Discounted Cash Flow Valuation

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BASIC PRINCIPAL

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

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 rateAlso know as the hurdle rate or the opportunity

cost of capital or the interest rate

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One Period Discounting

PV = Future Value / (1+ Discount Rate)V0 = C1 / (1+r)

Alternatively PV = Future Value * Discount Factor

V0 = C1 * (1/ (1+r))

Discount factor is 1/ (1+r)

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PV Example

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

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FV Example

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

Discount Rate Example

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

PV = FV =

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NPV NPV = PV of all expected cash flows

Represents the value generated by the projectTo compute we need: expected cash flows &

the discount rate Positive NPV investments generate value Negative NPV investments destroy value

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Net Present Value (NPV)

NPV = PV (Costs) + PV (Benefit)Costs: are negative cash flowsBenefits: are positive cash flows

One period exampleNPV = C0 + C1 / (1+r) For Investments C0 will be negative, and C1 will be

positiveFor Loans C0 will be positive, and C1 will be negative

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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?

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Net Present Value

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

At what price are we indifferent?

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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?

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Step 1: Draw out the cash flows

Today Year 1

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Step 2: Find the Discount Rate

Assume that the Starbucks offer is guaranteed US T-Bills are risk-free and currently pay 7%

interestThis is known as rf

Thus, the appropriate discount rate is 7%Why?

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Step 3: Find NPV

The NPV of the project is?

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If we are unsure about future?

What is the appropriate discount rate if we are unsure about the Starbucks offerrd = rf

rd > rf

rd < rf

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The Discount Rate

Should take account of two things:1. Time value of money

2. 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

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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?

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Calculations

Need to recalculate the NPV

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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

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Expectation Calculation The expected value is the weighted average of X’s

possible values, where the probability of any outcome is p

E(X) = p1X1 + p2X2 + …. psXs E(X) – Expected Value of X Xi Outcome of X in state i pi – Probability of state i s – Number of possible states

Note that = p1 + p2 +….+ ps = 1

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Risky Coffee Shop 2

Now the Starbucks offer depends on the state of the economy

Recession Normal BoomValue 300,000 400,000 700,000

Probability 0.25 0.5 0.25

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Calculations Discount Rate = 12% Expected Future Cash Flow =

NPV =

Do we still build the coffee shop?

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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

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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!!

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General Formula

PV0 = FVN/(1 + r)N OR FVN = PVo*(1 + r)N

Given any three, you can solve for the fourthPresent value (PV)Future value (FV)Time period Discount rate

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Four Related Questions

1. How much must you deposit today to have $1 million in 25 years? (r=12%)

2. If a $58,823.31 investment yields $1 million in 25 years, what is the rate of interest?

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

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

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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?

0 1 2 3 4 5

$10

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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%?

0 1 2 3 4 5

$20,000PV

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

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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

FV2 = (PV0 * (r) + PV0 *(r)) + PV0 = PV0 (1 + 2r) =• Compounded interest

FV2 = PV0 (1+r) (1+r)= PV0 (1+r)2 =

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Compounding Periods

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

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Non-Annual Compounding Cash flows are usually compounded over

periods shorter than a year The relationship between PV & FV when

interest is not compounded annuallyFVN = PV * ( 1+ r / M) M*N

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

M is number of compounding periods per year N is the number of years

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Compounding Examples

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

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

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?

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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 yearThe more frequent the compounding the higher the

Effective Annual Rate

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Compounding Example 2

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

Compounding Example 2: Alt. If you invest $50 for 3 years at 12% compounded

semi-annually, your investment will grow to: Calculate the EAR: EAR = (1 + R/m)m – 1

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

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$70.93

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EAR Example

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

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 =

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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 =

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Present Value Of a Cash Flow Stream

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

PVC

rC

rC

rCr

Cr

N

NN

t

tt

t

N

1

1

2

22

3

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1

1 1 1 1

1

( ) ( ) ( )...

( )

( )=

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Insight Example r = 10%

Year Project A Project B

1 100 300

2 400 400

3 300 100

PV

Which project is more valuable? Why?

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?

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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?

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Multiple Cash Flows (Given)0 1 2 3 4

200 400 600 800178.57

318.88

427.07

508.41

1,432.93

Don’t buy

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 + $200/1.032 – $200/1.033 + $200/1.034 PV = –$10.99. NPV = $10 – $10.99 = –$0.99. You would not take this project

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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

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Perpetuity A stream of cash flows that lasts forever

PV: = C/r What is PV if C=$100 and r=10%:

…0 1

C

2

C

3

C

32 )1()1()1( r

C

r

C

r

CPV

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?

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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?

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Growing Perpetuities Annual payments grow at a constant rate, g

PV= C1/(1+r) + C1(1+g)/(1+r)2 + C1(1+g)2(1+r)3 +… PV = C1/(r-g)

What is PV if C1 =$100, r=10%, and g=2%?

…

0 1 2 3

C1 C2(1+g) C3(1+g)2

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?

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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?

0

…1

$1.30

2

$1.30×(1.05)= $1.37

3

$1.30 ×(1.05)2

= $1.43

PV = 1.30 / (0.10 – 0.05) = $26

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ExampleAn 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?

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AnnuityA constant stream of cash flows with a fixed maturity

0 1

C

2

C

3

C

Tr

C

r

C

r

C

r

CPV

)1()1()1()1( 32

Trr

CPV

)1(

11

T

C

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Annuity Formula

TrrC

r

CPV

)1(

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

0 1

C

2

C

3

C

T

C

T+1

C

T+2

C

T+3

C

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Annuity Example 1

Compute the present value of a 3 year ordinary annuity with payments of $100 at r=10%

Answer:

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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

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Annuity Example 2 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

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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?

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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?

0 2 4 6 8 10

PV $600$600 $600$600$600

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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?

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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

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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?

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

2. Bring value in step 1 to t=0

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%?

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Alt: 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 if your discount rate is 5.5%?

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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?

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Growing AnnuityA growing stream of cash flows with a fixed maturity

0 1

C

T

T

r

gC

r

gC

r

CPV

)1(

)1(

)1(

)1(

)1(

1

2

T

r

g

gr

CPV

)1(

11

2

C×(1+g)

3

C ×(1+g)2

T

C×(1+g)T-1

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Growing Annuity: ExampleA 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%?

0 1

$20,000

2

$20,000×(1.03)

40

$20,000×(1.03)39

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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%?

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

0 1 2 3 4 5

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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:

g:

Price:

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Valuation Formulas

T

r

g

gr

CPV

)1(

11

1

Trr

CPV

)1(

11

gr

CPV

1

r

CPV

n

n

r

FVPV

)1( n

n rP VF V )1(*

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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

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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.

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Quick Quiz

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

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

3. What is the Net Present Value of an investment?

4. What is an EAR, and how is it computed?

5. What is a perpetuity? An annuity?

Why We Care

The Time Value of Money is the basis for all of finance

People will assume that you have this down cold

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