 / 1.15

5 = 916.48



Annuity Example 6

You win the $20 million Powerball. The lottery commission offers you $20 million dollars today or a nine payment annuity of $2,750,000, with the first payment being today. Which is more valuable is your discount rate is 5.5%?

N I/Y = 9 = 5.5

PV PMT = ????

= 2,750,000 FV = 0 PV = $19,118,536.94

When is the $19,118,536.94?

Year -1, so to bring it into today we?



Annuity Example 6

When is the $19,118,536.94?

Year -1, so to bring it into today we?

19118536.94 * 1.055 = 20,170,056.47

Take the annuity



Alt: Annuity Example 6

N I/Y = 8 = 5.5

PV PMT = ????

= 2,750,000 FV = 0 PV = $17420056.47

Then add today’s payment $2,750,000 20,170,056.47

Delayed first payment: Perpetuity

 What is the present value of a growing perpetuity, that pays $100 per year, growing at 6%, when the discount rate is 10%, if the first payment is in 12 years? 84

Delayed first payment: Perpetuity

 What is the present value of a growing perpetuity, that pays $100 per year, growing at 6%, when the discount rate is 10%, if the first payment is in 12 years?  Steps: 1.

Get value of perpetuity at t= 11 (year end)

Why year 11?

Delayed first payment: Perpetuity

  What is the present value of a growing perpetuity, that pays $100 per year, growing at 6%, when the discount rate is 10%, if the first payment is in 12 years? Steps: 1.

Get value of perpetuity at t= 11 (year end)

100/(0.10-0.06) = 2,500

Delayed first payment: Perpetuity

 What is the present value of a growing perpetuity, that pays $100 per year, growing at 6%, when the discount rate is 10%, if the first payment is in 12 years?  Steps: 1.

Get value of perpetuity at t= 11 (year end)

100/(0.10-0.06) = 2,500

Bring value in step 1 to t=0 87

Delayed first payment: Perpetuity

 What is the present value of a growing perpetuity, that pays $100 per year, growing at 6%, when the discount rate is 10%, if the first payment is in 12 years?  Steps: 1.

Get value of perpetuity at t= 11 (year end)

100/(0.10-0.06) = 2,500

Bring value in step 1 to t=0

2,500 / 1.1

11 = 876.23

Growing Annuity

A growing stream of cash flows with a fixed maturity

C C

×(1+

)

×(1+

) 2  C×(1+

)

-1

0 PV



1 2 3 T C

( 1 

) 

 ( 1  ( 1 

) 2

)   

 ( 1  ( 1 

g r

)

 1



r C



    1     1  ( 1 

g r

)   

    89

Growing Annuity: Example

A defined-benefit retirement plan offers to pay $20,000 per year for 40 years and increase the annual payment by 3% each year. What is the present value at retirement if the discount rate is 10%?

$20,000

$20,000×(1.03)

$20,000×(1.03) 39 

Growing Annuity: Example

A defined-benefit retirement plan offers to pay $20,000 per year for 40 years and increase the annual payment by 3% each year. What is the present value at retirement if the discount rate is 10%?

$20,000

$20,000×(1.03)

$20,000×(1.03) 39 



PV = (20,000/(.1-.03)) * [ 1- {1.03/1.1} 40 ] = 265,121.57

Growing Annuity: Example (Given)

You are evaluating an income generating property. Net rent is received at the end of each year. The first year's rent is expected to be $8,500, and rent is expected to increase 7% each year. What is the present value of the estimated income stream over the first 5 years if the discount rate is 12%?

0 1 2 3 4 5

PV = (8,500/(.12-.07)) * [ 1- {1.07/1.12} 5 ] = $34,706.26

Growing Perpetuity Example

 What is the value today a perpetuity that makes payments every other year, If the first payment is $100, the discount rate is 12%, and the growth rate is 7%?



Price:

Growing Perpetuity Example

 What is the value today a perpetuity that makes payments every other year, If the first payment is $100, the discount rate is 12%, and the growth rate is 7%?



r: is 12%/year so the 2-year is 25.44%





EB A R = (1 + 0.24/2) 2 -1



Price:

Growing Perpetuity Example

 What is the value today a perpetuity that makes payments every other year, If the first payment is $100, the discount rate is 12%, and the growth rate is 7%?



r: is 12%/year so the 2-year is 25.44%



EB A R = (1 + 0.24/2) 2 -1



g: is 7%/year so the 2-year is 14.49%



EB A GR = (1 + 0.14/2) 2 -1



What is half of infinity?



Infinity



Price:



100/(0.2544-0.1449) = $913.24

Valuation Formulas



FV n

( 1 

)

n PV



C r PV



C r

   1  ( 1  1

)

  

F V n



P V PV



r C

1 

* ( 1 

)

n PV



r C

1 

    1     1  ( 1 

g r

)   

    96

Valuation Formulas

Lump Sum Lump Sum



FV n

( 1 

)

n F V n



Growing Perpetuity Perpetuity

PV P V

* ( 1 

)



r C

1 

g PV



C r

Growing Annuity Annuity



C r

   1  ( 1  1

)

  



r C

1 

    1     1  ( 1 

g r

)   

    97

Remember

 That when you use one of these formula’s or the calculator the assumptions are that:  PV is right now  The first payment is next year 98

What Is a Firm Worth?

 Conceptually, a firm should be worth the present value of the firm’s cash flows.

 The tricky part is determining the size, timing, and

risk

of those cash flows.

Quick Quiz

How is the future value of a single cash flow computed?

How is the present value of a series of cash flows computed.

What is the Net Present Value of an investment?

What is an EAR, and how is it computed?

What is a perpetuity? An annuity?

100

Why We Care

 The Time Value of Money is the basis for all of finance  People will assume that you have this down cold 101