# Stock and Bond Valuation

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```					Lecture: 3 - Stock and Bond Valuation

How to Get a “k” to Discount Cash Flows - Two Methods

I.      Required Return on a Stock (k) - CAPM (Capital
Asset Pricing Model) or Other Securities

k = rf + B(rm - rf)

where     rf = Risk-Free Rate
B = Beta (Risk Measure)
rm = Expected Market Return

Example: k = .05 + 2(.12 - .05) = .19

kb = Rb = rreal + rrisk + E(I)
= Productivity Growth + Risk Premium +
Expected Inflation
= 2 - 4% + 0 - 10% + 2 - 6% (Estimates)
Lecture: 3 - Stock and Bond Valuation

III.   Bond Valuation

Bond    = Annuity Plus Single Par Payment
(often Semi-Annual Payments)

a. Par Value -face value, maturity value -
usually \$1000

b. Coupon Interest Rate - stated as a % of Par -
10% coupon on \$1000 par => coupon = \$100.

c. Maturity - length of time until Par value is paid off
Lecture: 3 - Stock and Bond Valuation
Bond Valuation
“A Bond is an Annuity Plus a Single Face Value Payment”
Annual Coupon
Lecture 3 - Stock and Bond Valuation

I.      Bond Valuation
General Formula

B0   = I[PVAk,n] + M[PVk,n]

B0= Bond Price
I = Interest (coupon) Payment
M = Par Value

II.     Example:
Suppose a bond offers a 10% coupon, on \$1000
par, for 3 years, and the expected inflation rate
is 2%, the real rate is 3% and the bond’s risk is
1%. What is its price?

B0   = \$100[PVA.06,3] + \$1000[PV.06,3]
= \$100(2.673) + \$1000(.84)
= \$1107
QUESTION: If the company only agrees to pay \$1000 at
maturity, won’t those who buy this bond lose \$107
at maturity?

QUESTION: Would you buy this bond? Why? - greater
coupon than par bonds.

A par bond would cost \$1000 but only pays a \$60 coupon.
The present value of the difference in coupons (100 -
60)(2.673) = 107 which is the difference in price between
this bond and a par bond.

Alternatively, a bond that offered a 2% coupon when rates
are 6% will have a price of B0 = 20[PVA.06, 3] + 1000[PV.06, 3]
= 893 or \$107 less than the par bond.
Bond Valuation
“A Bond is an Annuity Plus a Single Face Value Payment”
Semi-Annual Coupon
Lecture 3 - Stock and Bond Valuation

I.     Adjustments For Semi-Annual Coupon Bonds
a. n = the number of semi-annual payments
(maturity x 2)
b. k = one-half the bond’s annual yield
c. I = one-half the bond’s annual coupon

II.    Example:
Suppose a bond pays 10% coupon, semi-annually,
has 10 years until maturity and has a required
return (or Yield to Maturity) of 8%. What is its price?

B0   = \$50[PVA.04,20] + \$1000[PV.04,20]
= \$50(13.59) + \$1000(.456)
= \$1135.5

QUESTION: Consider two identical bonds except that one
pays an annual coupon and the other a semi-annual
coupon. Which should have the higher price?

Yield to Maturity and Realized Yield
Lecture 3 - Stock and Bond Valuation

YIELD TO MATURITY - The return one can expect on an
investment in a bond if the bond pays all its
coupons and par and yields do not change after
you purchase the bond.

PROBLEM: Suppose you observe a bond in the market
with a price of \$803 that pays a coupon of 10%
till maturity in 5 years. What is its implied yield
to maturity?

Try 16%
803       = 100(PVA?,5) + 1000(PV?,5)

= 100(3.274) + 1000(.476) = 803

REALIZED YIELD - The actual return one receives on the
initial investment in a bond.

QUESTION: If you buy a 20% coupon, par bond, with 3
years maturity and you hold it for three years are
you sure to earn 20%?

ANSWER: No because the calculation of YTM assumes
that the coupons are reinvested at 20%, if rates
change your realized yield will change because
you'll earn more or less than 20% on their
reinvested coupons.
Example: When you bought the bond, YTM was 20%. But
suppose rates fell to 5% the day after you bought and
stay there for three years. Your realized yield will be:

use PV = FV[PVk,n]

1000      = (200(1+.05)2 + 200(1+.05) + 1200)[PVk,3]

= 1630.5[PVk,3]

=> 1000/1630.5 = [PVk,3] = .6133

=> k = 17% realized yield falls when reinvestment rate falls

QUESTION: Then how can you truly lock-in a rate?

bonds.

QUESTION: How would you price a zero coupon bond?

ANSWER: Use the second term in the bond pricing formula.

QUESTION: Some find this attractive but is there a problem
with being locked-in?

earning extra interest on reinvested coupons.
Stock Valuation
“Valuation is Based Upon Expected Dividend Flow and the
Future Expected Market Value of the Stock”
Lecture 3 - Stock and Bond Valuation

I.         Common Stock Valuation
General Formula
P0   = E(D1)/(1+ke) + E(D2)/(1+ke)2 + ... +
E(Dn)/(1+ke)n + E(Pn)/(1+ke)n

where E means expectation, Dt means dividend at time t,
P means stock price, and ke is the cost of equity.

II.        Example:
Suppose a firm pays \$4 in Dividends, which will
increase by \$1 in each of the next 3 years and we
expect the price to be \$30 at the end of the 3rd year.
Assume stock beta = 1, the risk free rate is 10%, and
the expected market return is 15%. What is the stock
price?

Ke = .10 + 1(.15 - .10)
P0 = \$4/(1.15) + \$5/(1.15)2 + \$6/(1.15)3 +\$30/(1.15)3
= \$4(.87) + \$5(.756) + \$6(.658) + \$30(.658)
= \$30.94
Stock Valuation - Constant Growth
“Valuation is Based Upon Expected Dividends That Grow at
a Constant Rate For Ever”
Lecture 3 - Stock and Bond Valuation

Constant Growth Discounted Cash Flow Model, or DCF.
D0 (1  g )        D1
P0 =                  =
kg              kg

where      D0 = present dividend paid at time 0
D1 = dividend expected at time 1
g = constant future growth in dividends
k = required return (discount rate)

Note: Works only for k>g and dividend paying firm

PROBLEM: Suppose a company will pay a dividend of \$5 in
one year, has a required return of 10% and dividends
grow 5% per year. What is the stock price?
5      5
P =                    100
(.10.05) .05

Mixture of Dividends

PROBLEM: Suppose a firm will pay a dividend of \$5 per
year for 5 years and then increases the dividend
by 10% per year thereafter. The firm has a required
return of 15%. What is its price now?
.
5(11)
P   = 5(PVA.15, 5) +   (.15.10) [PV.15, 5]

= 5(3.352) + 110(.497) = 71.43
Stock Valuation - Implied Required Returns and
Growth Rates
“Just Manipulate the Constant Growth Formula”
Lecture 3 - Stock and Bond Valuation

PROBLEM: Suppose we know the price of the stock in the
market is \$80, and it pays a dividend of \$3 that
will grow by 10% per year. What is the return the
market requires on the stock?
D0 (1  g )
k =                g
P0

.
3(11)
=         .10
80
= .14

PROBLEM: If the market price of a stock is \$50, its required
return is 15 percent and next year’s dividend is
expected to be \$5, by what percent must the
market expect the company’s dividends to grow.

g = k - D1
P

so g = .15 - 5          = .05
50
PRICE-EARNINGS RATIOS
“PE’s Are Commonly Used to Compare Stocks”

Lecture 3 - Stock and Bond Valuation

PE Ratio - This is the number of dollars investors are willing
to pay for each dollar of a company’s earnings.

You can use the growth model to see why stocks have
different price-earnings ratios

D1
P =
(k  g)
dE1
=
(k  g)

P          d
=>        =
E       (k  g)

where     E         = earnings per share      D
d         = dividend payout ratio =
E

Clearly, a larger growth rate and payout ratio and a smaller
discount rate (k) makes for a larger price earnings ratio.

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 views: 3 posted: 10/5/2012 language: Latin pages: 12