Valuation of Common
Stocks and Bonds
How to apply the PV
concept
FIN 819: lecture 3
1
Today’s plan


Review what we have learned in the last
lecture
Valuing stocks
• Some terms about stocks
• Valuing stocks using dividends
• Valuing stocks using earnings
• Valuing stocks using free cash flows
FIN 819: lecture 3
2
Today’s plan (Continue)

Bond valuation and the term-structure of
interest rates
• Terminology about bonds
• The valuation of bonds
• The term structure of interest rates
• Use duration to measure the volatility of the
bond price
FIN 819: lecture 3
3
What have we learned in the
last lecture?

Payback rule

IRR rule

Free-cash flow calculation
• Shortcomings
• Shortcomings
FIN 819: lecture 3
4
Some specific questions in the
calculation of cash flows







Include all incidental effects
Do not forget working capital requirements
Forget sunk costs
Include opportunity costs
Be careful about inflation
Depreciation
Financing
FIN 819: lecture 3
5
Free cash flows calculation

Free cash flows = cash flows from
operations + cash flows from the change
in working capital + cash flows from
capital investment and disposal
FIN 819: lecture 3
6
Calculating cash flows from
operations

Method 1
• Cash flows from operations =revenue –cost
(cash expenses) – tax payment

Methods 2
• Cash flows from operations = accounting
profit + depreciation

Method 3
• Cash flows from operations =(revenue –
cost)*(1-tax rate) + depreciation *tax rate
FIN 819: lecture 4
7
A summary example 2

Now we can apply what we have
learned about how to calculate cash
flows to the IM&C’s Guano Project (in
the textbook), whose information is
given in the following slide.
FIN 819: lecture 4
8
IM&C’s Guano Project
Revised projections ($1000s) reflecting inflation
FIN 819: lecture 4
9
IM&C’s Guano Project
Cash flow analysis ($1000s)
FIN 819: lecture 4
10
IM&C’s Guano Project

NPV using nominal cash flows
1,630 2,381
6,205 10,685 10,136
NPV  12,600 




2
3
4
1.20 1.20
1.20 1.20 1.205
6,110
3,444


 3,519 or $3,519,000
1.206 1.207
FIN 819: lecture 4
11
New formula
In chapter 4, it is argued that
FCF=earnings –net investment
Net investment = total investment depreciation
 Do you agree with this formula? Why?

FIN 819: lecture 4
12
Example

A project costs $2,000 and is expected
to last 2 years, producing cash income of
$1,500 and $500 respectively. The cost
of the project can be depreciated at
$1,000 per year. If the tax rate is 50%,
what are the free cash flows?
FIN 819: lecture 4
13
One more question

Mr. Pool is now 40 years old and plans
to invest some fraction of his current
annual income of $40,000 in an account
with an annual real interest rate of 5%,
starting next year until he retires at the
age of 70 to accumulate $500,000 in real
terms. If the real growth rate of his
income is 2%, what fraction of his
income must be invested?
FIN 819: lecture 4
14
Some terms about stocks
Book Value – The value of the stocks
according to the balance sheet.
Liquidation Value - Net proceeds that
would be realized by selling the firm’s
assets and paying off its creditors.
Market Value Balance Sheet - Financial
statement that uses market value of
assets and liabilities.
FIN 819: lecture 4
15
Some terms about stocks
Secondary Market - market in which already
issued securities are traded by investors.
Dividend - Periodic cash distribution from the
firm to the shareholders.
P/E Ratio - Price per share divided by earnings
per share.
Dividend yield – Dividends per share over the
price of per share
FIN 819: lecture 4
16
Example



IBM has a trading price of $70 per share.
Its annual earnings per share is $5. Its
annual dividend per share is $3.5. What
are the P/E and the dividend yield?
P/E=70/5=14
Dividend yield=3.5/70 or 5%
FIN 819: lecture 4
17
Valuing Common Stocks using
dividends (first approach)
Dividend Discount Model - Computation of
today’s stock price which states that share
value equals the present value of all expected
future dividends plus the selling price of the
stock.
Div1
Div 2
Div H  PH
P0 

 ...
1
2
H
(1  r ) (1  r )
(1  r )
H - Time horizon for your investment.
FIN 819: lecture 4
18
Example

George has bought one IBM share in the
beginning of this year and decides to
hold this share until next year. The
expected dividend this year is $10 per
share and the stock is expected to sell at
$110 per share in the end of the year. If
the cost of the capital is 10%, what is the
current stock price?
FIN 819: lecture 4
19
Solution

P0=(110+10)/(1+0.1)=$109.1
FIN 819: lecture 4
20
Valuing common stocks using
dividends
Example
Current forecasts are for XYZ Company to pay
dividends of $3, $3.24, and $3.50 over the next
three years, respectively. At the end of three
years you anticipate selling your stock at a
market price of $94.48. What is the price of the
stock given a 12% expected return?
FIN 819: lecture 4
21
Solution
P0 
3.00
3.24
3.50  94.48


1
2
3
(1  .12) (1  .12)
(1  .12)
P0  $75.00
FIN 819: lecture 4
22
Valuing common stocks using
dividends
If we forecast no growth, and plan to hold out
stock indefinitely, we will then value the stock
as the PV of a PERPETUITY.
Div1 EPS1
PV ( perpetuity)  P0 
or
r
r
Assumes all earnings are
paid to shareholders.
FIN 819: lecture 4
23
Example

Suppose that a stock is going to pay a
dividend of $3 every year forever. If the
discount rate is 10%, what is the stock
price for the following cases:
• (a) you invest and hold it forever?
• (b) you invest and hold it for two years?
• (c) you invest and hold it for 20 years?
FIN 819: lecture 4
24
Solution
(a) P0=3/0.1=$30
 (b)P0=PV (annuity) + PV( the stock price at
year 2)
= 3/1.1 + 3/1.12+(3/0.1)/1.12
= 3/0.1=$30
(c) P0=PV (annuity of 20 years) +
PV (the stock price at the year of 20)
=$30

FIN 819: lecture 4
25
Valuing Common Stocks
Gordon Growth Model: A version of the
dividend growth model in which dividends
grow at a constant rate (Gordon Growth
Model).
Stocks can be valued as a perpetuity with a
growth rate, if you want to hold this stock
forever, that is
Div1
P0 
rg
FIN 819: lecture 4
26
Example

Suppose that a stock is going to pay a
dividend of $3 next year. Dividends grow
at a growth rate of 3%. If the discount
rate is 10%, what is the stock price for
the following cases:
• (a) you buy and hold it forever?
• (b) you buy and hold it for two years?
• (c) you buy and hold it for 20 years?
FIN 819: lecture 4
27
Solution
(a) P0=3/(0.1-0.03)=$42.86
 (b)P0=PV (annuity) + PV( the stock price at
year 2)
= 3/1.1 + 3*1.03/1.12+(3*1.032/(0.10.03))/1.12
= 3/(0.1-0.03)=$42.86
(c) P0=PV (annuity of 20 years) +
PV (the stock price at the year of 20)
=$42.86

FIN 819: lecture 4
28
Capitalization rate
Expected Return - The percentage yield that an
investor forecasts from a specific investment
over a set period of time. Sometimes called
the market capitalization rate.
Div1  P1  P0
Expected Return  r 
P0
FIN 819: lecture 4
29
Example
If Fledgling Electronics is selling for $100 per
share today and is expected to sell for $110
one year from now, what is the expected return
if the dividend one year from now is forecasted
to be $5.00?
FIN 819: lecture 4
30
Solution
According to the formula,
5  110  100
Expected Return 
 .15
100
FIN 819: lecture 4
31
Capitalization rate
The formula for the capitalization rate can
be broken into two parts.
Capital. Rate = Dividend Yield + Capital Appreciation
Div1 P1  P0
Expected Return  r 

P0
P0
FIN 819: lecture 4
32
Using dividends models to
derive the capitalization rate
Capitalization Rate can be estimated using the
perpetuity formula, given minor algebraic
manipulation.
Div1
P0 
rg
Div1
r
g
P0
FIN 819: lecture 4
33
Valuing Common Stocks
ExampleIf a stock is selling for $100 in the stock
market, the cost of capital is 12% and the next
year dividend is $3, what might the market be
assuming about the growth in dividends?
$3.00
$100 
.12  g
g .09
FIN 819: lecture 4
34
Some terms about dividend
growth rates
If a firm elects to pay a lower dividend, and
reinvest the funds, the stock price may
increase because future dividends may be
higher.
Payout Ratio - Fraction of earnings paid out as
dividends
Plowback Ratio - Fraction of earnings retained by
the firm.

FIN 819: lecture 4
35
Deriving the dividend growth
rate g
Growth can be derived from applying the
return on equity to the percentage of
earnings plowed back into operations.
ROE  Return on Equity
EPS
ROE 
Book Equity Per Share
g = return on equity X plowback ratio
FIN 819: lecture 4
36
Example
Our company forecasts to pay a
$5.00 dividend next year, which
represents 100% of its earnings.
This will provide investors with a
12% expected return. Instead, we
decide to plow back 40% of the
earnings at the firm’s current return
on equity of 20%. What is the
value of the stock before and after
the plowback decision?
FIN 819: lecture 4
37
Solution

Without growth

With growth
5
P0 
 $41.67
0.12
g  0.4 * 0.2  0.08
5 * 0.6
P0 
 $75
0.12  0.08
FIN 819: lecture 4
38
Example (continued)
If the company did not plowback some
earnings, the stock price would remain at
$41.67. With the plowback, the price rose to
$75.00.
The difference between these two numbers
(75.00-41.67=33.33) is called the Present
Value of Growth Opportunities (PVGO).
FIN 819: lecture 4
39
Valuing common stocks using
earnings

We often use earnings to value stocks
as
EPS1
P0 
 PVGO
r

What is the relationship between this
formula and the dividend growth
formula?
FIN 819: lecture 4
40
Example

Firm A has a market capitalization rate of 15%. The
earnings are expected to be $8.33 per share next year.
The plowback ratio is 0.4 and ROE is 25%. Every
investment in year i is to yield a simple perpetuity starting
in year (i+1) with each cash flow equal to total investment
times ROE. All the investments have the same
capitalization rate.
•
•
•
•
•
(a) Using the formula P=ESP1/r + PVGO to calculate the
stock price
(b) If ROE is increased, what will happen to the stock price?
Why?
(c) Use the dividend model to calculate the stock price?
(d) What have you found?
(e) Think about why you have this kind of result?
FIN 819: lecture 4
41
Simple Solution
(a)
g=10%, EPS1/r=8.33/0.15=$55.56
PVGO=NPV1/(r-g)=2.22/(0.150.1)=$44.44, P=$100
(b) The price will be increased
(c) P=Div1/(r-g)=5/(0.15-0.1)=$100
FIN 819: lecture 4
42
Valuing common stocks using
FCF (free cash flows)

The value of a business or stock is usually
computed as the discounted value of FCF out to
a valuation horizon (H).
The horizon value is sometimes called the
terminal value .
FCF1
FCF2
FCFH
PVH
PV 

 ... 

1
2
H
(1  r ) (1  r )
(1  r )
(1  r ) H
FIN 819: lecture 4
43
FCF and PV
FCF1
FCF2
FCFH
PVH
PV 

 ... 

1
2
H
(1  r ) (1  r )
(1  r )
(1  r ) H
PV (free cash flows)
FIN 819: lecture 4
PV (horizon value)
44
FCF and PV




Free Cash Flows (FCF) should be the
theoretical basis for all PV calculations.
FCF is a more accurate measurement of
PV than either Div or EPS.
The market price does not always reflect
the PV of FCF.
When valuing a business for purchase,
always use FCF.
FIN 819: lecture 4
45
FCF and PV
Example
Given the cash flows for Concatenator Manufacturing
Division, calculate the PV of near term cash flows, PV
(horizon value), and the total value of the firm. r=10% and
g= 6%
Year
1
2
3
4
5
6
Asset Value
10.00 12.00 14.40 17.28 20.74 23.43
Earnings
1.20 1.44 1.73 2.07 2.49 2.81
Investment
2.00 2.40 2.88 3.46 2.69 3.04
Free Cash Flow
- .80 - .96 - 1.15 - 1.39 - .20 - .23
.EPS growth (%) 20
20
20
20
20
13
FIN 819: lecture 4
7
8
9
10
26.47 28.05 29.73 31.51
3.18 3.36 3.57 3.78
1.59 1.68 1.78 1.89
1.59 1.68 1.79 1.89
13
6
6
6
46
FCF and PV
Solution
1  1.59 
PV(horizon value) 
  22.4
6 
1.1  .10  .06 
.80 .96
1.15 1.39
.20
.23
PV(FCF)  




2
3
4
5
1.1 1.1 1.1 1.1 1.1 1.16
 3.6
FIN 819: lecture 4
47
FCF and PV
PV(busines s)  PV(FCF)  PV(horizon value)
 -3.6  22.4
 $18.8
FIN 819: lecture 4
48
How to estimate the horizon
value?

It is very difficult to forecast or estimate
the horizon value. There are several
ideas that may be used to estimate the
horizon value.
• Competition
• Constant growth rate
FIN 819: lecture 4
49
Another example
Imagine Corporation has just paid a
dividend of $0.40 per share. The
dividends are expected to grow at 30%
per year for the next two years and at
5% per year thereafter. If the required
rate of return in the stock is 15% (APR),
calculate the current stock price.
FIN 819: lecture 4
50
Solution




Answer:
First: visualize the cash flow pattern;
• C1, C2 and P2
Then, you know what to do?
P0 = [(0.4 *1.3)/1.15] + [(0.4 *
1.3^2)/(1.15^2)] +
[(0.4 * 1.3^2*1.05)/((1.15^2 * (.15 -.05))]
= $6.33
FIN 819: lecture 4
51
Another cash flow problem!
Firm Excellent has an old packaging machine that can be used for another 3 years.
The remaining book value for this old machine is $15,000, which can be
depreciated in the next three years by using the straight-line depreciation approach.
If the old packaging machine is used, the annual total cost of operation, labor and
maintenance is $10,000. In the market, the new packaging machine is available
now at the price of $50,000, which will increase by 5% annually. Based on
Excellent’s experience, the new packaging machine will be used forever. The
capital investment cost of the new package machine will be depreciated in the next
10 years by using the straight-line depreciation approach. The annual total cost of
operation, labor and maintenance will be $8,000 when the new package machine is
used.
Questions:
a. What is the valuation horizon used in this problem?
b. Should Excellent invest in the new packaging machine now or
wait three years later?
FIN 819: Lecture 2
Bonds




Bond – a security or a financial instrument that
obligates the issuer (borrower) to make
specified payments to the bondholder during
some time horizon.
Coupon - The interest payments made to the
bondholder.
Face Value (Par Value, Principal or Maturity
Value) - Payment at the maturity of the bond.
Coupon Rate - Annual interest payment, as a
percentage of face value.
FIN 819: lecture 4
53
Bonds

A bond also has (legal) rights attached to
it:
• if the borrower doesn’t make the required
•
payments, bondholders can force bankruptcy
proceedings
in the event of bankruptcy, bond holders get
paid before equity holders
FIN 819: lecture 4
54
An example of a bond

A coupon bond that pays coupon of 10%
annually, with a face value of $1000, has a
discount rate of 8% and matures in three
years.
•
•
•
•
The coupon payment is $100 annually
The discount rate is different from the coupon rate.
In the third year, the bondholder is supposed to get
$100 coupon payment plus the face value of $1000.
Can you visualize the cash flows pattern?
FIN 819: lecture 4
55
Bonds
WARNING
The coupon rate IS NOT the discount
rate used in the Present Value
calculations.
The coupon rate merely tells us what cash flow
the bond will produce.
Since the coupon rate is listed as a %, this
misconception is quite common.
FIN 819: lecture 4
56
Bond Valuation
The price of a bond is the Present Value
of all cash flows generated by the bond
(i.e. coupons and face value) discounted
at the required rate of return.
PV 
cpn
(1  r )1

cpn
(1  r ) 2
 ... 
1,000  cpn
FIN 819: lecture 4
(1  r ) N
57
Zero coupon bonds



Zero coupon bonds are the simplest type of bond
(also called stripped bonds, discount bonds)
You buy a zero coupon bond today (cash outflow)
and you get paid back the bond’s face value at
some point in the future (called the bond’s maturity )
How much is a 10-yr zero coupon bond worth today
if the face value is $1,000 and the effective annual Face
value
rate is 8% ?
PV
Time=0
FIN 819: lecture 4
Time=t
58
Zero coupon bonds (continue)



P0=1000/1.0810=$463.2
So for the zero-coupon bond, the price is
just the present value of the face value
paid at the maturity of the bond
Do you know why it is also called a
discount bond?
FIN 819: lecture 4
59
Coupon bond
The price of a coupon bond is the
Present Value of all cash flows
generated by the bond (i.e. coupons and
face value) discounted at the required
rate of return.
PV 
cpn
(1  r )1

cpn
(1  r ) 2
 .... 
(cpn  par )
(1  r )t
1

1

  par  PV (annuity)  PV ( par )
 cpn 
 r r (1  r )t  (1  r )t


FIN 819: lecture 4
60
Bond Pricing
Example
What is the price of a 6 % annual coupon
bond, with a $1,000 face value, which matures
in 3 years? Assume a required return of 5.6%.
FIN 819: lecture 4
61
Bond Pricing
Example
What is the price of a 6 % annual coupon
bond, with a $1,000 face value, which matures
in 3 years? Assume a required return of 5.6%.
60
60
1,060
PV 


1
2
3
(1.056) (1.056) (1.056)
PV  $1,010.77
FIN 819: lecture 4
62
Bond Pricing
Example (continued)
What is the price of the bond if the required
rate of return is 6 %?
60
60
1,060
PV 


1
2
3
(1.06) (1.06) (1.06)
PV  $1,000
FIN 819: lecture 4
63
Bond Pricing
Example (continued)
What is the price of the bond if the required
rate of return is 15 %?
60
60
1,060
PV 


1
2
(1.15) (1.15) (1.15)3
PV  $794.51
FIN 819: lecture 4
64
Bond Pricing
Example (continued)
What is the price of the bond if the required
rate of return is 5.6% AND the coupons are
paid semi-annually?
FIN 819: lecture 4
65
Bond Pricing
Example (continued)
What is the price of the bond if the required
rate of return is 5.6% AND the coupons are
paid semi-annually?
30
30
30
1,030
PV 

 ... 

1
2
5
(1.028) (1.028)
(1.028) (1.028)6
PV  $1,010.91
FIN 819: lecture 4
66
Bond Pricing
Example (continued)
Q: How did the calculation change, given semiannual coupons versus annual coupon
payments?
FIN 819: lecture 4
67
Bond Yields


Current Yield - Annual coupon
payments divided by bond price.
Yield To Maturity (YTM)- Interest rate
for which the present value of the
bond’s payments equal the market
price of the bond.
P
cpn
(1  y )1

cpn
(1  y ) 2
 .... 
(cpn  par )
FIN 819: lecture 4
(1  y )t
68
An example of a bond

A coupon bond that pays coupon of 10%
annually, with a face value of $1000, has
a discount rate of 8% and matures in
three years. It is assumed that the
market price of the bond is the present
value of the bond at the discount rate of
8%.
• What is the current yield?
• What is the yield to maturity.
FIN 819: lecture 4
69
My solution





First, calculate the bond price
P=100/1.08+100/1.082+1100/1.083
=$1,051.54
Current yield=100/1051.54=9.5%
YTM=8%
FIN 819: lecture 4
70
Bond Yields
Calculating Yield to Maturity (YTM=r)
If you are given the market price of a
bond (P) and the coupon rate, the yield
to maturity can be found by solving for r.
P
cpn
1
(1  r )

cpn
(1  r )
2
 .... 
(cpn  par )
(1  r )t
FIN 819: lecture 4
71
Bond Yields
Example
What is the YTM of a 6 % annual coupon
bond, with a $1,000 face value, which matures
in 3 years? The market price of the bond is
$1,010.77
60
60
1,060
PV 


1
2
(1  r ) (1  r )
(1  r ) 3
PV  $1,010.77
FIN 819: lecture 4
72
Bond Yields



In general, there is no simple formula
that can be used to calculate YTM
unless for zero coupon bonds
Calculating YTM by hand can be very
tedious. We don’t have this kind of
problems in the quiz or exam
You may use the trial by errors
approach get it.
FIN 819: lecture 4
73
Bond Yields (3)

(a)
(b)
(c)
(d)

Can you guess which one is the
solution?
6.6%
7.1%
6.0%
5.6%
My solution is (d).
FIN 819: lecture 4
74
The rate of return on a bond
Coupon income + price change
Rate of return =
investment or bond price
Rate of return =
profit
cost of investment
Example: An 8 percent coupon bond has a
price of $110 dollars with maturity of 5 years
and a face value of $100. Next year, the
expected bond price will be $105. If you
hold this bond this year, what is the rate of
return?
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My solution

The expected rate of return for holing the
bond this year is (8-5)/110=2.73%
• Price change =105-110=-$5
• Coupon payment=100*8%=$8
• Profit=8-5=$3
• The investment cost or the initial price=$110
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Some new terms




So far, we consider one discount rate for all the
cash flows
In fact, the discount rate for one period cash
flows can be different from the discount rate for
two-period cash flows.
Spot interest rate: the actual interest rate
available today (t=0)
Future interest rate: the spot rate in the future
(t>0)
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Example






Spot rates (r)
Investment Horizon
1
2
3
4
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r
6%
6.5%
7%
7.2%
78
The Yield Curve
Term Structure of Interest Rates: is the
relationship between the spot rates and
their maturity dates
Yield Curve - Graph of the term structure.
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The term structure of interest
rates (Yield curve)
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Value the bond (revisit)



If we are given the term structure of interest
rates, we know the discount rates for cash
flows in different time periods.
Then
cpn
cpn
(cpn  par )
PV 

 .... 
1
2
(1  r1 ) (1  r2 )
(1  rt )t
Here r1, r2, …, rt are spot rates for period 1, 2,
…, t, respectively.
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Question


Which kind of the yield curve can make
you use a single discount rate for the
bond valuation?
For what kind of bonds, YTM is the same
as spot rates?
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Example

Please use the following information to
value a 10%, four years coupon bond, if
the spot rates are:
Year
Spot rate
1
5%
2
5.4%
3
5.7%
4
5.9%
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Solution

The interest payment is $100 every year.
PV 
100

100

100
(1  .05)1 (1  .054) 2 (1  .057)3
 $1,144.5
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
(100  1,000)
(1  .059) 4
84
A problem

A 6 percent six-year bond yields 12%
and a 10 percent six year bond yields 8
percent. Please calculate six-year spot
rate.
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Forward rate
Forward Rate - The interest rate, fixed
today, on a loan made in the future at a
fixed time according to the term structure
of the interest rates.
The forward rate is implied by the term
structure of interest rates and doesn’t
exist in a financial market
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Forward rate

Another way to look at the forward rate is
that bonds can be priced in the following
way:
PV 
PV 
cpn

cpn
(1  r1)1 (1  r2 ) 2
 .... 
(cpn  par )
(1  rt )t
cpn
(cpn  par )
 .... 
(1  r1)1 (1  r1)(1  f 2 )
(1  rt 1)t 1(1  ft )
cpn

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Forward rate calculation

One period forward rate can be calculated by
using the spot rates as follows:
1  fn 

(1  rn )
n
(1  rn 1 ) n 1
Where fn is the forward rate from period (n-1)
to period n, rn is the n-period spot rate and rn-1
is the spot rate for the (n-1)-period spot rate.
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Example

Using the information for spot rates
given in the previous example, what is
the forward rate(f2) from year 1 to year
2?
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Solution

Here n=2,
(1  r2 ) 2 1.0542
1  f2 

 1.058
1
1.05
(1  r1)
f 2  5.8%
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What moves the interest rate?





Nominal interest rate
•
What is it?
Real interest rate
•
What is it?
Inflation
•
What is it?
Can the nominal interest rate be less than
zero?
Can the real interest rate be less zero?
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What moves the interest rate?



Irving Fisher’s theory
•
•
•
Nominal r = (1+Real r)(1+ expected inflation)-1
Real r is theoretically somewhat stable
Change in inflation drives the change in the interest
rate
This theory doesn’t work bad in the past 50
years.
Why do we care about the movement of the
interest rate?
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Bond price volatility

When the interest rate changes, what
will happen to the bond price?
P

cpn

cpn
(1  y )1 (1  y ) 2
 .... 
( cpn  par )
(1  y )t
Then what decides the sensitivity of the
bond price to the interest rate change?
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Duration


To capture the sensitivity of the bond
price to the interest rate change,
financial economists have defined a
measure called Duration (or Macaulay
Duration)
Duration is a weighted average of the
maturity for the cash flows of the bond
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Example
A two-year 5% coupon bond with YTM of
10%.
Maturity
Cash flows
1
50
2
50+1000

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Duration

Mathematically
t
Duration   iwi
i 1
t
PV (Ci )
wi 
and  wi  1
P
i 1



Volatility (percent) =Duration/(1+YTM)
Change in bond price = volatility*change in
interest rates.
If YTM is small, Change in bond price =
Duration*change in interest rates.
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Example

Calculate the Duration, volatility and the
change of the bond price for 1% change
in the interest rate for a bond that is 6
7/8%, 5-year coupon bond with 4.9%
YTM.
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Solution
i
Ci
PV(Ci)
wi
__
1
68.75
65.54
.060
0.060
2
68.75
62.48
.058
0.115
3
68.75
59.56
.055
0.165
4
68.75
56.78
.052
0.209
5
68.75
841.39
1085.74
.775
1.00
3.875
4.424
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_i* wi _______
98
Solution


Volatility=4.424/(1.049)=4.22
Price change=4.22*1%=4.22%
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Example (practice question 8)

Please use the following information to
answer the questions in the next slide.
Year
Spot rate
1
5%
2
5.4%
3
5.7%
4
5.9%
5
6.0%
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Questions




(a) What are the discount factors for
each date?
(b) What is the forward rates for each
period?
(c) Calculate the PV of 5 percent and 10
percent five-year note?
(d) Which note is going to have a higher
YTM?
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Solution
(a)df1=1/1.05=0.95
df2=1/1.0542=0.90
(b) f2=1.0542/1.05-1=5.8%
f3=1.0573/1.0542-1=6.3%
(c) PV(5%)=$959.34
PV(10%)=$1171.43
(d) 5% note has a higher YTM.
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