Fixed Income Portfolios
Drake Fin 284
DRAKE UNIVERSITY
Overview
Setting Investment Objectives
Establishing investment policy
Selecting a portfolio strategy
Selecting assets
Measuring and Evaluating performance
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Setting Investment Objectives
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Varies with type of financial institution
pension fund -- generate cash flow sufficient to cover
obligations
life insurance -- meet obligations in insurance (long
term) and generate profit
banks earn spread over short term deposits, timing of
liabilities
Establishing Investment
Objectives
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Asset allocation
Match assets and liabilities based on goals of the
financial institution
Client and Regulatory constraints
limits on credit ratings
Tax and Financial Reporting implications
mutual funds are tax exempt so munis are not
attractive
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Selecting a Portfolio Strategy
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Active vs. passive strategies
Active - Attempts to forecast and exploit changes in
future rates and macro economic variables. Change
portfolio composition often in response to
expectations.
Passive - Closer to buy and hold. Goal is to replicate
or benchmark for example to an index.
Combinations of both
Selecting a Strategy
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Structured portfolio strategies
goal is to achieve a predetermined benchmark or goal
such as matching the timing of future liabilities.
Immunization - eliminating the impact of interest rate
changes in the cash flows received
Cash flow matching or horizon matching
Often include low risk active strategies within a
passive strategy
What Determines Strategy
Choice?
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Efficiency of market
If market is efficient, you cannot beat the market
return consistently. This implies indexing as the
strategy.
Liabilities
Must be able to meet future obligations of the firm
(think about a bank, pension fund or insurance firm)
Selecting Assets
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Identifying individual securities (identifying
mispriced securities if not indexing or matching
cash flows
Identifying cash flow characteristics
Measuring and Evaluating
Performance
Measuring against a benchmark
Meeting liability constraints
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Sources of Active Portfolio
Returns
1.
2.
3.
4.
5.
6.
Changes
Changes
Changes
sectors
Changes
Changes
Changes
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in the level of Interest Rates
in the shape of the Yield Curve
in the yield spreads among bond
in the option adjusted spread
in the yield spread of a particular bond
in asset allocation within bond sector
Manager Expectations vs.
Market Consensus
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The market consensus should be reflected in the
current market prices and yields.
This may or may not agree with the manager
expectations.
Interest rate expectations
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Expected change in interest rates will often force
manager to make a change in strategy.
This may not include actually changing the
underlying assets for example swaps may be
used to shorten or lengthen the duration of a
portfolio.
Problem – No reason to believe that you can
forecast accurately
Yield Curve Strategies
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Positioning your portfolio to capitalize on
expected changes in the shape of the Treasury
Yield Curve
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Parallel Shifts
Short
Short
Intermediate
Maturity
Intermediate
Maturity
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Long
Long
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Twists
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Flattening Twist
Short
Intermediate
Maturity
Long
Steepening Twist
Short
Intermediate
Maturity
Long
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Butterfly Shifts
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Positive Butterfly
Short
Intermediate
Maturity
Long
Negative Butterfly
Short
Intermediate
Maturity
Long
Common Shifts
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Most common shifts are combinations of the
types above
Downward shift combined with steepening
More likely to also be combined with a negative
butterfly
Upward shift combined with flattening
More likely to be combined with a positive butterfly
Portfolio Strategies
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Need to consider the timing of the cash flows
and therefore the duration of the portfolio and /
or maturity.
Look at expectations of future yield curve shifts
Match liabilities
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Ladder or Spaced Maturity
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Maturity is capped and then the portfolio is
spread out evenly across the range of maturities
Assume 5 year cap – then 20% of portfolio is in
each year.
20%
20%
1
20%
2
20%
3
20%
4
5
Ladder or Spaced
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Once a year matures it is assumed to be
reinvested in new 5 year bonds. Therefore the
trend continues
Advantages
Reduces Investment income fluctuations
Requires little investment expertise
Since it continues to roll over into cash it provides
flexibility
Front End Load (bullet) Strategy
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Place all of securities in a short period of time
20%
80%
1
2
3
4
5
Front End Load
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Uses the portfolio as a source of liquidity since it
is so short term
Advantages
Avoids large capital losses if rates increase since short
run securities are less sensitive to interest rate
changes.
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Back End (bullet) Strategy
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Places all of portfolio at the upper end of the
maturity
60%
20%
1
2
3
4
5
6
7
8
20%
9
10
Back End Strategy
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Stresses Investment income instead of liquidity
Advantages
Increases gain if interest rates decrease since long
term bonds are more sensitive to rate changes (but
also larger decline in value if rates increase)
Forces institution to depend upon money market
for short term returns.
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Barbell Strategy
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Combination of front end and back end load.
The goal is to balance the desire for liquidity and
income.
25%
25%
15%
10%
1
2
3
10%
4
5
6
7
8
15%
9
10
Barbell
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Combines both goals of liquidity and income
Advantages
Not as responsive to interest rates (either increase or
decrease) as back end load, more responsive than
front end load.
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Rate Expectations
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Aggressive strategy based on expected rates
Shift if rates are expected to decrease
Shift if rates are expected to increase
1
2
3
4
5
6
7
8
9
10
Rate Expectations
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Very aggressive, attempts to match portfolio to
rate expectations.
Advantages
If successful, capital gains will be increased and
capital losses will be decreased.
Analysis of the portfolios
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How the portfolios actually respond will be
dependent upon changes in the yield curve
(steepening etc.) Not just a static measure of
interest rates.
Given the duration of portfolio, and estimating
the value change is implicitly assuming that the
the yield on each of the assets in the portfolio
changes by the same amount.
Want to look at total return
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The best way to compare across portfolios is to
compare total return if a shift actually occurs.
Bond
Coup
Mat
Price
YTM
$Dur
$Conv
A
8.5
5
100
8.5
4.005 19.816
B
9.5
20
100
9.5
8.882 124.17
C
9.25
10
100
9.25
6.434
55.45
Compare two portfolios
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Bullet: 100% in Bond C
$ duration = 6.434
$ convexity = 55.4506
YTM 9.25%
Barbell: 50.2% in bond A and 49.8% in bond B
$ duration = (0.502)(4.005)+(.498)(8.882)
= 6.434
$ convexity = (0.502)(19.8164)+(.498)(124.17)
=71.7846
YTM = .502(.0850)+.498(.0950) = 8.998%
Cost of Convexity
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The barbell has a higher convexity but a lower
yield. The bullet has a yield 25.5 basis points
higher than the barbell. This is the cost of
convexity.
Which portfolio does better for a yield change?
It depends on the yield shift (parallel or twist etc)
Key Point
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Looking at just the duration, convexity, YTM etc.
does not provide a good indication of which
portfolio is “better.”
Measuring Yield Curve Risk
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Key Rate Duration, Calculating the change in
value for a security or portfolio after changing
one key interest rate keeping other rates
constant.
Each point on the spot yield curve has a separate
duration associated with it.
If you allowed all rates to change by the same
amount, you could measure the response to the
security or portfolio to a parallel shift in the yield
curve.
Key Rates and Portfolios
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By focusing on a group of key rates it is possible
to investigate the impact of changes in the shape
of the yield curve on specific parts of a portfolio,
we will cover this in more detail in the portfolio
section of the course.
Using Key Rate Durations*
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Assume you have three key rtes 2 years, 16
years and 30 years. Assume that you are
investing in zero coupon instruments at each
maturity (the duration will be equal to the
maturity).
Therefore each bond will respond to changes in
its portion of the yield curve.
From Fabozzi Fixed Income for the CFA
p310 - 312
Consider 3 portfolios
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Portfolio 1 (Barbell)
$50 in the 2 year, 0 in the 16 year, and $50 in the 30
year
Portfolio 2 (Bullet)
0 in the 2 year, $100 in the 16 year, and 0 in the 30
year
Portfolio 3 (Spread)
$33.33 in each of the possible bonds.
Portfolio Duration
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The weighted average of the key rate durations
similarly the effective duration will be the
weighted average of the durations of the
securities in the portfolio.
Key Rate Duration
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For each maturity (key rate) we need to find the
key rate duration.
Let D(1) be the duration for the 2 year part of the curve
Let D(2) be the duration for the 16 year part of the curve
Let D(3) be the duration for the 30 year part of the curve
Portfolio 1
For portfolio 1 the only portion of the portfolio that is
sensitive to a change in the 2 year rate is the two year
security, the similar result happens for each of the other
maturities.
Portfolio Key Rate Durations
Portfolio 1
D(1)=(50/100)2+(0/100)0+(50/100)0=1
D(2)=(50/100)0+(0/100)0+(50/100)0=0
D(3)=(50/100)0+(0/100)0+(50/100)30=15
Portfolio 1
D(1)=(0/100)0+(100/100)0+(0/100)0=0
D(2)=(0/100)0+(100/100)16+(0/100)0=16
D(3)=(0/100)0+(100/100)0+(0/100)30=0
Portfolio 1
D(1)=(33.3/100)2+(33.3/100)0+(33.3/100)0=.6666
D(2)=(33.3/100)0+(33.3/100)16+(33.3/100)0=5.333
D(3)=(33.3/100)0+(33.3/100)0+(33.3/100)30=10
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Effective Duration
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The effective duration of each portfolio would be
the weighted average of the securities durations
Portfolio 1
(50/100)2+(0/100)16+(50/100)30 = 16
Portfolio 2
(0/100)2+(100/100)16+(0/100)30 = 16
Portfolio 3
(33.3/100)2+(33.3/100)16+(33.3/100)30 = 16
A parallel shift in the yield curve
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Assume that all spot decrease by 10%
Given the key rate durations for portfolio 1
D(1)=1, D(2)=0, D(3)=15
For a 100 Bp decrease in the 2 year rate, the
portfolio should see a 1% increase in price, for a
10 Bp decrease price should increase by .1%
Similarly a 10 Bp decrease in the 30 year rate
should increase price by 1.5%
The total change in price is then .1%+ 1.5%
Three possible yield curve shifts
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Now lets consider the impact of three different
possible shifts in the yield curve on each of the
three portfolios
Scenario 1 Parallel Downward Shift
All maturities decrease by 10 Bp
Scenario 2
2-yr rate shifts up 10 Bp, 30-yr rate shifts down by 10Bp
Scenario 3
2-yr rate shifts down 10 Bp, 30-yr rate shifts up by 10Bp
Comparison of shifts
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Portfolio
Scenario1
Scenario 2
Scenario3
I
+1.6%
+1.4%
-1.4%
II
+1.6%
0%
0%
III
+1.6%
+0.94%
-0.94%
Yield Spread Strategies
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Positioning a portfolio to capitalized on expected
changes in yield spreads.
Intermarket Spread Swaps – exchanging one
bond for another between sectors of the bond
market based on the yield spread
Credit Spreads
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Credit spreads (Spread between treasury and
similar maturity non treasury) generally widen in
a declining economy and narrow during
expansion.
Yield Ratios vs. Spreads. As the level of rates
change so should the absolute spread.
Yield Spread Strategies
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Positioning a portfolio to take advantage of
changes in the spread between two
classifications of bonds.
One example would be an intermarket spread
swap.
May recognize differences in credit spreads, or
embedded options.
Example:
Credit Spreads Expected to
Widen
10% BBB rated Corp,
5 yrs to mat, YTM = .10
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8% Treasury, 5 yrs to Mat,
YTM = .09072458
Yield Spread = .10-.090724 = .009275742 (92.75742Bp)
What strategy should you undertake?
Purchase the Treasury and Short the Corp
1) Treasury yield falls - price of treasury increases
2) Corp. yield increases - price of corp decreases
Example continued
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Assume that you hold the positions for 1day. At that time the
treasury yield has decreased to 8.70%
Corp
Time 0
Receive $100
Next Day
Pay $100
Treasury
Time 0
Buy 1.044 of Treas =$100
Next Day
Sell 1.044 @ 97.21
Total = $100
Total = 101.50635
Importance of Duration
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When comparing spreads it is imperative to look
at positions that have the same duration.
If the duration of the new and old position are
not the same then you are accepting risk
associated with a change in the level of rates as
well as a change in the spread.
Example:
Credit Spreads Expected to
Widen
10% A rated Corp,
5 yrs to mat, YTM = .10
Mac Duration = 4.0539
Modified Duration =
4.0539/1.10 = 3.68537
$ duration =
3.68537(100)
=368.537
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8% Treasury
5 yrs to Mat, YTM = .090724
Mac Duration = 4.19
Modified Duration =
4.19/1.090724 = 3.84836
$ duration =
3.84836(95.7646)
=368.537
Example continued
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Assume that you hold the positions for 1day. At that time both
yields increased by 1 basis point
Corp
Treasury
Time 0
Time 0
Receive $100
Buy 1.044 of Treas =$100
Next Day
Next Day
Pay $99.9614
Sell 1.044 @ 95.726
Total = $99.961
Total = 99.957
The price change was basically the same for both!
Individual Security Selection
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Basic goal is to identify undervalued securities
Its yield is higher than other comparable securities
Its yield is expected to decline
In either case a substitution swap -- exchanging
a bond for another that offers a higher yield.
Allocation Within Sectors
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Within a broad sector (corporate for example) a
portfolio manager needs to decide how to
allocate within the sector (Across credit
categories).
Combination of past history and future
expectations.
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Rating Transition
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Aaa
Aa
A
Baa
Ba
Bb
C or
D
Aaa
91.9
7.38
0.72
0
0
0
0
100
Aa
1.13 91.26 7.09
0.31
0.21
0
0
100
A
0.10
2.56
91.2
5.33
0.61
0.2
0
100
Baa
0.00
0.21
.36
87.94 5.46
0.82
0.21
100
total
Next Step
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Given the rating transition, you then forecast
what the spreads will be at the end of the
holding period and the return based on the
spreads
Then use use the probabilities from the matrix to
find an expected return
Using Leverage
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Ability to use leverage to take out a larger
position will depend upon the guidelines of the
fund.
Basic goal is to earn a return greater than the
cost of the borrowed funds.
Allows the benefit of small price changes to be
magnified since relative size of position can be
increased.
Duration of levered portfolio
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The duration of the levered portfolio should be
calculated based on the “equity position” of the
portfolio. (the amount of non borrowed funds)
Calculating Duration
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1) Calculate the duration of the levered portfolio
2) Determine the dollar duration of the portfolio
for a given change in interest rates
3) Compute the ratio of the dollar value change in
2) to the initial unlevered portfolio
4) the duration is then:
5) (Ratio in 3))(100/rate change in 2 in bps)100
Creating Leverage
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Easiest way to create leverage is via the
repurchase market.
Repurchase agreement -- sale of security with
the agreement to repurchase it the following day
(overnight repo) or over a given short term (term
repo).
Repo Interest
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The dollar value of interest is calculated using a
360 day convention.
 $ Amount  Repo  Repo Term 


 

Interest  Borrowed  Rate 
360

Dollar
Reverse Repos
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You can also cover a short position with a
reverse repo (agreeing to buy the security and
then sell it back in the future).
Credit Risk
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Repo market credit risk can be reduced by over
collateralizing the repo transaction.
Repo margin - the amount by which the market
value of the collateral exceeds the dollar value of
the loan
Delivery of Collateral
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Direct delivery to the other party or the parties
agent causes transaction costs to be incurred.
The costs are figured into the interest.
An alternative to delivery is for the repo to be
held in custody (HIC repo)
Use of the lenders custodial account at the
borrowers clearing bank. Reduces transaction
costs and collateral risk.
Determining the Repo Rate
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The more difficult to obtain the collateral the
lower the repo rate (the party lending funds will
be willing to pay a lower rate to obtain the
collateral.
The higher the credit quality and the higher the
liquidity the lower the repo rate.
Structured Portfolios
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Structured portfolios are intended to satisfy an
investment objective, and are not based upon
interest rate expectations.
Indexing
Match Liabilities and Assets
Indexing
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Attempting to match the performance of a given
bond index.
Performance is measured in terms of total return
over a investment horizon.
Index Performance is determined relative to the
target index (An even split of treasuries and high
grade corporate bonds for example, or mortgage
backs, or global or….)
Popularity of Indexing
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Active bond management has traditionally
produced poor returns.
Indexed portfolio advisory fees are usually less
than actively managed portfolios.
Nonadvisory fees (custodial etc) are also lower.
May limit the risk by limiting the portfolio to
certain types of bonds (enhanced control by
sponsor).
Problems with Indexing
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Matching index performance, does not mean
optimal performance is achieved.
Indexing does not guarantee that return
objectives will be met. Even if index is matched,
it may not match other criteria.
Indexing may eliminate some profitable types of
investments
Institutional perspective
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My matching some form of “market” index the
institution can offer a return similar to the index.
Decreases the need for active management.
Fee income form management (even though the
fees are less)
Selecting an Index
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Return objectives -- look at both return and
variability
Risk factors -- match index to acceptable risk
levels
Broad Market Bond Indexes
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Lehman Brothers Aggregate Index, Salomon
Brothers Broad Investment Grade Bond
Index, and Merrill Lynch Domestic Market
Index
All have over 5,000 issues rated BBB or
better, The Salomon index is trader priced
while the others include some model priced
issues.
All exclude issues with less than one year to
maturity
Broad Market Indexes
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The three broad indexes produce very similar
returns with the correlation of the returns over
98% (Reilly, Kao and Wright (1992).
While the correlations of long run returns are
high, there is some variation on a month to
month basis.
Specific Indexes
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Each firm and many others also produce indexes
of specific markets such as the government
market or Mortgage backed securities.
Some also offer customized indexes such as the
Salomon Large Pension Fund Baseline Bond
Index which is designed to match the long
duration of pension fund liabilities.
Size of Portfolio
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Given an index that the manager is going to
attempt to match, decisions need to be made
concerning the construction of the portfolio.
Included in this decision is the number of
issues to use to attempt to match the index.
As issues are added variance decreases
The impact of portfolio size is very similar to
equity
The number of securities needed to eliminate
unsystematic risk may differ by sector.
Tracking Error
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Tracking error is the difference between the
indexed portfolio and the benchmark index.
Three sources of tracking error
Transaction Costs
Differences in index composition.
Difference in price used to construct the index and
those paid by the portfolio
Tracking Error Tradeoff
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The larger the number of issues in the portfolio
the greater the transaction cost and the greater
the associated tracking error.
The smaller the number of issues in the portfolio
the greater the differences in return based upon
composition and the greater the tracking error.
Indexing Methodologies
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Stratified Sampling (or Cell)
Optimization Approach
Variance Minimization Approach
In all three the goal is to minimize or eliminate
diversifiable risk leaving only the systematic risk
common to the sector.
Stratified Sampling (Cell)
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The index is split into cells representing different
characteristics of the index such as duration,
coupon, maturity, market sector, credit rating,
call features, and sinking fund features.
The total number of cells is then dependent upon
the partitions in each sector.
Stratified Sampling Example
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Characteristic 1 Duration: < 5 years & > 5 years
Characteristic 2 Maturity: < 7 years & > 7 years
Characteristic 3 Sector: Treasuries and Corporate
Then make cells out of each possible
combination of characteristics:
Cell 1: Duration < 5, Maturity < 7, Treasury
Cell 2: Duration < 5, Maturity < 7, Corporate
Cell 3: Duration < 5, Maturity > 7, Treasury etc…
Total cells = 2 x 2 x 2 = 8
Stratified Sampling
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For each cell select one or more issues from
the index that can represent the entire cell.
The total dollar amount form each cell is the
proportioned by the total dollar amount form
each cell in the index.
The number of cells will increase with the size
of the portfolio, since more cells require a
larger number of issues purchased, a small
portfolio should keep the number of cells
relatively small (but tracking error increases)
Optimization
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First the goal will be to match the cells as in
stratified sampling, but then add the goal of
optimizing an outcome subject to extra
constraints.
Outcome Examples: Maximize portfolio yield,
maximize convexity, Maximize total returns
Constraints: Limit the number of issues
purchased from a given issuer, overweighting a
cell
Optimization
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Given the objective and constraints mathematical
programming can then be used to determine
which issues to include in the portfolio.
Variance Minimization
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Requires historical data for each issue.
Based on the historical data a price function is
estimated for each issue.
The price function is then used to establish the
variance of the tacking error.
The goal is then to use mathematical
programming to minimize the variance of the
tracking error of the portfolio.
Problem in Implementation
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Published prices may not be executable. They
are often based on bid prices not ask prices.
Illiquidity of the market -- some of the issues
may not actually be available
Aggregation -- often generic issues are
established to look like a group of issues
(mortgage backs for example)
Enhanced Indexing
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The goal of enhanced indexing is to consistently
outperform the total return of a given index.
(this justifies higher advisory fees). It also
comes at the cost of a higher risk of under
performing the index.
The goal is accomplished by being more active in
management and accepting greater interest rate
risks and duration related risks.
Asset / Liability Management
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The goal of asset / liability management is to
match the timing and size of assets to the
expected cash flows associated with the
liabilities.
Nature of institution will determine the liabilities
and the associated management strategies.
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Liability Classification
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Liability
Type
Amount of
Cash Outlay
Timing of
Cash Outlay
I
Known
Known
II
Known
Uncertain
III
Uncertain
Known
IV
Uncertain
Uncertain
Liquidity Concerns
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Will depend upon the type of institution.
Banking -- depository withdraws
Life Insurance -- surrender and loan values
May also change the nature of expected cash
inflows.
Surplus Management
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Goals -- earn an adequate return and maintain a
surplus of assets beyond liabilities.
Three types of surpluses
Economic -- based on market value
Accounting -- based upon GAAP
Regulatory -- based upon regulatory accounting
principles
Economic Surplus
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Market Value of Assets - Market Value of Liabilities
Market value of Liabilities is simply the PV of the
expected cash flows.
The net effect of a change in interest rates will
depend upon the duration of both the assets and
liabilities.
In both cases an increase in rates will decrease
the value and vice versa.
Economic Surplus
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Drake University
Fin 284
Assuming that the $ value of assets is greater
than liabilities whether the surplus increases or
decreases will depend on duration and the
direction of an interest rate change.
If duration of assets > duration of liabilities:
An increase in rates implies an decrease in
surplus
A decrease in interest rates implies an increase in
surplus
Economic Surplus
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Drake University
Fin 284
What if the duration is the same?
If the market value of assets is greater than
market value of liabilities then then a decrease in
rates still will increase the surplus and vice versa.
Accounting Surplus
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Drake University
Fin 284
Three methods for reporting the value of assets
Amortized cost (historical cost)
Market value
Lower of cost or market value
FASB specifies how different types of assets must
be valued.
Drake
FASB 115
Account
Accounting
Classification
Method
Held to
maturity
Available
For Sale
Trading
Drake University
Fin 284
Will
Will
Affect
Affect
Reported
Surplus
Earnings
Amortized
Cost
No
No
Market Value
Yes
No
Market Value
Yes
Yes
Regulatory Surplus
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Drake University
Fin 284
Regulators require reports based upon separate
accounting principles (RAP).
Often the regulatory surplus will differ
significantly from the accounting or economic
surplus.
Immunization
Drake
Drake University
Fin 284
F.M. Reddington (1952): “The investment in
assets in such a way that the existing business is
immune to a general change in interest rates”
Immunization of a single liability
Drake
Drake University
Fin 284
Assume that an insurance co has offered a
guaranteed investment contract.
The guarantee is to pay a 6.25% return each 6
months (12.5% bond equivalent yield) for 5.5
years.
Invest $8,829,262 today and the buyer will have
$8,829,262(1.0625)11=$17,183,033 which is also
a liability for the insurance co.
Immunization attempt 1
Drake
Drake University
Fin 284
The life insurance firm uses the $8,829,262 to
purchase a 12.5% coupon bond selling at par
that matures in 5.5 years.
Will this immunize the portfolio?
NO -- you will only have the required
$17,183,033 if the coupons can be reinvested
at 6.25% each six months until the maturity
of the bond.
If rates increase (decrease) immediately total
value will be above (below) $17,183,033
Immunization attempt 2
Drake
Drake University
Fin 284
The life insurance firm uses the $8,829,262 to
purchase a 12.5% coupon bond selling at par
that matures in 15 years.
Will this immunize the portfolio?
NO -- you will only have the required
$17,183,033 if the coupons can be reinvested
at 6.25% each six months until 5.5 years
have passed
If rates increase immediately total value will
be below $17,183,033, and vice versa.
Immunization attempt 3
Drake
Drake University
Fin 284
The life insurance firm uses the $8,829,262 to
purchase a 12.5% coupon bond selling at par
that matures in 6 months.
Will this immunize the portfolio?
NO -- you will only have the required
$17,183,033 if the bond can be reinvested at
6.25% each six months until 5.5 years have
passed
If rates increase (decrease) immediately total
value will be above (below) $17,183,033.
Immunization attempt 4
Drake
Drake University
Fin 284
The life insurance firm uses the $8,829,262 to purchase
a 10.125% coupon bond selling to yield 12.5% that
matures in 8 years ($10,000,000 par value)
Will this immunize the portfolio?
Yes -- you will have the required $17,183,033
regardless of an immediate change in yield.
If rates increase the interest on interest offsets the
decline in value
If rates decrease the increase in value offsets the
decline in interest on interest.
Duration
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Drake University
Fin 284
The Macaulay duration of the liability is simply
the 5.5 years (a modified duration of 5.18).
The modified duration of the 8 year 10.125%
coupon bond is 5.18% (Macaulay duration of
5.5).
Immunization
Drake
Drake University
Fin 284
Two things to satisfy:
The Macaulay Duration of the portfolio is the
same as the liability.
The PV of the cash flows of the portfolio is the
same as the PV of the liability.
Note this assumes option free bond (if the
Macaulay duration is the same for both then
the modified duration will also be the same.)
If embedded options exist effective duration
must be used.
Changes over time
Drake
Drake University
Fin 284
The duration of the portfolio and of the liability
will change over time.
Fro immunization to remain intact the portfolio
should be rebalanced to keep the duration equal
to that of the liability.
Frequent rebalancing causes an increase in
transaction costs. Infrequent rebalancing causes
an increased risk of failing to meet the target.
Other complications
Drake
Drake University
Fin 284
Our example assumed that the yield curve is flat
and that any shifts in the yield curve are parallel
shifts.
Immunization Risk
Drake
Drake University
Fin 284
There are multiple portfolios that can be created
that satisfy the duration criteria.
Which one should be chosen?
Bierwag, Kaufman, and Toves (1981) If the
portfolio cash flows are more concentrated
around the liability due date it is less risky.
Immunization risk is closely tied to reinvestment
rate risk.
Measuring Immunization risk
Drake
Drake University
Fin 284
The product of two terms determine the impact
of a change in the shape of the yield curve.
The first term is based upon the characteristics
of the cash flows
The second term is based upon the change in the
shape of the yield curve which cannot be
predicted.
Therefore the first term can be used to measure
risk.
Measuring Immunization Risk
Drake
Drake University
Fin 284
CF1 (1  H)
CF2 (2  H)
CFn (n  H)

  
2
n
1 y
(1  y)
(1  y)
where :
CFt  cash flow of the portfolio at time t
2
2
H  length in years of the investment horizon
y  yield for the portfolio
n  time to receipt of the last csh flow
2
Immunizing with Zero Coupon
Bonds
Drake
Drake University
Fin 284
An alternative possibility is to invest in zero
coupon bonds that mature at the same time as
the investment horizon of the liability.
This satisfies the duration requirement however
the yield on the zero coupon is usually less than
on coupon instruments so it requires a greater
investment today
Portfolio Construction
Drake
Drake University
Fin 284
Credit Risk -- if a bond defaults the target yield
may not be reached
Call risk -- If callable issues are included then
there is a risk of the call being exercised and the
target yield not being reached.
Contingent Immunization
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Drake University
Fin 284
Actively managing the portfolio until a negative
outcome puts the potential total return (Realized
and immunized) down to a safety level. The
manager is then required to immunize the entire
portfolio to ensure the safety net level.
Satisfying Multiple Liabilities
Drake
Drake University
Fin 284
Multiperiod immunization. Just matching
duration will not guarantee matching the multiple
future liabilities.
Each liability must be immunized by a separate
cash flow stream of the portfolio.
Note: this requires decomposition of the
portfolios combined cash flow stream, not the
assets in the portfolio.
Satisfying Multiperiod Liabilities
Drake
Drake University
Fin 284
Cash Flow Matching - working backward through the
multiple cash flows.
Starting with the final liability, using a bond with the
same maturity as the final liability, an amount is
invested that will produce a final payment and
coupon equal to the liability.
The other cash flows are reduced by the coupons on
the bond and the process is repeated for the next to
last liability and so on.
Satisfying Multiperiod Liabilities
Drake
Drake University
Fin 284
Symmetric Cash Matching -- allows short term
borrowing of funds to satisfy a liability prior to
the liability due date, reducing the cost of
funding.
Active / Immunization
Combination
Drake
Drake University
Fin 284
Combining the two strategies (contingent
immunization is one or the other..).
A combined strategy might include immunizing a
portion of the portfolio and actively managing
the remainder of the portfolio.