Citibank: basic of corporate finance
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4-4 VALUING FINANCIAL ASSETS
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Coupon rate
2. Investment bankers set the coupon rate for the bonds.
The investment bankers attempt to gauge the interest rate
environment and set the coupon rate commensurate with other
bonds with similar risk and maturity. The coupon rate dictates
whether the bonds will be sold in the secondary market at face
value or at a discount or premium. If the coupon rate is higher than
the prevailing interest rate, the bonds will sell at a premium; if the
coupon rate is lower than the prevailing interest rate, the bonds will
sell at a discount.
Primary issue
3. Investment bankers find investors for the bonds and issue them in
the primary market.
The investment bankers use their system of brokers and dealers
to find investors to buy the bonds. When investment bankers
complete the sale of the bonds to investors, they turn over the
proceeds of the sale (less the fees for performing their services)
to the company to use for the purchase of equipment. The total
face value of the bonds appears as a liability on the company's
balance sheet.
Secondary
market
4. The bonds become available in the secondary market.
Once the bonds are sold in the primary market to investors, they
become available for purchase or sale in the secondary market.
These transactions usually take place between two investors – one
investor who owns bonds that are no longer needed for his/her
investment portfolio and another investor who needs those same
bonds.
Pricing Bonds
This section focuses on the calculations investors make to determine
the price at which they will sell or buy bonds in the secondary markets.
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Present value
of cash flows
Pricing a bond involves finding the present value of the cash flows
from the bond throughout its life. The formula for calculating the
present value of a bond is:
V = C[1 / (1+R)]
1
+ C[1 / (1+R)]
2
+ ... + C[1 / (1+R)]
T
+ F[1 / (1+R)]
T
Where:
V = Present value of the bond
C = Coupon payment (coupon rate multiplied by face value)
R = Discount rate (current prevailing rate)
F = Face value of the bond
T = Number of compounding periods until maturity
Similar to
present value of
an annuity
You may notice that this formula is similar to the formula for
calculating the present value of an annuity. In fact, it is the same
except the coupon payment (C) replaces the annuity payment (A).
There is also one additional term, F[1/(1+R)]
T
, at the end of the
formula. This term represents the discounting of the face value amount
that is received at the maturity of the bond. The formula can be
shortened to:
V = C x PVIFA(R,T) + F x PVIF(R,T)
Using this formula, we multiply the present value interest factor of an
annuity by the coupon payment and add the product of the face value and
the present value interest factor. Let's look at two examples to see how
this formula works.
Example one:
present value of
bond
What is the present value of a bond with a two-year maturity date,
a face value of $1,000, and a coupon rate of 6%? The current
prevailing rate for similar issues is 5%. To apply the formula, C = $60
($1,000 x 0.06), R = 0.05, T = 2, and F = $1,000.
V = C[1 / (1+R)]
1
+ C[1 / (1+R)]
2
+ F[1 / (1+R)]
2
V = $60[1 / (1+0.05)] + $60[1 / (1+0.05)]
2
+ $1,000[1 / (1+0.05)]
2
V = $60[0.95238] + $60[0.90703] + $1,000[0.90703]
V = $57.14 + $54.42 + $907.03
V = $1,018.59
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The present value of $1,018.59 is the price that the bond will trade for
in the secondary bond market. You will notice that the price is higher
than the face value of $1,000. In the time since these bonds were
issued, interest rates have fallen from 6% to 5%. Investors are willing
to pay more for the $60 interest payments when compared with new
bond issues that are only paying $50 in interest per $1,000 face value.
This inverse relationship is important.
As interest rates fall, bond prices rise;
as interest rates rise, bond prices fall.
A bond with a coupon rate that is the same as the market rate sells for
face value. A bond with a coupon rate that is higher than the prevailing
interest rate sells at a premium to par value; a bond with a lower rate
sells at a discount. Investment bankers attempt to set the coupon rate for
a newly issued bond at the market interest rate so that the bond sale will
yield the amount of capital their corporate clients need for operations.
Using the
financial
calculator
You can use a financial calculator to find the present value of the bond
in the example. With most financial calculators you input the number
of interest payments, the size of the payments, the face
value, and the discount rate. The present value key gives you the
appropriate answer. Check your owner's manual for the specifics on
how to calculate the present value of a bond.
Example two:
present value
of a bond
Let's look at one more example. If the current interest rate is 9%, how
much would an investor be willing to pay for a $10,000 face value bond
with 5 coupon payments of 7.5% left until maturity? Use these values
in the formula: C = $750, R = 0.09, T = 5, and F = $10,000.
V = C[1 / (1+R)]
1
+ C[1 / (1+R)]
2
+ ... + C[1 / (1+R)]
5
+ F[1 / (1+R)]
5
V = $750[1 / (1+0.09)]
1
+ $750[1 / (1+0.09)]
2
+ ... + $750[1 / (1+0.09)]
5
+
$10,000[1/(1+0.09)]
5
V = $750[0.91743] + $750[0.84168] + ... + $750[0.64993] + $10,000[0.64933]
V = $9,416.55
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As you can see, longer maturities require more calculations. Using
your calculator's cash flow functions can save a lot of time.
In this section, we have discussed the pricing of one type of debt
financing. You have seen that the present value of a bond is the price at
which it will trade in the secondary market. Now, let's see how the
market values equity financing.
COMMON STOCK
Ownership
claim on assets
In Unit One we defined common stock as a form of capital that
represents an ownership claim on the assets of a company. Common
stockholders have dividend and voting rights.
A common stock certificate may have a face (par) value printed on it
that represents the price of the stock at the time it was originally issued.
The current value of the stock may be greater than or less than the face
value. Since common stock has no maturity date, it is often treated as a
perpetuity when attempting to place a value on each share.
Valuing Common Stock
Dividend
valuation model
One method of valuing common stock is the dividend valuation model.
Remember, this is only one method of valuing stock. We will discuss
some of the other methods later in this course.
Stock with
constant
dividend
The most straight-forward application of the dividend valuation model
is to find the present value of stock that pays a constant dividend over
time and is held indefinitely by the shareholder. This is really a simple
perpetuity. As you learned in Unit Three, the formula for the present
value of a perpetuity is:
PV
p
= A x (1 / R)
Where:
PV
p
= Present value of the perpetuity
A = Amount of each payment
R = Discount rate (required rate of return)
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Dividend
valuation formula
For the dividend valuation formula, we replace A with D to represent the
dividend payment (the perpetuity) and use V to represent the value of the
stock. The dividend valuation formula is:
V = D x (1 / R)
Example
Let's look at an example. ABC Company's last common stock dividend
was $1.50, which is not expected to change in the foreseeable future. If
the investor has a required return of 10%, what is the present value of
this stock?
V = D x (1 / R)
V = $1.50 x (1 / 0.10)
V = $15.00
Dividend with
constant growth
rate
The next application of the dividend valuation model is one in which
the dividend grows at some constant rate. The formula is:
V = D
0
[ (1+G) / (1+R)]
1
+ D
0
[ (1+G) / (1+R)]
2
+ ... + D
0
[ (1+G) / (1+R)]
¥
Where:
V = Present value of the stock per share
D
0
= Most recent dividend per share
R = Discount rate (investor's required return)
G = Growth rate of the dividend
Condensed
formula
You may remember from our discussion on the present value of a
perpetuity, the sum of an infinite series can be reduced to a more
workable form. For the present value of a stock with constant dividend
growth, the condensed formula is:
V = D
0
[ (1+G) / (R - G)]
Sometimes the formula is presented in a slightly different format. The
expression D
1
replaces D
0
x (1+G) and represents the value of the next
dividend. The formula is:
V = D
1
/ (R - G)
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Both formulas arrive at the same result.
Three
assumptions to
consider
You must be aware of three important assumptions when using these
formulas.
1. The growth rate must be constant (G cannot change).
The formulas will not work if the analyst expects different
growth rates for the investment over its life.
2. The investor's required rate of return must be larger than the
growth rate.
As you can see by looking at the formula, if the growth rate is
larger than the discount rate, the present value of the stock will
be negative – not an acceptable solution.
3. The dividends earned on the stock are reinvested at the same
historical rate of return.
If the rate of return changes, the formulas must be adjusted to
allow for the differences.
Lets look at two examples. In the first example, we use the dividend
valuation model to solve for the present value of the stock, given the
investor's required rate of return. In the second example, we solve for
the investor's required rate of return, given the present value of the
stock.
Example one:
solve for
present value
The last common stock dividend of ABC Company was $1.00 and it is
expected to grow at an annual rate of 5%. If the required rate of return
is 10%, what is the present value of the common stock? We use our
dividend valuation model formula, with the following values: D
0
=
$1.00, G = 0.05, and R = 0.10.
V = D
0
[ (1+G) / (R - G)]
V = $1.00 [ (1 + 0.05) / (0.10 - 0.05)]
V = $1.00[(1.05) / (0.05)]
V = $1.00[21.00]
V = $21.00
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The present value of a share of ABC Company's common stock is
$21.00. We could have arrived at the same answer using the shorter
formula.
V = D
1
/ (R - G)
V = $1.05 / (0.10 - 0.05)
V = $1.05 / (0.05)
V = $21.00
As you may be able to infer from the formula, a larger growth rate
translates to a larger present value of the stock and a smaller growth rate
means a smaller present value.
Example two:
solve for
required rate of
return
Suppose that you buy a share of stock in a company for $50, which you
know is the appropriate price for the stock. The next dividend will be
$2, and you expect that the dividend will grow at a 10% annual rate.
What is your required rate of return?
R = (D
1
/ V) + G
R = ($2 / $50) + 0.10
R = (0.04) + 0.10
R = 0.14 or 14%
This formula automatically splits the rate of return into two
components: the return on the dividend (0.04) and the gain in the capital
investment (0.10). The dividend return is expressed as D
1
/ V, and the
capital gain is expressed as the growth rate, G.
It is important to remember that this is only one approach to valuing
common stock. Because of the three restrictive assumptions we
discussed on page 4-9, this model can be used only in a limited number
of situations. If a company does not pay any dividends, or if the growth
rate is not constant or is higher than the discount rate, then we must use
other valuation models. We will discuss some of these other approaches
in Unit 7: Corporate Valuation – Cost of Capital.
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PREFERRED STOCK
Common vs.
preferred stock
Preferred stock also represents an ownership claim on the assets of
a company. However, a company's preferred stockholders receive
dividends before the common stockholders, they have no voting rights,
and are given preference to company assets ahead of common
stockholders when a company is liquidated.
Pricing Preferred Stock
Behaves like a
perpetuity
We can take the same approach to valuing preferred stock as we did
for common stock. Since preferred stock has no maturity, it also
behaves like a perpetuity. Remember, the present value of a perpetuity
is the sum of an infinite series. The shortened formula for calculating
the present value of preferred stock is identical to the common stock
valuation formula for constant dividends.
V = D x (1 / R)
The characteristics of preferred stock are better suited for this
formula. Most companies that issue preferred stock do pay a constant
dividend. In fact, they have the obligation to pay the preferred
dividends before paying the common dividends. If business is slow, the
company may choose to reduce or eliminate the common dividend, but
it usually will try to continue paying the preferred dividend.
Example:
present value of
preferred stock
Let's look at an example using the formula to value preferred stock. If
Mega Company pays an annual dividend of $5 to its preferred
shareholders, and an investor's required rate of return is 10%, what is
the present value of the stock?
V = D x (1 / R)
V = $5 x (1/ 0.10)
V = $5 x (10.00)
V = $50.00
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An investor with a 10% required rate of return would be willing to pay
$50 for a share of Mega Company preferred stock that pays an annual
dividend of $5.
UNIT SUMMARY
In this unit, we discussed some bond terminology, the process for
issuing bonds, and some ways to value simple debt and equity
securities.
Calculating the present value of a bond is similar to calculating the
present value of an annuity.
V = C x PVIFA(R,T) + F x PVIF(R,T)
The dividend valuation model is one method for valuing common
stock. It is similar to the formula for calculating the present value of a
perpetuity. The dividend valuation formula for finding the present
value of stock that pays a constant dividend is:
V = D x (1 / R)
Another application of the dividend valuation model is one in which
the dividend grows at some constant rate. The formula is:
V = D
1
/ (R - G)
There are three important assumptions to consider when using this
formula.
1. The growth rate must be constant (G cannot change).
2. The investor's required rate of return must be larger than the
growth rate.
3. The dividends earned on the stock are reinvested at the same
historical rate of return.
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The same formula that is used to value common stock with constant
dividends is used to value preferred stock.
You have completed Unit Four: Valuing Financial Assets. Please complete the Progress
Check that follows. If you answer any questions incorrectly or have difficulty with the
calculations, please review the appropriate section in the text and then proceed to Unit Five:
Introduction to Capital Budgeting.
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