Valuation of shares & bonds NOTES @ BECDOMS

1. 03B
FIN211 Financial Management
Topic: Valuation of shares & bonds,
There are two distinct aspects of this topic: (1) pricing securities, & (2)
calculating a security’s expected rate of return. (1) involves calculating
a security’s intrinsic value (IV) based on the valuer’s required rate of
return. (2) involves calculating a security’s expected return based on
its current market price (P0). Note that: (1) a security’s current market
price may not be the same as its intrinsic value, and (2) a security’s
expected return may not be the same as its required rate of return.
These differences lead to the flowing decision rules:
If P0 < IV security is underpriced, therefore buy
If P0 > IV security is overpriced, therefore sell.
If expected return > required return, security is underpriced, therefore
buy
If expected return < required return, security is overpriced, therefore
sell.
The golden rule is ‘buy low, sell high’.
Pricing bonds
A bond has a: par value = face value = redemption value = nominal
value;
coupon rate of interest applied to its par value; maturity date .
Bond interest is typically paid twice-yearly.
PV of a bond = (1) PV of the interest payments (an annuity) +
(2) PV of the par value (future value).
See text pages 277-281.
Note that the price of a bond is inversely related to its yield or discount
rate; ie the higher the yield on a bond, the lower will be its price, and
vice-versa.

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When the yield on a bond is lower than its coupon rate, its price will be
higher than its par value.
When the yield on a bond is higher than its coupon rate, its price will
be lower than its par value.
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Example
A bond has a par value of $100, a coupon rate of 10.75% and matures
in 5 years. If interest is paid annually and the required rate of return is
10%, what is the bond’s IV?
Answer
PV of the annuity: $10.75 [1 − (1.1)-5] / 0.1 = $40.75
PV of par value: $100 / (1.1)5 = 62.09
Total PV = IV = $102.84
Pricing preference shares
IV = annual dividend ÷ required rate of return = Dp / RROR = Dp/Rp
Example
A preference share has a par value of $100 and a dividend rate of
10.75%. If the required rate of return is 10%, what is the share’s IV?
Answer
PV = IV = $10.75 / 0.1 = $107.50
Pricing ordinary shares
D1 = D0(1 + g)
g = expected growth rate. Can be defined in terms of growth in the
book value (BV) of shareholders’ funds
= BV1 / BV0 − 1
= return on equity (ROE) profit retention ratio.
Example
A share has just paid an annual dividend of $0.54. The book value of
shareholder’s funds is currently $11,020,801. One year ago the book
value of shareholders’ funds was $10,699,807. During the past year
there were no changes to paid-up capital. The risk-free rate of return
is currently 5%. A risk premium of 8% is required for investing in the
share. What is its PV?
Answer
Required rate of return = 5% + 8% = 13%
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Growth rate = 11,020,801/10,699,807 − 1 = 0.03
D0 = $0.54 D1 = $0.54 1.03 = $0.5562, or
D1 = $0.54 11,020,801/10,699,807 = $0.5562
PV = $0.5562 / (0.13 − 0.03) = $5.56
Expected rates of return
Shares
= D 1 / P0 + g
Example
If the share in the previous example has a current market price of
$5.02, what is its expected rate of return?
Answer
D 1 / P0 + g
= 0.5562/5.02 + 0.03
= 14.08%
Is this share worth buying?
Yes: its market price is below its intrinsic value and its expected rate of
return is higher than its required rate of return.
Bonds
Expected return = yield to maturity = internal rate of return (IRR -
interest symbol i on the calculator).
The IRR is the discount rate that equates the PV of a bond’s future cash
flows with its current market price. This rate can be identified either
through a process of iteration or by using an electronic device such as
a financial calculator or an Excel spreadsheet.
Note that if the price of a bond is below its par value, its yield will be
higher than its coupon rate, and vice-versa.
Example
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5. FIN211 03B
A bond has a par value of $100, a coupon rate of 10.5% payable
semi-annually and has 4.5 years to maturity. If its current price is
$117.52, what is its expected rate of return?
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Answer
Because the current market price is above par value, we know that the
expected return must be less than the coupon rate.
This calculation is best done using a financial calculator (or
spreadsheet).
The key strokes are:
9 n 117.52 +/- PV 5.25 PMT 100 FV COMP i x 2
= 6.00%
Calculating bonds yields without a financial calculator
If we know the price of a bond, we can calculate its yield to maturity
(YTM); i.e. the discount rate that equates the present value of the bond’
s future cash flows with its current price. The textbook sometimes
refers to a bond’s YTM as the bondholder’s expected rate of return.
However, whether or not a bond’s YTM will be the rate of return
actually realised by the bondholder will depend on the rate at which the
bond’s coupons can be reinvested. Remember that a bond YTM is an
annual percentage rate (APR), as opposed to an effective annual rate
(EAR).
Example
What is the YTM of a three-year bond with a face value of $100 and a
14% coupon paid twice per year, if the price of the bond is $110.15?
Solution
The time line is shown below.
-110.1 7 7 7 7 7 107
5
0 1 2 3 4 5 6
We need to find the value of i such that:
110.15 =
For equations of this type there is no algebraic solution for i when the
exponent is greater than 4. In all such cases an approximation method
needs to be used for determining the value of i. The routine is a
two-part process involving first using an equation for determining an
approximate value of i and then using a solution algorithm to bring
the approximate value of i closer and closer to its true value through
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successive iterations. This is the solution method used by electronic
devices and may also be used manually.
Using a financial calculator, the YTM in this example is easily found to
be 10.00%. Using the Sharp EL-735, the key-strokes are: 6 n 110.15
+/- PV 7 PMT 100 FV COMP i × 2 =
Without a financial calculator, the following equation defines an
approximate bond YTM:
Bond YTM (8)
Where a = Vb/M 1
Applied to the current example, a = 110.15/100 1 = 0.1015.
Equation (8) then defines an approximate yield of 10.007%, as follows:
Bond YTM
0.100072
10.0072%
In this case the approximation equation has yielded a reasonable result.
But if we do not know that 10.00% is the actual YTM, we cannot assess
the accuracy of 10.0072%. However, in all cases such as this we can
obtain a better approximation of the actual YTM by using an
appropriate solution algorithm, such as Newton's Method. Alternatively,
if we designate the first approximation as ia, a second approximation,
designated ib, can be defined in either of the following two ways:
(1) ib = ia ×
(2) ib =
where PVa = the present value of the bond’ s cash flows per dollar
of bond when discounted at ia
FVa = the future value of the bond’ s cash flows per dollar of
bond when compounded at ia
Using the first method, we find:
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PVa = $ + $1(1 + ia/m)-nm
=$ + $1(1 + 0.100072/2)-6
= $0.355257543 + $0.746061907
= $1.1013
ib = 10.0072% ×
= 10.0054%
Using the second method (which was developed by Chris Deeley in
2005), we find:
FVa = FV of coupons per dollar of bond + $1
=$ + $1
= $0.14 + $1
= $0.14 3.4012645 + $1
= $1.476177
ib =
= 0.10001466 = 10.001466%
Either of the foregoing methods can be repeated until the actual YTM is
identified when in = in 1 to the required degree of accuracy. The second
method (“ Deeley’ s Method” ) is recommended, despite its greater
complexity, on the grounds that it approaches the actual YTM far more
quickly than the first method.
Using Deeley’s Method, the second iteration gives us a YTM of
10.0006%, calculated as follows:
FVb = FV of coupons per dollar of bond + $1
=$ + $1
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= $0.14 + $1
= $0.14 3.401019135 + $1
= $1.47614268
ic =
= 0.100006 = 10.0006%
Repeating this routine with 10.0006% as the reinvestment rate will
confirm that the actual YTM is 10.00% (to two decimals). The following
equation also confirms 10.00% as the actual YTM:
Vb =
= $7
= $7 5.075692 + $100 0.7462
= $35.53 + $74.62
= $110.15 = the given price of the bond.
Note that although the method just demonstrated is of a type often
described as one of trial and error, it is nevertheless a systematic
routine of repetition or iteration. Each repetition of the routine will
always generate a more accurate approximation of the YTM until the
actual YTM is identified to the required degree of accuracy. It is, of
course, your understanding of the concepts that is important, not that
you can necessarily calculate the precise answer. Depending upon the
nature of the question, it is often sufficient to calculate a range within
which the YTM falls. Nevertheless, now that you’ve seen the hassles
involved in calculating a YTM without a financial calculator, you may
appreciate the value of having one, and knowing how to use it.
Preference shares
Expected return = D / P0
Example
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10. FIN211 03B
A preference share has a par value of $100 and a dividend rate of
10.5% payable annually. If its current price is $117.52, what is its
expected rate of return?
Answer
10.5 / 117.52 = 8.9%
Financial calculations without using a financial calculator
Rate of interest
If $1 grows to $10 over 10 years, what is the rate of interest?
Ans. i = 101/10 1 = 0.2589 = 25.89%
Check: $1 (1.2589)10 = $1 9.998 = $10.00
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Number of periods
If $1 grows to $10 at 10% pa, how long does it take?
Ans. n = log 10/ log 1.1 = 24.16 years
Check: $1 (1.1)24.16 = $1 10.0011 = $10.00
Calculating periodic payments when paying off a debt or saving for
a future amount, without using a financial calculator or tables
Paying off a debt
A debt which is paid off in equal periodic instalments may be referred
to as a credit foncier loan. It is a present value (PV) that needs to be
paid off and the amount of each instalment (or payment) can be
calculated by dividing the PV by equation 4-12a, which is on page 79
of the text. Note that this is a ‘double-decker’ equation in which i is
the denominator. Because the PV (the numerator) is to be divided by
this equation, the equation should be inverted and then multiplied by
the PV.
Example
See problem 4-15 at page 98.
Payment (PMT) = 60,000 0.09/(1 1.09-25)
= 60,000 0.10180625
= $6,108.38
Saving for a future amount
It is a future value (FV) that needs to be saved for and the amount of
each instalment (or payment) can be calculated by dividing the FV by
equation 4-11a, which is on page 81 of the text. Note that this is a
‘double-decker’ equation in which i is the denominator. Because the FV
(the numerator) is to be divided by this equation, the equation should
be inverted and then multiplied by the FV.
Example
See problem 4-18 at page 98.
Payment (PMT) = 100,000 1.0510 0.1/(1.110 − 1)
= 162,889.46 0.62745394
= $10,220.56
Note that the first part of this answer calculates the expected future
value of the property, which is expected to increase in value at the rate
of 5% per year for 10 years.
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Note also that with this sort of calculation, you should not round-off
until completion of the calculation.
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