Valuation of Long Term
– Important Terms
– Types of Bonds
– Valuation of Bonds
– Handling Semiannual Compounding
• A bond is a long-term
debt instrument issued
by a corporation or
• The maturity value
(MV) [or face value] of
a bond is the stated
value. In the case of a
US bond, the face
value is usually $1,000.
• The bond’s coupon
rate is the stated rate
of interest; the annual
divided by the bond
• The discount rate
(capitalization rate) is
dependent on the risk
of the bond and is
composed of the risk-
free rate plus a
premium for risk.
Different types of bonds
A perpetual bond is a bond that never matures. It
has infinite life.
V = I / kd
Bond P has a $1,000 face value and provides an
8% annual coupon. The appropriate discount rate
is 10%. What is the value of the perpetual bond?
Types of bonds:
• A non-zero coupon-paying bond is a coupon
paying bond with a finite life.
V = I (PVIFA kd, n) + MV (PVIF kd, n)
Bond C has a $1,000 face value and provides an
8% annual coupon for 30 years. The appropriate
discount rate is 10%. What is the value of the
• A zero coupon bond is a bond that pays no
interest but sells at a deep discount from its face
value; it provides compensation to investors in
the form of price appreciation
(1 + kd)n
= MV(PVIFkd, n)
Bond Z has a $1,000 face value and a 30 year life. The
appropriate discount rate is 10%. What is the value of the
A non-zero coupon bond adjusted for semi-annual
(1) Divide kd by 2
(2) (2) Multiply n by 2
(3) Divide I by 2
= I/2 (PVIFAkd /2 ,2*n) + MV (PVIFkd /2 ,2*n)
Bond C has a $1,000 face value and provides an
8% semi-annual coupon for 15 years. The
appropriate discount rate is 10% (annual rate).
What is the value of the coupon bond?
Preferred Stock Valuation
Common Stock Valuation
Preferred Stock is a type of stock that
promises a (usually) fixed dividend, but at
the discretion of the board of directors.
V = DivP / kP
(1 + kP)1 (1 + kP)2 (1 + kP)V = + + ... +
Stock PS has an 8%, $100 par value issue
outstanding. The appropriate discount rate
is 10%. What is the value of the preferred
• Common stock represents a residual
ownership position in the corporation.
• Dividend Valuation Model
Basic dividend valuation model accounts
for the PV of all future dividends.
t=1 (1 + ke)t
Divt: Cash Dividend
at time t
Dividend Growth Pattern
The dividend valuation model requires the
forecast of all future dividends. The
following dividend growth rate assumptions
simplify the valuation process.
The constant growth model assumes that dividends
will grow forever at the rate g.
The growth phases model assumes that
dividends for each share will grow at two or
more different growth rates.
The zero growth model assumes that
dividends will grow forever at the rate g = 0.
Zero Growth Model
Growth Phases Model
Constant Growth Model
James Consol Company currently pays a dividend
of $1.60 per share on its common stock. The
company expects to increase the dividend at a 20 %
annual rate for the first four years and at a 13 % rate
for the next four years, and then grow the dividend
at a 7 percent rate thereafter. This phased-growth
pattern is in keeping with the expected life cycle of
earnings. You require a 16 % return to invest in this
stock. What value should you place on a share of
PHASES 1 and 2: PRESENT VALUE OF DIVIDENDS TO BE RECEIVED OVER FIRST 8 YEARS
END OF PRESENT VALUE CALCULATION PRESENT VALUE
YEAR (Dividend × PVIF16%,t ) OF DIVIDEND
1 $1.60(1.20)1 = $1.92 × 0.862 = $ 1.66
2 1.60(1.20)2 = 2.30 × 0.743 = 1.71
3 1.60(1.20)3 = 2.76 × 0.641 = 1.77
4 1.60(1.20)4 = 3.32 × 0.552 = 1.83
5 3.32(1.13)1 = 3.75 × 0.476 = 1.79
6 3.32(1.13)2 = 4.24 × 0.410 = 1.74
7 3.32(1.13)3 = 4.79 × 0.354 = 1.70
8 3.32(1.13)4 = 5.41 × 0.305 = 1.65
PHASE 3: PRESENT VALUE OF CONSTANT GROWTH
Dividend at the end of year 9 = D8 ( 1 + g )
$5.41(1.07) = $5.79
Value of stock at the end of year 8
D9 $ 5.79 $64.33
ke – g (0.16 – 0.07)
Present value of $64.33 at end of year 8 = V8 = FV (PVIF16%,8)
= ($64.33)(0.305) = $19.62
PRESENT VALUE OF STOCK
V = $13.85 + $19.62 = $33.47
The expected rate of return on a
bond if bought at its current market price and
held to maturity.
Internal rate of return
It is also known as the bonds internal rate of
Calculating Rates of
Return (or Yields)
1. Determine the expected cash flows.
2. Replace the intrinsic value (V) with
market price (P0).
3. Solve for the market required rate of return
That equates the discount cash flows to the
Steps to calculate the rate of return (or Yield).
Determining Bond YTM
Determine the Yield-to-Maturity (YTM)
for the annual coupon paying bond with a finite
P0 = S
(1 + kd )t
= I (PVIFA kd , n) + MV (PVIF kd , n)
(1 + kd )n
kd = YTM
Bond Price - Yield
Discount Bond – The market required rate of
return exceeds the coupon rate (Par > P0 ).
Premium Bond – The coupon rate exceeds the
market required rate of return (P0 > Par).
Par Bond – The coupon rate equals the market
required rate of return (P0 = Par).
Yield to maturity
Interest rate (or yield) risk
Estimate an unknown number that lies
somewhere between to unknown numbers.
The variation in market price of
a security caused by changes in interest rates.
Determining the YTM
Julie Miller want to determine the YTM for an issue of
outstanding bonds at Basket Wonders (BW). BW has an
issue of 10% annual coupon bonds with 15 years left to
maturity. The bonds have a current market value of $1,250.
What is the YTM?