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程序代写案例-MANG 6222
时间：2022-05-07
MANG 6222
Fixed Income Securities Analysis
Bond Portfolio Management Strategies
Dr. Yun Luo
Learning Objectives
1. The Asset Allocation Decision
2. Variety of Bond Portfolio Strategies
3. The Use of Leverage
The Asset Allocation Decision
v Public pension funds have allocations of about 2/3 in
equities (which includes real estate and private
equity) and about 1/3 in fixed income.
v Regardless of the institutional investor, there are
two important decisions to be made by an
investor/client:
v “How much should be allocated to bonds?”
v “Who should manage the funds to be allocated to
bonds?”
How Much Should Be Allocated To Bonds?
• The decision as to how much to invest in the major asset
classes is referred to as the asset allocation decision.
• The asset allocation decision must be made in light of the
investor’s investment objective.
• For institutions such as pension funds, the investment
objective is to generate sufficient cash flow from investments
to satisfy pension obligations.
• For institutions such as banks and thrifts, funds are obtained
from the issuance of certificates of deposit, short-term
money market instruments, or floating-rate notes. These
funds are then invested in loans and marketable securities.
The objective in this case is to earn a return on invested
funds that exceeds the cost of acquiring those funds.
Who Should Manage the Bond Portfolio?
• Let’s assume that an investor has made the decision to
allocate a specified amount to the fixed income sector.
• The next decision that must be made is whether that
amount will be managed by
• internal managers or
• external managers or
• by a combination of internal and external managers.
• If external managers are hired, a decision must be made
as to which asset management firm (e.g. asset
management firm) to engage.
Who Should Manage the Bond Portfolio?
• In practice, the term asset allocation is used in two
contexts.
1） The first involves allocation of funds among major asset
classes that includes bonds, equities and alternative assets.
• Although we have mentioned bonds and equities as the
major asset classes, there is now accepted a group of
assets referred to as alternative assets. For example, for
the California Public Employees Retirement System
(CalPERS), the actual (as of January 31, 2011) and target
allocation (as of June 2009) asset allocation amongst the
asset classes defined by CalPERS is shown in Exhibit 22-
1.
Exhibit 22-1 Asset Allocation of CalPERS: Actual as
of January 31, 2011, and Target Allocation as of
June 2009
Asset Class
Market Value
($ billion)
Actual
Allocation
Target
Allocation
(%)
Cash Equivalents 4.50 2.0% 2.0%
Global Fixed Income 47.50 20.8% 20.0%
AIM 32.20 14.1% 14.0%
Equity 120.30 52.8% 49.0%
Total Global Equities 152.50 66.9% 63.0%
Real Estate Global 16.60 7.3% 10.0%
Inflation Linked Global 6.80 3.0% 5.0%
Total Fund* 227.90 100.0% 100.0%
Source: http://www.calpers.ca.gov/index.jsp?bc=/investments/assets/assetallocation.xml
Who Should Manage the Bond Portfolio?
2) The second way is how the funds should be allocated
amongst the different sectors within that asset class after a
decision has been made to invest in a specified asset
class.
• In the case of equities, equities are classified by
market capitalization and by other attributes such as
growth stocks value.
• E.g. companies may be categorized as large-,
mid-, or small-cap depending on their market
capitalization.
Who Should Manage the Bond Portfolio?
• The asset allocation among the different sectors of the
bond is made at two levels.
1. The first is where a client must make a decision as
to allocate among each sector and
2. then if an external money manager is to be hired,
deciding on the asset management and amount to
be allocated to each.
Portfolio Management Team
q We refer to the person making the investment decisions
as the “manager” or “portfolio manager.”
q In practice, while there is someone who will make the
ultimate decision about the composition and therefore
risk exposure of a portfolio, that decision is the result of
recommendations and research provided by the portfolio
management team.
q At the top of the investment organization chart of the
investment group is the chief investment officer (CIO)
who is responsible for all of the portfolios.
q A chief compliance officer (CCO) monitors portfolios to
make sure that the holdings comply with the fund’s
investment guidelines and that there are no activities
conducted by the managers of the fund that are in
violation of laws or investment policies.
Portfolio Management Team
q An asset management firm employs analysts and traders.
§ The analysts are responsible for the different sectors and
industries.
§ The traders are responsible for executing trades approved
by a portfolio manager.
§ The analysts and traders can support all of the portfolios
managed by the firm or just designated portfolios.
q A large firm may also employ an economist or an economic
staff that would support all portfolios managed by the firm.
q At the individual portfolio level there is either a lead or senior
portfolio manager or co-managers.
q It is the lead manager or co-managers who will make the
decision regarding the portfolio’s interest rate exposure and
the allocation of the fund’s assets among the countries,
sectors and industries.
Variety of Bond Portfolio Strategies
q The bond portfolio strategy selected by an investor
or client depends on the investment objectives and
policy guidelines.
q In general, bond portfolio strategies can be
categorized into the following three groups:
1) bond benchmark-based strategies,
2) absolute return strategies, and
3) liability-driven strategies.
Bond Benchmark-Based Strategies
§ There is a wide range of bond portfolio management
strategies for an investor or client who has selected a
bond index as a benchmark.
§ Traditional bond benchmark-based strategies can be
classified as:
1) pure bond index matching;
2) enhanced indexing: matching primary risk factors;
3) enhanced indexing: minor risk-factor mismatches;
4) active management: larger risk-factor mismatches;
and
5) active management: full-blown active.
Bond Benchmark-Based Strategies
• These strategies are categorized based on how much
the portfolio manager departs from the primary risk
factors.
• Basically, one can view a benchmark (bond index) as a
package of risk factors.
• The classification of a strategy into one of the
categories above depends on the degree to which a
portfolio manager is allowed to depart from the
quantity of risk in the benchmark.
• It is not only important to understand what the risk
factors are, but also how to quantify them.
The Primary Risk Factors
• The primary risk factors can be divided into two general
types:
1) systematic risk factors and
2) non-systematic risk factors.
• Systematic risk factors are forces that affect all
securities in a certain category in the benchmark.
• Non-systematic risk factors are the risks that are not
attributable to the systematic risk factors.
– Non-systematic factor risks are classified as non-
systematic risks associated with a particular
issuer, issuer-specific risk, and those associated
with a particular issue, issue-specific risk.
The Primary Risk Factors (cont.)
Systematic risk factors, in turn, are divided into two
categories: term structure risk factors and non-term structure
risk factors.
1) Term structure risk factors are risks associated with
changes in the shape of the term structure.
2) Non-term structure risk factors include sector risk,
credit risk and optionality risk.
§ Sector risk is the risk associated with exposure to the
sectors of the benchmark.
§ Credit risk, also referred to as quality risk, is the risk
associated with exposure to the credit rating of the
securities in the benchmark.
§ Optionality risk is the risk associated with an adverse
impact on the embedded options of the securities in
the benchmark.
Absolute Return Strategies
§ In an absolute return strategy, the portfolio manager
seeks to earn a positive return over some time frame
irrespective of market conditions.
§ Few restrictions are placed on the exposure to the primary
risk factors.
§ Absolute return strategies are typically pursued by hedge
fund managers using leverage.
§ Other absolute return managers set as their target as
earning a return from 150 to 400 basis points per annum
over the return on cash and hence such strategies are
referred to as cash-based absolute return strategies.
§ A bond portfolio strategy that calls for structuring a
portfolio to satisfy future liabilities is called a liability-
driven strategy.
§ When the portfolio is constructed so as to generate
sufficient funds to satisfy a single future liability
regardless of the course of future interest rates, a
strategy known as immunization is often used.
§ When the portfolio is designed to funding multiple
future liabilities regardless of how interest rates
change, strategies such as immunization, cash flow
matching (or dedication), or horizon matching can
be employed.
Liability-Driven Strategies
Top-Down Versus Bottom-Up Portfolio
Construction and Management
q There are two general approaches to construction and
management of a bond portfolio: top-down and bottom-
up.
q In the top-down approach, a bond portfolio manager
looks at the major macro drivers of bond returns (hence
this approach is also referred to as a macro approach)
and obtains a view (forecast) about these drivers in the
form of a macroeconomic forecast.
q Among the major variables considered in obtaining a
macroeconomic forecast are monetary policy, fiscal
policy, tax policy, political developments, regulatory
matters, exchange-rate movements, trade policy,
demographic trends, and credit market conditions.
q Given the amount of the portfolio's funds to be allocated
to each sector of the bond market, the manager must
then decide how much to allocate to each industry within
a sector.
§ In the case of bond portfolio manager who is entitled
to invest in both U.S. and non-U.S. bonds, the first
decision is the allocation among countries, then
sectors within a country, and then industries.
Top-Down Versus Bottom-Up Portfolio
Construction and Management (Cont.)
q The bottom-up approach to active bond portfolio
management focuses on the micro analysis of individual
bond issues, sectors, and industries.
§ The primary research tools used in this form of
investing is credit analysis, industry analysis, and
relative value analysis.
§ To control the portfolios risk, risk modeling is used.
Top-Down Versus Bottom-Up Portfolio
Construction and Management (Cont.)
q Types of Shifts in the Yield Curve
§ A shift in the yield curve refers to the relative change
in the yield for each Treasury maturity.
§ A parallel shift in the yield curve is a shift in
which the change in the yield on all maturities is the
same.
§ A nonparallel shift in the yield curve indicates
that the yield for maturities does not change by the
same number of basis points.
§ Historically, two types of nonparallel yield curve
shifts have been observed: a twist in the slope of
the yield curve and a change in the humpedness of
the yield curve.
Active/Passive Portfolio Strategies
- Yield curve strategies
q Impact on Historical Returns
§ In practice, the slope of the yield curve is measured
by the spread between some long-term Treasury
yield and some short-term Treasury yield.
§ A flattening of the yield curve indicates that the
yield spread between the yield on a long-term and a
short-term Treasury has decreased;
§ A steepening of the yield curve indicates that the
yield spread between a long-term and a short-term
Treasury has increased.
§ The other type of nonparallel shift, a change in the
humpedness of the yield curve, is referred to as a
butterfly shift.
Yield curve strategies
q Frank Jones analyzed the types of yield curve shifts
that occurred between 1979 and 1990.
q He found that the three types of yield curve shifts are
not independent, with the two most common types of
yield curve shifts being
q a downward shift in the yield curve combined with a
steepening of the yield curve.
q an upward shift in the yield curve combined with a
flattening of the yield curve.
q These two types of shifts in the yield curve are
depicted in Exhibit 22-3.
Yield curve strategies empirical
evidence (Jones 1991)
• In portfolio strategies that seek to capitalize on
expectations based on short-term movements in
yields.
• This means that the maturity of the securities in the
portfolio will have an important impact on the
portfolio’s return.
• For example: the total return over a one-year
investment horizon for a portfolio consisting of
securities all maturing in 30 years will be sensitive
to how the yield curve shifts because one year
from now the value of the portfolio will depend on
the yield offered on 29-year securities.
Yield curve strategies
q The key point is that for short-term investment
horizons, the spacing of the maturity of bonds in the
portfolio will have a significant impact on the total
return.
There are three yield curve strategies
q In a bullet strategy, the portfolio is constructed so
that the maturities of the securities in the portfolio are
highly concentrated at one point on the yield curve.
q In a barbell strategy, the maturities of the
securities in the portfolio are concentrated at two
extreme maturities.
q In a ladder strategy the portfolio is constructed to
have approximately equal amounts of each maturity.
Yield curve strategies
Yield curve strategies
Example Question
Two portfolios with a market value of $1000 million. The bonds in both
portfolios are trading at par value. The dollar duration of the two
portfolios is the same.
Issue
Years to
Maturity
Par Value
(in millions)
Bonds Included in Portfolio I
A 4.0 $240
B 5.0 $260
C 40.0 $300
D 41.0 $200
Bonds Included in Portfolio II
E 19.4 $400
F 20.0 $460
G 20.2 $ 140
Which portfolio can be characterized as a bullet portfolio?
Which portfolio can be characterized as a barbell portfolio?
Answer Question
Which portfolio can be characterized as a bullet portfolio?
In a bullet strategy, the portfolio is constructed so that the maturities
of the securities in the portfolio are highly concentrated at one point
on the yield curve. Thus, Portfolio II can be characterized as a bullet
portfolio because the maturities of its securities are concentrated
around one maturity (twenty years).
Which portfolio can be characterized as a barbell portfolio?
In a barbell strategy, the maturities of the securities included in the
portfolio are concentrated at two extreme maturities. Thus, Portfolio I
can be characterized as a barbell portfolio because the maturities of
its securities are concentrated at two extreme maturities (four years
and forty years).
Duration and Yield Curve Shifts
• Duration is a measure of the sensitivity of the price of a bond
or the value of a bond portfolio to changes in market yields.
• A bond with a duration of 4 means that if market yields
change by 100 basis points, the bond will change by
approximately 4%.
• However, if a three-bond portfolio has a duration of 4, the
statement that the portfolio’s value will change by 4% for a
100-basis-point change in yields actually should be stated as
follows:
• The portfolio’s value will change by 4% if the yield on five-,
10-, and 20-year bonds all change by 100 basis points. That
is, it is assumed that there is a parallel yield curve shift.
Analyzing Expected Yield Curve Strategies
• The proper way to analyze any portfolio strategy is to look at its
potential total return.
• If a manager wants to assess the outcome of a portfolio for any
assumed shift in the Treasury yield curve, this should be done
by calculating the potential total return if that shift actually
occurs.
• This can be illustrated by looking at the performance of two
hypothetical portfolios of Treasury securities assuming different
shifts in the Treasury yield curve.
Exhibit 22-5 Three Hypothetical Treasury
Securities
Bond
Coupon
(%)
Maturity
(years)
Price Plus
Accrued
Yield to
Maturity
(%)
Dollar
Duration
Dollar
Convexity
A 8.50 5 100 8.50 4.005 19.8164
B 9.50 20 100 9.50 8.882 124.1702
C 9.25 10 100 9.25 6.434 55.4506
• The three hypothetical Treasury securities shown in Exhibit 22-
5 are considered for inclusion in our two portfolios.
• For our illustration, the Treasury yield curve consists of these
three Treasury securities: a short-term security (A, the five-year
security), an intermediate-term security (C, the 10-year
security), and a long-term security (B, the 20-year security).
Analyzing Expected Yield Curve Strategies
• Consider the following two yield curve strategies: a bullet
strategy and a barbell strategy. We will label the portfolios
created based on these two strategies as the “bullet
portfolio” and the “barbell portfolio” and they comprise the
following:
• Bullet portfolio: 100% bond C
• Barbell portfolio: 50.2% bond A and 49.8% bond B
Analyzing Expected Yield Curve Strategies
• The dollar duration for the bullet portfolio per 100-basis-
point change in yield is 6.434.
• For the barbell portfolio, the dollar duration is just the
weighted average of the dollar duration of the two bonds:
• dollar duration of barbell portfolio = 0.502(4.005)
+0.498(8.882) = 6.434
• The dollar duration of the barbell portfolio is the same as
that of the bullet portfolio.
Analyzing Expected Yield Curve Strategies
• Duration is just a first approximation of the change in price
resulting from a change in interest rates. Convexity
provides a second approximation.
• Although we did not discuss dollar convexity, it has a
meaning similar to convexity, in that it provides a second
approximation to the dollar price change.
• For two portfolios with the same dollar duration, the
greater the convexity, the better the performance of a
bond or a portfolio when yields change.
Analyzing Expected Yield Curve Strategies
• The dollar convexity of the bullet portfolio is 55.4506.
• The dollar convexity for the barbell portfolio is a weighted
average of the dollar convexity of the two bonds.
• That is, dollar convexity of barbell portfolio =
0.502(19.8164) + 0.498(124.1702) = 71.7846
• Therefore, the dollar convexity of the barbell portfolio is
greater than that of the bullet portfolio.
Analyzing Expected Yield Curve Strategies
• Similarly, the yield for the two portfolios is not the same.
The yield for the bullet portfolio is simply the yield to
maturity of bond C, 9.25%.
• The traditional yield calculation for the barbell portfolio,
which is found by taking a weighted average of the yield
to maturity of the two bonds included in the portfolio, is
8.998%:
• Portfolio yield for barbell portfolio = 0.502(8.50%) +
0.498(9.50%) = 8.998%
• This approach suggests that the yield of the bullet
portfolio is 25.2 basis points greater than that of the
barbell portfolio (9.25% - 8.998%).
Analyzing Expected Yield Curve Strategies
• Although both portfolios have the same dollar duration,
the yield of the bullet portfolio is greater than the yield of
the barbell portfolio.
• However, the dollar convexity of the barbell portfolio is
greater than that of the bullet portfolio.
• The difference in the two yields is sometimes referred to
as the cost of convexity (i.e., giving up yield to get better
convexity).
Analyzing Expected Yield Curve Strategies
• Now suppose that a portfolio manager with a six-month
investment horizon has a choice of investing in the bullet
portfolio or the barbell portfolio.
• Which one should he choose? The manager knows that
(1) the two portfolios have the same dollar duration, (2)
the yield for the bullet portfolio is greater than that of the
barbell portfolio, and (3) the dollar convexity of the barbell
portfolio is greater than that of the bullet portfolio.
• Actually, this information is not adequate in making the
decision. What is necessary is to assess the potential total
return when the yield curve shifts.
Exhibit 22-6 Six-Month Investment Horizon
bullet portfolio’s total return – barbell portfolio’s total return
Yield
Change
Parallel
Shift (a)
Nonparallel
Shift (b)
Nonparallel
Shift (c)
−5.000 −7.19 −10.69 −3.89
−4.750 −6.28 −9.61 −3.12
−4.500 −5.44 −8.62 −2.44
−4.250 −4.68 −7.71 −1.82
−4.000 −4.00 −6.88 −1.27
−3.750 −3.38 −6.13 −0.78
−3.500 −2.82 −5.44 −0.35
… … … …
3.750 −1.39 −1.98 −0.85
4.000 −1.57 −2.12 −1.06
4.250 −1.75 −2.27 −1.27
4.500 −1.93 −2.43 −1.48
4.750 −2.12 −2.58 −1.70
5.000 −2.31 −2.75 −1.92
(b) Change in yield for bond C. Nonparallel shift as follows (flattening of yield curve):
yield change bond A = yield change bond C + 25 basis points
yield change bond B = yield change bond C - 25 basis points
(c) Change in yield for bond C. Nonparallel shift as follows (steepening of yield curve): yield
change bond A = yield change bond C - 25 basis points
yield change bond B = yield change bond C + 25 basis points
Analyzing Expected Yield Curve Strategies
• Let’s focus on the second column which is labeled
“parallel shift.”
• In this case parallel movement of the yield curve means
that the yields for the short-term bond (A), the
intermediate-term bond (C), and the long-term bond (B)
change by the same number of basis points, shown in
the “yield change” column of the table.
• Which portfolio is the better investment alternative if the
yield curve shifts in a parallel fashion and the
investment horizon is six months? The answer depends
on the amount by which yields change.
• Notice that when yields change by less than 100 basis
points, the bullet portfolio outperforms the barbell
portfolio. The reverse is true if yields change by more
than 100 basis points.
Analyzing Expected Yield Curve Strategies
• This illustration makes two key points.
• First, even if the yield curve shifts in a parallel fashion, two
portfolios with the same dollar duration will not give the
same performance. The reason is that the two portfolios
do not have the same dollar convexity.
• The second point is that although with all other things
equal it is better to have more convexity than less, the
market charges for convexity in the form of a higher price
or a lower yield. But the benefit of the greater convexity
depends on how much yields change.
Analyzing Expected Yield Curve Strategies
• Now let’s look at what happens if the yield curve does
not shift in a parallel fashion.
• Specifically, the first nonparallel shift column assumes
that if the yield on bond C (the intermediate-term bond)
changes by the amount shown in the first column,
• bond A (the short-term bond) will change by the
same amount + 25 basis points,
• bond B (the long-term bond) will change by the
same amount shown in the first column - 25 basis
points.
• The spread between the long-term yield (yield on bond
B) and the short-term yield (yield on Bond A), the spread
has decreased by 50 basis points =》 flattening of the
yield curve
• The barbell outperforms the bullet (always negative)
Analyzing Expected Yield Curve Strategies
• In the last column, the nonparallel shift assumes that for
a change in bond C’s yield,
• The yield on bond A will change by the same
amount - 25 basis points,
• The bond B will change by the same amount + 25
points
• Thus, the spread between the long-term yield and the
short-term yield has increased by 50 basis points, and
the yield curve has steepened
• In this case, the bullet portfolio outperforms the barbell
portfolio as long as the yield on bond C does not rise by
more than 250 basis points or fall by more than 325
basis points.
Analyzing Expected Yield Curve Strategies
• The key point here is that looking at measures such as
yield (yield to maturity or some type of portfolio yield
measure), duration, or convexity tells us little about
performance over some investment horizon, because
performance depends on the magnitude of the change in
yields and how the yield curve shifts.
• Therefore, when a manager wants to position a portfolio
based on expectations as to how he might expect the
yield curve to shift, it is important to perform total return
analysis.
The Use of Leverage
q If permitted by investment guidelines a manager
may use leverage in an attempt to enhance portfolio
returns.
q A portfolio manager can create leverage by
borrowing funds in order to acquire a position in the
market that is greater than if only cash were
invested.
q For example, a manager may have cash to invest in
the bond market of $50 million but wants an
exposure of $200 million.
The Use of Leverage
q The funds available to invest without borrowing are
referred to as the “equity.”
q A portfolio that does not contain any leverage is
called an unlevered portfolio.
q A levered portfolio is a portfolio in which a manager
has created leverage.
§ The basic principle in using leverage is that a manager
wants to earn a return on the borrowed funds that is
greater than the cost of the borrowed funds.
§ The return from borrowing funds is produced from a
higher income and/or greater price appreciation relative
to a scenario in which no funds are borrowed.
• If a manager can invest $50 million earning 5% for a
year but can borrow funds at a cost of 4.5% for a year,
then the manager can generate for the year 50 basis
points in income over its funding cost. By borrowing
greater amounts, the manager can magnify the 50 basis
points. This is a benefit of leveraging.
Motivation for Leverage
• The return from investing the funds comes from two
sources.
i. interest income
ii. change in the value of the security (or securities) at
the end of the borrowing period
§ There are some managers who use leverage in the
hopes of benefiting primarily from price changes.
§ Small price changes will be magnified by using
leveraging.
• For example, if a manager expects interest rates to
fall, the manager can borrow funds to increase price
exposure to the market.
• Effectively, the manager is increasing the duration of
the portfolio.
Motivation for Leverage
§ The risk associated with borrowing funds：
§ Cost of the borrowed funds > Borrowed funds are
invested may earn
§ Due to failure to generate interest income plus
capital appreciation as expected when the funds
were borrowed.
Motivation for Leverage
Motivation for Leverage
• Leveraging is a necessity for depository institutions
• Depository institutions refers to such as banks and
savings and loan associations
• Because the spread over the cost of borrowed funds is
typically small.
• The magnitude of the borrowing (i.e., the degree of
leverage) is what produces an acceptable return for the
institution.
Duration of a Leveraged Portfolio
In general, the procedure for calculating the duration of a
portfolio that uses leverage is as follows:
Step 1: Calculate the duration of the levered portfolio.
Step 2: Determine the dollar duration of the portfolio of
the levered portfolio for a change in interest rates.
Step 3: Compute the ratio of the dollar duration of the
levered portfolio to the value of the initial unlevered
portfolio (i.e., initial equity).
Step 4: The duration of the unlevered portfolio is then
found as follows:
Duration of a Leveraged Portfolio
Suppose that a portfolio
• Market value: $100 million
• Duration:3
• This means that the manager would expect that for a 100-
basis-point change in interest rates
• The portfolio’s value would change by approximately $3
million.
• For this unlevered fund, the duration of the portfolio is 3.
Duration of a Leveraged Portfolio
Suppose now that the manager of this portfolio:
• Borrow an additional $300 million.
• This means that the levered fund will have $400 million to
invest
• consisting of $100 million that the manager has
available before borrowing (i.e., the equity) and $300
million borrowed.
• All of the funds are invested in a bond with a duration of 3.
• Now let’s look at what happens if interest rates change by
100 basis points. The levered portfolio’s value will
change?
Duration of a Leveraged Portfolio
• All of the funds are invested in a bond with a duration of 3.
• If interest rates change by 100 basis points.
• The levered portfolio’s value will change by $12 million
(3% times $400 million).
• This means that on an investment of $100 million, the
portfolio’s value changes by $12 million.
• The proper way to measure the portfolio’s duration is
relative to the unlevered amount or equity because the
manager is concerned with the risk exposure relative to
equity
Duration of a Leveraged Portfolio
Step 1: Calculate the duration of the levered portfolio.
Assume that calculation of the duration of the levered
portfolio finds that the duration is 3.
Step 2: Determine the dollar duration of the portfolio of the
levered portfolio for a change in interest rates.
Let’s use a 50-basis-point change in interest rates to
compute the dollar duration. If the duration of the levered
portfolio is 3, then the dollar duration for a 50-basis-point
change in interest rates is $6 million (1.5% (3*0 change
for a 50-basis-point move times $400 million).
Duration of a Leveraged Portfolio
Step 3: Compute the ratio of the dollar duration of the
levered portfolio to the value of the initial unlevered
portfolio (i.e., initial equity).
The ratio of the dollar duration for a 50-basis-point
change in interest rates to the $100 million initial market
value of the unlevered portfolio is 0.06 ($6 million divided
by $100 million).
Step 4: The duration of the unlevered portfolio is then found
as follows:
Duration of a Leveraged Portfolio
Step 4: The duration of the unlevered portfolio is then found
as follows:
• If interest rates change by 100 basis points.
• This means that on an investment of $100 million, the
portfolio’s value changes by $12 million.
• The proper way to measure the portfolio’s duration is
relative to the unlevered amount or equity because the
manager is concerned with the risk exposure relative to
equity
Previous Lecture Tutorial
Question
The YTM on 1-year zeros is currently 7%; the
YTM on 2-year zeros is 8%. The treasury plans
to issue a 2-year maturity coupon bond, paying
coupon once per year with a coupon rate of 9%.
The face value of the bond is $100.
a. At what price will the bond sell?
b. What will the YTM on the bond be?
c. If the expectation theory of the yield curve
is correct, what is the market expectation of the
price that the bond will sell for next year?
d. Recalculate your answer to c) if you
believe in the liquidity preference theory and
you believe that the liquidity premium is 1%.
Previous Lecture Tutorial Question
Fixed Income Securities Analysis
Bond Portfolio Management Strategies
Dr. Yun Luo
Learning Objectives
1. The Asset Allocation Decision
2. Variety of Bond Portfolio Strategies
3. The Use of Leverage
The Asset Allocation Decision
v Public pension funds have allocations of about 2/3 in
equities (which includes real estate and private
equity) and about 1/3 in fixed income.
v Regardless of the institutional investor, there are
two important decisions to be made by an
investor/client:
v “How much should be allocated to bonds?”
v “Who should manage the funds to be allocated to
bonds?”
How Much Should Be Allocated To Bonds?
• The decision as to how much to invest in the major asset
classes is referred to as the asset allocation decision.
• The asset allocation decision must be made in light of the
investor’s investment objective.
• For institutions such as pension funds, the investment
objective is to generate sufficient cash flow from investments
to satisfy pension obligations.
• For institutions such as banks and thrifts, funds are obtained
from the issuance of certificates of deposit, short-term
money market instruments, or floating-rate notes. These
funds are then invested in loans and marketable securities.
The objective in this case is to earn a return on invested
funds that exceeds the cost of acquiring those funds.
Who Should Manage the Bond Portfolio?
• Let’s assume that an investor has made the decision to
allocate a specified amount to the fixed income sector.
• The next decision that must be made is whether that
amount will be managed by
• internal managers or
• external managers or
• by a combination of internal and external managers.
• If external managers are hired, a decision must be made
as to which asset management firm (e.g. asset
management firm) to engage.
Who Should Manage the Bond Portfolio?
• In practice, the term asset allocation is used in two
contexts.
1） The first involves allocation of funds among major asset
classes that includes bonds, equities and alternative assets.
• Although we have mentioned bonds and equities as the
major asset classes, there is now accepted a group of
assets referred to as alternative assets. For example, for
the California Public Employees Retirement System
(CalPERS), the actual (as of January 31, 2011) and target
allocation (as of June 2009) asset allocation amongst the
asset classes defined by CalPERS is shown in Exhibit 22-
1.
Exhibit 22-1 Asset Allocation of CalPERS: Actual as
of January 31, 2011, and Target Allocation as of
June 2009
Asset Class
Market Value
($ billion)
Actual
Allocation
Target
Allocation
(%)
Cash Equivalents 4.50 2.0% 2.0%
Global Fixed Income 47.50 20.8% 20.0%
AIM 32.20 14.1% 14.0%
Equity 120.30 52.8% 49.0%
Total Global Equities 152.50 66.9% 63.0%
Real Estate Global 16.60 7.3% 10.0%
Inflation Linked Global 6.80 3.0% 5.0%
Total Fund* 227.90 100.0% 100.0%
Source: http://www.calpers.ca.gov/index.jsp?bc=/investments/assets/assetallocation.xml
Who Should Manage the Bond Portfolio?
2) The second way is how the funds should be allocated
amongst the different sectors within that asset class after a
decision has been made to invest in a specified asset
class.
• In the case of equities, equities are classified by
market capitalization and by other attributes such as
growth stocks value.
• E.g. companies may be categorized as large-,
mid-, or small-cap depending on their market
capitalization.
Who Should Manage the Bond Portfolio?
• The asset allocation among the different sectors of the
bond is made at two levels.
1. The first is where a client must make a decision as
to allocate among each sector and
2. then if an external money manager is to be hired,
deciding on the asset management and amount to
be allocated to each.
Portfolio Management Team
q We refer to the person making the investment decisions
as the “manager” or “portfolio manager.”
q In practice, while there is someone who will make the
ultimate decision about the composition and therefore
risk exposure of a portfolio, that decision is the result of
recommendations and research provided by the portfolio
management team.
q At the top of the investment organization chart of the
investment group is the chief investment officer (CIO)
who is responsible for all of the portfolios.
q A chief compliance officer (CCO) monitors portfolios to
make sure that the holdings comply with the fund’s
investment guidelines and that there are no activities
conducted by the managers of the fund that are in
violation of laws or investment policies.
Portfolio Management Team
q An asset management firm employs analysts and traders.
§ The analysts are responsible for the different sectors and
industries.
§ The traders are responsible for executing trades approved
by a portfolio manager.
§ The analysts and traders can support all of the portfolios
managed by the firm or just designated portfolios.
q A large firm may also employ an economist or an economic
staff that would support all portfolios managed by the firm.
q At the individual portfolio level there is either a lead or senior
portfolio manager or co-managers.
q It is the lead manager or co-managers who will make the
decision regarding the portfolio’s interest rate exposure and
the allocation of the fund’s assets among the countries,
sectors and industries.
Variety of Bond Portfolio Strategies
q The bond portfolio strategy selected by an investor
or client depends on the investment objectives and
policy guidelines.
q In general, bond portfolio strategies can be
categorized into the following three groups:
1) bond benchmark-based strategies,
2) absolute return strategies, and
3) liability-driven strategies.
Bond Benchmark-Based Strategies
§ There is a wide range of bond portfolio management
strategies for an investor or client who has selected a
bond index as a benchmark.
§ Traditional bond benchmark-based strategies can be
classified as:
1) pure bond index matching;
2) enhanced indexing: matching primary risk factors;
3) enhanced indexing: minor risk-factor mismatches;
4) active management: larger risk-factor mismatches;
and
5) active management: full-blown active.
Bond Benchmark-Based Strategies
• These strategies are categorized based on how much
the portfolio manager departs from the primary risk
factors.
• Basically, one can view a benchmark (bond index) as a
package of risk factors.
• The classification of a strategy into one of the
categories above depends on the degree to which a
portfolio manager is allowed to depart from the
quantity of risk in the benchmark.
• It is not only important to understand what the risk
factors are, but also how to quantify them.
The Primary Risk Factors
• The primary risk factors can be divided into two general
types:
1) systematic risk factors and
2) non-systematic risk factors.
• Systematic risk factors are forces that affect all
securities in a certain category in the benchmark.
• Non-systematic risk factors are the risks that are not
attributable to the systematic risk factors.
– Non-systematic factor risks are classified as non-
systematic risks associated with a particular
issuer, issuer-specific risk, and those associated
with a particular issue, issue-specific risk.
The Primary Risk Factors (cont.)
Systematic risk factors, in turn, are divided into two
categories: term structure risk factors and non-term structure
risk factors.
1) Term structure risk factors are risks associated with
changes in the shape of the term structure.
2) Non-term structure risk factors include sector risk,
credit risk and optionality risk.
§ Sector risk is the risk associated with exposure to the
sectors of the benchmark.
§ Credit risk, also referred to as quality risk, is the risk
associated with exposure to the credit rating of the
securities in the benchmark.
§ Optionality risk is the risk associated with an adverse
impact on the embedded options of the securities in
the benchmark.
Absolute Return Strategies
§ In an absolute return strategy, the portfolio manager
seeks to earn a positive return over some time frame
irrespective of market conditions.
§ Few restrictions are placed on the exposure to the primary
risk factors.
§ Absolute return strategies are typically pursued by hedge
fund managers using leverage.
§ Other absolute return managers set as their target as
earning a return from 150 to 400 basis points per annum
over the return on cash and hence such strategies are
referred to as cash-based absolute return strategies.
§ A bond portfolio strategy that calls for structuring a
portfolio to satisfy future liabilities is called a liability-
driven strategy.
§ When the portfolio is constructed so as to generate
sufficient funds to satisfy a single future liability
regardless of the course of future interest rates, a
strategy known as immunization is often used.
§ When the portfolio is designed to funding multiple
future liabilities regardless of how interest rates
change, strategies such as immunization, cash flow
matching (or dedication), or horizon matching can
be employed.
Liability-Driven Strategies
Top-Down Versus Bottom-Up Portfolio
Construction and Management
q There are two general approaches to construction and
management of a bond portfolio: top-down and bottom-
up.
q In the top-down approach, a bond portfolio manager
looks at the major macro drivers of bond returns (hence
this approach is also referred to as a macro approach)
and obtains a view (forecast) about these drivers in the
form of a macroeconomic forecast.
q Among the major variables considered in obtaining a
macroeconomic forecast are monetary policy, fiscal
policy, tax policy, political developments, regulatory
matters, exchange-rate movements, trade policy,
demographic trends, and credit market conditions.
q Given the amount of the portfolio's funds to be allocated
to each sector of the bond market, the manager must
then decide how much to allocate to each industry within
a sector.
§ In the case of bond portfolio manager who is entitled
to invest in both U.S. and non-U.S. bonds, the first
decision is the allocation among countries, then
sectors within a country, and then industries.
Top-Down Versus Bottom-Up Portfolio
Construction and Management (Cont.)
q The bottom-up approach to active bond portfolio
management focuses on the micro analysis of individual
bond issues, sectors, and industries.
§ The primary research tools used in this form of
investing is credit analysis, industry analysis, and
relative value analysis.
§ To control the portfolios risk, risk modeling is used.
Top-Down Versus Bottom-Up Portfolio
Construction and Management (Cont.)
q Types of Shifts in the Yield Curve
§ A shift in the yield curve refers to the relative change
in the yield for each Treasury maturity.
§ A parallel shift in the yield curve is a shift in
which the change in the yield on all maturities is the
same.
§ A nonparallel shift in the yield curve indicates
that the yield for maturities does not change by the
same number of basis points.
§ Historically, two types of nonparallel yield curve
shifts have been observed: a twist in the slope of
the yield curve and a change in the humpedness of
the yield curve.
Active/Passive Portfolio Strategies
- Yield curve strategies
q Impact on Historical Returns
§ In practice, the slope of the yield curve is measured
by the spread between some long-term Treasury
yield and some short-term Treasury yield.
§ A flattening of the yield curve indicates that the
yield spread between the yield on a long-term and a
short-term Treasury has decreased;
§ A steepening of the yield curve indicates that the
yield spread between a long-term and a short-term
Treasury has increased.
§ The other type of nonparallel shift, a change in the
humpedness of the yield curve, is referred to as a
butterfly shift.
Yield curve strategies
q Frank Jones analyzed the types of yield curve shifts
that occurred between 1979 and 1990.
q He found that the three types of yield curve shifts are
not independent, with the two most common types of
yield curve shifts being
q a downward shift in the yield curve combined with a
steepening of the yield curve.
q an upward shift in the yield curve combined with a
flattening of the yield curve.
q These two types of shifts in the yield curve are
depicted in Exhibit 22-3.
Yield curve strategies empirical
evidence (Jones 1991)
• In portfolio strategies that seek to capitalize on
expectations based on short-term movements in
yields.
• This means that the maturity of the securities in the
portfolio will have an important impact on the
portfolio’s return.
• For example: the total return over a one-year
investment horizon for a portfolio consisting of
securities all maturing in 30 years will be sensitive
to how the yield curve shifts because one year
from now the value of the portfolio will depend on
the yield offered on 29-year securities.
Yield curve strategies
q The key point is that for short-term investment
horizons, the spacing of the maturity of bonds in the
portfolio will have a significant impact on the total
return.
There are three yield curve strategies
q In a bullet strategy, the portfolio is constructed so
that the maturities of the securities in the portfolio are
highly concentrated at one point on the yield curve.
q In a barbell strategy, the maturities of the
securities in the portfolio are concentrated at two
extreme maturities.
q In a ladder strategy the portfolio is constructed to
have approximately equal amounts of each maturity.
Yield curve strategies
Yield curve strategies
Example Question
Two portfolios with a market value of $1000 million. The bonds in both
portfolios are trading at par value. The dollar duration of the two
portfolios is the same.
Issue
Years to
Maturity
Par Value
(in millions)
Bonds Included in Portfolio I
A 4.0 $240
B 5.0 $260
C 40.0 $300
D 41.0 $200
Bonds Included in Portfolio II
E 19.4 $400
F 20.0 $460
G 20.2 $ 140
Which portfolio can be characterized as a bullet portfolio?
Which portfolio can be characterized as a barbell portfolio?
Answer Question
Which portfolio can be characterized as a bullet portfolio?
In a bullet strategy, the portfolio is constructed so that the maturities
of the securities in the portfolio are highly concentrated at one point
on the yield curve. Thus, Portfolio II can be characterized as a bullet
portfolio because the maturities of its securities are concentrated
around one maturity (twenty years).
Which portfolio can be characterized as a barbell portfolio?
In a barbell strategy, the maturities of the securities included in the
portfolio are concentrated at two extreme maturities. Thus, Portfolio I
can be characterized as a barbell portfolio because the maturities of
its securities are concentrated at two extreme maturities (four years
and forty years).
Duration and Yield Curve Shifts
• Duration is a measure of the sensitivity of the price of a bond
or the value of a bond portfolio to changes in market yields.
• A bond with a duration of 4 means that if market yields
change by 100 basis points, the bond will change by
approximately 4%.
• However, if a three-bond portfolio has a duration of 4, the
statement that the portfolio’s value will change by 4% for a
100-basis-point change in yields actually should be stated as
follows:
• The portfolio’s value will change by 4% if the yield on five-,
10-, and 20-year bonds all change by 100 basis points. That
is, it is assumed that there is a parallel yield curve shift.
Analyzing Expected Yield Curve Strategies
• The proper way to analyze any portfolio strategy is to look at its
potential total return.
• If a manager wants to assess the outcome of a portfolio for any
assumed shift in the Treasury yield curve, this should be done
by calculating the potential total return if that shift actually
occurs.
• This can be illustrated by looking at the performance of two
hypothetical portfolios of Treasury securities assuming different
shifts in the Treasury yield curve.
Exhibit 22-5 Three Hypothetical Treasury
Securities
Bond
Coupon
(%)
Maturity
(years)
Price Plus
Accrued
Yield to
Maturity
(%)
Dollar
Duration
Dollar
Convexity
A 8.50 5 100 8.50 4.005 19.8164
B 9.50 20 100 9.50 8.882 124.1702
C 9.25 10 100 9.25 6.434 55.4506
• The three hypothetical Treasury securities shown in Exhibit 22-
5 are considered for inclusion in our two portfolios.
• For our illustration, the Treasury yield curve consists of these
three Treasury securities: a short-term security (A, the five-year
security), an intermediate-term security (C, the 10-year
security), and a long-term security (B, the 20-year security).
Analyzing Expected Yield Curve Strategies
• Consider the following two yield curve strategies: a bullet
strategy and a barbell strategy. We will label the portfolios
created based on these two strategies as the “bullet
portfolio” and the “barbell portfolio” and they comprise the
following:
• Bullet portfolio: 100% bond C
• Barbell portfolio: 50.2% bond A and 49.8% bond B
Analyzing Expected Yield Curve Strategies
• The dollar duration for the bullet portfolio per 100-basis-
point change in yield is 6.434.
• For the barbell portfolio, the dollar duration is just the
weighted average of the dollar duration of the two bonds:
• dollar duration of barbell portfolio = 0.502(4.005)
+0.498(8.882) = 6.434
• The dollar duration of the barbell portfolio is the same as
that of the bullet portfolio.
Analyzing Expected Yield Curve Strategies
• Duration is just a first approximation of the change in price
resulting from a change in interest rates. Convexity
provides a second approximation.
• Although we did not discuss dollar convexity, it has a
meaning similar to convexity, in that it provides a second
approximation to the dollar price change.
• For two portfolios with the same dollar duration, the
greater the convexity, the better the performance of a
bond or a portfolio when yields change.
Analyzing Expected Yield Curve Strategies
• The dollar convexity of the bullet portfolio is 55.4506.
• The dollar convexity for the barbell portfolio is a weighted
average of the dollar convexity of the two bonds.
• That is, dollar convexity of barbell portfolio =
0.502(19.8164) + 0.498(124.1702) = 71.7846
• Therefore, the dollar convexity of the barbell portfolio is
greater than that of the bullet portfolio.
Analyzing Expected Yield Curve Strategies
• Similarly, the yield for the two portfolios is not the same.
The yield for the bullet portfolio is simply the yield to
maturity of bond C, 9.25%.
• The traditional yield calculation for the barbell portfolio,
which is found by taking a weighted average of the yield
to maturity of the two bonds included in the portfolio, is
8.998%:
• Portfolio yield for barbell portfolio = 0.502(8.50%) +
0.498(9.50%) = 8.998%
• This approach suggests that the yield of the bullet
portfolio is 25.2 basis points greater than that of the
barbell portfolio (9.25% - 8.998%).
Analyzing Expected Yield Curve Strategies
• Although both portfolios have the same dollar duration,
the yield of the bullet portfolio is greater than the yield of
the barbell portfolio.
• However, the dollar convexity of the barbell portfolio is
greater than that of the bullet portfolio.
• The difference in the two yields is sometimes referred to
as the cost of convexity (i.e., giving up yield to get better
convexity).
Analyzing Expected Yield Curve Strategies
• Now suppose that a portfolio manager with a six-month
investment horizon has a choice of investing in the bullet
portfolio or the barbell portfolio.
• Which one should he choose? The manager knows that
(1) the two portfolios have the same dollar duration, (2)
the yield for the bullet portfolio is greater than that of the
barbell portfolio, and (3) the dollar convexity of the barbell
portfolio is greater than that of the bullet portfolio.
• Actually, this information is not adequate in making the
decision. What is necessary is to assess the potential total
return when the yield curve shifts.
Exhibit 22-6 Six-Month Investment Horizon
bullet portfolio’s total return – barbell portfolio’s total return
Yield
Change
Parallel
Shift (a)
Nonparallel
Shift (b)
Nonparallel
Shift (c)
−5.000 −7.19 −10.69 −3.89
−4.750 −6.28 −9.61 −3.12
−4.500 −5.44 −8.62 −2.44
−4.250 −4.68 −7.71 −1.82
−4.000 −4.00 −6.88 −1.27
−3.750 −3.38 −6.13 −0.78
−3.500 −2.82 −5.44 −0.35
… … … …
3.750 −1.39 −1.98 −0.85
4.000 −1.57 −2.12 −1.06
4.250 −1.75 −2.27 −1.27
4.500 −1.93 −2.43 −1.48
4.750 −2.12 −2.58 −1.70
5.000 −2.31 −2.75 −1.92
(b) Change in yield for bond C. Nonparallel shift as follows (flattening of yield curve):
yield change bond A = yield change bond C + 25 basis points
yield change bond B = yield change bond C - 25 basis points
(c) Change in yield for bond C. Nonparallel shift as follows (steepening of yield curve): yield
change bond A = yield change bond C - 25 basis points
yield change bond B = yield change bond C + 25 basis points
Analyzing Expected Yield Curve Strategies
• Let’s focus on the second column which is labeled
“parallel shift.”
• In this case parallel movement of the yield curve means
that the yields for the short-term bond (A), the
intermediate-term bond (C), and the long-term bond (B)
change by the same number of basis points, shown in
the “yield change” column of the table.
• Which portfolio is the better investment alternative if the
yield curve shifts in a parallel fashion and the
investment horizon is six months? The answer depends
on the amount by which yields change.
• Notice that when yields change by less than 100 basis
points, the bullet portfolio outperforms the barbell
portfolio. The reverse is true if yields change by more
than 100 basis points.
Analyzing Expected Yield Curve Strategies
• This illustration makes two key points.
• First, even if the yield curve shifts in a parallel fashion, two
portfolios with the same dollar duration will not give the
same performance. The reason is that the two portfolios
do not have the same dollar convexity.
• The second point is that although with all other things
equal it is better to have more convexity than less, the
market charges for convexity in the form of a higher price
or a lower yield. But the benefit of the greater convexity
depends on how much yields change.
Analyzing Expected Yield Curve Strategies
• Now let’s look at what happens if the yield curve does
not shift in a parallel fashion.
• Specifically, the first nonparallel shift column assumes
that if the yield on bond C (the intermediate-term bond)
changes by the amount shown in the first column,
• bond A (the short-term bond) will change by the
same amount + 25 basis points,
• bond B (the long-term bond) will change by the
same amount shown in the first column - 25 basis
points.
• The spread between the long-term yield (yield on bond
B) and the short-term yield (yield on Bond A), the spread
has decreased by 50 basis points =》 flattening of the
yield curve
• The barbell outperforms the bullet (always negative)
Analyzing Expected Yield Curve Strategies
• In the last column, the nonparallel shift assumes that for
a change in bond C’s yield,
• The yield on bond A will change by the same
amount - 25 basis points,
• The bond B will change by the same amount + 25
points
• Thus, the spread between the long-term yield and the
short-term yield has increased by 50 basis points, and
the yield curve has steepened
• In this case, the bullet portfolio outperforms the barbell
portfolio as long as the yield on bond C does not rise by
more than 250 basis points or fall by more than 325
basis points.
Analyzing Expected Yield Curve Strategies
• The key point here is that looking at measures such as
yield (yield to maturity or some type of portfolio yield
measure), duration, or convexity tells us little about
performance over some investment horizon, because
performance depends on the magnitude of the change in
yields and how the yield curve shifts.
• Therefore, when a manager wants to position a portfolio
based on expectations as to how he might expect the
yield curve to shift, it is important to perform total return
analysis.
The Use of Leverage
q If permitted by investment guidelines a manager
may use leverage in an attempt to enhance portfolio
returns.
q A portfolio manager can create leverage by
borrowing funds in order to acquire a position in the
market that is greater than if only cash were
invested.
q For example, a manager may have cash to invest in
the bond market of $50 million but wants an
exposure of $200 million.
The Use of Leverage
q The funds available to invest without borrowing are
referred to as the “equity.”
q A portfolio that does not contain any leverage is
called an unlevered portfolio.
q A levered portfolio is a portfolio in which a manager
has created leverage.
§ The basic principle in using leverage is that a manager
wants to earn a return on the borrowed funds that is
greater than the cost of the borrowed funds.
§ The return from borrowing funds is produced from a
higher income and/or greater price appreciation relative
to a scenario in which no funds are borrowed.
• If a manager can invest $50 million earning 5% for a
year but can borrow funds at a cost of 4.5% for a year,
then the manager can generate for the year 50 basis
points in income over its funding cost. By borrowing
greater amounts, the manager can magnify the 50 basis
points. This is a benefit of leveraging.
Motivation for Leverage
• The return from investing the funds comes from two
sources.
i. interest income
ii. change in the value of the security (or securities) at
the end of the borrowing period
§ There are some managers who use leverage in the
hopes of benefiting primarily from price changes.
§ Small price changes will be magnified by using
leveraging.
• For example, if a manager expects interest rates to
fall, the manager can borrow funds to increase price
exposure to the market.
• Effectively, the manager is increasing the duration of
the portfolio.
Motivation for Leverage
§ The risk associated with borrowing funds：
§ Cost of the borrowed funds > Borrowed funds are
invested may earn
§ Due to failure to generate interest income plus
capital appreciation as expected when the funds
were borrowed.
Motivation for Leverage
Motivation for Leverage
• Leveraging is a necessity for depository institutions
• Depository institutions refers to such as banks and
savings and loan associations
• Because the spread over the cost of borrowed funds is
typically small.
• The magnitude of the borrowing (i.e., the degree of
leverage) is what produces an acceptable return for the
institution.
Duration of a Leveraged Portfolio
In general, the procedure for calculating the duration of a
portfolio that uses leverage is as follows:
Step 1: Calculate the duration of the levered portfolio.
Step 2: Determine the dollar duration of the portfolio of
the levered portfolio for a change in interest rates.
Step 3: Compute the ratio of the dollar duration of the
levered portfolio to the value of the initial unlevered
portfolio (i.e., initial equity).
Step 4: The duration of the unlevered portfolio is then
found as follows:
Duration of a Leveraged Portfolio
Suppose that a portfolio
• Market value: $100 million
• Duration:3
• This means that the manager would expect that for a 100-
basis-point change in interest rates
• The portfolio’s value would change by approximately $3
million.
• For this unlevered fund, the duration of the portfolio is 3.
Duration of a Leveraged Portfolio
Suppose now that the manager of this portfolio:
• Borrow an additional $300 million.
• This means that the levered fund will have $400 million to
invest
• consisting of $100 million that the manager has
available before borrowing (i.e., the equity) and $300
million borrowed.
• All of the funds are invested in a bond with a duration of 3.
• Now let’s look at what happens if interest rates change by
100 basis points. The levered portfolio’s value will
change?
Duration of a Leveraged Portfolio
• All of the funds are invested in a bond with a duration of 3.
• If interest rates change by 100 basis points.
• The levered portfolio’s value will change by $12 million
(3% times $400 million).
• This means that on an investment of $100 million, the
portfolio’s value changes by $12 million.
• The proper way to measure the portfolio’s duration is
relative to the unlevered amount or equity because the
manager is concerned with the risk exposure relative to
equity
Duration of a Leveraged Portfolio
Step 1: Calculate the duration of the levered portfolio.
Assume that calculation of the duration of the levered
portfolio finds that the duration is 3.
Step 2: Determine the dollar duration of the portfolio of the
levered portfolio for a change in interest rates.
Let’s use a 50-basis-point change in interest rates to
compute the dollar duration. If the duration of the levered
portfolio is 3, then the dollar duration for a 50-basis-point
change in interest rates is $6 million (1.5% (3*0 change
for a 50-basis-point move times $400 million).
Duration of a Leveraged Portfolio
Step 3: Compute the ratio of the dollar duration of the
levered portfolio to the value of the initial unlevered
portfolio (i.e., initial equity).
The ratio of the dollar duration for a 50-basis-point
change in interest rates to the $100 million initial market
value of the unlevered portfolio is 0.06 ($6 million divided
by $100 million).
Step 4: The duration of the unlevered portfolio is then found
as follows:
Duration of a Leveraged Portfolio
Step 4: The duration of the unlevered portfolio is then found
as follows:
• If interest rates change by 100 basis points.
• This means that on an investment of $100 million, the
portfolio’s value changes by $12 million.
• The proper way to measure the portfolio’s duration is
relative to the unlevered amount or equity because the
manager is concerned with the risk exposure relative to
equity
Previous Lecture Tutorial
Question
The YTM on 1-year zeros is currently 7%; the
YTM on 2-year zeros is 8%. The treasury plans
to issue a 2-year maturity coupon bond, paying
coupon once per year with a coupon rate of 9%.
The face value of the bond is $100.
a. At what price will the bond sell?
b. What will the YTM on the bond be?
c. If the expectation theory of the yield curve
is correct, what is the market expectation of the
price that the bond will sell for next year?
d. Recalculate your answer to c) if you
believe in the liquidity preference theory and
you believe that the liquidity premium is 1%.
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