Unformatted text preview: Class Roadmap
Bonds
Stocks
Conclusion Bond Pricing,
the Term Structure of Interest Rates
and Stock Pricing
Ding Ding
University of Toronto
September 28, 2011 D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Class Roadmap Bonds
Bonds and Bond Markets
Bond Pricing
The Yield Curve
Forward Rates Stocks
Stocks and Equity Financing
Valuation D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure Bond Characteristics – What is a Bond?
A bond is a borrowing arrangement in which the borrower issues
(sells) IOU to the investor.
Issuer receives a certain amount of money from the
Purchaser(Creditor) in return for the bond
Issuer is obligated to repay coupons + the principal at the end
of the lifetime of the bond (maturity)
It speciﬁes
Maturity
Face/Par Value
Coupon Rate (Coupon = CouponRate ×FaceValue
No .ofCouponPaymentsPerYear ) D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure Bond Characteristics – Issuers Governments/Treasuries
E.g., Tbills/notes/bonds Municipalities
Agencies
E.g., CMHC’s Canada Mortgage Bond Program, Ginnie Mae’s
Mortgage Bonds, etc Corporations
Notes, Debentures, Mortgage/Assetbacked D.Ding Bonds and Stocks Ca
C Class Roadmap
Bonds
Stocks
Conclusion 24.73% Characteristics
Bond Pricing quality
Very poor
Bond Yields
Yield to Maturity and Term Structure Imminent default or
in default Bond Ratings Equivalent Credit Ratings
Credit Risk Moody's Standard & Poor's Fitch IBCA Duff & Phelps INVESTMENT GRADE
Highest quality Aaa AAA AAA AAA High quality (very strong) Aa AA AA AA Upper medium grade (strong) A A A A Baa BBB BBB BBB Lower medium grade (somewhat speculative) Ba BB BB BB Low grade (speculative) B B B B Poor quality (may default) Caa CCC CCC CCC Most speculative Ca CC CC CC No interest being paid or bankruptcy petition filed C C C C In default C D D D Medium grade
NOT INVESTMENT GRADE Source: The Bond Market Association D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure Bond Ratings  Average Default Rates
Moody's
Rating Average Default Rate Within One
Year of Rating (19702001) Definition Notes Aaa 0.00% Highest Rating
Available Investment grade bonds. Aa 0.02% Very High Quality A 0.01% High Quality Baa 0.15% Minimum
Investment Grade Ba 1.21% Low grade B 6.53% Very speculative Caa
Ca Substantial Risk
Very poor quality 24.73% C Imminent default or
in default D.Ding Bonds and Stocks Below investment grade.
"Junk Bonds" Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure Bond Characteristics – Zeros Pure discount bonds (zerocoupon bonds)
For instance, a Treasury bill (Tbill)
It speciﬁes
The issuer (government) The face value/par value (e.g.
$1, 000, or $10, 000) Maturity (e.g., 3 months, 6 months, with
a date)
Sold for less than their face value (at a discount) D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure Bond Characteristics – Straight Bonds Couponpaying Bonds
E.g., government of Canada bonds, corporate bonds.
Usually, pay a coupon of the face value (e.g. 8%) at regular
periods, e.g., annually or semiannually, etc.
The maturity, the face value, and the issuer are also quoted.
At maturity, the face value is repaid. D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure Cashﬂow, Time to Maturity, Interest Rate – Key to Pricing The value of a bond depends on the size of its coupon payments,
the length of time remaining until the bond matures and the
current level of interest rates. The value of a bond = present value of its cash ﬂows (coupons +
principal) discounted at a suitable interest rate(s) D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure Example
A zero coupon bond with face value of $100 is trading for $80. It
matures in six years from now. The current interest rate is 7%.
In bond pricing notation for the class,
P = $80
F = $100
r = 7%
T=6 D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure Example
A 8% coupon bond with a par value of $100 is trading for $110. It
matures in three years from now and pays the coupon
semiannually. The current interest rate is 6%.
In bond pricing notation for the class,
P = $110
F = $100
r/m = 3% (eﬀective periodic interest rate)
m = 2 (the number of coupon payments per year)
N = m × T = 2 × 3 = 6 (the total number of periods) D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure Bond Pricing I — The Zero Coupon Bond P= F
(1+r /m)N Note that, as r ↑, price of the zero ↓ D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure Bond Pricing — The Law of One Price (No Arbitrage)
The Law of One Price (LOP) indicates that two assets with
identical cash ﬂows must trade at the same price.
Suppose the interest rate on bank deposits is 9%.
There are two ways to get $10, 000 in one year:
1 Deposit amount C in the bank, which gives you $10, 000 in
one year:
F
$10, 000
C=
=
= $9, 174
1+r
1 + 9% 2 Buy a Treasury bill Since both strategies produce $10, 000 in one year, they
should cost the same today.
The TBill should trade for $9, 174.
D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure Bond Pricing II – More on Pure Discount Bonds (1)
For the pricing, the basis for the discount rate is 360 days.
With n days to maturity, and d % the discount from face
value, the $price P typically quoted as per $100 is
P = 100 × 1 − n
d%
360 For instance, a 91day Tbill’s price = 4.36% means
P = 100 1 − 91
4.36%
360 = 98.898% of face value So if the face value is F = $10, 000, you have to pay
P = $9, 889.80 for the Tbill.
D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure Bond Pricing II – More on Pure Discount Bonds (2) d is also called bank discount yield
d is calculated as:
d% = 100 − P
360
×
100
n D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure Bond Pricing II – More on Pure Discount Bonds (3)
Bond equivalent yield is used when computing Tbill yields.
One uses 365 days as a basis.
$price P is typically quoted as per $100.
Given $price P and n days to maturity the formula for the
bond equivalent yield y for a TBill is
y= 100 − P
365
×
P
n D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure Bond Pricing II – More on Pure Discount Bonds (4) y= 365
100 − P
×
P
n Note that this is not the eﬀective annual rate but merely a
simple annualization.
Use simple interest rather than compound interest
One can understand this rate to be the rate of return that
investors require for the TBill investment. D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure Bond Pricing III — Coupon Bonds (1)
Pay a coupon of the face value (e.g. 8%), either annual or
semiannual (etc.).
The coupon payment is quoted annual.
A $1, 000 face value bond with 8% semiannual coupons pays
1
2 × 8% × $1, 000 = $40 every half year.
Note: The coupon rate is not the interest rate!
interest rate is set by the market according to demand and
supply,
the coupon rate is determined by the bondissuer.
D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure Bond Pricing III — Coupon Bonds (2)
Bond value = PV of coupons + PV of par value.
Let maturity date be T , the prevailing market interest rate be
r,
T
coupon
par value
Bond value =
.
t+
(1 + r )
(1 + r )T
t =1
or if there is m > 1 coupon payments per year, let N = m × T
N Bond value =
t =1 par value
coupon
.
t+
(1 + r /m)
(1 + r /m)N So the bond price is
P = coupon × 1
r 1− 1
T (1 + r ) D.Ding + par value × Bonds and Stocks 1
(1 + r )T . Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure Bond Pricing III — Coupon Bonds (3)
Example
Calculate the price of a 7% coupon, 30 year maturity bond making
semiannual coupon payment with par value of $1, 000. Assume
the annual interest rate is 8%.
N par value
coupon
t+
(1 + r /m)
(1 + r /m)N Bond value = t =1
30×2 =
t =1
60 =
t =1 $1, 000 ×
1+ 8%
2 7%
2
t + $1, 000
1+ 8%
2 30×2 $1, 000 × 3.5%
$1, 000
+
t
(1 + 4%)
(1 + 4%)60 = $886.88
D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure Bond Pricing III — Coupon Bonds (4)
Example
Calculate the price of a 7% coupon, 30 year maturity bond making
semiannual coupon payment with par value of $1, 000. Assume the
annual interest rate increases to 1) 10%, 2) 20%.
60 P1 =
t =1 = $716.06
60 P2 =
t =1 = $1, 000
$1, 000 × 3.5%
+
t
60
(1 + 5%)
(1 + 5%) $1, 000 × 3.5%
$1, 000
+
t
60
(1 + 10%)
(1 + 10%) $352.13
D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure Bond Pricing — Prices and Interest Rates (1)
Prices and Interest Rates have an inverse relationship
When interest rates get very high the price of the bond will be
very low
When interest rates approach zero, the price of the bond
approaches the sum of the cash ﬂows D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure Bond Pricing — Prices and Interest Rates (2) A bond sells at par only if its coupon rate equals the interest
rate.
A bond sells at a premium only if its coupon rate above the
interest rate.
A bond sells at a discount only if its coupon rate below the
interest rate. D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure Bond Pricing between Coupon Dates
When a bond is bought between two coupon payments, the
buyer has to pay more than just the price (ﬂat/clean price).
Implicitly, whoever held the bond for some time between the
two coupons is entitled to receive a reward, that is a share of
the upcoming coupon payment.
This is referred to as accrued interest and computed as
follows:
AI = days since last coupon date
× coupon amount
days between two coupon dates Invoice/Dirty/Full price = ﬂat price + accrued interest
D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure Bonds are quoted on a ﬂat price basis in units of 100. Fractions of
1
a dollar are quoted in units of 32nds. So for example, 100 − 07 4
means 100 + 7.25/32 = 100.226563. D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion D.Ding Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure Bond yields I
Current yield
ycurrent = Cash income of a bond (i.e., Annual coupon receipts)
Bond price Example
a bond with a par value of $1, 000 sell for $886.88, mature in 30
years, and has a 7% annual coupon rate paid semiannually.
ycurrent = 7% × $1, 000
= 7.89%
$886.88 Ignores any prospective capital gains or loss
Ignores reinvest value of coupon payment
D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure Bond yields — Yield to Maturity (YTM) The YTM is a measure of average rate of return that will be
earned on a bond if it is bought now and held until maturity.
It is the one discount rate that equates the PV of future
payments to the current bond price.
YTM is the interest rate that investors are using to value the
bond.
It is an implicit measure that one can obtain only from the
current bondprice. D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure Example
A 20year bond with a face value of $1, 000 pays an annual coupon
of 8% and trades at 90.871%. What is its YTM?
we need to solve:
20 $908.71 =
t =1 $1, 000
8% × $1, 000
+
→ y = 9%
t
(1 + y )
(1 + y )20 D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure YTM More generally Price and YTM are negatively related
If YTM increases, price drops
Intuition: Higher yield = more discounting = less willing to
pay for fardistant payments. D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure YTM More generally
The bonds features:
F face value
C annual coupon
m per annum coupon payments —
n period remaining
y yield to maturity/interest rate C
m are physical payments Price of the bond will be
n P=
t =1 = C
y C
m 1+
1− yt
m + 1
y
1+ m F
y
1+ m
n + n F
y
1+ m Every variable is known, except for y , solve for y .
D.Ding Bonds and Stocks n Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure Precise Pricing between Coupon Dates I
The formula applies only when there are exactly n payment periods.
Between two periods the formula must be adjusted as follows:
Assume there are n full periods and the length between
payment periods is 365/m. There are k days to an (n + 1)st
payment. For k = 0 (the n + 1 coupon payment is about to
happen), the present value of this stream is
n
t =1 C
m 1+ yt
m + F
y
1+ m n + C
m thus for k > 0, the present value of this stream is
1
y
1+ m k /(365/m) n
t =1 D.Ding C
m 1+ yt
m + Bonds and Stocks F
y
1+ m n + C
m Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure Precise Pricing between Coupon Dates II However, an investor, will not see all of the upcoming
payment as he has to pay accrued interest! Taking this into
account, the precise formula is
1
y
1+ m p=
− k /(365/m) n
t =1 C
m 1+ yt
m 365/m − k C
365/m m D.Ding Bonds and Stocks + F
y
1+ m n + C
m Class Roadmap
Bonds
Stocks
Conclusion D.Ding Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure Spot Rates, The Term Structure, The Yield Curve I A Spot Rate is the interest rate on a Tyear loan to be made
today.
y1 = 5% means the current rate for a oneyear loan today is
5%
y2 = 7.01% means the current rate for a twoyear loan today
is 7.01%
Spot rates ≡ YTM on riskfree zero coupon bonds D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure Spot Rates, The Term Structure, The Yield Curve II
The Term Structure of Interest Rates is the series of spot rates
y1 , y2 , y3 , ...
Essentially linking interest rate to investment term.
The Yield Curve is a plot of the term structure: Interest rate/spot
rates against investment term/maturity
ZeroCoupon Yield Curve: zerocoupon bond yields (STRIPS)
Coupon Yield Curve: coupon bond yields (Treasuries)
Corporate Yield Curve: corporate bond yields of the same
class (i.e., similar risks or the same credit rating) D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure The US yield curve in May D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure The Current US yield curve D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure The Spot Curve I
The spot curve is based on the YTM of zerocoupon bonds
(or the price of Tbills).
If F is the principal repayment, and there is maturity n then
P= F
(1 + yn )n The spot rate curve consists of the sequence of rates y1 ,
y2 , ..., yT .
Naturally, with the spot rates, one can compute the present
value of a large variety of cash ﬂows.
E.g., cashﬂows are CF1 , CF2 , ..., CFt , then
PV = CF1
CF2
CFT
+ ... +
+
2
(1 + y1 ) (1 + y2 )
(1 + yT )T
D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure The Spot Curve II
YTM computations typically assume that the same interest
rate is applied for discounting. With spot curves, there can be
diﬀerences.
However, the noarbitrage principle must apply! Thus a bond
investment based on spotrates cannot give you a diﬀerent PV
than the current bondprice: = CF
CF
CF
F
+ ... +
+
+
∗ )T
(1 + y ∗ ) (1 + y ∗ )2
(1 + y
(1 + y ∗ )T
CF
CF
F
CF
+
+ ... +
+
2
T
(1 + y1 ) (1 + y2 )
(1 + yT )
(1 + yT )T where yt are the spot rates and y ∗ is the yield to maturity
(YTM) for this bond with maturity T
D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure The Spot Curve III What if not, e.g. if bond price is smaller?
Buy bond.
Strip the coupons
Sell zerocoupon bonds with face value CFt and maturity t ,
(t = 1, 2, ...T − 1) and one with face value CF + F and
maturity T . D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure The Spot Curve IV Spot rate
Yield to maturity on zerocoupon bonds
It is the rate that prevails today for a time period
corresponding to the zero’s maturity Short rate
The interest rate available for a given time interval (e.g., 1
year) at diﬀerent points in time. D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure Short rates vs. Spot rates D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure Forward Rates
The forward rates signiﬁes a commitment rate for a future
investment/borrowing.
It derives from the spot rate: Suppose you know y1 and y2 and
you care to know the forward rate f2 , i.e. the rate at which
you can precontract to borrow a period from now. Then
(1 + y2 )2 = (1 + y1 ) (1 + f2 )
The more distant ft is computed in the same manner
(1 + yt )t = (1 + yt −1 )t −1 (1 + ft )
f2 , ..., fT is then the forward curve
Claim: forward rates are unbiased estimates of expected
future interest rates.
D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion D.Ding Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure Spot rates, short rates and forward rates D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure Spot rates, short rates and forward rates D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure Interest rate uncertainty and forward rates
In a certain world, diﬀerent investment strategies with
common terminal dates must provide equal rates of return,
e.g.,
two consecutive 1 year investments in zeros would need to
oﬀer the same total return as an equal sized investment in a 2
year zero.
(1 + r1 ) (1 + r2 ) = (1 + y2 )2
However, future shortrate is unknown
(1 + r1 ) (1 + E (r2 )) = (1 + y2 )2 D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure Example
Suppose r1 = 5% , E (r2 ) = 6%. Now consider a shortterm
investor who wishes to invest only for one year. In a certain world,
she should be indiﬀerent between i ) buying a 1 year bond and ii )
buying a 2year bond then selling it after 1 year.
However, if we take risk into account,
Option i ) rate of return = 5%
Option ii ) At the end of year 1
if r2 > E (r2 ) , bonds price decreases, rate of return < 5%;
if r2 < E (r2 ) , bonds price increases, rate of return > 5%;
So, Option ii ) is more risky
Clearly, this investor will not hold the 2year bond unless the
oﬀered price of 2year bond is lower!
D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Bond Pricing
Bond Yields
Yield to Maturity and Term Structure Theories of the term structure
The expectation theory:
fn = E (rn )
The liquidity preference theory:
fn = E (rn ) + risk premium
A upward sloping yield curve because
Investors expect rising interest rate (expectation hypothesis)
Large risk premium for holding longerterm bond (liquidity
preference theory) D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Stock Valuation Common Stocks Issuer: Corporations.
Ownership with residual claim and limited liability D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Stock Valuation Valuation of Common Stocks
The value of any asset is the present value of its expected
future cash ﬂows.
Stock ownership produces cash ﬂows from:
Dividends
Capital Gains The price you are willing to pay depends on size and timing of
future dividends, and risks of the stock.
Discount rate, r , of the stock is the rate of return investors
can expect to earn on stocks with similar risk.
Valuation of Diﬀerent Types of Stocks
Zero Growth
Constant Growth
Diﬀerential Growth
D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Stock Valuation Valuation of Diﬀerent Types of Stocks I Zero growth
P0 = Div
r Constant growth
P0 = D.Ding Div
r −g Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Stock Valuation Valuation of Diﬀerent Types of Stocks II 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
Compute the total present value of the estimated future
dividends and future stock price at the appropriate discount
rate. D.Ding Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Stock Valuation Valuation of Diﬀerent Types of Stocks III Suppose dividends will grow at rate g1 for N years and grow
at rate g2 thereafter
To value a Diﬀerential Growth Stock, we can use
P= Div1
(1 + g1 )N
1−
r − g1
(1 + r )N D.Ding + DivN +1 / (r − g2 ) Bonds and Stocks (1 + r )N Class Roadmap
Bonds
Stocks
Conclusion Characteristics
Stock Valuation 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.
If stocks of similar risk earn 12% eﬀective annual return, what is
the stock worth? P= (1 + g1 )N
Div1
1−
r − g1
(1 + r )N + 2 × 1.08
1.083
p=
1−
+
0.12 − 0.08
1.123
= 28.89
D.Ding DivN +1
(r −g2 ) (1 + r )N
2×1.083 ×1.04
0.12−0.04
1.123 Bonds and Stocks Class Roadmap
Bonds
Stocks
Conclusion Recap Pricing is just the PV of all cash ﬂows that you expect to receive.
Bonds
Bond Pricing: Zeros and Coupon bonds
YTM
The Yield Curve and Theories Stocks
Valuation  Discounted Dividend Model D.Ding Bonds and Stocks ...
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This note was uploaded on 10/08/2011 for the course ECO 258 taught by Professor Dingding during the Fall '11 term at University of Toronto.
 Fall '11
 DingDing
 Economics, Interest Rates

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