INTRODUCTION TO DERIVATIVES MARKETS (INVESTMENTS BACKGROUND) SPRING 2006

1
INTRODUCTION TO DERIVATIVES MARKETS (INVESTMENTS
BACKGROUND) SPRING 2006
2
REAL VS. FINANCIAL ASSETS

• A. REAL ASSETS
• Plant and EquipmentPhysical Capital
• Growth Opportunities e.g. RD, Patents, New
  Ventures
• Human CapitalExpertise, Labor Services
• Contribute Directly to the Productive Capacity of
  the Economy (i.e. to GNP Growth)

3
REAL VS. FINANCIAL ASSETS

• 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 FINANCIAL 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 BETWEEN CAPITAL MARKETS AND FIRMS
OPERATIONS
2 .CASH INVESTED IN FIRM
1. CASH RAISED FROM INVESTORS
FIRMS OPERATIONS
CAPITAL MARKETS
FINANCIAL MANAGER
5.CASH RE- INVESTED
3. CASH GENERATED BY OPERATIONS
4. CASH RETURNED TO INVESTORS

6
MONEY MARKET INSTRUMENTS

• U.S. TREASURY BILLS
• FEDERAL FUNDS
• EURODOLLARS
• REPOS AND REVERSES
• BROKER CALLS
• THE LIBOR MARKET
• COMMERCIAL PAPER
• BANKERS ACCEPTANCES

7
TREASURY BILL PRICING CONVENTIONS

• 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 TERMINOLOGY

• PCURRENT PRICE
• FFACE VALUE
• NNUMBER OF DAYS TO MATURITY
• BDYBANK DISCOUNT YIELD
• BEYBOND EQUIVALENT YIELD

9
PRICING U.S. TREASURY 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(BDYFN)/360
• STEP 3. THE CURRENT PRICE
• PF-D
• STEP 4. CALCULATE THE HOLDING PERIOD YIELD,
• HPYD/P
• STEP 5 CALCULATE THE BOND EQUIVALENT YIELD,
• BEYHPY365/N

10
TREASURY BILL PRICING FORMULAE

• CURRENT PRICE,
• PF(1-BDYN/360)
• BOND EQUIVALENT YIELD (BEY)
• BEY365BDY/(360-BDYN)

11

U.S. TREASURY BILLS PRICE QUOTE (SOURCE
www.bloomberg.com)
12
MONEY RATES (SOURCE WSJ 01/06/03)
See BKM Text p. 32 Figure 2.1
13
MEDIUM TO LONG-TERM FIXED INCOME INSTRUMENTS

• U.S. TREASURY NOTES AND BONDS
• FEDERAL AGENCY DEBT
• MUNICIPAL BONDS (MUNIS)
• CORPORATE BONDS
• MORTGAGES
• MORTGAGE-BACKED SECURITIES

14
TREASURY BOND PRICING CONVENTIONS

• TREASURY BONDS ARE QUOTED IN DOLLARS PLUS 32NDS
  PER FACE VALUE. THE LATTER ARE CALLED BOND POINTS
• E.G. A BOND POINT (1/32) TRANSLATES INTO
  1,000/3231.25 FOR EACH 1,000 OF FACE VALUE
• BOND YIELD TO MATURITY (YTM) IS THE BONDS IRR
  BASED ON A 365 DAY YEAR

15
US TREASURY BOND PRICE QUOTATIONS

• (SOURCEWSJ(01/06/03)
• See Text BKM Figure 2.3 Page 37)
• 4.750 Nov08n 10725 10726 1 3.27

16
U.S. T-BOND CALCULATIONS

• 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 10725
• NOTE this means 107 25/32 per each 100 of face
  value.
• Ask Price 10726 or107 26/32 per 100 of face
  value
• 1 ask price up by 1/32 from previous days ask
  price.
• Ask yield the yield to maturity (IRR) of the
  bond based on the asked price3.27

17
CORPORATE BOND QUOTATIONS
See text BKM Figure 2.7, page 42
18
READING CORPORATE BOND 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 MARKET QUOTATIONS(SOURCE WSJ
(09/08/97)
See text BKM Figure 2.9, page 46
20
READING STOCK MARKET QUOTES

• HIGHLIGHTED FIRM (GE CORP.)
• 52 week high and low stock price per share41.24
  and 21.40 respectively.
• Dollar Dividends .76 /share annually.
• Dividend Yield annual dividend/current price3.0
  .76/25.40
• PE price earnings ratio16.
• Volume 100s of shares traded that day 148191
• High and low for that trading day see
  www.nyse.com
• Closing Price25.40 per share.
• Net change -.08 per share from previous days
  close.

21
STOCK AND BOND MARKET INDICES

• STOCK INDICES
• DJIA
• SP 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 INDICES 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.
• SP 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 STOCK MARKET INDICES

• PRICE-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 THE 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
CONTINUED

• CURRENT YEAR
• Stock Price Shares
• A 22 20,000,000 B
  55 20,000,000
• C 66 30,000,000
• 1. 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-CLASS PROBLEM

• 1. A price-weighted index simply adds up the
  prices of the individual stocks underlying the
  Indexs construction and divides by the number
  of such stocks.
• Therefore, the initial value of the Index is
• 405060/350.
• If we did the same in the current year we would
  obtain
• 225566/347.67
• which represents a -4.67 decline
• in the index. But did it decline ?

27
IN-CLASS SOLUTION (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
• 205060/x405060/3
• x2.6

28
IN-CLASS SOLUTION (CONT.)

• In computing the new value of the Index we use
  the adjusted divisor 2.6 instead of 3.0
• Index in current year
• 225566/2.655
• The percentage change in the Index (representing
  the true increase in the market) is
• 55-50/5010

29
IN-CLASS SOLUTION (CONT.)

• 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
• 4010mm5020mm60303200mm
• 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
IN-CLASS SOLUTION (CONT.)

• Value of the Index in the current year
• 2220mm5520mm66303520mm
• Note the automatic adjustment for the stock
  split. The value of the Index in the base year is
  3520/32110
• Clearly the Index increased by
• 110-100/10010

31
IN-CLASS SOLUTION (CONT.)

• 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
  (60402400), 48 shares of stock B
  (48502400), and 40 shares of stock C
  (40602400).
• The adjustment for stock splits
• occurs naturally because in the current year you
  own 120 shares

32
IN-CLASS SOLUTION (CONT.)

• The value of the Index in the base-year is just
  the value of the dollars invested in it
• 2400240024007200
• Normalize to 100 Index points by dividing by 72 .
• The value of the Index in the current year is
  12022485540667920
• 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 EQUITY POSITIONS

• INITIAL MARGINS
• SET BY THE FEDERAL RESERVE
• CURRENTLY EQUALS 50
• INITIAL MARGININVESTORS 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 MARGIN6,000/10,00060.

34
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 CUSTOMERS
  EQUITY, HENCE THE ACTUAL MARGIN1,000/5,00020
• 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 OF SHORT SALES

• The way its 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 PRICE ON LONG POSITIONS

• 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 PRICE ON SHORT POSITIONS

• Let N the number of shares sold short,
• P0the price per share at the time of the short
  sale,
• P1the price per share when the short sale is
  covered, I.e. the shares are bought back.
• IMthe initial margin
• M the maintenance margin level
• Then Pm(Nx P0IM)/(Nx(M1))

38
THE MARGIN CALL PRICE ON SHORT POSITIONS 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
• (10,0005000)/(100x(0.31))
• 115.38

39
DEFINING INVESTMENTS A GENERAL 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 SOLVED BY INVESTMENTS

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

41
ONE MAIN REASON FOR 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 CHOICES
Consumption later
2.5
2.2
Invest in tennis facility
1.4
1.1
Invest in the bank
Consumption now (millions)
Villa in Spain
2.0
2.3
2.5
43
BORROWING AND LENDING ENLARGE CHOICES
Dollars, period 1
H
Interest rate lines shows cash flows from
borrowing or lending
F
O
Dollars, period 0
B
D
By borrowing OF, an individual can consume an
extra BD today by lending OB, he can consume an
extra FH tomorrow.
44
THE EFFECT OF INVESTMENT IN REAL ASSETS
Consumption, period 1
Investment opportunities line shows cash flows
from investing in real assets
Consumption, period 1
Notice the diminishing return on additional units
of investment
45
HOW INVESTMENT IN REAL ASSETS IMPROVES WELFARE
Consumption, period 1
The miser can spend more today and the next period
M
H
L
G
... and so can the prodigal
O
J
D
K
Consumption, period 0
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.
46
KEY QUESTIONS ADDRESSED BY INVESTMENT ANALYSIS

• 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 INVESTMENT PROGRAMS

• 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
OBJECTIVES OF INVESTMENT ANALYSIS

• 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
  CANT BE DONE SO AS TO SATISFY THEIR SUBJECTIVE
  RISK/RETURN PREFERENCES.

49
TYPES OF INVESTMENT STRATEGIES

• 1. MARKET TIMING.
• 2. STATIC PORTFOLIO DIVERSIFICATION.
• 3. DYNAMIC PORTFOLIO DIVERSIFICATION.
• 4. ASSET ALLOCATION.

50
BASIC ASSET ALLOCATION STRATEGIES

• 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 ALLOCATION LINES
E
p
E
1
R
F
p
1
52
THE EQUATION OF THE CAL

• E(RP ) RF(E1-RF ) /s1xsp
• WHERE E(RP ) IS THE EXPECTED RATE OF RETURN OF
  THE PORTFOLIO.
• AND sp IS THE STANDARD DEVIATION OF THE RATE OF
  RETURN OF THE PORTFOLIO.

53
CAPITAL ALLOCATION LINES(REWARD TO RISK RATIO)

• THE SLOPE OF THE CAPITAL
• ALLOCATION LINE IS THE (EXCESS) REWARD TO
  RISK RATIO (E1-RF ) /s1
• 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,
• s1 IS A MEASURE OF RISK

54
CAPITAL ALLOCATION LINES
E
p
CAL
2
E
CAL
1
2
E
1
R
F
s
s
p
1
55
MORE EFFICIENT CALS

• 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 PORTFOLIO MANAGER

• OFFER MORE AND MORE EFFICIENT CAPITAL ALLOCATION
  LINES TO INVESTORS RATHER THAN
• ATTEMPTING TO SATISFY THEIR SUBJECTIVE
  RISK/RETURN PREFERENCES DIRECTLY.

57
DIFFERENT INVESTORS HAVE DIFFERENT INDIFFERENCE
CURVES
E
p
Investor As indifference curves
Investor Bs indifference curves
NOTE B IS LESS RISK-AVERSE THAN A.
?
p
58
PORTFOLIO CHOICES FOR DIFFERENT INVESTORS ARE
DIFFERENT
E
p
Investor As indifference curves
Investor Bs indifference curves
Bs choice
CAL
As choice
NOTE since B is less risk-averse than A, B will
choose a riskier portfolio from the CAL.
?
p
59
PORTFOLIO ANALYSIS

• 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 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 PORTFOLIO WEIGHTS

• 1. xigt0 for i1,2,N
• (No short sales allowed.)
• 2. ? xi1.0
• (Portfolio wealth is fully allocated.)

62
THE SP 500 UNDERLYING PORTFOLIO

• 1. The list of securities is all current Fortune
  500 companies.
• xithe market value of company is equity divided
  by the aggregate market value of all companys
  equities.
• xiNi 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 EXAMPLE OF WHAT A PORTFOLIO DOES
64
GRAPHICAL ILLUSTRATION OF WHAT A PORTFOLIO DOES
11
10
60
50
00
00
100
118
30
34.5
00
00
10
12.5
65
CALCULATING THE RATE OF RETURN OF A PORTFOLIO

• The holding period rate of return of the
  portfolio in the last example is clearly
• 118-100/10018
• But it is also x1 R1 x2 R2 x3 R3 x4R4
  x5 R5 x6 R6
• The general formula emerges
• A portfolios rate of return is the
  portfolio-weighted average of the individual
  securities returns.

66
CALCULATING THE EXPECTED RATE OF RETURN OF A
PORTFOLIO

• 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 RISK
Portfolio variance is the sum of the boxes
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.
68
PORTFOLIO RISK AN EXAMPLE
.4
.6
.6
.4
where ?12 .30, ?1 20 ?2 30 X1 .6 and
X2 .4 ?PSQRT(144144(2x43.2))19.35
69
EFFECT OF DIVERSIFICATION
For a correlation coefficient of ?120.3
E
p
20
10
p
30
20
70
THE DIVERSIFICATION EFFECT IN AN EXTREME CASE
71
PORTFOLIO VARIANCE THE GENERAL CASE ADD UP ALL
THE BOXES
Portfolio Weights
x 3
x 2
x N
x 1
x 1
THE SHADED BOXES CONTAIN VARIANCE TERMS
THE REMAINDER CONTAIN COVARIANCE TERMS
1
x 2
2
x 3
3
x 4
4
x 5
A typical variance term x i2 ?i2
5
x 6
6
A typical COvariance term x i x j ?i?j ?ij
x N
N
1
2
3
4
5
6
N
STOCK
72
PORTFOLIO VARIANCE AS A FUNCTION OF THE NUMBER OF
SECURITIES IN THE PORTFOLIO
Portfolio
standard
deviation
UNIQUE RISK
MARKET RISK
Number of
5
10
securities
73
FOUNDATIONS OF PORTFOLIO ANALYSIS

• 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 MODELS

• 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 SP500
  Index.
• Riaibi RMei
• 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 MODELS (CONT.)

• ai is the actual rate of return that the security
  can earn on its own, i.e. independently of the
  Market,
• bi is the beta of the securitys rate of return,
  i.e. a measure of its comovement with the market
  as a percentage of the total volatility of the
  market,
• ei is a pure noise term, I.e. a random variable
  that is independent of the Markets rate of
  return.

76
SINGLE-INDEX MODELS (CONT.)

• KEY PROPERTIES OF ei
• a. E(ei)0 (zero mean, I.e no systematic bias in
  any direction)
• b.Cov(ei, RM )0 (noise is not a fundamental
  economic factor, it is not correlated with any
  such factor).

77
SINGLE FACTOR INDEX MODELS VS. 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 INDEX MODELS VS. THE CAPM

• By taking expected values of the single-factor
  index model one notes that
• E(Ri)aibi E(RM)E(ei)
• aibi E(RM)
• by property(a) of the noise term.
• Then equating corresponding terms in the SML one
  notes that the following equality must hold
• ai (1- bi)RF
• Thus the CAPM is a significantly stronger
• statement than the single factor Index model.

79
PORTFOLIO CHOICES OF DIFFERENT INVESTORS

• 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 CAPM 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 CONDITIONS

• 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 b of the security. b
  measures the extent to which the stock returns
  respond to the market returns.

82
DERIVATION OF THE CAPM

• The Reward-to-Variability Ratio of the CML
• E(RM) - RF / sM
• 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 bi ( E(RM ) -RF)

83
The Number of Estimates Needed for Standard
Portfolio Analysis Vs. the Single Factor Index
Model

• 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 for Standard
Portfolio Analysis Vs. the Single Factor Index
Model

• 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 APT

• 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 ATTRIBUTION PROCEDURES

• 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 ATTRIBUTION 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 ATTRIBUTION PROCEDURES (CONT.)

• Compare your equity performance to the SP 500
  Index.
• Compare your fixed income performance to the
  Shearson-Lehman Index .
• Compare sector weights in your portfolio to the
  sector weights in the SP 500 Index.

89
RISK-ADJUSTED MEASURES OF PORTFOLIO PERFORMANCE

• SHARPE MEASURE
• E(RP) - RF / sP
• TREYNOR MEASURE
• E(RP) - RF / bP
• JENSEN MEASURE
• aP
• E(RP) -RF bi ( E(RM ) -RF)
• APPRAISAL RATIO
• aP/s(eP)

90
INVESTOR ClASSIFICATIONS

• INDIVIDUAL INVESTORS
• PERSONAL TRUSTS
• MUTUAL FUNDS
• PENSION FUNDS
• ENDOWMENT FUNDS
• LIFE INSURANCE COMPANIES
• NONLIFE INSURANCE COMPANIES
• BANKS

91
CONSTRAINTS ON INVESTING

• LIQUIDITY
• INVESTMENT HORIZON
• REGULATIONS
• TAX CONSIDERATIONS
• UNIQUE NEEDS