drake drake university fin 284 fixed income portfolios

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Drake DRAKE UNIVERSITY Fin 284 Fixed Income Portfolios

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DrakeDRAKE UNIVERSITY

Fin 284

Fixed Income Portfolios

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Fin 284Overview

Setting Investment Objectives

Establishing investment policy

Selecting a portfolio strategy

Selecting assets

Measuring and Evaluating performance

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Fin 284

Setting Investment Objectives

Varies with type of financial institutionpension fund -- generate cash flow sufficient to cover obligationslife insurance -- meet obligations in insurance (long term) and generate profitbanks earn spread over short term deposits, timing of liabilities

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Fin 284

Establishing Investment Objectives

Asset allocationMatch assets and liabilities based on goals of the financial institution

Client and Regulatory constraintslimits on credit ratings

Tax and Financial Reporting implicationsmutual funds are tax exempt so munis are not attractive

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Fin 284

Selecting a Portfolio Strategy

Active vs. passive strategiesActive - 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

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Fin 284Selecting a Strategy

Structured portfolio strategiesgoal 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 receivedCash flow matching or horizon matching Often include low risk active strategies within a passive strategy

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Fin 284

What Determines Strategy Choice?

Efficiency of marketIf market is efficient, you cannot beat the market return consistently. This implies indexing as the strategy.

LiabilitiesMust be able to meet future obligations of the firm (think about a bank, pension fund or insurance firm)

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Fin 284Selecting Assets

Identifying individual securities (identifying mispriced securities if not indexing or matching cash flowsIdentifying cash flow characteristics

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Fin 284

Measuring and Evaluating Performance

Measuring against a benchmark

Meeting liability constraints

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Fin 284

Sources of Active Portfolio Returns

1. Changes in the level of Interest Rates2. Changes in the shape of the Yield Curve3. Changes in the yield spreads among bond

sectors4. Changes in the option adjusted spread5. Changes in the yield spread of a particular

bond6. Changes in asset allocation within bond

sector

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Fin 284

Manager Expectations vs.Market Consensus

The market consensus should be reflected in the current market prices and yields.

This may or may not agree with the manager expectations.

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Fin 284Interest rate expectations

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

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Fin 284Yield Curve Strategies

Positioning your portfolio to capitalize on expected changes in the shape of the Treasury Yield Curve

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Fin 284Parallel Shifts

Short Intermediate LongMaturity

Short Intermediate LongMaturity

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Fin 284Twists

Short Intermediate LongMaturity

Short Intermediate LongMaturity

Flattening Twist

Steepening Twist

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Fin 284Butterfly Shifts

Short Intermediate LongMaturity

Short Intermediate LongMaturity

Positive Butterfly

Negative Butterfly

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Fin 284Common Shifts

Most common shifts are combinations of the types aboveDownward shift combined with steepening

More likely to also be combined with a negative butterfly

Upward shift combined with flatteningMore likely to be combined with a positive butterfly

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Fin 284Portfolio Strategies

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 shiftsMatch liabilities

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Fin 284Ladder or Spaced Maturity

Maturity is capped and then the portfolio is spread out evenly across the range of maturitiesAssume 5 year cap – then 20% of portfolio is in each year.

20% 20% 20% 20% 20%

1 2 3 4 5

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Fin 284Ladder or Spaced

Once a year matures it is assumed to be reinvested in new 5 year bonds. Therefore the trend continuesAdvantages

Reduces Investment income fluctuationsRequires little investment expertiseSince it continues to roll over into cash it provides flexibility

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Fin 284

Front End Load (bullet) Strategy

Place all of securities in a short period of time

20% 80%

1 2 3 4 5

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Fin 284Front End Load

Uses the portfolio as a source of liquidity since it is so short termAdvantages

Avoids large capital losses if rates increase since short run securities are less sensitive to interest rate changes.

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Fin 284Back End (bullet) Strategy

Places all of portfolio at the upper end of the maturity

20% 20%

1 2 3 4 5 6 7 8 9 10

60%

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Fin 284Back End Strategy

Stresses Investment income instead of liquidityAdvantages

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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Fin 284Barbell Strategy

Combination of front end and back end load. The goal is to balance the desire for liquidity and income.

10%

25%

1 2 3 4 5 6 7 8 9 10

15%

25%

15%

10%

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Fin 284Barbell

Combines both goals of liquidity and incomeAdvantages

Not as responsive to interest rates (either increase or decrease) as back end load, more responsive than front end load.

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Fin 284Rate Expectations

Aggressive strategy based on expected rates

1 2 3 4 5 6 7 8 9 10

Shift if rates are expected to decrease

Shift if rates are expected to increase

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Fin 284Rate Expectations

Very aggressive, attempts to match portfolio to rate expectations. Advantages

If successful, capital gains will be increased and capital losses will be decreased.

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Fin 284Analysis of the portfolios

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.

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Fin 284Want to look at total return

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 4.00519.816

B 9.5 20 100 9.5 8.882124.17

C 9.25 10 100 9.25 6.434 55.45

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Fin 284Compare two portfolios

Bullet: 100% in Bond C$ duration = 6.434$ convexity = 55.4506YTM 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)

=71.7846YTM = .502(.0850)+.498(.0950) = 8.998%

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Fin 284Cost of Convexity

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)

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Fin 284Key Point

Looking at just the duration, convexity, YTM etc. does not provide a good indication of which portfolio is “better.”

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Fin 284Measuring Yield Curve Risk

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.

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Fin 284Key Rates and Portfolios

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.

From Fabozzi Fixed Income for the CFA p310 - 312

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Fin 284Using Key Rate Durations*

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.

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Fin 284Consider 3 portfolios

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.

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Fin 284Portfolio Duration

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.

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Fin 284Key Rate Duration

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 curveLet D(2) be the duration for the 16 year part of the curveLet D(3) be the duration for the 30 year part of the curve

Portfolio 1For 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.

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Fin 284Portfolio Key Rate Durations

Portfolio 1D(1)=(50/100)2+(0/100)0+(50/100)0=1D(2)=(50/100)0+(0/100)0+(50/100)0=0D(3)=(50/100)0+(0/100)0+(50/100)30=15

Portfolio 1D(1)=(0/100)0+(100/100)0+(0/100)0=0D(2)=(0/100)0+(100/100)16+(0/100)0=16D(3)=(0/100)0+(100/100)0+(0/100)30=0

Portfolio 1D(1)=(33.3/100)2+(33.3/100)0+(33.3/100)0=.6666D(2)=(33.3/100)0+(33.3/100)16+(33.3/100)0=5.333D(3)=(33.3/100)0+(33.3/100)0+(33.3/100)30=10

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Fin 284Effective Duration

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 = 16Portfolio 2(0/100)2+(100/100)16+(0/100)30 = 16Portfolio 3(33.3/100)2+(33.3/100)16+(33.3/100)30 = 16

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Fin 284

A parallel shift in the yield curve

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%

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Fin 284

Three possible yield curve shifts

Now lets consider the impact of three different possible shifts in the yield curve on each of the three portfolios

Scenario 1 Parallel Downward ShiftAll maturities decrease by 10 BpScenario 22-yr rate shifts up 10 Bp, 30-yr rate shifts down by 10BpScenario 32-yr rate shifts down 10 Bp, 30-yr rate shifts up by 10Bp

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Fin 284Comparison of shifts

Portfolio Scenario1 Scenario 2 Scenario3

I +1.6% +1.4% -1.4%

II +1.6% 0% 0%

III +1.6% +0.94% -0.94%

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Fin 284Yield Spread Strategies

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

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Fin 284Credit Spreads

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.

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Fin 284Yield Spread Strategies

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.

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Fin 284

Example: Credit Spreads Expected to

Widen

10% BBB rated Corp, 5 yrs to mat, YTM = .10

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 increases2) Corp. yield increases - price of corp decreases

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Fin 284Example continued

CorpTime 0

Receive $100Next Day Pay $100

Total = $100

TreasuryTime 0

Buy 1.044 of Treas =$100

Next DaySell 1.044 @ 97.21

Total = 101.50635

Assume that you hold the positions for 1day. At that time the treasury yield has decreased to 8.70%

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Fin 284Importance of Duration

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.

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Fin 284

Example: Credit Spreads Expected to

Widen

10% A rated Corp, 5 yrs to mat, YTM = .10Mac Duration = 4.0539

Modified Duration = 4.0539/1.10 = 3.68537

$ duration = 3.68537(100)

=368.537

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

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Fin 284Example continued

CorpTime 0

Receive $100Next Day

Pay $99.9614

Total = $99.961

TreasuryTime 0

Buy 1.044 of Treas =$100Next Day

Sell 1.044 @ 95.726

Total = 99.957

Assume that you hold the positions for 1day. At that time both yields increased by 1 basis point

The price change was basically the same for both!

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Fin 284Individual Security Selection

Basic goal is to identify undervalued securities

Its yield is higher than other comparable securitiesIts yield is expected to decline

In either case a substitution swap -- exchanging a bond for another that offers a higher yield.

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Fin 284Allocation Within Sectors

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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Fin 284Rating Transition

Aaa Aa A Baa Ba Bb

C or D

total

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

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Fin 284Next Step

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 spreadsThen use use the probabilities from the matrix to find an expected return

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Fin 284Using Leverage

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.

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Fin 284Duration of levered portfolio

The duration of the levered portfolio should be calculated based on the “equity position” of the portfolio. (the amount of non borrowed funds)

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Fin 284Calculating Duration

1) Calculate the duration of the levered portfolio2) Determine the dollar duration of the portfolio

for a given change in interest rates3) Compute the ratio of the dollar value change in

2) to the initial unlevered portfolio4) the duration is then:5) (Ratio in 3))(100/rate change in 2 in bps)100

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Fin 284Creating Leverage

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).

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Fin 284Repo Interest

The dollar value of interest is calculated using a 360 day convention.

360

Term Repo

Rate

Repo

Borrowed

Amount $

Interest

Dollar

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Fin 284Reverse Repos

You can also cover a short position with a reverse repo (agreeing to buy the security and then sell it back in the future).

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Fin 284Credit Risk

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

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Fin 284Delivery of Collateral

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.

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Fin 284Determining the Repo Rate

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.

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Fin 284Structured Portfolios

Structured portfolios are intended to satisfy an investment objective, and are not based upon interest rate expectations.

IndexingMatch Liabilities and Assets

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Fin 284Indexing

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….)

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Fin 284Popularity of Indexing

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).

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Fin 284Problems with Indexing

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

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Fin 284Institutional perspective

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)

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Fin 284Selecting an Index

Return objectives -- look at both return and variability

Risk factors -- match index to acceptable risk levels

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Fin 284Broad Market Bond Indexes

Lehman Brothers Aggregate Index, Salomon Brothers Broad Investment Grade Bond Index, and Merrill Lynch Domestic Market IndexAll 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

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Fin 284Broad Market Indexes

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.

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Fin 284Specific Indexes

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.

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Fin 284Size of Portfolio

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 decreasesThe impact of portfolio size is very similar to equityThe number of securities needed to eliminate unsystematic risk may differ by sector.

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Fin 284Tracking Error

Tracking error is the difference between the indexed portfolio and the benchmark index.Three sources of tracking error

Transaction CostsDifferences in index composition.Difference in price used to construct the index and those paid by the portfolio

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Fin 284Tracking Error Tradeoff

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.

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Fin 284Indexing Methodologies

Stratified Sampling (or Cell)Optimization ApproachVariance Minimization Approach

In all three the goal is to minimize or eliminate diversifiable risk leaving only the systematic risk common to the sector.

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Fin 284Stratified Sampling (Cell)

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.

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Fin 284Stratified Sampling Example

Characteristic 1 Duration: < 5 years & > 5 yearsCharacteristic 2 Maturity: < 7 years & > 7 yearsCharacteristic 3 Sector: Treasuries and CorporateThen make cells out of each possible combination of characteristics:Cell 1: Duration < 5, Maturity < 7, TreasuryCell 2: Duration < 5, Maturity < 7, CorporateCell 3: Duration < 5, Maturity > 7, Treasury etc…Total cells = 2 x 2 x 2 = 8

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Fin 284Stratified Sampling

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)

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Fin 284Optimization

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 returnsConstraints: Limit the number of issues purchased from a given issuer, overweighting a cell

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Fin 284Optimization

Given the objective and constraints mathematical programming can then be used to determine which issues to include in the portfolio.

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Fin 284Variance Minimization

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.

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Fin 284Problem in Implementation

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 availableAggregation -- often generic issues are established to look like a group of issues (mortgage backs for example)

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Fin 284Enhanced Indexing

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.

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Fin 284

Asset / Liability Management

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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Fin 284Liability Classification

LiabilityType

Amount ofCash Outlay

Timing ofCash Outlay

I Known Known

II Known Uncertain

III Uncertain Known

IV Uncertain Uncertain

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Fin 284Liquidity Concerns

Will depend upon the type of institution.Banking -- depository withdrawsLife Insurance -- surrender and loan valuesMay also change the nature of expected cash inflows.

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Fin 284Surplus Management

Goals -- earn an adequate return and maintain a surplus of assets beyond liabilities.Three types of surpluses

Economic -- based on market valueAccounting -- based upon GAAPRegulatory -- based upon regulatory accounting principles

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Fin 284Economic Surplus

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.

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Fin 284Economic Surplus

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 surplusA decrease in interest rates implies an increase in surplus

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Fin 284Economic Surplus

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.

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Fin 284Accounting Surplus

Three methods for reporting the value of assets

Amortized cost (historical cost)Market valueLower of cost or market value

FASB specifies how different types of assets must be valued.

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Fin 284FASB 115

Account Classification

Accounting Method

Will Affect

Surplus

Will Affect

Reported Earnings

Held to maturity

Amortized Cost

No No

Available For Sale

Market Value Yes No

Trading Market Value Yes Yes

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Fin 284Regulatory Surplus

Regulators require reports based upon separate accounting principles (RAP).Often the regulatory surplus will differ significantly from the accounting or economic surplus.

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Fin 284Immunization

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”

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Fin 284

Immunization of a single liability

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.

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Fin 284Immunization attempt 1

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

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Fin 284Immunization attempt 2

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 passedIf rates increase immediately total value will be below $17,183,033, and vice versa.

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Fin 284Immunization attempt 3

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 passedIf rates increase (decrease) immediately total value will be above (below) $17,183,033.

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Fin 284Immunization attempt 4

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 valueIf rates decrease the increase in value offsets the decline in interest on interest.

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Fin 284Duration

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).

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Fin 284Immunization

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.

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Fin 284Changes over time

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.

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Fin 284Other complications

Our example assumed that the yield curve is flat and that any shifts in the yield curve are parallel shifts.

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Fin 284Immunization Risk

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.

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Fin 284

Measuring Immunization risk

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 flowsThe 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.

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Fin 284

Measuring Immunization Risk

flowcsh last theofreceipt to timen

portfolio for the yield y

horizon investment theof yearsin length H

tat time portfolio theof flowcash CF

:where

y)(1

H)(nCF

y)(1

H)(2CF

y1

H)(1CF

t

n

2n

2

22

21

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Fin 284

Immunizing with Zero Coupon Bonds

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

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Fin 284Portfolio Construction

Credit Risk -- if a bond defaults the target yield may not be reachedCall risk -- If callable issues are included then there is a risk of the call being exercised and the target yield not being reached.

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Fin 284Contingent Immunization

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.

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Fin 284Satisfying Multiple Liabilities

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.

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Fin 284

Satisfying Multiperiod Liabilities

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.

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Fin 284

Satisfying Multiperiod Liabilities

Symmetric Cash Matching -- allows short term borrowing of funds to satisfy a liability prior to the liability due date, reducing the cost of funding.

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Fin 284

Active / Immunization Combination

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.