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1 Discounted Cash Flow Valuation Chapter 4 You want to retire at the age of 60 and you estimate that you will need $100,000 a year for the rest of your life – HOW MUCH DO YOU NEED TO SAVE PER MONTH STARTING TODAY? If I can afford to pay $1000 a month, WHAT PRICE OF A HOUSE SHOULD I CONSIDER? When buying a new car, which offer is better – the $3000 rebate or the 1.9% interest rate over 60 months? How can I value a business/firm? What about corporate or government bonds?

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Page 1: Discounted Cash Flow Valuation - Indiana University · 1 Discounted Cash Flow Valuation Chapter 4 ! You want to retire at the age of 60 and you estimate that you will need $100,000

1

Discounted Cash Flow Valuation

Chapter 4

! You want to retire at the age of 60 and you estimate that you will need $100,000 a year for the rest of your life – HOW MUCH DO YOU NEED TO SAVE PER MONTH STARTING TODAY?

! If I can afford to pay $1000 a month, WHAT PRICE OF A HOUSE SHOULD I CONSIDER?

! When buying a new car, which offer is better – the $3000 rebate or the 1.9% interest rate over 60 months?

! How can I value a business/firm? What about corporate or government bonds?

Page 2: Discounted Cash Flow Valuation - Indiana University · 1 Discounted Cash Flow Valuation Chapter 4 ! You want to retire at the age of 60 and you estimate that you will need $100,000

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}  Describe and compute the future value and/or present value of a single cash flow or series of cash flows

}  Define and calculate the return on an investment }  Recognize and compute the impact of

compounding periods on the true return of stated interest rates

}  Use a financial calculator and spreadsheets to solve time value problems

}  Comprehend and calculate time value metrics for perpetuities and annuities

}  Familiarization with loan types and amortization

To Use DCF We Need To Know Three Things:

1.  The Amount Of The Projected Cash Flows 2.  The Timing Of The Cash Flows 3.  The Proper Discount (Interest) Rate, r

(r should reflect current capital market conditions, and generally can include a premium for risk)

Page 3: Discounted Cash Flow Valuation - Indiana University · 1 Discounted Cash Flow Valuation Chapter 4 ! You want to retire at the age of 60 and you estimate that you will need $100,000

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}  Individuals and institutions have different income streams and different intertemporal consumption preferences.

}  An individual can alter his consumption across time periods through borrowing and lending. ◦  The job of balancing the supply of and demand for

loanable funds is taken by the money market. ◦  When the quantity supplied equals the quantity

demanded, the market is in equilibrium at the equilibrium price.

}  Because of this, a market has arisen for money. The price of money is the interest rate.

Page 4: Discounted Cash Flow Valuation - Indiana University · 1 Discounted Cash Flow Valuation Chapter 4 ! You want to retire at the age of 60 and you estimate that you will need $100,000

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}  A dollar today is more valuable than a dollar to be received in the future

}  Why? ◦  A dollar today is more valuable because: �  It can be invested to make more dollars �  It can be immediately consumed �  There is no doubt about its receipt

4-7

}  If you know your required rate of return and the length of time before cash is harvested, you can calculate some critical metrics:

◦  The value today of a payment to be received in the future �  This measure is called a “Present Value” ◦  The value in the future of a sum invested today �  This measure is called a “Future Value”

◦  Present and Future Values can be calculated over single and multiple periods

Page 5: Discounted Cash Flow Valuation - Indiana University · 1 Discounted Cash Flow Valuation Chapter 4 ! You want to retire at the age of 60 and you estimate that you will need $100,000

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}  If you were to invest $10,000 at 5-percent interest for one year, your investment would grow to $10,500.

$500 would be interest ($10,000 × .05) $10,000 is the principal repayment ($10,000 ×1) $10,500 is the total due. It can be calculated as:

$10,500 = $10,000×(1.05)

q The total amount due at the end of the investment is call the Future Value (FV).

}  If you were to be promised $10,000 due in one year when interest rates are 5-percent, your investment would be worth $9,523.81 in today’s dollars.

05.1000,10$81.523,9$ =

The amount that a borrower would need to set aside today to be able to meet the promised payment of $10,000 in one year is called the Present Value (PV).

Note that $10,000 = $9,523.81×(1.05).

Page 6: Discounted Cash Flow Valuation - Indiana University · 1 Discounted Cash Flow Valuation Chapter 4 ! You want to retire at the age of 60 and you estimate that you will need $100,000

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}  In the one-period case, the formula for FV can be written as:

FV = C0×(1 + r) Where C0 is cash flow today (time zero), and r is the appropriate interest rate

}  In the one-period case, the formula for PV can be written as:

r

CPV+

=1

1

Where C1 is the cash flow at date one, and r is the appropriate interest rate

}  The Net Present Value (NPV) of an investment is the present value of the expected cash flows, less the cost of the investment.

}  Suppose an investment that promises to pay $10,000 in one year is offered for sale for $9,500. Your interest rate is 5%. Should you buy?

Page 7: Discounted Cash Flow Valuation - Indiana University · 1 Discounted Cash Flow Valuation Chapter 4 ! You want to retire at the age of 60 and you estimate that you will need $100,000

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81.23$81.523,9$500,9$

05.1000,10$500,9$

=

+−=

+−=

NPVNPV

NPV

The present value of the cash inflow is greater than the cost. In other words, the Net Present Value is positive, so the investment should be purchased.

In the one-period case, the formula for NPV can be written as:

NPV = –Cost + PV If we had not undertaken the positive NPV project considered on the last slide, and instead invested our $9,500 elsewhere at 5 percent, our FV would be less than the $10,000 the investment promised, and we would be worse off in FV terms :

$9,500×(1.05) = $9,975 < $10,000

Page 8: Discounted Cash Flow Valuation - Indiana University · 1 Discounted Cash Flow Valuation Chapter 4 ! You want to retire at the age of 60 and you estimate that you will need $100,000

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}  The general formula for the future value of an investment over many periods can be written as:

FV = C0×(1 + r)T

Where C0 is cash flow at date 0,

r is the appropriate interest rate, and T is the number of periods over which the cash is invested.

}  Suppose a stock currently pays a dividend of $1.10, which is expected to grow at 40% per year for the next five years.

}  What will the dividend be in five years?

FV = C0×(1 + r)T

$5.92 = $1.10×(1.40)5

Page 9: Discounted Cash Flow Valuation - Indiana University · 1 Discounted Cash Flow Valuation Chapter 4 ! You want to retire at the age of 60 and you estimate that you will need $100,000

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}  Notice that the dividend in year five, $5.92, is considerably higher than the sum of the original dividend plus five increases of 40-percent on the original $1.10 dividend:

$5.92 > $1.10 + 5×[$1.10×.40] = $3.30

This is due to compounding.

0 1 2 3 4 5

10.1$

3)40.1(10.1$ ×

02.3$

)40.1(10.1$ ×

54.1$

2)40.1(10.1$ ×

16.2$

5)40.1(10.1$ ×

92.5$

4)40.1(10.1$ ×

23.4$

Page 10: Discounted Cash Flow Valuation - Indiana University · 1 Discounted Cash Flow Valuation Chapter 4 ! You want to retire at the age of 60 and you estimate that you will need $100,000

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}  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

$20,000 PV

5)15.1(000,20$53.943,9$ =

}  Examples thus far have offered the time and interest rate and solved for PV or FV

}  Keep in mind that there are four variables: ◦  PV ◦  FV ◦  T ◦  R

}  If you have any three you can solve for the fourth }  The math can become cumbersome ◦  Financial Calculators and Spreadsheets are very helpful

Page 11: Discounted Cash Flow Valuation - Indiana University · 1 Discounted Cash Flow Valuation Chapter 4 ! You want to retire at the age of 60 and you estimate that you will need $100,000

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If we deposit $5,000 today in an account paying 10%, how long does it take to grow to $10,000?

TrCFV )1(0 +×= T)10.1(000,5$000,10$ ×=

2000,5$000,10$)10.1( ==T

)2ln()10.1ln( =T

years 27.70953.06931.0

)10.1ln()2ln(

===T

Assume the total cost of a college education will be $50,000 when your child enters college in 12 years. You have $5,000 to invest today. What rate of interest must you earn on your investment to cover the cost of your child’s education?

TrCFV )1(0 +×= 12)1(000,5$000,50$ r+×=

10000,5$000,50$)1( 12 ==+ r 12110)1( =+ r

2115.12115.1110 121 =−=−=r

About 21.15%.

Page 12: Discounted Cash Flow Valuation - Indiana University · 1 Discounted Cash Flow Valuation Chapter 4 ! You want to retire at the age of 60 and you estimate that you will need $100,000

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}  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?

0 1 2 3 4

200 400 600 800 178.57

318.88

427.07

508.41

1,432.93 Present Value < Cost → Do Not Purchase

Page 13: Discounted Cash Flow Valuation - Indiana University · 1 Discounted Cash Flow Valuation Chapter 4 ! You want to retire at the age of 60 and you estimate that you will need $100,000

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First, set your calculator to 1 payment per year. Then, use the cash flow menu:

1,432.93

CF2

CF1

F2

F1

CF0

1

200

1

0

400

I

NPV

12

CF4

CF3

F4

F3 1

600

1

800

}  All examples thus far have assumed annual compounding

}  Instances of other compounding schedules abound: ◦  Banks compound interest quarterly, monthly or

daily ◦  Mortgage companies compound interest monthly

}  Yet, almost all interest rates are expressed annually }  If a rate is expressed annually, but compounded

more frequently, then the effective rate is higher than the stated rate

}  This concept is called the Effective Annual Rate or EAR

Page 14: Discounted Cash Flow Valuation - Indiana University · 1 Discounted Cash Flow Valuation Chapter 4 ! You want to retire at the age of 60 and you estimate that you will need $100,000

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Compounding an investment m times a year for T years provides for the future value of wealth:

mT

mrCFV ⎟⎠

⎞⎜⎝

⎛ +×= 10

q  For example, if you invest $50 for 3 years at 12% compounded semi-annually, your investment will grow to

93.70$)06.1(50$212.150$ 6

32

=×=⎟⎠

⎞⎜⎝

⎛ +×=×

FV

Page 15: Discounted Cash Flow Valuation - Indiana University · 1 Discounted Cash Flow Valuation Chapter 4 ! You want to retire at the age of 60 and you estimate that you will need $100,000

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A reasonable question to ask in the above example is “what is the effective annual rate of interest on that investment?”

The Effective Annual Rate (EAR) of interest is the annual rate that would give us the same end-of-investment wealth after 3 years:

93.70$)06.1(50$)212.1(50$ 632 =×=+×= ×FV

93.70$)1(50$ 3 =+× EAR

So, investing at 12.36% compounded annually is the same as investing at 12% compounded semi-annually.

93.70$)1(50$ 3 =+×= EARFV

50$93.70$)1( 3 =+ EAR

1236.150$93.70$ 31

=−⎟⎠

⎞⎜⎝

⎛=EAR

Page 16: Discounted Cash Flow Valuation - Indiana University · 1 Discounted Cash Flow Valuation Chapter 4 ! You want to retire at the age of 60 and you estimate that you will need $100,000

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

}  What we have is a loan with a monthly interest rate rate of 1½%.

}  This is equivalent to a loan with an annual interest rate of 19.56%.

1956.1)015.1(1218.11 12

12

==⎟⎠

⎞⎜⎝

⎛ +=⎟⎠

⎞⎜⎝

⎛ +m

mr

Stated And Effective Rates

■  Be Careful with Effective Rates – remember that the annuity and perpetuity formulas all require an “Effective rate” to be used properly

11 −⎟

⎞⎜⎝

⎛ +=t

mQuotedraterateEffective

Where m = the number of compounding periods in the stated or quoted rate

t= the number of times you receive or pay the “interest” in the “effective rate” period

The Effective Annual Rate is a special case of the effective rate formula where m=t!

Page 17: Discounted Cash Flow Valuation - Indiana University · 1 Discounted Cash Flow Valuation Chapter 4 ! You want to retire at the age of 60 and you estimate that you will need $100,000

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}  Make sure you understand that simplifying formulas require an “effective rate” for the period of time between payments

}  APR vs APY – for additional reading and how banks use this number, read the APR vs. APY article at: APR versus APY

}  Perpetuity ◦  A constant stream of cash flows that lasts

forever }  Growing perpetuity ◦  A stream of cash flows that grows at a constant

rate forever }  Annuity ◦  A stream of constant cash flows that lasts for a

fixed number of periods }  Growing annuity ◦  A stream of cash flows that grows at a constant

rate for a fixed number of periods

Page 18: Discounted Cash Flow Valuation - Indiana University · 1 Discounted Cash Flow Valuation Chapter 4 ! You want to retire at the age of 60 and you estimate that you will need $100,000

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A constant stream of cash flows that lasts forever

0

…1

C

2

C

3

C

!++

++

++

= 32 )1()1()1( rC

rC

rCPV

rCPV =

What is the value of a British consol that promises to pay £15 every year for ever?

The interest rate is 10%.

0

…1

£15

2

£15

3

£15

£15010.£15

==PV

Page 19: Discounted Cash Flow Valuation - Indiana University · 1 Discounted Cash Flow Valuation Chapter 4 ! You want to retire at the age of 60 and you estimate that you will need $100,000

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A growing stream of cash flows that lasts forever

0

…1

C

2

C×(1+g)

3

C ×(1+g)2

!++

+×+

+

+×+

+= 3

2

2 )1()1(

)1()1(

)1( rgC

rgC

rCPV

grCPV−

=

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)

3

$1.30 ×(1.05)2

00.26$05.10.30.1$

=−

=PV

Page 20: Discounted Cash Flow Valuation - Indiana University · 1 Discounted Cash Flow Valuation Chapter 4 ! You want to retire at the age of 60 and you estimate that you will need $100,000

20

38

A stream of constant cash flows that lasts for a fixed number of periods

0 1

C

2

C

3

C

TrC

rC

rC

rCPV

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

++

++

+= !

⎥⎦

⎤⎢⎣

+−= Trr

CPV)1(

11

T

C

!

⎥⎦

⎤⎢⎣

+−= Trrr

CPV)1(*

11*

Page 21: Discounted Cash Flow Valuation - Indiana University · 1 Discounted Cash Flow Valuation Chapter 4 ! You want to retire at the age of 60 and you estimate that you will need $100,000

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If you can afford a $400 monthly car payment, how much car can you afford if interest rates are 7% on 36-month loans?

0 1

$400

2

$400

3

$400

59.954,12$)1207.1(

1112/07.

400$36 =⎥⎦

⎤⎢⎣

+−=PV

36

$400

!

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%?

22.297$09.197.323$

0 ==PV0 1 2 3 4 5

$100 $100 $100 $100 $323.97 $297.22

97.323$)09.1(

100$)09.1(

100$)09.1(

100$)09.1(

100$)09.1(

100$4321

4

11 =+++==∑

=ttPV

Page 22: Discounted Cash Flow Valuation - Indiana University · 1 Discounted Cash Flow Valuation Chapter 4 ! You want to retire at the age of 60 and you estimate that you will need $100,000

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A growing stream of cash flows with a fixed maturity

0 1

C

T

T

rgC

rgC

rCPV

)1()1(

)1()1(

)1(

1

2 +

+×++

+

+×+

+=

!

⎥⎥⎦

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

+

+−

−=

T

rg

grCPV

)1()1(1

!2

C×(1+g)

3

C ×(1+g)2

T

C×(1+g)T-1

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

0 1

$20,000

57.121,265$10.103.11

03.10.000,20$ 40

=⎥⎥⎦

⎢⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛−−

=PV

!2

$20,000×(1.03)

40

$20,000×(1.03)39

Page 23: Discounted Cash Flow Valuation - Indiana University · 1 Discounted Cash Flow Valuation Chapter 4 ! You want to retire at the age of 60 and you estimate that you will need $100,000

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}  Pure Discount Loans are the simplest form of loan. The borrower receives money today and repays a single lump sum (principal and interest) at a future time.

}  Interest-Only Loans require an interest payment each period, with full principal due at maturity.

}  Amortized Loans require repayment of principal over time, in addition to required interest.

}  Each payment covers the interest expense plus reduces principal

}  Consider a 4 year loan with annual payments. The interest rate is 8% ,and the principal amount is $5,000. ◦  What is the annual payment?

�  4 N �  8 I/Y �  5,000 PV �  CPT PMT = -1,509.60

}  Click on the Excel icon to see the amortization table

Page 24: Discounted Cash Flow Valuation - Indiana University · 1 Discounted Cash Flow Valuation Chapter 4 ! You want to retire at the age of 60 and you estimate that you will need $100,000

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}  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.

}  You can solve time value problems in any of four ways: ◦  Math (Formulae given above) ◦  Tables (See Appendix A) ◦  Financial Calculator ◦  Spreadsheet Software

}  Financial calculators and spreadsheet software are the most common methods now.

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