Chapter 4 Discounted Cash Flow Valuation You want to retire at the - - PDF document
Chapter 4 Discounted Cash Flow Valuation 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
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Discounted Cash Flow Valuation
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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} 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:
(r should reflect current capital market conditions, and generally can include a premium for risk)
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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.
loanable funds is taken by the money market.
demanded, the market is in equilibrium at the equilibrium price.
} Because of this, a market has arisen for
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} A dollar today is more valuable than a dollar
to be received in the future
} Why?
It can be invested to make more dollars It can be immediately consumed There is no doubt about its receipt
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} If you know your required rate of return and the
length of time before cash is harvested, you can calculate some critical metrics:
the future
This measure is called a “Present Value”
This measure is called a “Future Value”
single and multiple periods
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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
your investment would be worth $9,523.81 in today’s dollars.
05 . 1 000 , 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).
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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 C PV + = 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?
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81 . 23 $ 81 . 523 , 9 $ 500 , 9 $ 05 . 1 000 , 10 $ 500 , 9 $ = + − = + − = NPV NPV 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
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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
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} Notice that the dividend in year five, $5.92,
is considerably higher than the sum of the
percent on the original $1.10 dividend: $5.92 > $1.10 + 5×[$1.10×.40] = $3.30 This is due to compounding. 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 $
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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%? 1 2 3 4 5
$20,000 PV
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) 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:
} If you have any three you can solve for the fourth } The math can become cumbersome
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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?
T
r C FV ) 1 ( + × =
T
) 10 . 1 ( 000 , 5 $ 000 , 10 $ × = 2 000 , 5 $ 000 , 10 $ ) 10 . 1 ( = =
T
) 2 ln( ) 10 . 1 ln( =
T
years 27 . 7 0953 . 6931 . ) 10 . 1 ln( ) 2 ln( = = = 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?
T
r C FV ) 1 ( + × =
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) 1 ( 000 , 5 $ 000 , 50 $ r + × = 10 000 , 5 $ 000 , 50 $ ) 1 (
12
= = + r
12 1
10 ) 1 ( = + r 2115 . 1 2115 . 1 1 10 12
1
= − = − = r
About 21.15%.
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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?
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
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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 1 400
I NPV
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CF4 CF3 F4 F3
1 600 1 800
} All examples thus far have assumed annual
compounding
} Instances of other compounding schedules abound:
daily
} 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
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Compounding an investment m times a year for T years provides for the future value of wealth:
mT
m r C FV ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + × = 1
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 $ 2 12 . 1 50 $
6 3 2
= × = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + × =
×
FV
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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 $ ) 2 12 . 1 ( 50 $
6 3 2
= × = + × =
×
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 =
+ × = EAR FV 50 $ 93 . 70 $ ) 1 (
3 =
+ EAR 1236 . 1 50 $ 93 . 70 $
3 1
= − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = EAR
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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 ( 12 18 . 1 1
12 12
= = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ +
m
m r
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 1 1 − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + =
t
m Quotedrate rate Effective Where m = the number of compounding periods in the stated
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!
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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
forever
} Growing perpetuity
rate forever
} Annuity
fixed number of periods
} Growing annuity
rate for a fixed number of periods
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A constant stream of cash flows that lasts forever
1 C 2 C 3 C
+ + + + + =
3 2
) 1 ( ) 1 ( ) 1 ( r C r C r C PV r C PV =
What is the value of a British consol that promises to pay £15 every year for ever? The interest rate is 10%.
1 £15 2 £15 3 £15
£150 10 . £15 = = PV
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A growing stream of cash flows that lasts forever
1 C 2 C×(1+g) 3 C ×(1+g)2
+ + × + + + × + + =
3 2 2
) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( r g C r g C r C PV g r C PV − =
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
1 $1.30 2 $1.30×(1.05) 3 $1.30 ×(1.05)2
00 . 26 $ 05 . 10 . 30 . 1 $ = − = PV
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A stream of constant cash flows that lasts for a fixed number of periods
1 C 2 C 3 C
T
r C r C r C r C PV ) 1 ( ) 1 ( ) 1 ( ) 1 (
3 2
+ + + + + + + =
⎦ ⎤ ⎢ ⎣ ⎡ + − =
T
r r C PV ) 1 ( 1 1
T C
⎦ ⎤ ⎢ ⎣ ⎡ + − =
T
r r r C PV ) 1 ( * 1 1 *
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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? 1 $400 2 $400 3 $400
59 . 954 , 12 $ ) 12 07 . 1 ( 1 1 12 / 07 . 400 $
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= ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + − = PV
36 $400
per year that makes its first payment two years from today if the discount rate is 9%?
22 . 297 $ 09 . 1 97 . 323 $ = = PV
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 $
4 3 2 1 4 1 1
= + + + = = ∑
= t t
PV
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A growing stream of cash flows with a fixed maturity 1 C
T T
r g C r g C r C PV ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 (
1 2
+ + × + + + + × + + =
−
⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + − − =
T
r g g r C PV ) 1 ( ) 1 ( 1
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?
1 $20,000
57 . 121 , 265 $ 10 . 1 03 . 1 1 03 . 10 . 000 , 20 $
40
= ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − = PV
$20,000×(1.03) 40 $20,000×(1.03)39
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} Pure Discount Loans are the simplest form
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.
4 N 8 I/Y 5,000 PV CPT PMT = -1,509.60
} Click on the Excel icon to see the
amortization table
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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:
} Financial calculators and spreadsheet
software are the most common methods now.
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