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CHAPTER
5How to Value
Bonds and Stocks
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Chapter Outline
5.1 Definition and Example of a Bond5.2 How to Value Bonds5.3 Bond Concepts5.4 The Present Value of Common Stocks5.5 Estimates of Parameters in the Dividend-Discount Model5.6 Growth Opportunities5.7 The Dividend Growth Model and the NPVGO Model
(Advanced)5.8 Price Earnings Ratio5.9 Stock Market Reporting5.10 Summary and Conclusions
5.1 Definition and Example of a Bond5.2 How to Value Bonds5.3 Bond Concepts5.4 The Present Value of Common Stocks5.5 Estimates of Parameters in the Dividend-Discount Model5.6 Growth Opportunities5.7 The Dividend Growth Model and the NPVGO Model
(Advanced)5.8 Price Earnings Ratio5.9 Stock Market Reporting5.10 Summary and Conclusions
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Valuation of Bonds and Stock
First Principles:Value of financial securities = PV of expected future cash flows
To value bonds and stocks we need to: Estimate future cash flows:
Size (how much) and Timing (when)
Discount future cash flows at an appropriate rate:The rate should be appropriate to the risk presented by the security.
First Principles:Value of financial securities = PV of expected future cash flows
To value bonds and stocks we need to: Estimate future cash flows:
Size (how much) and Timing (when)
Discount future cash flows at an appropriate rate:The rate should be appropriate to the risk presented by the security.
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5.1 Definition and Exampleof a Bond
A bond is a legally binding agreement between a borrower and a lender:
Specifies the principal amount of the loan.
Specifies the size and timing of the cash flows:In dollar terms (fixed-rate borrowing)
As a formula (adjustable-rate borrowing)
A bond is a legally binding agreement between a borrower and a lender:
Specifies the principal amount of the loan.
Specifies the size and timing of the cash flows:In dollar terms (fixed-rate borrowing)
As a formula (adjustable-rate borrowing)
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5.1 Definition and Exampleof a Bond
Consider a U.S. government bond listed as 6 3/8 of December 2009.
The Par Value of the bond is $1,000.
Coupon payments are made semi-annually (June 30 and December 31 for this particular bond).
Since the coupon rate is 6 3/8 the payment is $31.875.
On January 1, 2005 the size and timing of cash flows are:
Consider a U.S. government bond listed as 6 3/8 of December 2009.
The Par Value of the bond is $1,000.
Coupon payments are made semi-annually (June 30 and December 31 for this particular bond).
Since the coupon rate is 6 3/8 the payment is $31.875.
On January 1, 2005 the size and timing of cash flows are:
05/1/1
875.31$
05/30/6
875.31$
05/31/12
875.31$
09/30/6
875.031,1$
09/31/12
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5.2 How to Value Bonds
Identify the size and timing of cash flows.
Discount at the correct discount rate.If you know the price of a bond and the size and timing of cash flows, the yield to maturity is the discount rate.
Identify the size and timing of cash flows.
Discount at the correct discount rate.If you know the price of a bond and the size and timing of cash flows, the yield to maturity is the discount rate.
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Pure Discount BondsInformation needed for valuing pure discount bonds:
Time to maturity (T) = Maturity date - today’s dateFace value (F)Discount rate (r)
Information needed for valuing pure discount bonds:Time to maturity (T) = Maturity date - today’s dateFace value (F)Discount rate (r)
Tr
FPV
)1(
Present value of a pure discount bond at time 0:
0
0$
1
0$
2
0$
1T
F$
T
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Pure Discount Bonds: Example
Find the value of a 30-year zero-coupon bond with a $1,000 par value and a YTM of 6%.
Find the value of a 30-year zero-coupon bond with a $1,000 par value and a YTM of 6%.
11.174$)06.1(
000,1$
)1( 30
Tr
FPV
0
0$
1
0$
2
0$
29
000,1$
30
0
0$
1
0$
2
0$
29
000,1$
30
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Pure Discount Bonds: Example
Find the value of a 30-year zero-coupon bond with a $1,000 par value and a YTM of 6%.
Find the value of a 30-year zero-coupon bond with a $1,000 par value and a YTM of 6%.
PMT
I/Y
FV
PV
N
PV 174.11
6
1,000
30
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Level-Coupon BondsInformation needed to value level-coupon bonds:
Coupon payment dates and time to maturity (T) Coupon payment (C) per period and Face value (F) Discount rate
Information needed to value level-coupon bonds:Coupon payment dates and time to maturity (T) Coupon payment (C) per period and Face value (F) Discount rate
TT r
F
rr
CPV
)1()1(
11
Value of a Level-coupon bond= PV of coupon payment annuity + PV of face value
0
C$
1
C$
2
C$
1T
FC $$
T
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Level-Coupon Bonds: ExampleFind the present value (as of January 1, 2004), of a 6-3/8 coupon T-
bond with semi-annual payments, and a maturity date of December 2009 if the YTM is 5-percent.
On January 1, 2004 the size and timing of cash flows are:
Find the present value (as of January 1, 2004), of a 6-3/8 coupon T-bond with semi-annual payments, and a maturity date of December 2009 if the YTM is 5-percent.
On January 1, 2004 the size and timing of cash flows are:
04/1/1
875.31$
04/30/6
875.31$
04/31/12
875.31$
09/30/6
875.031,1$
09/31/12
52.070,1$)025.1(
000,1$
)025.1(
11
205.
875.31$1212
PV
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Level-Coupon Bonds: Example
PMT
I/Y
FV
PV
N
PV
31.875 =
5
1,000
– 1,070.52
12
1,000×0.06375
2
Find the present value (as of January 1, 2004), of a6-3/8 coupon T-bond with semi-annual payments, anda maturity date of December 2009 if the YTM is5-percent.
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5.3 Bond Concepts1. Bond prices and market interest rates move in opposite
directions.
2. When coupon rate = YTM, price = par value.When coupon rate > YTM, price > par value (premium bond)When coupon rate < YTM, price < par value (discount bond)
3. A bond with longer maturity has higher relative (%) price change than one with shorter maturity when interest rate (YTM) changes. All other features are identical.
4. A lower coupon bond has a higher relative price change than a higher coupon bond when YTM changes. All other features are identical.
1. Bond prices and market interest rates move in opposite directions.
2. When coupon rate = YTM, price = par value.When coupon rate > YTM, price > par value (premium bond)When coupon rate < YTM, price < par value (discount bond)
3. A bond with longer maturity has higher relative (%) price change than one with shorter maturity when interest rate (YTM) changes. All other features are identical.
4. A lower coupon bond has a higher relative price change than a higher coupon bond when YTM changes. All other features are identical.
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YTM and Bond Value
When the YTM < coupon, the bond trades at a premium.
When the YTM = coupon, the bond trades at par.
When the YTM > coupon, the bond trades at a discount.
800
1000
1100
1200
1300
$1400
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Discount Rate
Bon
d V
alu
e
6 3/8
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Maturity and Bond Price Volatility
Consider two otherwise identical bonds.
The long-maturity bond will have much more volatility with respect to
changes in the discount rate
C Discount Rate
Bon
d V
alu
e
Par
Short Maturity Bond
Long Maturity Bond
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Coupon Rate and Bond Price Volatility
Consider two otherwise identical bonds.
The low-coupon bond will have much more volatility with respect to changes in the
discount rate
Discount Rate
Bon
d V
alu
e
High Coupon Bond
Low Coupon Bond
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5.4 The Present Valueof Common Stocks
Dividends versus Capital Gains
Valuation of Different Types of StocksZero Growth
Constant Growth
Differential Growth
Dividends versus Capital Gains
Valuation of Different Types of StocksZero Growth
Constant Growth
Differential Growth
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Case 1: Zero GrowthAssume that dividends will remain at the same level foreverAssume that dividends will remain at the same level forever
rP
rrrP
Div
)1(
Div
)1(
Div
)1(
Div
0
33
22
11
0
321 DivDivDiv
Since future cash flows are constant, the value of a zero growth stock is the present value of a perpetuity:
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Case 2: Constant Growth
)1(DivDiv 01 g
Since future cash flows grow at a constant rate forever, the value of a constant growth stock is the present value of a growing perpetuity:
grP
1
0
Div
Assume that dividends will grow at a constant rate, g, forever. i.e.
2012 )1(Div)1(DivDiv gg
..
3023 )1(Div)1(DivDiv gg
.
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Case 3: Differential Growth
Assume that dividends will grow at different rates in the foreseeable future and then will grow at a constant rate thereafter.To value a Differential Growth Stock, we need to:
Estimate future dividends in the foreseeable future.Estimate the future stock price when the stock becomes a Constant Growth Stock (case 2).Compute the total present value of the estimated future dividends and future stock price at the appropriate discount rate.
Assume that dividends will grow at different rates in the foreseeable future and then will grow at a constant rate thereafter.To value a Differential Growth Stock, we need to:
Estimate future dividends in the foreseeable future.Estimate the future stock price when the stock becomes a Constant Growth Stock (case 2).Compute the total present value of the estimated future dividends and future stock price at the appropriate discount rate.
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Case 3: Differential Growth
)(1DivDiv 101 g
Assume that dividends will grow at rate g1 for N years and grow at rate g2 thereafter
210112 )(1Div)(1DivDiv gg
NNN gg )(1Div)(1DivDiv 1011
)(1)(1Div)(1DivDiv 21021 ggg NNN
...
...
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Case 3: Differential Growth Dividends will grow at rate g1 for N years
and grow at rate g2 thereafter
)(1Div 10 g 210 )(1Div g
…0 1 2
Ng )(1Div 10 )(1)(1Div
)(1Div
210
2
gg
gN
N
…N N+1
…
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Case 3: Differential GrowthWe can value this as the sum of:
an N-year annuity growing at rate g1
T
T
Ar
g
gr
CP
)1(
)1(1 1
1
plus the discounted value of a perpetuity growing at rate g2 that starts in year N+1
NBr
grP
)1(
Div
2
1N
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Case 3: Differential Growth
To value a Differential Growth Stock, we can use
NT
T
r
gr
r
g
gr
CP
)1(
Div
)1(
)1(1 2
1N
1
1
Or we can cash flow it out.
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A Differential Growth Example
A common stock just paid a dividend of $2. The dividend is expected to grow at 8% for 3 years, then it will grow at 4% in perpetuity.
What is the stock worth? The discount rate is 12%.
A common stock just paid a dividend of $2. The dividend is expected to grow at 8% for 3 years, then it will grow at 4% in perpetuity.
What is the stock worth? The discount rate is 12%.
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With the Formula
NT
T
r
gr
r
g
gr
CP
)1(
Div
)1(
)1(1 2
1N
1
1
3
3
3
3
)12.1(04.12.
)04.1()08.1(2$
)12.1()08.1(
108.12.
)08.1(2$
P
3)12.1(
75.32$8966.154$ P
31.23$58.5$ P 89.28$P
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A Differential Growth Example (continued)
08).2(1$ 208).2(1$…
0 1 2 3 4
308).2(1$ )04.1(08).2(1$ 3
16.2$ 33.2$
0 1 2 3
08.
62.2$52.2$
89.28$)12.1(
75.32$52.2$
)12.1(
33.2$
12.1
16.2$320
P
75.32$08.
62.2$3 P
The constant growth phase
beginning in year 4 can be valued as a
growing perpetuity at time 3.
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A Differential Growth Example (continued)
PMT
I/Y
FV
PV
N
$2 =
3.70 =
0
– 5.58
3
PV
A common stock just paid a dividend of $2. The dividend is expected to grow at 8% for 3 years, then it will grow at 4% in perpetuity. What is the stock worth?
$28.89 = 5.58 + 23.31First find the PV of the supernormal dividend stream then find
the PV of the steady-state dividend stream.
2×1.08
1.08
1.12
1.08–1 ×100
PMT
I/Y
FV
PV
N
0
12
32.75 =
3
PV – 23.31
2×(1.08)3 ×(1.04).08
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5.5 Estimates of Parameters in the Dividend-Discount Model
The value of a firm depends upon its growth rate, g, and its discount rate, r.
Where does g come from?
Where does r come from?
The value of a firm depends upon its growth rate, g, and its discount rate, r.
Where does g come from?
Where does r come from?
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Where does g come from?
g = Retention ratio × Return on retained earningsg = Retention ratio × Return on retained earnings
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Where does r come from?
The discount rate can be broken into two parts. The dividend yield
The growth rate (in dividends)
In practice, there is a great deal of estimation error involved in estimating r.
The discount rate can be broken into two parts. The dividend yield
The growth rate (in dividends)
In practice, there is a great deal of estimation error involved in estimating r.
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5.6 Growth Opportunities
Growth opportunities are opportunities to invest in positive NPV projects.
The value of a firm can be conceptualized as the sum of the value of a firm that pays out 100-percent of its earnings as dividends and the net present value of the growth opportunities.
Growth opportunities are opportunities to invest in positive NPV projects.
The value of a firm can be conceptualized as the sum of the value of a firm that pays out 100-percent of its earnings as dividends and the net present value of the growth opportunities.
NPVGOr
EPSP
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5.7 The Dividend Growth Model and the NPVGO Model (Advanced)
We have two ways to value a stock:The dividend discount model.
The price of a share of stock can be calculated as the sum of its price as a cash cow plus the per-share value of its growth opportunities.
We have two ways to value a stock:The dividend discount model.
The price of a share of stock can be calculated as the sum of its price as a cash cow plus the per-share value of its growth opportunities.
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The Dividend Growth Model and the NPVGO Model
Consider a firm that has EPS of $5 at the end of the first year, a dividend-payout ratio of 30%, a discount rate of 16-percent, and a return on retained earnings of 20-percent.The dividend at year one will be $5 × .30 = $1.50 per share.
The retention ratio is .70 ( = 1 -.30) implying a growth rate in dividends of 14% = .70 × 20%
From the dividend growth model, the price of a share is:
Consider a firm that has EPS of $5 at the end of the first year, a dividend-payout ratio of 30%, a discount rate of 16-percent, and a return on retained earnings of 20-percent.The dividend at year one will be $5 × .30 = $1.50 per share.
The retention ratio is .70 ( = 1 -.30) implying a growth rate in dividends of 14% = .70 × 20%
From the dividend growth model, the price of a share is:
75$14.16.
50.1$Div10
grP
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The NPVGO Model First, we must calculate the value of the firm
as a cash cow. First, we must calculate the value of the firm
as a cash cow.
25.31$16.
5$Div10
rP
Second, we must calculate the value of the growth opportunities.
75.43$14.16.
875$.16.20.50.3
50.3
0
gr
P
Finally, 75$75.4325.310 P
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5.8 Price Earnings RatioMany analysts frequently relate earnings per share to price.
The price earnings ratio is a.k.a. the multipleCalculated as current stock price divided by annual EPS
The Wall Street Journal uses last 4 quarter’s earnings
Many analysts frequently relate earnings per share to price.
The price earnings ratio is a.k.a. the multipleCalculated as current stock price divided by annual EPS
The Wall Street Journal uses last 4 quarter’s earnings
EPS
shareper Priceratio P/E
Firms whose shares are “in fashion” sell at high multiples. Growth stocks for example.
Firms whose shares are out of favor sell at low multiples. Value stocks for example.
Firms whose shares are “in fashion” sell at high multiples. Growth stocks for example.
Firms whose shares are out of favor sell at low multiples. Value stocks for example.
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Other Price Ratio AnalysisMany analysts frequently relate earnings per share to variables other than price, e.g.:
Price/Cash Flow Ratiocash flow = net income + depreciation = cash flow from operations or operating cash flow
Price/Salescurrent stock price divided by annual sales per share
Price/Book (a.k.a. Market to Book Ratio)price divided by book value of equity, which is measured as assets – liabilities
Many analysts frequently relate earnings per share to variables other than price, e.g.:
Price/Cash Flow Ratiocash flow = net income + depreciation = cash flow from operations or operating cash flow
Price/Salescurrent stock price divided by annual sales per share
Price/Book (a.k.a. Market to Book Ratio)price divided by book value of equity, which is measured as assets – liabilities
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5.9 Stock Market Reporting52WEEKS YLD VOL NET
HI LO STOCKSYMDIV % PE 100s HI LOCLOSE CHG52.75 19.06 Gap Inc GPS 0.09 0.5 15 65172 20.50 19 19.25 -1.75
Gap has been as high as $52.75 in the last year.
Gap has been as low as $19.06 in the last year.
Gap pays a dividend of 9 cents/share
Given the current price, the dividend yield is ½ %
Given the current price, the PE ratio is 15 times earnings
6,517,200 shares traded hands in the last day’s trading
Gap ended trading at $19.25, down $1.75 from yesterday’s close
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5.9 Stock Market Reporting52WEEKS YLD VOL NET
HI LO STOCKSYMDIV % PE 100s HI LOCLOSE CHG52.75 19.06 Gap Inc GPS 0.09 0.5 15 65172 20.50 19 19.25 -1.75
Gap Incorporated is having a tough year, trading near their 52-week low. Imagine how you would feel if within the past year you had paid $52.75 for a share of Gap and now had a share worth $19.25! That 9-cent dividend wouldn’t go very far in making amends.
Yesterday, Gap had another rough day in a rough year. Gap “opened the day down” beginning trading at $20.50, which was down from the previous close of $21.00 = $19.25 + $1.75
Looks like cargo pants aren’t the only things on sale at Gap.
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5.10 Summary and Conclusions
In this chapter, we used the time value of money formulae from previous chapters to value bonds and stocks.
1. The value of a zero-coupon bond is
2. The value of a perpetuity is
In this chapter, we used the time value of money formulae from previous chapters to value bonds and stocks.
1. The value of a zero-coupon bond is
2. The value of a perpetuity is
Tr
FPV
)1(
r
CPV
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5.10 Summary and Conclusions (continued)
3. The value of a coupon bond is the sum of the PV of the annuity of coupon payments plus the PV of the par value at maturity.
3. The value of a coupon bond is the sum of the PV of the annuity of coupon payments plus the PV of the par value at maturity.
TT r
F
rr
CPV
)1()1(
11
The yield to maturity (YTM) of a bond is that single rate that discounts the payments on the bond to the purchase price.
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5.10 Summary and Conclusions (continued)
5. A stock can be valued by discounting its dividends. There are three cases:
5. A stock can be valued by discounting its dividends. There are three cases:
NT
T
r
gr
rg
grC
P)1(
Div
)1()1(
1 2
1N
1
1
rP
Div0Zero growth in dividends
grP
1
0
DivConstant growth in dividends
Differential growth in dividends
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5.10 Summary and Conclusions (continued)
6. The growth rate can be estimated as:g = Retention ratio × Return on retained earnings
7. An alternative method of valuing a stock was presented, the NPVGO values a stock as the sum of its “cash cow” value plus the present value of growth opportunities.
6. The growth rate can be estimated as:g = Retention ratio × Return on retained earnings
7. An alternative method of valuing a stock was presented, the NPVGO values a stock as the sum of its “cash cow” value plus the present value of growth opportunities.
NPVGOr
EPSP