1 introduction to derivatives markets (investments background) spring 2006
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1
INTRODUCTION TO INTRODUCTION TO DERIVATIVES DERIVATIVES
MARKETS MARKETS (INVESTMENTS (INVESTMENTS BACKGROUND) BACKGROUND)
SPRING 2006 SPRING 2006
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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)
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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
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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.
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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
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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
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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
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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
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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
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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)
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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
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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
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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
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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
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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%
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CORPORATE BOND CORPORATE BOND QUOTATIONSQUOTATIONS
See text BKM Figure 2.7, page 42
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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.
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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
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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.
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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
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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.
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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.
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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
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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?
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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 ?
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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
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IN-CLASS SOLUTION IN-CLASS SOLUTION (CONT.)(CONT.)
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%
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IN-CLASS SOLUTION IN-CLASS SOLUTION (CONT.)(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:
$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.
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IN-CLASS SOLUTION IN-CLASS SOLUTION (CONT.)(CONT.)
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%
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IN-CLASS SOLUTION IN-CLASS SOLUTION (CONT.)(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 (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
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IN-CLASS SOLUTION IN-CLASS SOLUTION (CONT.)(CONT.)
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%.
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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
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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.
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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))
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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
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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
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THE PROBLEM THE PROBLEM SOLVED BY SOLVED BY
INVESTMENTSINVESTMENTS
Re-allocating consumption claims (certain and uncertain) across time and under conditions of uncertainty
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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)
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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
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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
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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
Consumption, period 1
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.
O
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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.
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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.
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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
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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.
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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.
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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.
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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).
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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.
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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.
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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.
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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.
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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.
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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)
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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
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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
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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
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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.
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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.
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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.
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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)
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INVESTOR INVESTOR ClASSIFICATIONSClASSIFICATIONS
INDIVIDUAL INVESTORS PERSONAL TRUSTS MUTUAL FUNDS PENSION FUNDS ENDOWMENT FUNDS LIFE INSURANCE
COMPANIES NONLIFE INSURANCE
COMPANIES BANKS
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CONSTRAINTS ON CONSTRAINTS ON INVESTINGINVESTING
LIQUIDITY
INVESTMENT HORIZON
REGULATIONS
TAX CONSIDERATIONS
UNIQUE NEEDS