Chapter 6: Bond Primer

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CHAPTER 6
Bond Primer
This chapter explores the basis of bond pricing and
portfolio management. This chapter is organized into the
following sections:
1. Yield Concepts
2. Bond Market Instruments
3. Principles of Bond Price Movements
4. Duration
5. Term Structure of Interest Rates
6. Bond Portfolio Immunization
Chapter 6
1
Yield Concepts
Pure Discount Bond (Zero Coupon Bond)
A pure discount bond promises to pay a certain amount at a
specified time in the future (par value). The instrument is
sold for less than this promised future payment.
There is no payment between the original issue of the bond
and the maturity of the bond. The investor’s return is the
difference between the amount that he/she paid for the bond
and the promised future value (yield).
The general bond pricing formula for all bonds can be stated
as:
M
Pi = 
Ct
t
t = 1 (1+ r i )
where:
Pi
Ct
ri
t
= the price of the bond I
= cash flow from the bond I at time m
= the annualized yield to maturity on bond I
= the time in years until the bond matures
Chapter 6
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Yield Concepts
Bond Discount
The bond discount is the difference between the par value
and the selling price.
Yield to Maturity
The yield is the promised payment that the bond holder will
realize after buying and holding the bond until maturity.
The appropriate current price of a bond is determined by
calculating the present value of the par value.
In order to calculate the present value of the par value, we
must first know the appropriate interest rate.
The interest rate is dependent upon general interest rate
levels in the economy, and the riskiness of the bond. That
is, the probability that the investor receives the par value
as promised.
Chapter 6
3
Pure Discount Bond
Suppose you have a pure discount bond that agrees to
pay $1,000 five years from today. The bond discount rate
is 12%.
What is the appropriate price for this bond?
0
5
??
P i=
P=
$1,000
Cm
(1+ r i ) t
$1000
 $567.43
( 1.12 )5
Chapter 6
4
Pure Discount Bond
0
5
-$567.43
$1,000
Price
Par Value
Bond Discount = Face Value –Current Price
Bond Discount = $1,000-$567.43
Bond Discount = $432.57
Chapter 6
5
Coupon Bonds
Coupon bonds are longer term bonds making regularly
scheduled payments between the original date of issue
and the maturity date.
Suppose you have a bond with a $1,000 face value that
matures 1 year from today. The coupon rate is 12% and
the bond makes semi-annual coupon payments of $60.
The bond yield is 13%.
0
??
1
$60
Chapter 6
$1,000
$60
6
Coupon Bonds
Par Value =
$1,000
Yield
=
13% annual (13/2 =6.5% semiannual)
Coupon =
12% with semiannual payment of $60
Maturity =
1 year
M
Pi = 
Ct
t
t = 1 (1+ r i )
P= $
60
$1,060
+
= 56.34+ 934.56= $990.90
1.065 1.0652
0
-$990.90
1
$60
$1,000
$60
Chapter 6
7
Accrued Interest
Accrued interest is portion of the next coupon payment that
has been earned at the time of purchase.
The accrued interest (AI) is calculated using the following
formula:
 days since last couponpayment

AI = Coupon Payment
days
between
coupon
payments


Chapter 6
8
Money Market Yield
Many money market securities are quoted in terms of the
discount yield. The formula for the discount yield ( d) is:
d=
360  DISC 


t  FV 
where:
DISC = the dollar discount from the face value
FV
= the face value of the instrument
t
= the number of days until the instrument
matures
Rearranging the formula allows us to solve for the
discount as:
 d 
DISC = FV 1 - t 
 360
The actual dollar price, P, depends on the face value
and the amount of the dollar discount, DISC.
P = FV - DISC
Chapter 6
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Money Market Yield
Example: 90-Day Money Market
Par value
=
$1,000,000
Discount yield
=
11%
 .11 (90)
DISC = $1,000,000
 = $27,500
 360 
P = FV - DISC = $1,000,000 - $27,500 = $972,500
Chapter 6
10
Major Money and Bond Market
Instruments
The bond market is divided into the money market and the
bond market.
– The money market trades debt instruments issued with an
original maturity of one year or less.
– The bond market trades debt instruments issued with
longer maturities.
In this section, the following instruments are considered:
– Treasury bills
– Eurodollar deposits
– Repurchase agreements (Repos)
– Treasury bonds
– Treasury notes
Chapter 6
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Treasury Bills and Erodollar CDs
Treasury bills (T-bills)
T-bills are obligations of the U. S. Treasury issued weekly
with a maturity of 91 and 182 days. Bills with a 52-week
maturity are offered monthly.
T-bill yields are expressed on a discount basis. T-bills have
a minimum face value of $10,000 and face value
increments of $5,000.
Erodollar Certificates of Deposit (Erodollar CDs)
A Eurodollar is a dollar denominated bank deposit held in a
bank outside the United States. Many investors prefer
Eurodollar CDs to domestic CDs because Eurodollar CDs
pay a somewhat higher rate.
Higher Erodollar CD rates are the result of stringent
reserve requirements for U.S. banks. Consequently, the
cost of operating many foreign banks is lower than the cost
of operating a U.S. bank. As a result, foreign banks are
able to pay higher interest rates.
Chapter 6
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Repurchase Agreement
Repurchase agreements arise when one party sells a
security to another party and agrees to buy it back
(repurchase it) at a specified time and at a specified price.
The Wall Street Journal, Money Rates section has the yield
quotations on money market instruments.
Insert Figure 6.1 here
Money Rates
Chapter 6
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Treasury Bond and Treasury Notes
T-bonds, and T-notes are issued by the Treasury
Department. Both T-bonds and T-notes pay semiannual
coupon payments. Most T-bonds are callable, with the first
call date coming five years before the bond matures (a call
feature allows the issuer to pay off the loan early).
T-bonds and T-notes are referenced as follows:
Ten-and-three-eights of 2007 to 2012.
This notation refers to a bond that has:
Coupon Rate =
10-3/8%
Maturity
=
2012
Callable
=
Beginning 2007
T-bonds are quoted as in 32nds of par. Thus, a bond having
a quoted price of 98:20 is trading for 98.625% of par value.
Figure 6.2 illustrates T-bond and T-notes quotations.
Chapter 6
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Treasury Bond and Treasury Notes
Insert Figure 6.2 here
Chapter 6
15
Bond Price Changes Due to The
Passing of Time
At maturity the price of a bond must equal its par value.
Because of this, bond prices change due merely to the
passage of time. Bonds can be classified as:
Premium Bond:
Price exceeds its face value
Price must fall over its life
Par Bond:
Price equals its face value
Price may remain the same over time
Discount Bond
Prices below its face value
Price must rise to reach par value at
maturity.
To examine the price path of bonds over time, we consider
three bonds. Assume that the interest rates are a constant
10%, so bond 1 is a premium bond, bond 2 is a par bond,
and bond 3 is a discount bond.
Interest rates:
Bond 1:
Bond 2:
Bond 3
10% constant
20-year 12% coupon
20-year 10% coupon
20-year 8% coupon
Chapter 6
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Bond Price Changes due to The Passing
of Time
Figures 6.3 illustrates the time path of these bond prices.
Insert figure 6.3 here
Chapter 6
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Bond Prices and Interest Rate Changes
The effect of a given change in interest rates on the price
of a bond depends upon three key variables:
1. The maturity of the bond.
2. The coupon rate.
3. The prevailing interest rates at the time of interest rates
change.
Based on the above variables, five principles of bond
pricing have been developed:
1. Bond prices move inversely with interest rates.
2. Bonds with longer maturities experience greater
percentage price changes for a given change in interest
rates.
3. The price sensitivity of bonds increases with maturity, but
it increases at a decreasing rate.
4. Bonds with lower coupon rates experience greater
percentage price changes for a given change in interest
rates.
5. For a given bond, the absolute dollar price increase
caused by a fall in bond yields will exceed the price
decrease caused by an increase in bond yields of the
same magnitude.
Chapter 6
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Bond Prices and Interest Rate Changes
The following example illustrates the sensitivity of
bonds to interest rates changes.
Consider two bonds. Both bonds initially have a 10%
discount rates.
Notice that when the discount rate increases to 12%,
the price of Bond B changes more than the price of
Bond A.
Bond
Maturity
Bond A 5 years
Bond B 30 years
Coupon
Rate
10%
6%
Price at
10%
$1,000
$ 661.41
Price at
12%
$926.40
$515.16
% of Change
7.36%
17.10%
This example illustrates three important bond
characteristics:
1. Bond prices move inversely with interest rates.
2. Bonds with longer maturities experience greater percentage
price changes for a given change in interest rates.
3. Bonds with lower coupon rates experience greater percentage
price changes for a given change in interest rates.
Chapter 6
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Bond Prices and Interest Rate Changes
Table 6.1 examines two bonds that are similar in all
respect except bond C has 15 years to maturity while
Bond D has 20 years to maturity.
Table 6.1
Price Sensitivity of Bonds C and D
Bond C
15Byear, 8% Coupon
Bond D
20BYear 8% Coupon
Yield Change
Yield
(%)
Price
($)
% Price
Change
Yield
(%)
Price
($)
% Price
Change
B2%
9.94
86.68
+16.01
11.73
73.38
+16.01
Before Change
11.94
74.66
0.00
13.73
63.25
0.00
+2%
13.94
65.06
B12.87
15.73
55.15
B12.80
This example shows two important points:
– Holding coupon and maturity constant, a bond with a
higher yield has a lower price sensitivity.
– Coupon and maturity are generally not equal across
bonds.
Chapter 6
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Duration
As we have noted, there are three factors that affect the
way the price of a bond reacts to changes in interest rates.
These three factors are:
– The coupon rate.
– Term to maturity.
– Yield to maturity.
Duration measures the combined effect of all the factors
that affect bond’s price sensitivity to changes in interest
rates.
Duration is a weighted average of the present values of the
bond's cash flows, where the weighting factor is the time at
which the cash flow is to be received.
Duration tells us the sensitivity of the bond price to one
percent change in interest rates.
Note: Each time the discount rate changes, the duration
must be recomputed to identify the effect of the
change.
Chapter 6
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Duration
Duration (D) can be calculated using the following formula:
M
Di=
Where:
Pi
Ct
ri
t
=
=
=
=
tCt

t
t = 1 (1+ r i )
Pi
the bond's price
the cash flow from the bond occurring at time t
the yield-to-maturity on bond I
the time measured from the present until a
payment is made
Chapter 6
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Duration
To illustrate the calculations of duration, consider a 5-year
bond paying an annual coupon rate of 10%. The bond
yields 14% and has a part value of $1,000. Currently the
bond is priced at $862.69.
Table 6.2 shows the cash flows and calculations of this
bond’s duration.
Table 6.2
The Calculation of Duration
t
Ct
t
Ct /(1 + r)
t
t[Ct /(1 + r) ]
1
2
3
4
5
$100.00
87.72
87.72
$100.00
76.95
153.90
$100.00
67.50
202.50
$100.00
59.21
236.84
$1,100.00
571.31
2,856.55
Di = (87.72 + 153.90 + 202.50 + 236.84 + 2,856.55)/862.69 = 3,537.51/862.69 = 4.10
Chapter 6
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Duration
The following equation expresses duration as the negative
of elasticity of the bond's price with respect to a change in
the discount factor (1 + r).
dP
D P
d 1  r 
1  r 
Where:
dP
= change in the price
d(1 + r) = change in the interest rate
Duration gives a single measure of the bond price change
for a change in the discount factor (1 + r).
Rearranging the above equation give us:
 d (1+ r i ) 
 P i
dP i = - D i 
 (1+ r i ) 
The change in bond price associated with a one time shift
in the interest rates.
Chapter 6
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Duration
Using a duration of 4.10, assume that yields suddenly fell
from 14 to 12 percent. The bond's price would rise. The
amount of the change in price is computed as:
 - .02 
dP = - 4.10
 $862.69= + $62.05
 1.14
With this change in price, the new price should be the old
price plus the price change.
New Price = 862.69 + 62.05 = $924.74
Bond investors can compare the price movement
sensitivities of different bonds by simply comparing their
durations.
Chapter 6
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Term Structure of Interest Rates
The term structure of interest rates is the relationship
between term-to-maturity and yield-to-maturity for bonds
that are similar in all respects except maturity.
Term structure analysis helps us understand the
differences in bond yields stemming only from differences
in maturity.
The yield curve shows the relationship between interest
rates and the time to maturity graphically.
Bonds used in a yield curve should have similar:
– Risk level
– Call provisions
– Sinking fund characteristics
– Tax status
Treasury securities all have the same level of default risk
and tend to be alike in their tax status and other features
as well. Thus, it is customary to focus on the term structure
of Treasury securities.
Chapter 6
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Treasury Yield Curve
The Treasury yield curve as displayed in Figure 6.4
provides the basic yield curve.
Insert figure 6.4 here
Chapter 6
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Treasury Yield Curve
At various times, the yield curve takes on different shapes
as illustrated in Figure 6.5.
Insert figure 6.5 here
Chapter 6
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Forward Rates
Forward rates of interest are rates which cover future time
periods. Forward rates occur if the time covered begins after
time = 0.
A spot rate is a yield prevailing at a given moment in time on
a security. Forward rates are implied by currently available
spot rates.
Bond Yield Notation:
Rx, y  the rate to prevail on a bond for the period
beginning at time “x” and maturing at time “y”
Present time is = 0
If today is Jan 1, 2007, we would denote a bond
covering a period from January 1, 2008 through
January 1, 2009 as R1,2.
Chapter 6
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Forward Rates
Principle of calculation for Forward Rates
Examples
r0,5= spot rate for an instrument maturing in 5 years.
r2,5 = forward rate covering a 3-year period beginning
2 years from now.
0
2
5
Spot rate instrument expires 2 yrs.
Rate known today
Spot rate instrument expires 5 yrs.
Rate known today
Future rate instrument expires 3 yrs?
Rate unknown today
Forward rates are calculated on the assumption that returns
over a given period of time are all equal, no matter which
maturities of bonds are held over that span of time.
Chapter 6
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Forward Rates
For a 5-year period, this principle implies that forward
rates can be calculated assuming that any of the
following strategies would earn the same returns:
A. Buy a five-year bond and hold it to maturity.
0
5
B. Buy a one-year bond and hold it until it matures.
When it matures, buy another one-year bond,
following this procedure for the entire five years.
0
1
2
3
4
5
C. Buy a two-year bond and hold it until it matures.
When it matures, use the proceeds to buy a
three-year bond and hold it to maturity.
0
2
5
Chapter 6
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Forward Rates Forward Rates
These three strategies can be expressed as:
1.
Hold one five-year bond for five years:
Total Return = (1 + r0,5)5
2.
Hold a sequence of one-year bonds:
Total Return = (1 + r0,1)(1 + r1,2)(1 + r2,3)(1 + r3,4)(1 + r4,5)
3.
Hold a two-year bond followed by a three-year bond:
Total Return = (1+r0,2)2(1+r2,5)3
Chapter 6
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Forward Rates
According to the Principle of Calculation the total returns
on these bond combinations should all be equal:
(1 + r0,5)5 = (1 + r0,1)(1 + r1,2)(1 + r2,3)(1 + r3,4)(1 + r4,5) = (1 + r0,2)2(1 + r2,5)3
Based on the third strategy and using the following
treasury securities spot yields, calculate the forward rate.
Spot Rate
R0,1
R0,2
R0,3
R0,4
R0,5
Yield
.08
.088
.09
.093
.095
Maturity
1 year
2 years
3 years
4 years
5 years
Chapter 6
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Forward Rates
Comparing strategies 1 to strategy 3, we have:
(1 + r0,5)5 = (1 + r0,2)2(1 + r2,5)3
Using the spot rates given above:
(1.095)5 = (1.088)2 (1 + r2,5)3
Solving for the forward rate:
r2,5 = .0997 = 9.97%
Notice the small difference in yield between holding the 5year bond and the combination of the 2-year and 3-year
bonds.
Chapter 6
34
Theories of The Term Structure
This section explores three theories of the term structure:
1. The Pure Expectations Theory
2. The Liquidity Premium Theory
3. The Market Segmentation Theory
Chapter 6
35
The Pure Expectations Theory
The Pure Expectations Theory states that forward rates
are unbiased estimators of future interest rates, or that
forward rates equal expected future spot rates of interest.
In other words, the Pure Expectations Theory claims that,
on average, today's forward rate equals the future spot rate
for the period corresponding to the forward rate.
This theory assumes that many investors don’t have strong
maturity preferences.
In our previous example, the forward rate was 9.97%
The Pure Expectations Theory would say that 9.97% is a
good estimate of the spot rate that will prevail on a threeperiod bond two periods from now.
Chapter 6
36
The Liquidity Premium Theory
The Liquidity Premium Theory holds that forward rates are
upwardly biased estimators of expected future spot rates.
The theory asserts that the estimates are too high and that
forward rates exceed expected future spot rates. This
theory rejects the claim that there are numerous investors
who are indifferent about the maturities of the bonds that
they hold.
Forward rates exceed expected future spot rates of interest
by the amount of the liquidity premium. Therefore, forward
rates are biased predictors of future spot rates.
The defenders of this theory assert that investors prefer to
hold short-term bonds rather than long-term bonds, and
that investors are willing to pay more for short-term bonds.
The extra amount paid is referred to as a liquidity premium.
Thus, long-term bonds must pay a greater return than
short-term bonds to induce investors to commit their funds
to the long-term bonds.
Chapter 6
37
The Liquidation Premium Theory
Example
Spot Rate
R0,1
R0,2
R0,3
R0,4
R0,5
Yield
.08
.088
.09
.093
.095
Maturity
1 year
2 years
3 years
4 years
5 years
Assume that a 5-year bond must pay 1/10 of one percent
greater yield than a series of 1-year bonds. Because the 5year bond is yielding 9.5%, the series of 1-year bonds
must produce an average yield of 9.4%. Your plan is to
hold five one-year bonds in succession.
In addition to observing the 5-year bond yield today, we
can observe the 1-year today today. The 5-year yields
9.5% and the 1-year yields 8%.
We can not observe the yield on the four following oneyear bonds. However, combined with the current one year,
they must produce an average 9.4% yield over 5 years.
(1.094)5 = (1.08) (1 + X)4 X = 9.753%
The forward rate r1,5 can be calculated as well:
(1 + r0,5)5 = (1 + r0,1)1 (1 + r1,5)4
r1,5 = 9.88%
Chapter 6
38
The Market Segmentation Theory
The Market Segmentation Theory says the yield curve that
exists at any one time reflects the actions and preferences
of certain major participants in the bond market.
These major participants have preferred maturity ranges,
but attractive yields in different maturities lead these
participants to accept maturities outside the preferred
range.
Chapter 6
39
Bond Portfolio Maturity Strategies
When an investor holds only a single bond, it is easy to
understand and manage the maturity of the investment.
However, when an individual holds multiple bonds,
management of the average maturity can be difficult. There
are two basic strategies for managing portfolio maturity:
The Laddered Strategy: funds in the bond portfolio are
distributed approximately evenly over the range of maturities.
Advantage
–
It is easy to maintain the same kind of maturity distribution with
very little transaction cost.
Disadvantage
–
Difficulty in changing the maturity composition of the portfolio.
The Dumbbell or Barbell Strategy: the funds in the bond
portfolio are committed just to short maturity and to very long
maturity bonds.
Advantages
–
It is easy to change the maturity composition of the portfolio.
Disadvantages
–
Requires greater management effort and higher transaction costs.
Chapter 6
40
Portfolio Immunization Techniques
In recognition of the inability to predict interest rates, many
bond managers protect their portfolios from undesirable
effects due to changes in interest rates by immunizing their
portfolios.
A bond portfolio is immunized if its investment performance
is not sensitive to changes in interest rates. A bond
portfolio is immunized if its duration is equal to zero.
Immunization techniques fall into two categories which rely
on the idea of duration:
1. The bank immunization case
2. The planning period case
Chapter 6
41
Bank Immunization Case
Commercial banks borrow money by accepting deposits
and use those funds to make loans. The portfolio of
deposits and the portfolio of loans may both be viewed as
bond portfolios, with the deposit portfolio constituting the
liability portfolio and the loan portfolio constituting the asset
portfolio.
If a bank’s deposits and loans have different maturities, the
bank may lose money in the event of an overall change in
interest rate levels.
To eliminate this risk, banks may wish to immunize their
portfolio. A portfolio is immunized if the value of the
portfolio is not affected by a change in interest rates.
Immunization is accomplished by managing the duration of
the portfolio.
Chapter 6
42
Bank Immunization Case
Table 6.3 illustrates the impact of interest rate changes for
a bank with no immunization.
Table 6.3
Balance Sheet of Simple National Bank
Original Position
Assets
Loan Portfolio Value
Portfolio Duration
Interest Rate
$1,000
5 years
10%
Liabilities
Deposit Portfolio Value
Portfolio Duration
Owners' Equity
Interest Rate
$1,000
1 year
$0
10%
Liabilities
Deposit Portfolio Value
Owners' Equity
$982
B $72
Following Rise in Rates to 12 Percent
Assets
Loan Portfolio Value
$909
Notice that the duration of the assets is 5 years and the
duration of the liabilities is 1 year.
Chapter 6
43
Bank Immunization Case
Using the data from Table 6.3. Assume that interest rates
rise from 10% to 12% on both deposit and loan
portfolios.
What is the change in value of the deposit and loan
portfolios?
Applying the following duration formula:
 d (1+ r i ) 
 P i
dP i = - D i 
 (1+ r i ) 
Deposit Portfolio
dP = -1 (.02/1.10) $1,000 = -$18.18
Loan Portfolio
dP = -5 (.02/1.10) $1,000 = - $90.91
So the deposits (liabilities) have decreased in value by
$18.18 and the assets have decreased in value by $90.91.
The combined effect is equal to a $72 reduction in equity.
Chapter 6
44
Bank Immunization Case
Table 6.4 illustrates the impact of interest rates changes for
a bank with immunization. Notice that both the liabilities
and assets have a duration of 3 years.
Estimate the price change using the duration formula:
dP = -3 (.02/1.10) $1,000 = - $54.55
Because the bank is immunized against a change in
interest rates, the change in rates have an equal and
offsetting effect on the liabilities and assets of the bank
leaving the equity position of the bank unchanged.
Table 6.4
Immunized Balance Sheet of Simple National Bank
Original Position
Assets
Loan Portfolio Value
Portfolio Duration
Interest Rate
Liabilities
$1,000
3 years
10%
Deposit Portfolio Value
Portfolio Duration
Owners' Equity
Interest Rate
$1,000
3 years
$0
10%
Following Rise in Rates to 12 Percent
Assets
Loan Portfolio Value
Liabilities
$945
Deposit Portfolio Value
Owners' Equity
Chapter 6
$945
$0
45
The Planning Period Case
The planning period involves managing a portfolio toward
a horizon date with the goal of achieving a target value for
the portfolio at the end of the planning period (e.g. pension
fund management). This approach is also useful for an
individual wishing to have a certain amount of funds
available by the time he/she retires.
Bond managers deal with issues concerning the effect of
changing interest rates on the immediate value of the bond
portfolio and on the reinvestment rate.
The reinvestment rate is the rate at which cash thrown off
by the bond portfolio can be reinvested. That is, when you
receive a coupon payment from a bond, you will need to
reinvest the funds. The funds are invested at the
reinvestment rate.
Chapter 6
46
The Planning Period Case
A bond manager investing toward a future horizon date
tries to maximize the value of the portfolio on that future
date, subject to risk constraints. Thus, the manager might
attempt to maximize the Realized Compound Yield to
Maturity (RCYTM).
RCYTM is the compound yield realized on an investment
over n periods.
RCYT M = n
T erminalValue
-1
InitialValue
Chapter 6
47
The Planning Period Case
Example
For a bond portfolio with an initial value of $500,000 that is
managed for 8 years with a resulting terminal value of
$900,000. What is the RCYTM?
RCYT M = n
T erminalValue
$900,000
1 = 8
- 1 = .0762
InitialValue
$500,000
Or equivalently
1
1
Terminal Value n
900 ,000 8
RCYTM 
1 
 1  0.0762
Initial Value
500 ,000
(
)
(
)
What would be the impact on RCYTM of a change in
interest rates during the planning period?
Chapter 6
48
The Planning Period Case
To demonstrate how a change in interest rates can affect
the RCYTM, we will examine the following scenarios:
1. Calculate the RCYTM for single bond.
Recalculate the RCYTM after interest rates drop from
10% to 8%.
2. Calculate the RCYTM for two bonds with different
maturities.
Recalculate the RCYTM after interest rates drop from
10% to 8%.
Chapter 6
49
The Planning Period Case
Scenario 1A: assume that a bond portfolio consists of one
10% coupon bond, with a face value of $1000 and 5 years
to maturity. Interest rates are 10% and will remain at this
level for the next 5 years, so coupons payment received
can be reinvested at a 10% rate.
What is the future value of all cash flows at maturity?
What is the RCYTM?
A. The future value of the first four coupons
$100(1.10)4 + $100(1.10)3 + $100(1.10)2 + $100(1.10) = $
510.51
B. The future value of the last coupon plus the face value
$1,000+ $100 = $1,100
C. The total future value
$510.51 +$1,100 = $1,610.51
D. The RCYTM is
($1,610.51/$1,000.00).2 - 1 = 10%
Chapter 6
50
The Planning Period Case
Scenario1B: now assume that interest rates drop from 10%
to 8%.
What is the future value of all cash flows at maturity?
What is the RCYTM?
A. The future value of the first four coupons
$100(1.08)4 + $100(1.08)3 + $100(1.08)2 + $100(1.08) =
$486.66
B. The future value of the last coupon plus the face value
$1,000 +$100 = $1,100
C. The total future value
$486.66 + $1,100.00 = $1,586.66
D. The RCYTM is
($1,586.66/$1,000.00)
.2
-1 = 9.67%
Notice the drop in RCYTM associated with the drop in
interest rates.
Chapter 6
51
The Planning Period Case
The problem with scenario 1B occurred because the
duration of the bond was less than the planning period.
The duration (D): 4.17
The planning period: 5 years
If the duration of the portfolio equals the number of years in
the planning period, the portfolio will be immunized. Thus,
a shift in interest rates will not affect the RCYTM or the
terminal value.
Chapter 6
52
The Planning Period Case
Scenario 2A: assume that a second bond with 8-year
maturity, 10% coupon rate and a yield-to-maturity of 10% is
now available. The bond price will be $1,000 and the
duration 5.87. Now the bond manager has two bonds.
How would the bond manager create a portfolio with
average duration of 5 utilizing some combination of both
bonds?
A. Allocate the funds between 5-year and the 8-year bonds
Bond
Duration (D)
5-year 4.17
8-year 5.87
Total funds allocated
% Allocated
Total $
51.18%
48.82%
$511.80
$488.20
$1,000
B. Determine the new portfolio duration
New portfolio duration = .5118(4.17) +.4882(5.87) = 5
By investing 51.18% of the funds in the 5-year bond and
48.82%in the 8-year bond, the manager can achieve a
portfolio with duration of 5.0.
Chapter 6
53
The Planning Period Case
What is the future value of all cash flows at maturity? And
What is the RCYTM?
A. The future value of the bonds
The total future value at year 5 will come from 4 sources.
1. Future value of reinvested coupons received year 1-4.
$486.66
2. Future value of payoff on the 5-year bond
.5118($1,000+$100) =$562.98
3. The coupon payment at year 5 on the 8-year loan
$100(.4882) = $48.82
4. Sale at year 5 of the 8-year bond
($1,100-$48.82).4882 = $513.36
The total future value
$486.66 +$562.98+$48.82+$513.36 = $1,611.82
Chapter 6
54
The Planning Period Case
B. RCYTM
RCYTM = ($1,611.82/$1,000).2-1 = 10.02%
Scenario 2B: now assume that interest rates drop from
10% to 8%.
What is the new value of the portfolio, today?
What is the future value of all cash flows at maturity?
What is the RCYTM?
Chapter 6
55
The Planning Period Case
A. New value of the portfolio, today.
With a 8% drop in interest rates, the new value will be:
5-year Bond
dP = -4.17(-.02/1.10)$511.80 = $38.80
8-year Bond
dP = -5.18(-.02/1.10)$488.20 = $52.10
Total New Value
New Value = $1000+$38.80 +$52.10 = $1,090.90
B. Future value of all cash flows at maturity
($1,090.90)(1.08)5= $1,602.89
C. RCYTM
RCYTM = ($1,602.89/$1,000)2-1 = 9.90%
Chapter 6
56
Immunization Issues
First, immunization assumes that interest rates change by
the same amount for all different maturities. This is the
same as assuming that any change in the yield curve is
merely a parallel shift.
Second, immunization results hold only for a single change
in interest rates, even when rates change by the same
amount for all maturities. Thus, each time interest rates
change, you must re-compute the duration.
Chapter 6
57

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