0 Portfolio Management 3-228-07 Albert Lee Chun Equity Portfolio Management Strategies Lecture 9 2...

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Portfolio ManagementPortfolio Management3-228-073-228-07

Albert Lee ChunAlbert Lee Chun

Equity Portfolio Equity Portfolio Management StrategiesManagement Strategies

Lecture 9Lecture 9

2 Dec 2007

2Albert Lee Chun Portfolio Management

Passive Portfolio ManagementPassive Portfolio Management


Management FeesManagement Fees

Malkiel (2001) reports that on average that:Malkiel (2001) reports that on average that:

Costs of managing a passive fund oscillate between 10 and 20 Costs of managing a passive fund oscillate between 10 and 20 basis points (Vanguard S&P 500: 20 b.p.)basis points (Vanguard S&P 500: 20 b.p.)

For active funds the average management fees are 140 b.p. For active funds the average management fees are 140 b.p.

(fees for research, analyzing information, transaction costs).(fees for research, analyzing information, transaction costs).

40 billion dollars are spent each year on management fees. 40 billion dollars are spent each year on management fees.

Passive strategies have a tend to also minimize taxes (Malkiel Passive strategies have a tend to also minimize taxes (Malkiel 2001) 2001)


Managing and Index PortfolioManaging and Index Portfolio

Select a benchmark portfolio index to replicate. S&P 500, Select a benchmark portfolio index to replicate. S&P 500, TSX, etc.TSX, etc.

Determine an acceptable Determine an acceptable tracking errortracking error. Which depends on . Which depends on the the return differentialreturn differential or or total return of the replicating total return of the replicating portfolioportfolio minus minus the return of the benchmark indexthe return of the benchmark index

t,bt,pt RR

N

1i,i, w titp RR

where the return of the tracking portfolio is given bywhere the return of the tracking portfolio is given by


Portfolio Tracking ErrorPortfolio Tracking Error

Return Differential Return Differential

Average Return DifferentialAverage Return Differential

Variance in Return DifferentialVariance in Return Differential

Tracking ErrorTracking Error

Annualize Tracking ErrorAnnualize Tracking Error

t,bt,pt RR

T

ttT 1

1

2

1

2

1

1

T

ttT

2

p where P is equal to the number of periods in a year.


Portfolio Tracking ErrorPortfolio Tracking Error

ObjectiveObjective: Minimize the expected tracking error by : Minimize the expected tracking error by optimizing overoptimizing over

1. The number of securities in the portfolio1. The number of securities in the portfolio

2. The securities to include in the portfolio2. The securities to include in the portfolio


Expected Tracking ErrorExpected Tracking Error

500 400 300 200 100 0

2.0

1.0

3.0

4.0

Expected Tracking Error (Percent)

Number of Stocks


Portfolio IndexationPortfolio Indexation

The number of securities used in the replication determines a The number of securities used in the replication determines a tradeoff between transaction costs and tracking errors. tradeoff between transaction costs and tracking errors.

A smaller number of securities will result in lower transaction A smaller number of securities will result in lower transaction costs but higher tracking errors and visa versa. costs but higher tracking errors and visa versa.

The presence of tracking errors is inevitable. The presence of tracking errors is inevitable.

1. The replication implies irregular lots.1. The replication implies irregular lots.2. The composition of securities in the index may change.2. The composition of securities in the index may change.3. The modification of the index: entry and exit of securities, 3. The modification of the index: entry and exit of securities, merges, defaults, etc.merges, defaults, etc.


Techniques for Replicating an IndexTechniques for Replicating an Index


1. The simplest method1. The simplest method

Purchase all the securities in the index in proportion Purchase all the securities in the index in proportion to the weights in the index.to the weights in the index.

This helps ensure close trackingThis helps ensure close tracking

Advantage: Minimizes the tracking errorsAdvantage: Minimizes the tracking errors Disadvantage: High transaction costs and reinvesting Disadvantage: High transaction costs and reinvesting

dividends results in high adjustment fees. dividends results in high adjustment fees.


2. Method of Market Capitalization2. Method of Market Capitalization

Consider the stocks with the largest market Consider the stocks with the largest market capitalization in the index and purchase them in capitalization in the index and purchase them in proportion to their importance in the index.proportion to their importance in the index.

Fewer stocks means lower commissionsFewer stocks means lower commissions Reinvestment of dividends is less difficultReinvestment of dividends is less difficult Will not track the index as closely, so there will be Will not track the index as closely, so there will be

some some tracking errortracking error..


3. Method of Stratified Sampling:3. Method of Stratified Sampling:

Purchase only the most representative securities in Purchase only the most representative securities in the index portfolio.the index portfolio.

Classify the securities in the index into homogeneous Classify the securities in the index into homogeneous categories (by industry or activity sector, beta, total categories (by industry or activity sector, beta, total risk, stock market capitalization, etc....). risk, stock market capitalization, etc....).

Select from each category, a few titles which best Select from each category, a few titles which best represents that group, thus forming a representative represents that group, thus forming a representative portfolio for each category. portfolio for each category.

The replication portfolio is composed by balancing The replication portfolio is composed by balancing the portfolios for each category according to their the portfolios for each category according to their importance in the index.importance in the index.


4. Quadratic Optimization4. Quadratic Optimization

01

N

1ii

N

1ii

N

1ii

),...,(~~w~~

1w

such that

~~w Error Tracking ,...,1

RwwCrrErrE

rrMin

Nmimp

miw Nii

Transaction costs Required minimum

Expected Excess Return


Quadratic OptimizationQuadratic Optimization

Historical information on price changes and Historical information on price changes and correlations between securities are used to determine correlations between securities are used to determine the composition of a portfolio that will minimize the composition of a portfolio that will minimize tracking error with the benchmarktracking error with the benchmark

This relies on historical correlations, which may This relies on historical correlations, which may change over time, leading to a failure to track the change over time, leading to a failure to track the index.index.


Expected Tracking ErrorExpected Tracking Error

500 400 300 200 100 0

2.0

1.0

3.0

4.0

Expected Tracking Error (Percent)

Number of Stocks


Reference PortfolioReference Portfolio

Constructing a Reference PortfolioConstructing a Reference Portfolio

- Value Weighted- Value Weighted- Price Weighted- Price Weighted- Equal Weighted- Equal Weighted

It may be necessary to rebalance the portfolio when:It may be necessary to rebalance the portfolio when:

- Mergers and Acquisitions: Companies disappear from the - Mergers and Acquisitions: Companies disappear from the market.market.- Changes to the composition of the index- Changes to the composition of the index- Stock splits and dividend payments- Stock splits and dividend payments- New stock issues- New stock issues- Stock repurchases- Stock repurchases


Replicating an Index PortfolioReplicating an Index Portfolio

Small investors often find it more practical and less Small investors often find it more practical and less expensive to choose a "pre-made" a fund for expensive to choose a "pre-made" a fund for replicating an index. replicating an index.

- Mutual Funds- Mutual Funds

- Exchange Traded Funds- Exchange Traded Funds


Exchange Traded Funds (ETFs)Exchange Traded Funds (ETFs)

Exchange Traded Funds are less expensive than Exchange Traded Funds are less expensive than mutual funds but more diversified than individual mutual funds but more diversified than individual stocks. A cross between stocks and mutual funds.stocks. A cross between stocks and mutual funds.

ETFs seek returns of a broad market index or a sector ETFs seek returns of a broad market index or a sector index. They are index-linked rather than actively index. They are index-linked rather than actively managed.managed.

ETFs are exchange listed and can be bought and sold ETFs are exchange listed and can be bought and sold throughout the trading day: they are "funds that trade throughout the trading day: they are "funds that trade like stocks."like stocks."

Example: SPY, Cubes, Diamonds, Spiders, Webs, VIPERS, iShares, Ultra Sectors, etc.


Active Portfolio ManagementActive Portfolio Management


Active Portfolio ManagementActive Portfolio Management

Active Portfolio Management StrategiesActive Portfolio Management Strategies Goal is to earn a portfolio return that exceeds the Goal is to earn a portfolio return that exceeds the

return of a passive benchmark portfolio, net of return of a passive benchmark portfolio, net of transaction costs, on a risk-adjusted basis.transaction costs, on a risk-adjusted basis.

Practical difficulties of active managerPractical difficulties of active manager Transactions costs must be offset Risk can exceed passive benchmark


Active Management StrategiesActive Management StrategiesThe chose between using and active or passive portfolio management The chose between using and active or passive portfolio management

strategy depends on 2 factors:strategy depends on 2 factors:

1. Belief in the efficiency of the markets. An investor who rejects 1. Belief in the efficiency of the markets. An investor who rejects the Efficient Market Hypothesis will tend to adopt an active the Efficient Market Hypothesis will tend to adopt an active strategy with the goal of obtaining “abnormal” returns:strategy with the goal of obtaining “abnormal” returns:

where,where, ARARit it : is the abnormal return of security i in period t : is the abnormal return of security i in period t

RRitit: is the return of security i in period t: is the return of security i in period t

E(RE(Ritit): is the expected return ): is the expected return

2. Degree of risk aversion of the investor.2. Degree of risk aversion of the investor.

)R(ER = AR ititit


Performance of Active Mutual FundsPerformance of Active Mutual Funds

The average fund manager is not able to outperform the index!


Broad Overview of Investment StrategiesBroad Overview of Investment Strategies

Passive Management StrategiesPassive Management Strategies 1. 1. Efficient Markets HypothesisEfficient Markets Hypothesis - Buy and Hold- Buy and Hold - Indexing- Indexing Active Management StrategiesActive Management Strategies 2. 2. Fundamental AnalysisFundamental Analysis

““Top Down” (asset class rotation, sector rotation)Top Down” (asset class rotation, sector rotation) “ “Bottom Up” (stock undervaluation/overvaluation)Bottom Up” (stock undervaluation/overvaluation) 3. 3. Technical AnalysisTechnical Analysis Contrarian (e.g. overreaction)Contrarian (e.g. overreaction) Continuation (e.g. price momentum)Continuation (e.g. price momentum) 4. 4. Anomalies and AttributesAnomalies and Attributes

Calendar effects (Weekend, January)Calendar effects (Weekend, January) Security Characteristics (P/E,P/B, earnings momentum, firm size)Security Characteristics (P/E,P/B, earnings momentum, firm size) Investment Style (value, growth)Investment Style (value, growth)


Investment Style and Tracking ErrorInvestment Style and Tracking Error


Fundamental StrategiesFundamental Strategies

Top down approach involves analysis of broad Top down approach involves analysis of broad country and asset class allocations and progresses country and asset class allocations and progresses down through sector allocation decisions to the down through sector allocation decisions to the bottom level where individual securities are selected.bottom level where individual securities are selected.

Bottom-up approach emphasizes security selection Bottom-up approach emphasizes security selection without any initial market or sector analysis.without any initial market or sector analysis.


Top Down ApproachTop Down Approach

Top Down ApproachTop Down Approach

- Evaluate and forecast the future economy- Evaluate and forecast the future economy

- Chose the proportions to invest in each country or - Chose the proportions to invest in each country or economic region.economic region.

- Identify the sectors and industries that would profit - Identify the sectors and industries that would profit based on your economic outlook and choose based on your economic outlook and choose proportions to invest in each industy or sector.proportions to invest in each industy or sector.

- Choose the best securities in each sector selected.- Choose the best securities in each sector selected.


Bottom-Up ApproachBottom-Up Approach

Security selection is places less importance on the economic cycle.

Securities are selected based on well defined characteristic of individual stocks such as price-dividend ratios, book-to-market ratios, market capitalizations, etc.


Fundamental StrategiesFundamental Strategies

Tactical Asset AllocationTactical Asset Allocation - Asset Class Rotation: Shifts funds between stocks, bonds and - Asset Class Rotation: Shifts funds between stocks, bonds and

other securities depending on market forecasts and estimated other securities depending on market forecasts and estimated returns.returns.

Sector, Industry or Style Rotation StrategySector, Industry or Style Rotation Strategy- Shifts funds between different equity sectors and industries - Shifts funds between different equity sectors and industries (financial stocks, technology stocks, consumer cyclicals, (financial stocks, technology stocks, consumer cyclicals, durable goods) or among investment styles (e.g., large durable goods) or among investment styles (e.g., large capitalization, small capitalization, value growth)capitalization, small capitalization, value growth)

Individual Stock SelectionIndividual Stock Selection- Buy low, Sell High- Buy low, Sell High


Sector, Industry and Style AnalysisSector, Industry and Style Analysis

How should we choose which sector, industry or style How should we choose which sector, industry or style to rotate into next?to rotate into next?

Important to look to the underlying nature of the Important to look to the underlying nature of the economy. Security markets reflect the strength and economy. Security markets reflect the strength and weakness of the economy. weakness of the economy.

Most of the variables that determine security market Most of the variables that determine security market value are economic variables: monetary policy, value are economic variables: monetary policy, interest rates, aggregate output, inflation, etc. interest rates, aggregate output, inflation, etc.

Macro-analysisMacro-analysis links up industry effects to links up industry effects to business business cyclescycles and economic variables. and economic variables.


Asset and Sector PerformanceAsset and Sector Performance


A strong relationship exists between the economy and A strong relationship exists between the economy and the stock marketthe stock market

Security markets reflect what is expected to go on in Security markets reflect what is expected to go on in the economy because the value of an investment is the economy because the value of an investment is determined by determined by its expected cash flows required rate of return (i.e., the discount rate)

Macro-market AnalysisMacro-market Analysis


Is It A Worm?Is It A Worm?


Is It A Wave?Is It A Wave?


It’s the Business Cycle!It’s the Business Cycle!


Business CyclesBusiness Cycles

The aggregate economy expands and contracts in The aggregate economy expands and contracts in discernable periods.discernable periods.

Economic trends affect industry performance.Economic trends affect industry performance. Cyclical or Structural Changes?Cyclical or Structural Changes?

Cyclical changes in the economy arise from the ups and downs of the business cycle

Structure changes occur when the economy undergoes a major change in organization or how it functions


The Stock Market and The Stock Market and the Business Cyclethe Business Cycle

trough

peak

E

How can we predict the next peak or trough?


Business Cycle IndicatorsBusiness Cycle Indicators

National Bureau of Economic Research National Bureau of Economic Research (NBER)(NBER)

Cyclical indicator categoriesCyclical indicator categories leading indicators coincident indicators lagging indicators

Composite series and ratio of seriesComposite series and ratio of series


Business Cycle IndicatorsBusiness Cycle Indicators

Leading indicatorsLeading indicators – economic series that usually – economic series that usually reach peaks or troughs before corresponding peaks or reach peaks or troughs before corresponding peaks or troughs in aggregate economy activitytroughs in aggregate economy activity

Coincident indicatorsCoincident indicators – economic series that have – economic series that have peaks and troughs that roughly coincide with the peaks and troughs that roughly coincide with the peaks and troughs in the business cycle peaks and troughs in the business cycle

Lagging indicatorsLagging indicators – economic series that experience – economic series that experience their peaks and troughs after those of the aggregate their peaks and troughs after those of the aggregate economyeconomy

Selected seriesSelected series – economic series that do not fall into – economic series that do not fall into one of the three main groups.one of the three main groups.


CONFERENCE BOARD DATA Standardization Leading Index Factor 1. BCI-01 Average weekly hours, manufacturing .1946 2. BCI-05 Average weekly initial claims for unemployment insurance .0268 3. BCI-08 Manufacturers' new orders, consumer goods and materials .0504 4. BCI-32 Vendor performance, slower deliveries diffusion index .0296 5. BCI-27 Manufacturers' new orders, non-defense capital goods .0139 6. BCI-29 Building permits, new private housing units .0205 7. BCI-19 Stock prices, 500 common stocks .0309 8. BCI-106 Money supply, M2 .2775 9. BCI-129 Interest rate spread, 10-year Treasury bonds less federal funds .3364 10. BCI-83 Index of consumer expectations .0193 Coincident Index 1. BCI-41 Employees on nonagricultural payrolls .5186 2. BCI-51 Personal income less transfer payments .2173 3. BCI-47 Industrial production .1470 4. BCI-57 Manufacturing and trade sales .1170 Lagging Index 1. BCI-91 Average duration of unemployment .0368 2. BCI-77 Inventories to sales ratio, manufacturing and trade .1206 3. BCI-62 Labor cost per unit of output, manufacturing .0693 4. BCI-109 Average prime rate .2692 5. BCI-101 Commercial and industrial loans .1204 6. BCI-95 Consumer installment credit to personal income ratio .1951 7. BCI-120 Consumer price index for services .1886


Other Indicator Sources... Other Indicator Sources...

I ndicator Released By

Beige Book Federal Reserve Board

Consumer Confidence Index Consumer Confidence Board

Consumer Price Index Bureau of Labor and Statistics

Employee Cost Index Bureau of Labor and Statistics

Employment Situation Report Bureau of Labor and Statistics

Gross Domestic Product Commerce Department

Housing Starts Department of Commerce

Philadelphia Fed Index Federal Reserve Bank of Philadelphia

Producer Price Index Bureau of Labor and Statistics

Retail Sales Data Census Bureau


Stock Markets are a Leading IndicatorStock Markets are a Leading Indicator

Stock prices consistently turn before the economy Stock prices consistently turn before the economy does.does.

Stock prices are forward looking.Stock prices are forward looking. Stock prices reflect expectations of earnings, Stock prices reflect expectations of earnings,

dividends, and interest ratesdividends, and interest rates Stock market reacts to various leading indicator seriesStock market reacts to various leading indicator series


Forecasting Business CyclesForecasting Business Cycles

The current state of the business cycle has already The current state of the business cycle has already been incorporated into asset prices.been incorporated into asset prices.

Investors need to make decisions based on future Investors need to make decisions based on future economic conditions. economic conditions.

To invest properly, it is important to forecast changes To invest properly, it is important to forecast changes in economic variables.in economic variables.

High inflation: high interest rates, bad for stocks in High inflation: high interest rates, bad for stocks in general. general.


Sector Rotation StrategySector Rotation Strategy

Certain industries make attractive investments over Certain industries make attractive investments over the course of the business cycle.the course of the business cycle.

A A sector rotationsector rotation strategy strategy is when one switches from is when one switches from one industry group to another over the course of a one industry group to another over the course of a business cycle.business cycle.



Financial Stocks Excel

trough

peak


Financial StocksFinancial Stocks

Adversely impacted by interest rates, difficult to pass on to their customers.

Toward the end of a recession, financial stocks rise as investors trade securities, businesses issue debt and equity, increase in merger activity during recovery.

Expecting increases in loan demand, housing construction and companies going public.




trough

peakConsumer Durables

Excel


Consumer Durables Consumer Durables

Consumer Durables are cars, PCs, Miele washing machines, GE refrigerators, John Deer lawn machinery, cooking ranges, etc.

As the economy begins to come out of the recession, consumer confidence and income increase.




trough


Excel

Capital Goods Excel


Capital GoodsCapital Goods

As the economy moves out of recession, businesses As the economy moves out of recession, businesses begin to modernize, renovate and purchase new begin to modernize, renovate and purchase new equipment.equipment.

Heavy equipment manufactures, machine tool Heavy equipment manufactures, machine tool makers, airplane manufacturers become attractive.makers, airplane manufacturers become attractive.




trough


Excel

Capital Goods Excel

Basic Industries

Excel


Basic IndustriesBasic Industries

End of the business cycle coincides with increases in End of the business cycle coincides with increases in inflation, as demand outstrips supply.inflation, as demand outstrips supply.

Inflation doesn’t influence the cost of extracting these Inflation doesn’t influence the cost of extracting these products, whereas prices increase. Increasing profit products, whereas prices increase. Increasing profit margins.margins.

Basic material industries as oil, metal and lumber are Basic material industries as oil, metal and lumber are attractive.attractive.




trough


Excel

Capital Goods Excel

Basic Industries

Excel

Consumer Staples Excel


Consumer StaplesConsumer Staples

Consumer staples are food, beverages and Consumer staples are food, beverages and pharmaceuticals. pharmaceuticals.

People still need to eat, drink, be merry and get sick.People still need to eat, drink, be merry and get sick.




trough


Excel

Capital Goods Excel

Basic Industries

Excel

Consumer Staples Excel


Central Banks and Interest RatesCentral Banks and Interest Rates

By far the most visible and obvious power of many modern By far the most visible and obvious power of many modern central banks is to influence market interest rates.central banks is to influence market interest rates.

When interest rates go down, money supply increases. When interest rates go down, money supply increases. Businesses and consumers have a lower cost of capital and can Businesses and consumers have a lower cost of capital and can increase spending and capital improvement projects. This increase spending and capital improvement projects. This encourages growth.encourages growth.

When interest rates go up, the money supply falls and slows When interest rates go up, the money supply falls and slows the economy. Increases in interest rate flight inflation.the economy. Increases in interest rate flight inflation.

The US central-bank lending rate is known as the Fed funds The US central-bank lending rate is known as the Fed funds rate. rate.

Bank of Canada sets a target overnight rate, and a band of plus Bank of Canada sets a target overnight rate, and a band of plus or minus 0.25%.or minus 0.25%.


Technical StrategiesTechnical Strategies

Contrarian investment strategyContrarian investment strategy- Best time to buy a stock is when the majority of other Best time to buy a stock is when the majority of other

investors are selling.investors are selling.- Buy low, sell high. Hope asset prices are mean Buy low, sell high. Hope asset prices are mean

reverting.reverting.- Overreaction hypothesis.Overreaction hypothesis. Price momentum strategyPrice momentum strategy Earnings momentum strategy Earnings momentum strategy


Market OverreactionMarket Overreaction


Price and Earnings MomentumPrice and Earnings Momentum


Anomalies and AttributesAnomalies and Attributes

The Weekend EffectThe Weekend Effect The January EffectThe January Effect Firm SizeFirm Size P/E and P/BV ratiosP/E and P/BV ratios


Large and Small Cap ReturnsLarge and Small Cap Returns


Standard Deviation of ReturnsStandard Deviation of Returns


P/E Ratios and PerformanceP/E Ratios and Performance


Value vs. Growth StocksValue vs. Growth Stocks

Over time value stocks have offered somewhat higher returns than growth Over time value stocks have offered somewhat higher returns than growth stocks.stocks.

However, growth stocks will outperform value stocks from time to time.However, growth stocks will outperform value stocks from time to time. Growth-oriented investor will:Growth-oriented investor will:

focus on EPS and its economic determinants look for companies expected to have rapid EPS growth assumes constant P/E ratio

Value-oriented investor will: Value-oriented investor will: focus on the price component not care much about current earnings Assume that P/E ratio is below its natural level for these stock and that

the market will soon correct this situation.


Value vs. Growth: Mutual FundsValue vs. Growth: Mutual Funds


Value and Growth StocksValue and Growth Stocks


Russell 100 PerformanceRussell 100 Performance


Value vs. GrowthValue vs. Growth


StyleStyle

Construct a portfolio to capture one or more of the Construct a portfolio to capture one or more of the characteristics of equity securitiescharacteristics of equity securities

Small-capitalization stocks, low-P/E stocks, etc…Small-capitalization stocks, low-P/E stocks, etc… Value stocks appear to be underpricedValue stocks appear to be underpriced

price/book or price/earnings Growth stocks enjoy above-average earnings per Growth stocks enjoy above-average earnings per

share increasesshare increases


Style GridStyle Grid

Style grid: Style grid: firm size (large cap, mid cap, small cap) Relative value (value, blend, growth)

characteristics Variations in returns among mutual funds are largely Variations in returns among mutual funds are largely

attributable to differences in stylesattributable to differences in styles Different styles tend to move at different times in the Different styles tend to move at different times in the

business cyclebusiness cycle


Style Analysis GridStyle Analysis Grid


Investment StyleInvestment Style


Optimal Portfolio Selection Optimal Portfolio Selection


Next WeekNext Week Optional Readings for this week:Optional Readings for this week: (Course Reader)(Course Reader) Value and Growth Investing: Review and Update (Chan and Value and Growth Investing: Review and Update (Chan and

Lakonishok)Lakonishok) (Course Reader)(Course Reader) Hedge Funds: Risk and Return (Maikiel ad Saha) Hedge Funds: Risk and Return (Maikiel ad Saha)

For Next Week:For Next Week: Rappel des concepts de durée et de convexitéRappel des concepts de durée et de convexité La durée des taux clésLa durée des taux clés Stratégies Stratégies Barbell Barbell etet Bullet Bullet Immunisation de portefeuilleImmunisation de portefeuille

Review of Duration and Convexity (Rappel des concepts de durée et de Review of Duration and Convexity (Rappel des concepts de durée et de convexité)convexité)

Key Rate Duration (La durée des taux clés) Key Rate Duration (La durée des taux clés) Barbell and Bullet Strategies (Stratégies Barbell and Bullet Strategies (Stratégies Barbell Barbell etet Bullet) Bullet) Portfolio Immunization (Immunisation de portefeuille)Portfolio Immunization (Immunisation de portefeuille)

(BKMR) (BKMR) chapter 15 chapter 15 (Course Reader)(Course Reader) Fabozzi, Fabozzi, Bonds Markets, Analysis and StrategiesBonds Markets, Analysis and Strategies, 4th edition., , 4th edition.,

p.410-416. p.410-416. (Course Reader)(Course Reader) Tuckman, Bruce W. Tuckman, Bruce W. Key Rate and Bucket Exposures, Key Rate and Bucket Exposures, Fixed Income Fixed Income

Securities: Tools for Today's Markets, 2nd edition, New York, J. Wiley, 2002, p. 133-Securities: Tools for Today's Markets, 2nd edition, New York, J. Wiley, 2002, p. 133-137137

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