CHAPTER
VALUATION CONCEPTS
BOND AND STOCK VALUATION
1. Valuation of Bonds
• A bond is a long-term debt instrument issued by a
corporation or government.
• A bond pays a stated amount of interest to the investor,
period after period, until it is fully retired by the issuing
company.
• A bond has a face value – this is a stated value of a an
asset. In the case of a bond, the face value is usually
$1,000.
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• A bond has a stated maturity which is the time
when the company is obliged to pay the bond
holder the face value of the instrument.
• The coupon rate or nominal annual rate of interest,
is stated on the face of the bond.
 If for example, the coupon rate is 12% on a $1,000
face value bond, the company pays the holder $120
each year.
2
• Calculating the value of bonds is relatively easy
because size and time pattern of their cash flow
over their life are known.
• Bond cash flows comes in two basic forms;
i. Interest payments every six months equals to
one-half the coupon rate times the face value
of the bond.
ii. The payment of the principle on bond’s
maturity date.
3
Perpetual Bonds
• Perpetual bond is a bond that never matures e.g. a
consol – originally issued by Great Britain after the
Napoleonic wars to consolidate debt issues.
• The present value of a perpetual bond is given by;
I
V=
kd
or
V = I(PVIFkd∞ )
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Example 1
Suppose you bought a bond that paid $50 a year forever. Assuming that
your required rate of return for this type of bond is 12%, what is the
present value of this security?
Solution
I
50
V= =
= $416.67
kd 0.12
 This is the maximum amount you would be willing to pay for the
bond.
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Bond with a Finite Maturity
None Zero Coupon Bonds
• If a bond has a finite maturity, then we must consider
not only the interest stream but also the terminal or
maturity (face) value in valuing the bond.
• The valuation equation of such a bond that pays
interest at the end of each year is;
6
n
v=
t=1
I
MV
+
t
(1 + k d )
(1 + k d )n
or
v = I (PVIFAkd,n ) + MV (PVIFkd,n )
• Where;
n = number of years until final maturity
mv = maturity value of the bond
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Example 2
What is the value of a $1,000 par value bond with a 10%
coupon and 9 years to maturity and the required rate of
return of 12%.
Solution
Coupon rate = $1,000 x 10% = $100
100
100
100
1,000
V=
+
+. … +
+
= $893.80
1
2
9
9
(1.12)
(1.12)
1.12
(1.12)
or
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v = I (PVIFA12%,9 ) + MV (PVIF12%,9 )
= 100(5.328) + 1000(0.361)
= 532.80 + 361.00
= $893.80
Therefore, interest payments have a present value of
$532.80, whereas the principal payment at maturity
has a present value of $361.00
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 Note: the intrinsic value of an interest-bearing bond
with a finite maturity is equal to the present value of
the interest payments plus the present value of
principal payment at maturity, all discounted.
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Zero-Coupon Bond
• 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.
MV
V=
(1+ kd )n
or
V = MV(PVIFKd,n )
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Example 3
Espinosa Enterprises issues a zero-coupon having a 10 year maturity and
$1,000 face value. If the required rate of return is 12%, what is the value?
Solution:
MV
$1,000
V=
(1+ kd
)n
=
(1.12)10
= $322
or
V = MV(PVIF12%,10 ) = $1,000 (0.332) = $322
12
Semi-Annual Compounding of Interest
• Although some bonds make interest payments once a
year, most bonds issued especially in the united states
pay interest twice a year.
• As a result, it is necessary to modify our bond
valuation equation to account for compounding twice
a year. the formula changes to;
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2n
v=
t=1
i
MV
2
+
2t
(1 + k d )
(1 + k d )2n
2
v = (i 2) (PVIFAkd
or
2,2n
2
) + MV (PVIFkd
2,2n
)
Where;
k d = nominal annual required rate of interest
i
= semi-annual coupon payment
2
2n = number of semi-annual periods until maturity
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Example 4
If the 10% coupon bond of US Blivet Corporation have 12 years to maturity and our
nominal required rate of return is 14% and $1,000 par value. What is the value of
the bond?
Solution
2n
v=
t=1
100
100
i
MV
2
+
2t
(1 + k d )
(1 + k d )2n
2
2
100
1,000
V = 0.14 2x1 + 0.14 2x2 +. … + 0.14 2x12 +
0.14 )2x12 ≅ $770.45
(1+
)
(1+
)
(1+
)
(1+
2
2
2
2
2
2
2
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Or
v = (i 2) (PVIFAkd
2,2n
) + MV (PVIFkd
2,2n
)
v = 50(PVIFA7%,24 ) + 1,000(PVIF7%,24 )
= 50(11.469) + 1,000(0.197)
= $770.45
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Preferred Stock
• A preferred stock is a type of stock that promises fixed
dividend, but at the discretion of the board of directors.
• Preferred stock enjoy preference over common stock in the
payment of dividends and claims on assets.
• Preferred stock have no stated maturity date and to a larger
extent they are like perpetual bonds due to the fixed nature
of payments.
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• Valuation of a preferred stock is given by;
𝐷𝑝
V=
𝑘𝑝
where;
stock
𝐷𝑝 = stated annual dividend per share of preferred
𝑘𝑝 = appropriate discount rate
Example 5
Hughes Corporation had a 9% $100 par value preferred stock
issue outstanding and your required rate of return is 14% on this
investment. What is the value of this security?
Solution
𝐷𝑝
$9
V=
=
= $64.29
𝑘𝑝
0.14
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• However, there are times when preferred stock
may be issued with a terminal value and time. In
this case the value of the preferred stock is
arrived by;
V=
Dp
Call price
+
t
(1+ kp )
(1+ kp )n
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Common Stock Valuation
• Common stock are securities that represent the
ultimate ownership and risk position in a corporation.
• Unlike bonds and preferred stock cash-flows, which
are contractually stated, much more uncertainty
surrounds the future stream of returns connected
with common stock.
• The valuation of common stock can be viewed as the
discounted value of all expected cash dividends
provided by the issuing firm until the end of time.
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• The value of common stock is given by;
D1
D2
D∞
V=
+
+ ⋯+
1
2
(1+ ke )
(1+ ke )
(1+ ke )∞
where;
Dt = the cash dividend at the end of time period t
and
k e = the investor’s required return or capitalisation
rate for this equity investment
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• If the investor plans to own stock for only two years,
the model becomes;
D1
D2
P2
V=
+
+
1
2
(1+ ke )
(1+ ke )
(1+ ke )2
Where;
P2 = expected sales price of stock at the end of two
years
• Among the models used to value common stock are
called dividend discount models.
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• Dividend discount models are designed to compute
the intrinsic value of a share of common stock under
specific assumptions such as the expected growth
pattern of future dividends and the appropriate
discount rate to employ.
• The models are basically in three models – constant
growth (Gordon dividend valuation) model, no growth
model and growth phase model.
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Constant Growth (Gordon Dividend Valuation) Model
• Critical assumptions in this valuation model are;
1) Dividends per share are expected to grow perpetually at a
compound rate of g (constant rate).
2) k e is greater than g
• Therefore, valuation is given by;
D1
V=
(ke −𝑔)
and required rate of return can be given by;
D1
ke = + g
V
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Example 6
Suppose that LKN Limited’s dividend per share at the
end of year one is expected to be K4, it’s expected
dividend growth at 6% rate forever and that the
appropriate discount rate is 14%. What is the value of
LKN stock?
Solution:
4
V=
= K50
(0.14 −0.06)
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No Growth Model
• A special case of the constant growth dividend model
calls for an expected growth rate, g, of zero.
• Here the assumption is that dividends will be
maintained at their current level forever.
• In this case the valuation is reduced to;
𝐷1
V=
𝑘𝑒
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 Note: not always are stocks expected simply to
maintain a constant dividend forever. However,
when a stable dividend is expected to be
maintained for long period of time the equation
can provide a good approximation of value.
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Growth Phase Model
• When dividends growth is expected to differ
during various phases of the firm’s development,
the present value of dividends for various
growth phases can be determined and summed
up to produce the stock’s intrinsic value.
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Example 7
If a company’s dividends per share are expected
to grow at 10% compounded rate for 5 years and
thereafter at a 6%. If the current dividend (D0 ) is
K2 per share and the required rate of return (k e )
is 14%. What is the value of the stock?
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Solution
Phase 1 – Present value of dividends to be received over first 5 years
Year
Dividend
𝑃𝑉𝐼𝐹14%,𝑡
Present Value (K)
1
2(1.10)1 = 2.20
0.877
1.93
2
2(1.10)2 = 2.42
0.769
1.86
3
2(1.10)3 = 2.66
0.675
1.80
4
2(1.10)4 = 2.93
0.592
1.73
5
2(1.10)5 = 3.22
0.519
1.67
8.99
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Phase 2 – Present value of constant growth component
Dividend D6 = 3.22(1.06) = K3.41
• Value of stock at end of year 5
D6
3.41
V5 =
=
= K42.63
(ke −𝑔) (0.14 −0.06)
• PV of K42.63 at year 5 = K42.63(PVIF14%,5 )= K42.63(0.519) = K22.13
• PV of stock = V= K8.99 + K22.13 = K31.12
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END
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