1 INTRODUCTION TO DERIVATIVES MARKETS (INVESTMENTS BACKGROUND) SPRING 2006.

1

INTRODUCTION TO INTRODUCTION TO DERIVATIVES DERIVATIVES

MARKETS MARKETS (INVESTMENTS (INVESTMENTS BACKGROUND) BACKGROUND)

SPRING 2006 SPRING 2006

2

REAL VS. FINANCIAL REAL VS. FINANCIAL ASSETSASSETS

A. REAL ASSETS Plant and Equipment=Physical

Capital Growth Opportunities: e.g. R&D,

Patents, New Ventures Human Capital=Expertise, Labor

Services Contribute Directly to the Productive

Capacity of the Economy (i.e. to GNP Growth)

3

RREAL VS. FINANCIAL EAL VS. FINANCIAL ASSETSASSETS

B. FINANCIAL ASSETS

Stocks, Bonds, Hybrid Securities Are Claims to the After-Tax Earnings

Streams Generated by Real Assets Provide an Incentive to Invest in Real

Assets by Providing Liquidity Establishes a Pricing (Valuation)

Mechanism for Real Assets Thereby Contribute Indirectly to the

Productive Capacity of the Economy

4

CLIENTS OF THE CLIENTS OF THE FINANCIAL SYSTEMFINANCIAL SYSTEM THE HOUSEHOLD SECTOR

(INDIVIDUALS)The financial assets households desire to hold

depend on their tax status, investment horizons, need for liquidity, cash-flow needs, and risk

preferences.

THE BUSINESS SECTOR (CORPORATIONS) Raises money by debt and equity issues in primary capital markets. The business sector raises money

efficiently by using investment bankers and by keeping securities simple.

THE GOVERNMENT SECTOR (STATE, FEDERAL, AND MUNICIPAL AGENCIES )

Can only borrow through debt issues and taxation,but regulates the financial sector.

5

FLOW OF CASH FLOW OF CASH BETWEEN CAPITAL BETWEEN CAPITAL

MARKETS AND FIRM’S MARKETS AND FIRM’S OPERATIONSOPERATIONS

FINANCIAL MANAGER

FIRM’SOPERATIONS

CAPITALMARKETS

1. CASH RAISED FROM INVESTORS

2 .CASH INVESTED IN

FIRM

3. CASH GENERATED BY

OPERATIONS

4. CASH RETURNED

TO INVESTORS

5.CASH RE-INVESTED

6

MONEY MARKET MONEY MARKET INSTRUMENTSINSTRUMENTS

U.S. TREASURY BILLS

FEDERAL FUNDS

EURODOLLARS

REPOS AND REVERSES

BROKER CALLS

THE LIBOR MARKET

COMMERCIAL PAPER

BANKER’S ACCEPTANCES

7

TREASURY BILL TREASURY BILL PRICING PRICING

CONVENTIONSCONVENTIONS FOR PURPOSES OF

DISCOUNTING, THE TREASURY USES 360 DAYS AS ITS YEAR

BOND YIELDS, ON THE OTHER HAND, ARE QUOTED ON THE BASIS OF A 365 DAY YEAR

HENCE ADJUSTMENTS MUST BE MADE

8

TREASURY BILL TREASURY BILL TERMINOLOGYTERMINOLOGY

P=CURRENT PRICE

F=FACE VALUE

N=NUMBER OF DAYS TO MATURITY

BDY=BANK DISCOUNT YIELD

BEY=BOND EQUIVALENT YIELD

9

PRICING U.S. PRICING U.S. TREASURY BILLSTREASURY BILLS

STEP #1. DETERMINE THE NUMBER OF DAYS TO MATURITY: N.

STEP #2. CALCULATE THE DOLLAR DISCOUNT CORRESPONDING TO N. THIS IS CALLED THE DOLLAR BANK DISCOUNT YIELD;

D=(BDY*F*N)/360

STEP #3. THE CURRENT PRICE

P=F-D

STEP #4. CALCULATE THE HOLDING PERIOD YIELD,

HPY=D/P STEP #5 CALCULATE THE BOND

EQUIVALENT YIELD,

BEY=HPY*365/N

10

TREASURY BILL TREASURY BILL PRICING PRICING

FORMULAEFORMULAE

CURRENT PRICE,

P=F(1-BDY*N/360)

BOND EQUIVALENT YIELD (BEY)

BEY=365*BDY/(360-BDY*N)

11

U.S. TREASURY BILLS ‘PRICE’ QUOTE

(SOURCE: www.bloomberg.com)

12

MONEY RATES (SOURCE: MONEY RATES (SOURCE: WSJ 01/06/03)WSJ 01/06/03)

See BKM Text p. 32

Figure 2.1

13

MEDIUM TO LONG-MEDIUM TO LONG-TERM FIXED INCOME TERM FIXED INCOME

INSTRUMENTSINSTRUMENTS U.S. TREASURY NOTES AND

BONDS

FEDERAL AGENCY DEBT

MUNICIPAL BONDS (MUNIS)

CORPORATE BONDS

MORTGAGES

MORTGAGE-BACKED SECURITIES

14

TREASURY BOND TREASURY BOND PRICING PRICING

CONVENTIONSCONVENTIONS TREASURY BONDS ARE

QUOTED IN DOLLARS PLUS 32ND’S PER FACE VALUE. THE LATTER ARE CALLED BOND POINTS

E.G. A BOND POINT (1/32) TRANSLATES INTO $1,000/32=$31.25 FOR EACH $1,000 OF FACE VALUE

BOND YIELD TO MATURITY (YTM) IS THE BOND’S IRR BASED ON A 365 DAY YEAR

15

US US TREASURY TREASURY BOND PRICE BOND PRICE QUOTATIONSQUOTATIONS

(SOURCE:WSJ(01/06/03)

See Text BKM Figure 2.3 Page 37)

4.750 Nov08n 107:25 107:26 1 3.27

16

U.S. T-BOND U.S. T-BOND CALCULATIONSCALCULATIONS

HIGHLIGHTED BOND (06/01/03): Coupon Rate = 4.75 %; coupon payment;

4.75 % of face value paid annually; coupon payments are paid every six months (i.e. semi-annually)

Maturity = November 2008. Bid Price = 107:25

NOTE: this means 107 25/32 per each $100 of face value.

Ask Price = 107:26 or107 26/32 per $100 of face value

1 = ask price up by 1/32 from previous day’s ask price.

Ask yield = the yield to maturity (IRR) of the bond based on the asked price=3.27%

17

CORPORATE BOND CORPORATE BOND QUOTATIONSQUOTATIONS

See text BKM Figure 2.7, page 42

18

READING CORPORATE READING CORPORATE BOND QUOTATIONSBOND QUOTATIONS

HIGHLIGHTED BOND: Bond =ATT, 73/4% coupon, maturing

in 2007. Interest paid semiannually; $77.50 per

$1,000 of face value. Current yield = $77.50/$1060 =7.3 %

= annual coupon current bond price. Trading volume = 54 $1000 face

value bonds traded that day. Closing price =$1060 per $1,000 of

face value (i.e. a premium bond). Net change =closing price 1/2% up

from closing price on the previous day.

19

READING STOCK READING STOCK MARKET QUOTATIONSMARKET QUOTATIONS(SOURCE WSJ (09/08/97)(SOURCE WSJ (09/08/97)

See text BKM Figure 2.9, page 46

20

READING STOCK READING STOCK MARKET QUOTESMARKET QUOTES

HIGHLIGHTED FIRM (GE CORP.) 52 week high and low stock price per share:$41.24

and $21.40 respectively. Dollar Dividends: $.76 /share annually. Dividend Yield: annual dividend/current price=3.0

%=.76/25.40 PE: price earnings ratio=16. Volume: 100’s of shares traded that day =148191 High and low for that trading day : see

www.nyse.com Closing Price=$25.40 per share. Net change: -$.08 per share from previous day’s

close.

21

STOCK AND BOND STOCK AND BOND MARKET INDICESMARKET INDICES

STOCK INDICES : DJIA S&P 500 NYSE AMEX NASDAQ WILSHIRE 5000 VALUELINE CRSP VW* CRSP EW**

BOND INDICES: SOLOMON BROTHERS LEHMAN BROTHERS* Center for Research on Security Prices, value-weighted

** Center for Research on Security Prices, equally-weighted

22

STOCK MARKET STOCK MARKET INDICES: EXAMPLESINDICES: EXAMPLES

DJIA: 30 “blue chip” stocks; NYSE traded: price weighted: divisor adjustment produces a large number average with large movements; overly influenced by higher priced stocks; oldest; most frequently quoted.

S&P 500: 500 stocks - industrials, transportation, utilities, financials -- NYSE and NASDAQ traded, value weighted..

NYSE: All NYSE-listed stocks; value weighted. NASDAQ: All stocks listed on NASDAQ;

value weighted.. WILSHIRE 5000: Value weighted; all exchange

listed and NASDAQ listed stocks; most comprehensive, readily available stock index.

VALUELINE: 1,700 stocks; price weighted, no divisor manipulation; geometric average.

23

THREE TYPES OF THREE TYPES OF STOCK MARKET STOCK MARKET

INDICESINDICESPRICE-WEIGHTED

implies one share of each stock is purchased,

therefore overweights the higher priced stocks in the index,

VALUE-WEIGHTED

implies that stocks are held in the index in proportion to their relative market values,

EQUALLY-WEIGHTED

implies that equal dollar amounts of each stock are purchased.

24

IN-CLASS PROBLEM ON IN-CLASS PROBLEM ON THE TYPES OF INDICESTHE TYPES OF INDICES

Use the following information to answer questions 1-4:

BASE YEAR

Stock Price Shares

A $40 10,000,000

B $50 20,000,000

C $60 30,000,000

25

CONTINUEDCONTINUEDCURRENT YEAR

Stock Price Shares

A $22 20,000,000 B $55 20,000,000

C $66 30,000,0001. What is the percentage change in a

price-weighted index ?

2. What is the percentage change in a market value-weighted index ?

3. What is the percentage change in an equally-weighted index ?

4. What is the geometric average of the returns?

26

SOLUTION TO IN-SOLUTION TO IN-CLASS PROBLEMCLASS PROBLEM

1. A price-weighted index simply adds up the prices of the individual stocks underlying the Index’s construction and divides by the number of such stocks.

Therefore, the initial value of the Index is:

$40+$50+$60/3=$50. If we did the same in the current

year we would obtain:

$22+$55+$66/3=47.67

which represents a -4.67% decline

in the index. But did it decline ?

27

IN-CLASS SOLUTION IN-CLASS SOLUTION (CONT.)(CONT.)

Since there are double the number of shares outstanding in the current year compared to the base year, the stock must have split 2 for 1. Part of the decline in the Index was caused by this stock split and therefore does not represent a true decline in the market. To account for this, the divisor used in calculating the Index must be adjusted: let x be the new value of the divisor. Then x is given as the solution to:

$20+$50+$60/x=$40+$50+$60/3 x=2.6

28


In computing the new value of the Index we use the adjusted divisor 2.6 instead of 3.0

Index in current year=

$22+$55+$66/2.6=$55 The percentage change in the

Index (representing the true increase in the market) is

$55-$50/$50=10%

29


2. A value-weighted Index multiplies each price by the number of shares outstanding and therefore automatically adjusts for stock splits.

Value of the Index in the base year:

$40*10mm+$50*20mm+$60*30=3200mm

Usually, this is set to a standard number in the base year, e.g. 100 Index points by dividing by 32. The value of the Index in the base year is 100.

30


Value of the Index in the current year:

$22*20mm+$55*20mm+$66*30=3520mm

Note the automatic adjustment for the stock split. The value of

the Index in the base year is 3520/32=110

Clearly the Index increased by

110-100/100=10%

31


3. An equally- weighted Index requires that the same dollar investment be placed in each stock in the Index. The least common divisor of the stock prices in the base-year $40, $50, and $60 is $ 2400.

$2400 purchases 60 shares of stock A (60*$40=$2400), 48 shares of stock B (48*$50=$2400), and 40 shares of stock C (40*$60=$2400).

The adjustment for stock splits occurs naturally because in the

current year you own 120 shares

32


The value of the Index in the base-year is just the value of the dollars invested in it:

$2400+$2400+$2400=$7200 Normalize to 100 Index points by

dividing by 72 . The value of the Index in the current

year is 120*$22+48*$55+40*$66=$7920 Divide by 72 to obtain 110. This represents a $10% increase as before.

4. Stock A increased by 10% after adjusting for the stock split ($20 to $22), Stock B by 10% ($50 to $55) and Stock C by 10% ($60 to $66). The geometric average is 10%.

33

MARGINING OF LONG MARGINING OF LONG EQUITY POSITIONSEQUITY POSITIONS

INITIAL MARGINS

SET BY THE FEDERAL RESERVE

CURRENTLY EQUALS 50% INITIAL MARGIN=INVESTOR’S

EQUITY/MARKET VALUE OF SECURITIES HELD

E.G. AN INVESTOR PURCHASES $10,000 WORTH OF COMMON

STOCK BY PUTTING $6,000 DOWN AND BORROWING $4,000

HIS INITIAL MARGIN=$6,000/$10,000=60%.

34

MARGINS (CONT.)MARGINS (CONT.)MAINTENANCE MARGINS

SET BY BROKERS CURRENTLY 30% E.G. SUPPOSE THAT THE MARKET

VALUE OF THE STOCKS HELD FALLS TO $5,000.

THE LOSS COMES OUT OF THE CUSTOMER’S EQUITY, HENCE THE ACTUAL MARGIN=$1,000/$5,000=20%

THIS REQUIRES AN ADDITIONAL $5,00 FROM THE INVESTOR TO RESTORE THE MAINTENANCE MARGIN LEVEL TO 30%

OR THE BROKER CAN SELL OFF $1,667 OF THE INVESTMENT

35

THE MECHANICS THE MECHANICS OF SHORT SALESOF SHORT SALES

The way it’s supposed to work:

STEP 1: BORROW STOCK FROM BROKER, STEP 2: SELL STOCK AT CURRENT PRICE (SAY, 100

DOLLARS A SHARE), STEP 3: HOPEFULLY, BUY BACK STOCK AT LOWER

PRICE (SAY, 80 DOLLARS PER SHARE, STEP 5: ENJOY 20 DOLLAR PROFIT.

The way it could work:

STEP 1: BORROW STOCK FROM BROKER, STEP 2: SELL STOCK AT CURRENT PRICE (SAY, 100

DOLLARS A SHARE),

STEP 3. THE STOCK PRICE KEEPS GOING UP. SO YOU GIVE UP AND BUY STOCK AT HIGHER PRICES (SAY, 120 DÓLLARS PER SHARE),

STEP 4 .RETURN SHARES TO BROKER, STEP 5. WEEP OVER 20 DOLLAR LOSS.

36

THE MARGIN CALL THE MARGIN CALL PRICE ON LONG PRICE ON LONG

POSITIONSPOSITIONS How low can the security price fall

before the investor receives a margin call ?

Let L= the amount borrowed from the broker.

Let N= the number of shares purchased

Let M= the maintenance margin level Then Pm=(L/N(1-M)) E.g. Pm=(4000/(100x(1-0.30))=57.14

37

THE MARGIN CALL THE MARGIN CALL PRICE ON SHORT PRICE ON SHORT

POSITIONSPOSITIONS Let N = the number of shares sold

short, P0=the price per share at the time of

the short sale, P1=the price per share when the short

sale is covered, I.e. the shares are bought back.

IM=the initial margin M= the maintenance margin level Then Pm=(Nx P0+IM)/(Nx(M+1))

38

THE MARGIN CALL THE MARGIN CALL PRICE ON SHORT PRICE ON SHORT

POSITIONS: EXAMPLEPOSITIONS: EXAMPLE Suppose that you sell short 100

shares at 100 dollars per share. You post 5,000 in initial margin,

The maintenance margin requirement is 30 %,

Then the margin call price is (10,000+5000)/(100x(0.3+1))=

115.38

39

DEFINING DEFINING INVESTMENTS: A INVESTMENTS: A

GENERAL DEFINITIONGENERAL DEFINITION

We need a definition of ‘investment’ sufficiently general to encompass investments in real assets and investment in financial assets. Further, it should apply to explaining the connection between the two. The following definition serves:

THE SACRIFICE OF (CERTAIN) PRESENT CONSUMPTION FOR FUTURE (GENERALLY UNCERTAIN) CONSUMPTION

40

THE PROBLEM THE PROBLEM SOLVED BY SOLVED BY

INVESTMENTSINVESTMENTS

Re-allocating consumption claims (certain and uncertain) across time and under conditions of uncertainty

41

ONE MAIN REASON ONE MAIN REASON FOR INVESTINGFOR INVESTING

IN ORDER TO REALLOCATE CONSUMPTION CLAIMS IN THE PRESENT AND IN THE FUTURE FROM GIVEN PATTERNS INTO PREFERRED PATTERNS.

THE PRICING MECHANISM GIVES THE RATES AT WHICH THIS IS POSSIBLE IN THE MARKET THROUGH A VARIETY OF FINANCIAL VEHICLES.

42

CONSUMPTION CONSUMPTION CHOICESCHOICES

2.5

2.2

1.4

1.1

2.0 2.3 2.5

Invest in tennis facility

Invest inthe bank

Consumptionlater

Villa in Spain

Consumptionnow

(millions)

43

BORROWING AND BORROWING AND LENDING ENLARGE LENDING ENLARGE

CHOICESCHOICESDollars, period 1

Dollars, period 0

Interest rate lines showscash flows from borrowing

or lending

By borrowing OF, an individual can consume an extra BD today; bylending OB, he can consume an extra FH tomorrow.

F

H

B DO

44

THE EFFECT OF THE EFFECT OF INVESTMENT IN REAL INVESTMENT IN REAL

ASSETSASSETSConsumption, period 1

Consumption, period 1

Investment opportunities lineshows cash flows frominvesting in real assets

Notice the diminishing return on additional units of investment

45

HOW INVESTMENT IN HOW INVESTMENT IN REAL ASSETS IMPROVES REAL ASSETS IMPROVES

WELFAREWELFARE

... and so can the prodigal

The miser can spend moretoday and the next periodM

HL

G

J D K



The miser and prodigal have initial wealth of OD. Both are better off if they invest JD in real assets and then borrow or lend in the capital markets.

O

46

KEY QUESTIONS KEY QUESTIONS ADDRESSED BY ADDRESSED BY INVESTMENT INVESTMENT

ANALYSISANALYSIS 1. WHAT TYPES OF RE-

ALLOCATIONS ARE AVAILABLE IN THE MARKETS FOR FIXED INCOME, EQUITIES, HYBRIDS, ETC. ?

2. WHAT ARE THE RISK/EXPECTED RETURN CHARACTERISTICS OF THESE MECHANISMS (OPPORTTUNITY COSTS) ?

3. HOW CAN THESE INVESTMENT VEHICLES BE RISK-MANAGED ?

E.G. THROUGH PORTFOLIO DIVERSIFICATION, AND THE CORRECT USES OF DERIVATIVES .

47

DEFINING VIABLE DEFINING VIABLE INVESTMENT INVESTMENT

PROGRAMSPROGRAMS

1. THE SET OF AVAILABLE ‘RISK-FREE’ INVESTMENT ALTERNATIVES.

2. THE SET OF AVAILABLE RISKY INVESTMENT ALTERNATIVES.

3. SUBJECTIVE PREFERENCES FOR THE RISK/EXPECTED RETURN TRADEOFFS EMBODIED IN FINANCIAL INSTRUMENTS AS INVESTMENT VEHICLES.

48

OBJECTIVESOBJECTIVES OF OF INVESTMENT INVESTMENT

ANALYSISANALYSIS 1. MAP OUT THE

RISK/RETURN CHARACTERISTICS OF ALTERNATIVE INVESTMENT STRATEGIES.

2. SIFT OUT WHAT CAN ACTUALLY BE DONE BY PORTFOLIO MANAGERS FOR THEIR CLIENTS FROM WHAT CAN’T BE DONE SO AS TO SATISFY THEIR SUBJECTIVE RISK/RETURN PREFERENCES.

49

TYPES OF TYPES OF INVESTMENT INVESTMENT STRATEGIESSTRATEGIES

1. MARKET TIMING.

2. STATIC PORTFOLIO DIVERSIFICATION.

3. DYNAMIC PORTFOLIO DIVERSIFICATION.

4. ASSET ALLOCATION.

50

BASIC ASSET BASIC ASSET ALLOCATION ALLOCATION STRATEGIESSTRATEGIES

ALLOCATING FUNDS BETWEEN CASH EQUIVALENTS, BONDS, AND EQUITIES.

E.G. CAPITAL ALLOCATION LINE STRATEGIES--HOW MUCH IN THE BANK =, HOW MUCH IN A SINGLE RISKY ASSET MUTUAL FUND

51

CAPITAL CAPITAL ALLOCATION ALLOCATION

LINESLINES

R F

E p

p

1

E1

52

THE EQUATION OF THE EQUATION OF THE CAL THE CAL

E(RP )= RF+[(E1-RF ) /xp

WHERE E(RP ) IS THE EXPECTED RATE OF RETURN OF THE PORTFOLIO.

AND p IS THE STANDARD DEVIATION OF THE RATE OF RETURN OF THE PORTFOLIO.

53

CAPITAL CAPITAL ALLOCATION LINESALLOCATION LINES(REWARD TO RISK (REWARD TO RISK

RATIO)RATIO) THE SLOPE OF THE CAPITAL

ALLOCATION LINE IS THE (EXCESS)

REWARD TO RISK RATIO= (E1-RF ) /

NOTE THAT (E1-RF ) IS THE EXCESS

EXPECTED RETURN OFFERED BY SECURITY OR PORTFOLIO 1 ABOVE THAT OFFERED BY CASH EQUIVALENTS REPRESENTED BY THE SURE RATE OF RETURN,

IS A MEASURE OF ‘RISK’

54

CAPITAL CAPITAL ALLOCATION ALLOCATION

LINESLINES

R F

E p

p1

E1

E2

CAL

CAL

55

MORE EFFICIENT MORE EFFICIENT CAL’SCAL’S

THE REWARD TO RISK RATIO OF CAL2 IS GREATER THAN THE REWARD TO RISK RATIO OF CAL1.

THEREFORE CAL2 PROVIDES MORE EFFICIENT RISK-RETURN OPPORTUNITIES THAN DOES CAL1.

56

ROLE OF THE ROLE OF THE PORTFOLIO PORTFOLIO MANAGERMANAGER

OFFER MORE AND MORE EFFICIENT CAPITAL ALLOCATION LINES TO INVESTORS RATHER THAN:

ATTEMPTING TO SATISFY THEIR SUBJECTIVE RISK/RETURN PREFERENCES DIRECTLY.

57

DIFFERENT INVESTORS DIFFERENT INVESTORS HAVE DIFFERENT HAVE DIFFERENT

INDIFFERENCE CURVESINDIFFERENCE CURVES

E

p

Investor A’s indifference

curves Investor B’s indifference curves

NOTE: B IS LESS RISK-AVERSE THAN A.

p

58

PORTFOLIO CHOICES PORTFOLIO CHOICES FOR DIFFERENT FOR DIFFERENT INVESTORS ARE INVESTORS ARE

DIFFERENT DIFFERENT

E p

p

Investor A’s indifference

curvesInvestor B’s indifference curves

NOTE: since B is less risk-averse than A, B will choose a riskier portfolio from the CAL.

A’s choice

B’s choice

CAL

59

PORTFOLIO PORTFOLIO ANALYSISANALYSIS

1. What is a portfolio ?

2. Calculating two parameters of paramount importance to risk-averse investors:

(a) Expected rate of return of a portfolio: E(RP ).

(b) Standard deviation of the

rate of return of a portfolio: P.

60

PORTFOLIO PORTFOLIO ANALYSIS (CONT.)ANALYSIS (CONT.)

Suppose that there are N securities traded in the market.

A portfolio is an asset allocation scheme for distributing your capital among the available securities traded in the market.

In order to define a portfolio, you need to have :

1. A list of the securities that you want to include in the portfolio.

An asset allocation scheme defined by a set of portfolio weights: x1, x2, x3, …….,xN.

61

PROPERTIES FOR PROPERTIES FOR PORTFOLIOPORTFOLIO

WEIGHTSWEIGHTS

1. xi>0 for i=1,2,…N

(No short sales allowed.)

2. xi=1.0

(Portfolio wealth is fully allocated.)

62

THE S&P 500 THE S&P 500 UNDERLYING UNDERLYING PORTFOLIOPORTFOLIO

1. The list of securities is all current Fortune 500 companies.

xi=the market value of company i’s equity divided by the aggregate market value of all company’s equities.

xi=Ni P i/ Ni P i

Checking the properties is easy

(a) Insofar as companies have equity, the weights are positive,

(b) If we add up the portfolio weights, we get the sum of the equity values of all companies divided by aggregate market value which is clearly 1.0.

63

NUMERICAL NUMERICAL EXAMPLE OF WHAT EXAMPLE OF WHAT A PORTFOLIO DOESA PORTFOLIO DOES

SECURITY xi Ri

1 .10 .50

2 .50 .20

3 .00 .05

4 .30 .15

5 .00 .07

6 .10 .25

TOTAL 1.00

64

GRAPHICAL GRAPHICAL ILLUSTRATION OF ILLUSTRATION OF

WHAT A WHAT A PORTFOLIO DOESPORTFOLIO DOES

$100

$10$50$00$30$00

$10

$11$60

$00

$34.5$00

$12.5

$118

65

CALCULATING THE CALCULATING THE RATE OF RETURN OF A RATE OF RETURN OF A

PORTFOLIOPORTFOLIO The holding period rate of return of

the portfolio in the last example is clearly:

$118-$100/$100=18%

But it is also= x1 R1 + x2 R2+ x3 R3

+x4R4 + x5 R5 + x6 R6

The general formula emerges:

A portfolio’s rate of return is the portfolio-weighted average of the individual securities’ returns.

66

CALCULATING THE CALCULATING THE EXPECTED RATE OF EXPECTED RATE OF

RETURN OF A RETURN OF A PORTFOLIOPORTFOLIO

Calculating the expected rate of return of any portfolio, in general, is easy:

Just take the expected value of the random rate of return:

E(Rp)=

x1 E(R1)+ x2 E(R2)+…. +xNE(RN)

67

PORTFOLIO RISKPORTFOLIO RISK

X1

X1X2

XX X

where 12 is the correlation coefficient between the

return on security 1 and the return on security 2, 1

is the standard deviation of the rate of return of security 1 and 2 is the standard deviation of the

rate of return of security 2.

Portfolio variance is the sum of the boxes:

68

PORTFOLIO RISK: PORTFOLIO RISK: AN EXAMPLE AN EXAMPLE

.6 x

.6x.4xx20x30

.6x.4xx20x30 .4

x

where 12 =.30,

1 = 20%

2 = 30 %

X1 =.6 and X2 =.4

P=SQRT(144+144+(2x43.2))=19.35

.6 .4

.6

.4

69

EFFECT OF EFFECT OF DIVERSIFICATIONDIVERSIFICATION

E p

p

30

10

20

20

For a correlation coefficient of

12=0.3

70

THE DIVERSIFICATION THE DIVERSIFICATION EFFECT IN AN EFFECT IN AN

EXTREME CASEEXTREME CASE

STATE OFTHE

ECONOMY

PROBABILI-TY

OF STATE

STRATEGY#1:

INVESTALL IN

TAXICABSTOCK

STRATEGY#2:

INVESTALL IN

BUSSTOCK

STRATEGY #3:INVEST ½ IN EACH OF TAXICAB

STOCK AND BUS STOCK

BOOM 0.5 40% -20% 10%

RECESSION 0.5 -20% 40% 10%

71

THE

SHADED BOXES

CONTAIN VARIANCE TERMS;

THE REMAINDER

CONTAIN COVARIANCE

TERMS:

1

2

3

4

5

6

N

1 2 3 4 5 6 N

STOCK

PORTFOLIO VARIANCE: THE PORTFOLIO VARIANCE: THE GENERAL CASE : ADD UP ALL THE GENERAL CASE : ADD UP ALL THE

BOXES BOXES

x 1 x 2 x 3 x N

x 1

x 2

x 3

x 4

x 5

x N

x 6

Portfolio Weights

A typical variance term=

x i2

i2

A typical COvariance

term=

x i x j ij

ij

72

PORTFOLIO PORTFOLIO VARIANCE AS A VARIANCE AS A

FUNCTION OF THE FUNCTION OF THE NUMBER OF NUMBER OF

SECURITIES IN THE SECURITIES IN THE PORTFOLIO PORTFOLIO

PORTFOLIO PORTFOLIO VARIANCE AS A VARIANCE AS A

FUNCTION OF THE FUNCTION OF THE NUMBER OF NUMBER OF

SECURITIES IN THE SECURITIES IN THE PORTFOLIO PORTFOLIO

deviation

standard

Portfolio

UNIQUE RISK

MARKET RISK

Number of

securities

5 10

73

FOUNDATIONS OF FOUNDATIONS OF PORTFOLIO PORTFOLIO ANALYSISANALYSIS

The efficient frontier of risky assets:

Identify the efficient risk-expected return combinations from among the simply feasible ones,

Choosing the optimal risky asset portfolio from the efficient frontier:

Find the optimal portfolio that supports the highest CAL.

74

SINGLE-INDEX SINGLE-INDEX MODELSMODELS

The objective here is to define a return-generating model for security returns.

The simplest way to do this is in terms of a single factor which can be thought of as an aggregate stock market index: e.g. the S&P500 Index.

Ri=iiRMi

Here Ri is the random holding period rate of return of the security over a chosen holding period, RM is the random holding period rate of return of the Market over a chosen holding period.

75

SINGLE-INDEX SINGLE-INDEX MODELS (CONT.)MODELS (CONT.)

i is the actual rate of return that the security can earn on its own, i.e. independently of the Market,

i is the beta of the security’s rate of return, i.e. a measure of its comovement with the market as a percentage of the total volatility of the market,

i is a pure noise term, I.e. a random variable that is independent of the Market’s rate of return.

76

SINGLE-INDEX SINGLE-INDEX MODELS (CONT.)MODELS (CONT.)

KEY PROPERTIES OF i:

a. E(i)=0 (zero mean, I.e no systematic bias in any direction)

b.Cov(i, RM )=0 (noise is not a fundamental economic factor, it is not correlated with any such factor).

77

SINGLE FACTOR SINGLE FACTOR INDEX MODELS INDEX MODELS VS. THE CAPMVS. THE CAPM

The first note is that the CAPM in the form of the Security Market Line (SML) describes expected rates of return (not actual rates of return).

The Index model describes actual rates of return.

However, the two types of models are consistent with each other.

78

SINGLE FACTOR SINGLE FACTOR INDEX MODELS INDEX MODELS VS. THE CAPMVS. THE CAPM

By taking expected values of the single-factor index model one notes that:

E(Ri)=iiRM)i)

= iiRM)

by property(a) of the noise term. Then equating corresponding terms

in the SML one notes that the following equality must hold:

i =(1- i)RF

Thus the CAPM is a significantly stronger

statement than the single factor Index model.

79

PORTFOLIO CHOICES PORTFOLIO CHOICES OF DIFFERENT OF DIFFERENT

INVESTORSINVESTORS The optimal final portfolio and

the Separation Property:

Mix the optimal risky portfolio with cash equivalents to get the final portfolio for the given investor.

80

SINGLE-PERIOD SINGLE-PERIOD CAPM ASSUMPTIONSCAPM ASSUMPTIONS

1. There is a risk-free rate, RF at which investors can borrow and lend as much as they wish without affecting that rate (e.g. T-Bills).

2. All investors make their investment decisions solely on the basis of the mean and the variance of their portfolios. Further, in making their portfolio decisions, they maximize the expected utility of their final wealth positions.

3. All investors have homogenous expectations regarding the relevant parameters underlying their portfolio decisions.

81

CAPM EQUILIBRIUM CAPM EQUILIBRIUM CONDITIONSCONDITIONS

1. The market portfolio will be on the efficient frontier and will be the optimal risky asset portfolio to be combined with riskless borrowing or lending in building their final, personal, optimal portfolios.

That is, all investors hold the same risky portfolio(M), adding T-bills to their portfolios to obtain desired risk levels.

2. The CML is therefore the best obtainable CAL.

3. The risk premium on individual assets is proportional to the risk premium on the market portfolio and to the of the security. measures the extent to which the stock returns respond to the market returns.

82

DERIVATION OF DERIVATION OF THE CAPMTHE CAPM

The Reward-to-Variability Ratio of the CML :

[E(RM) - RF] / M

The risk premium for security I is in proportion to its contribution of the risky asset portfolio in which it is held. This is the Market portfolio according to the CAPM.

Setting the two values equal to each other produces the SML:

E(Ri) = RF + i ( E(RM ) -RF)

83

The Number of Estimates Needed The Number of Estimates Needed for Standard Portfolio Analysis for Standard Portfolio Analysis

Vs. the Single Factor Index Vs. the Single Factor Index ModelModel

STANDARD ANALYSIS (50 Stocks):

N = 50 Estimates of expected returns

N = 50 Estimates of variances

(N2 - N)/2 = 1,225 Estimates of covariances

1,325 Estimates in Total

84

The Number of Estimates Needed The Number of Estimates Needed for Standard Portfolio Analysis for Standard Portfolio Analysis

Vs. the Single Factor Index Vs. the Single Factor Index ModelModel

SINGLE-INDEX ANALYSIS (50 Stocks):

N = 50 Estimates of expected excess returns

N = 50 Estimates of betas

N = 50 Estimates of firm-specific variances

1 Estimate of the variance of the common macro-economic factor

151 Estimates (3n + 1) in Total

85

THE CAPM VS. THE THE CAPM VS. THE APTAPT

1. The CAPM assumes an unobservable “market” portfolio,

2. The APT is based on the assumption of no arbitrage profits in well-diversified portfolios,

3. However, the APT admits the possibility of arbitrage profits on a “few” individual securities,

4. The APT provides no guidance for identification of the various market factors and appropriate risk premiums for these factors

86

PERFORMANCE PERFORMANCE ATTRIBUTION ATTRIBUTION PROCEDURESPROCEDURES

First, decide on the proportions of equity, fixed income, and money market funds in the portfolio.

Secondly, decide on the proportions of particular industries (sectors) within each market.

Third, decide on the particular securities in an industry to be included in the portfolio.

Use a benchmark or “bogey” portfolio as the standard of a passive strategy.

87

PERFORMANCE PERFORMANCE ATTRIBUTION ATTRIBUTION

PROCEDURES (CONT.)PROCEDURES (CONT.) For allocation comparisons, compare

the bogey portfolio returns to the returns on your portfolio which has different allocations.

Subtract the allocation differential returns from the total return differential to get the security return difference.

88

PERFORMANCE PERFORMANCE ATTRIBUTION ATTRIBUTION PROCEDURES (CONT.)PROCEDURES (CONT.)

Compare your equity performance to the S&P 500 Index.

Compare your fixed income performance to the Shearson-Lehman Index .

Compare sector weights in your portfolio to the sector weights in the S&P 500 Index.

89

RISK-ADJUSTED RISK-ADJUSTED MEASURES OF MEASURES OF

PORTFOLIO PORTFOLIO

PERFORMANCEPERFORMANCE

SHARPE MEASURE

=[E(RP) - RF] / P

TREYNOR MEASURE

=[E(RP) - RF] / P

JENSEN MEASURE

= P

=E(RP) -[RF + i ( E(RM ) -RF)] APPRAISAL RATIO

=P/P)

90

INVESTOR INVESTOR ClASSIFICATIONSClASSIFICATIONS

INDIVIDUAL INVESTORS PERSONAL TRUSTS MUTUAL FUNDS PENSION FUNDS ENDOWMENT FUNDS LIFE INSURANCE

COMPANIES NONLIFE INSURANCE

COMPANIES BANKS

91

CONSTRAINTS ON CONSTRAINTS ON INVESTINGINVESTING

LIQUIDITY

INVESTMENT HORIZON

REGULATIONS

TAX CONSIDERATIONS

UNIQUE NEEDS

1 INTRODUCTION TO DERIVATIVES MARKETS (INVESTMENTS BACKGROUND) SPRING 2006.

Documents

Transcript of 1 INTRODUCTION TO DERIVATIVES MARKETS (INVESTMENTS BACKGROUND) SPRING 2006.