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chapter 1 chapter 2 chapter 3 chapter 5 chapter 6 chapter 7 chapter 8 chapter 9
ECON 4751
Financial Economics
Instructor: Zoe Xie
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Part I: Introduction
Chapter 1. The investment environment
Chapter 2. Asset classes and financial instruments
Chapter 3. How securities are traded
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Chapter 1. The Investment Environment
Learning Objectives
Understand the differences between real assets and financial assets
Know how to classify assets into three asset classes: debt, equityand derivative
Understand the purpose of financial market and know who the majorplayers are
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Investment
The Saving decision is between consumption and accumulation of(financial) assets (Econ 3101)
The Investment decision is how to allocate savings into differentassets
choice between real assets (e.g. buy machine, build factory) andfinancial assets (e.g. buy Google stock)this course focuses on investment in financial assets
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Real Assets vs. Financial Assets
Real assets
determine the productive capacity and net income of the economy
examples: land, machinery, buildings, knowledge (human capital),technology, commodities
Financial assets
are claims on real assets or income
define the allocation of income or wealth among investors
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Financial Assets1 debt securities (fixed income)
short-term debtlong-term debt
2 equity securities (common stock)3 derivative securities
*note: a security is a tradable financial asset. a “debt security” is a tradable debt, i.e. it can be
bought and sold in the open market. sometimes I will use “debt”, “equity”, and “derivative” for
convenience, just know we’re talking about a security.
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Debt Securities
also called ”fixed income securities” because they promise either afixed stream of income or a stream of income determined by aspecified formula
income is paid unless the borrower defaults
examples: corporate bond, floating-rate bond (TIPS, I-Bonds),money market
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Equity Securities
represents part ownership in a corporation (shares of stock)
firms may choose to issue dividend to shareholders or re-investprofits in the company
dividend amount is not fixed, determined by management
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Derivative Securities
value derived from other assets (real and financial)
provide payoffs that are determined by the prices of other assets
can be used to hedge risk and distribute risk to those willing toaccept it
examples: options, futures, swaps, index
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Purposes of Financial Markets
information: capital flows to companies with best prospects
consumption timing: store wealth and shift consumption to thefuture
allocation of risk: shift risk (and return) from more to lessrisk-averse investor
separation of ownership and management: allow owners to sellshares to other investors
stabilityagency problem
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The Investment Process
Portfolio is a collection of assets
Choice of a portfolio can be broken down into
asset allocation: choice among broad asset classessecurity selection: choice of which securities to hold within asset classsecurity analysis to value securities and determine investmentattractiveness
Different portfolio choice strategies
top-down portfolio choice starts with asset allocation, then proceedsto security selection example: I want 50% in stocks, 30% in bonds,20% in cash.bottom-up strategy results from selection of individual securities withless concern for overall asset allocation This stock looks good, let‘sbuy 200 shares.
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chapter 1 chapter 2 chapter 3 chapter 5 chapter 6 chapter 7 chapter 8 chapter 9
The Investment Process
Portfolio is a collection of assets
Choice of a portfolio can be broken down into
asset allocation: choice among broad asset classessecurity selection: choice of which securities to hold within asset classsecurity analysis to value securities and determine investmentattractiveness
Different portfolio choice strategies
top-down portfolio choice starts with asset allocation, then proceedsto security selection example: I want 50% in stocks, 30% in bonds,20% in cash.bottom-up strategy results from selection of individual securities withless concern for overall asset allocation This stock looks good, let‘sbuy 200 shares.
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Does it matter how I invest?
Active vs. Passive Management: spend time analyzing securities andlooking for mispriced assets or just own a highly diversified portfolio?
under competitive markets, in equilibrium, there will be a certainamount of active investing and analysis but bargains are not easy tofind
so does it still matter how I invest?
yes, because Risk-Return tradeoff: higher returns are generallyaccompanied by higher risk. Investment strategy depends on risktolerance and return target
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Does it matter how I invest?
Active vs. Passive Management: spend time analyzing securities andlooking for mispriced assets or just own a highly diversified portfolio?
under competitive markets, in equilibrium, there will be a certainamount of active investing and analysis but bargains are not easy tofind
so does it still matter how I invest?
yes, because Risk-Return tradeoff: higher returns are generallyaccompanied by higher risk. Investment strategy depends on risktolerance and return target
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Players in Financial Market
Firms
sell equity and issue debt to raise capital (i.e. borrow money forprojects)pay investors dividends (for equity) and interest (for debt)
Households
save and invest in securities issued by firms and government
Government
issue debt to borrow
Financial intermediaries: banks, investment companies, insurancecompanies, mutual funds, hedge funds, . . .
facilitate transactions among players and make profit doing it
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Recent Trends
Globalization
more investment and funding choices
Securitization
pool a group of loans into tradable securities, e.g. mortgage-backedsecurities
Financial Engineering
e.g. tranching (i.e. slicing) debt pool into difference risk-returnsecurities to cater to different investors (read up on housing financeand crisis if interested)
information and computer networks
information available to public sooner, and so even less “easy money”
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Chapter 2. Asset Classes and Financial Instruments
Learning Objectives
Know the main instruments in money market and capital market
For debt securities
calculate (annualized) discount and yield, issue price and par value ofa T-billcalculate coupon payment, current yield and accrued interest of abondcalculate equivalent taxable yield of a tax-exempt bondknow some of the factors influencing bond value
For equity securities
know the differences between common stock and preferred stockunderstand how a stock split works
For derivative securities
understand how a call option and a put option workcalculate profit from an optionunderstand how market index is computedcalculate price-weighted index and value-weighted index
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Asset ClassesTwo segments of financial market: money markets and capital markets
Money market instrumentsshort-term (1yr or less) debt securities: liquid, less risky
Capital market instrumentslong-term debt securities: less liquid, more riskyequity securitiesderivative securities
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Debt Security Basics
Par Value: the face value of a debt security, or its worth at maturity
Maturity: the length of time from issue until the debt securityreaches its par value and is repaid in full
Coupon Rate: percentage of par value paid annually in interest
Zero-Coupon Bond: makes no interest payment
Discount: percentage by which debt security is priced under par
Yield: annualized rate of return
For the secondary market
bid & ask prices: amount at which dealer is willing to buy & sell abondbid-ask spread: difference between ask and bid price
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Money Markets
money market is a sub-sector of the fixed-income market, consists ofvery short-term (under 1 year) debt securities that are usually highlymarketable.
often have large denominations, money market mutual funds allowindividuals to access the money market
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Money Market Instruments
U.S. Treasury bills (T-bills)
short-term (maturities of 4, 13, 26, or 52 weeks) debt of government
Commercial Paper
short-term (maturity up to 270 days) debt of a company, issued in$100K multiples
Certificates of Deposit
time deposit with a bank, issued in $100K multiples
Banker’s Acceptance
promise by bank to pay a debt on behalf of a customer of a bank
Repos and Reverses
government securities sold on short-term (1day+) basis withagreement to buy back at a later date
Fed Funds
very short-term inter-bank loans
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Money Market Instruments: T-bills
a way for the U.S. government (assumed risk-free) to borrow forshort-terms (maturities of 4, 13, 26, or 52 weeks)
zero-coupon payment; at maturity, investor receives a payment equalto the face value
investors buy the bills at a discount from the face value
issued in multiples of $100
price determined in public auction, income is exempt from tax atstate and local levels
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annualizing discounts and yields
annualized yield
Par Value − Price
Price× 365
term of maturity in days× 100%
annualized discount
Par Value − Price
Par Value× 365
term of maturity in days× 100%
note: 360-day year approximation (bank-discount method) can beused for convenience and is still used in some settings due toconvention
issue price
par value
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Treasury Bills: Exercise
Example 1: A 4-week T-bill has issue price $98.5 and par value$100. Calculate its annualized yield and discount
Example 2: A 1-year T-bill has a par value of $100, and a yield of3.25%. Calculate its price at issue
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Capital Market Instruments
Long-term debt securities (bonds)
U.S. Treasury notes and bondsInflation-Protected Treasury bonds (TIPS)Municipal BondsCorporate BondsFederal Agency DebtInternational BondsMortgages & MBS
Equity securities (stock)
Derivative securities
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U.S. Treasury Notes and Bonds
issued by U.S. government
treasury notes maturities 1-10 years
treasury bonds maturities 10-30 years
coupon payments twice a year
each coupon payment
Par Value× Coupon Rate
Number Payments per Year
current yieldTotal Coupon Payment
Sale Price× 100%
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accrued interest
notes and bonds accrue interest in between coupon payments. this ismoney owed to investors that hasn’t been paid yet
Par Value × Coupon Rate × days since last coupon payment
365
any accrued interest is added to the purchase price. the investorreceives the full coupon payment at the next scheduled payout
invoice price: the quoted price plus any accrued interest
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Basic Bond Math: Exercise
Example 1: A 10-year corporate bond has a Par Value of $10,000,coupon rate 6.25% and coupon payments are made twice a year.Calculate each coupon payment and current yield if sold at par.
Example 2: A corporate bond has a Par Value of $5,000, a couponrate of 6%, and an issue date of January 1, 2014. Coupon paymentsare made every six months.
Calculate the amount of accrued interest on July 15, 2014.Suppose the quoted price of the bond on July 15 is $5,015. Whatwould be the invoice price?
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TIPS (Treasury Inflation-Protected Securities)
the principal of a TIPS bond is adjusted by the Consumer PriceIndex (CPI)
the interest rate is fixed but the interest payment varies from oneperiod to the next
at maturity, the buyer receives the higher of the original principaland the adjusted principal
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Municipal Bonds
issued by state and local governments
interest income is tax-exempt from federal tax and state and localtax in the issuing state
because interest income is tax-exempt, investors expect lower yields
to compare a fully-taxable bond to a tax-exempt bond
equivalent taxable bond yield
r =rm
1 − t
equivalent municipal bond yield
rm = r(1 − t)
where t is combined tax rate applied to a taxable bond; rm ismunicipal bond yield; r is an equivalent taxable bond yield
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Tax-Exempt Bonds: Exercise
Example: An investor is subject to a 20% federal income tax and10% state income tax. If a corporate bond has an interest rate of7.5%, what is the equivalent interest rate for a tax-free municipalbond?
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Corporate Bonds
issued by private firms
pay interest every 6 months
risk of default is generally higher than government securities
secured bonds are backed by specific collateral
unsecured bonds are not backed by collateral
options in corporate bonds
callable bonds can be repurchased by the firm from the holder at aspecified priceconvertible bonds can be exchanged by the holder for shares of stock
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Factors influencing value of bonds
coupon rate and par value
maturity
risk of default
liquidity: how easy to switch to another investment or consumption
return on other investments
interest rates, inflation, taxes
expectations
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Equity Securities
Common stock: ownership
residual claim: last in claim orderlimited liability: maximum downside is money investedhave voting privilegesdividend usually paid quarterly, not fixed
Preferred stock: perpetuity
no voting privilegesclaim order ranks over common stock and below bondholdersdividends are fixed and cumulative (i.e. unpaid dividends are owed)
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Stock Split
a stock split is when existing stock shares are increased in numberwhile adjusting the price downward so the total value of theoutstanding shares remains the same
example
holding 100 shares of Xcel Energy stock worth $100 eachwith a 2-for-1 split, the stockholder has 200 shares worth $50 each
why do companies do stock split?
usually happens when stock price grows very high to increase liquidityit is often claimed that stock splits lead to higher stock prices; stillan open research question
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Derivative Securities: Optionscall option: gives the holder right to buy the underlying asset at aspecified price up to a specified quantity until a given expiration date
put option: gives the holder right to sell . . .the specified price is called exercise price or strike pricethe price paid for the option is called premiuman option is exercised when the holder uses the right
in the money: can make a profit right away (disregarding premium)call: strike price < current market priceput: strike price > current market price
out of the money: makes a loss right away (disregarding premium)
profit (loss) from exercisingcall option
market price of underlying asset − strike price − premium
put option
strike price − market price of underlying asset − premium
profit (loss) from not exercising by expiration date: −premium
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An ExampleCurrent stock price $543 per share
which of these are in/out of the money
any patterns in premium?
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More on Options
a put can be bought as insurance for stockholders, i.e. if price fallscan still sell the stock at strike price to minimize downside risk
American options can be exercised at any point before expiration,while European options can only be exercised on the expiration date
price of an option is quoted for one share of a stock, but options aresold in lots of 100
can resell options before expiration
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Derivative Securities: Market Index
used to measure performance or change in price of a group of assetsin a market
e.g. Dow Jones Industrials (DJIA), S&P 500
price-weighted index: measures the price of a portfolio initiallycomprised of one share of each stock
tracks simple average of stock pricesadjusts for stock splits
value-weighted index: measures price of portfolio initially weightedby total market value of the companies
tracks how total value of companies change
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An Example
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Other Derivative Securities
Forward contract
an agreement for the sale of an asset at some specified time in thefuture at a set price, determined in the presentthis is in contrast to a spot contract which is a sale in the present
Futures contract
standardized contract where parties agree to delivery of an asset at acertain time and pricefutures are traded on exchanges such as Chicago MercantileExchangeone contract calls for delivery of a standardized amount of the asset,e.g. 500 bushels of corn
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Chapter 3. How Securities are Traded
Learning Objectives
Understand the difference between primary and secondary markets
Security Issuance
Understand how a firm issues stocks and bondsUnderstand how Initial Public Offering (IPO) works
Security Trading
Know the different types of trading markets and how they workKnow how market order, limit order, and stop order workknow how bond, equity, derivatives are commonly tradedKnow some of the trading costs
Understand how buying on margin works and calculate margins andborrowing allowance
Understand how short sale works and calculate profit/loss from ashort sale
Know some of the regulatory issues in the U.S. security markets
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Primary vs. Secondary Markets
Primary Market
firms issue new securities through underwriter (investment bank) topublicinvestors get new securities; firm gets funding
Secondary Market
investors trade previously issued securities among themselves
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How different securities are issued
Stocks
Initial Public Offering (IPO): first time a firm issues stockSeasoned Offering: firms that already are public and want to issuemore equity
Bonds
Public offeringPrivate placement: sale of securities to a small group of investors
not traded in secondary marketsliquidity is low but cheaper than public offerings
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Initial Public Offering (IPO)
Used by private company that sells stock to the public for the firsttime
preliminary prospectus (filing with SEC)road show to publicize new offeringbook building to determine demand for new issue: investor interestprovides valuable pricing information
Correctly pricing equity offering
Underpricing: issuing firm loses moneyOverpricing: underwriter may lose money and have problems sellingthe securitiesreputation is also at stake
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How Securities are Traded: Types of Markets
Direct search: buyers and sellers seek each other
Brokered markets: brokers act like middleman between buyers andsellers
Dealer markets: hold inventories of assets to buy and sell
Specialist dealer market:“market makers” (specialist dealer) grantedmonopoly position and responsible for making a market (i.e. providebid and ask prices)
Electronic Communication Networks (ECNs): electronic interfaceamong traders, bypass traditional dealership function
Auction markets: traders converge at one place to buy and sell
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How Securities are Traded: Bid and Ask Prices
Bid price: offers to buy
in dealer markets, bid price is the price at which the dealer is willingto buy
Ask price: offers to sell
in dealer markets, ask price is the price at which the dealer is willingto sellinvestors must pay the ask price to buy the security
Bid-ask spread: the difference between ask and bid prices (mosttimes bid<ask)
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How Securities are Traded: Types of OrdersMarket order: buy and sell immediately at the current market price(ask or bid)Limit order: triggered by a specified price
buy when price drops below limit (limit-buy)sell when price rises above limit (limit-sell)
Stop orderbuy when price rises to stop (stop-buy)sell when price drops to stop (stop-loss)
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How different securities are traded
Stock trading
traditional and electronic trading platforms, e.g. NASDAQ, NYSEbroker service, e.g. ScottTrade, Fidelitylarge orders negotiated through broker/dealer
Bond trading
most bond trading takes place in the over-the-counter (OCT) marketamong bond dealersmost bond markets are “thin” – not as much volume as stock
Derivative trading
some are traded in an exchange, others are traded over-the-counter(OTC)e.g. stock option, bond optoin, market index options areexchange-traded; swaps are traded OTC
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Trading Costs
Explicit costs
Brokerage commission: fee paid to broker for making the transaction(per transaction)Account maintenance fees (full-service brokers)Management fees (mutual funds)Interest (if borrowing from broker, i.e. “buying on margin”)
Implict cost of trading: Bid-ask spread
e.g. a stock has a bid of $20 and an ask of $21, you would expect tolose $1.00 or 4.8% of your money if you bought at the ask of $21and then immediately sold at the bid of $20. If you had bought 100shares, you lose $100 on the bid ask spreadmore liquid securities have lower bid-ask spread
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Buying on Margin
Investor borrowing some money from broker to purchase security
Investor pays service fee + interest to broker for borrowed moneyInvestor’s own money is called “equity” (because it is residual, andbears all the loss from security)
Margin refers to the percentage of total investment that comes fromthe investor
margin rises when security price rises; falls when security price falls
Initial margin: margin when account first opened
set by the fed: currently 50%
Maintenance margin: minimum margin required to be held inaccount
usually 30%
Margin call: call from broker to investor when margin falls belowmaintenance margin, i.e. investor needs to add cash or securities totrading account
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Margin Example
Example 1: Investor puts $10,000 in trading account, borrows$5,000 from broker. Calculate the initial margin. If maintenancemargin is 30%, how far can portfolio value fall from $15,000 before amargin call from broker?
Example 2: Investor puts $5,000 into trading account, initial marginis 50%. What is the most the investor can borrow from broker?
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Short Sale
Sale of a security before owning it, with intent of buying it later.Investor profits from a decline in price of security
How it works
borrow security from a broker/dealersell it and deposit proceeds and margin (value of borrowed security)in an accountclose out position: buy security and return to the broker/dealer (buyto cover)
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Short Sale Example
Investor borrows and sells 100 shares of Google at $700 per share
Scenario 1: price drops to $650. investor buys to cover. Calculateprofit
Scenario 2: price rises to $750. investor buys to cover. Calculateprofit (loss)
What is the maximum potential profit and the maximum potentialloss?
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U.S. Gov‘t Regulation of Security Markets
Policies designed to limit common problemsConflict of Interest
Glass-Steagall (Banking Act of 1933) called for separation ofcommercial and investment bankingRestrictions were removed in Financial Services Modernization Act(1999).
Asymmetric Information / Lack of Transparency, e.g. Insider Trading
Officers, directors, and major stockholders must report transactionsSarbanes-Oxley Act (2002): requires auditing, more accountability
Issue and Trade of Questionable Securities
Securities Act (1933) requires registration and prospectusCommodity Futures Modernization Act (2000) opened up market
Dodd-Frank Wall Street Reform and Consumer Protection Act(2010)
Designed to deal with some of the issues related to recent recession
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Part II: Risk, Return, Risk Aversion, and Capital Allocation
Chapter 5. Risk, Return, and the Historical Record
Chapter 6. Risk Aversion and Capital Allocation to Risky Assets
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Chapter 5. Risk, Return, and the Historical Record
Learning Objectives
Understand real and nominal rates of interest
Calculate real, nominal interest rates, and real after-tax interest rates
Calculate annualized rate of return
effective annual rate of return (EAR)annual percentage rate (APR)
Calculate Expected return, variance and standard deviation ofreturns
Calculate Risk premium and excess return
Understand the meaning of Sharpe ratio and calculate it
Know some basic properties of U.S. historical returns on riskyporfolios
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How to measure return on securities
stocks: rate of return depends on change in value and dividends
rate of return =P1 − P0 + D
P0
P0 is price at purchase, P1 is price at sale, D is total dividends paidin holding period
bonds: rate of return depends on price, par value and couponpayments
rate of return =P1 − P0 + C
P0
P0 is price at purchase, P1 is price at sale (or Par Value if held tomaturity), C is total coupon payments paid in holding period(C = 0 for zero-coupon bond)
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Real vs. Nominal Interest Rate
Nominal (“in name”) interest rate, R, of a financial asset measuresthe money rate of return from investment
this is the rate quoted for investment instruments
Real interest rate, r, measures the rate of return in purchasing powerfrom investment
Purchasing power is reduced by inflation, i. So relationship betweenreal and nominal interest rates
1 + r =1 + R
1 + i, or r =
R − i
1 + i
in approximation, i ≈ R − i (Fisher Equation)
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Real vs. Nominal Interest Rate
Since future inflation is unknown, even “risk-free” investments carrythe “risk” of inflation.
When investors expect higher inflation in the future, they demandhigher nominal rates. The nominal rate investors demand
R ≈ r + ie
where ie is expected inflation
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Taxes and Interest Rate
Taxes are based on nominal interest income
Given tax rate (t), and nominal interest rate (R), the after-tax realinterest rate is
rafter−tax =R(1− t)− i
1 + i
in approximation, rafter−tax ≈ R(1− t)− i
The after-tax real interest rate falls as inflation rate rises
Example: A corporate bond pays interest rate of 5%. If the (actual)inflation rate is 1%, what is the real interest rate? In addition, theholder of the bond is subject to 30% income tax. What is the realafter-tax interest rate of the bond?
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Annualized Rate of Return
Suppose a zero-coupon bond with $100 par value
maturity price at issue rate of return if held to maturityhalf-year $97.36 100−97.36
97.36 = 2.71%
1-year $95.52 100−95.5295.52 = 4.69%
25 years $23.30 100−23.3023.30 = 329.18%
How to compare return for different terms of investment?
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Annualized Rate of ReturnWe want a measure over 1-year horizon such that
(1 + measure)term of investment = 1 + total rate of return
e.g. a 6-month T-bill with 3% total return, would be able to buythis bond twice in a year, so
annual rate = (1 + 3%)2 − 1 = 6.09%
on the other hand, a 2-year bond with 3% total return, it’s likebuying a 1-year bond twice
(1 + annual rate)2 − 1 = 3%
solve for annual rate gives 1.49%
Effective annual rate (EAR)
EAR = (1 + total rate of return)1T − 1
where T is terms of investment in years.
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Annualized Rate of Return
Another way to annualize return is the Annual Percentage Rate(APR)
APR =total rate of return
T
A 6-month bond gives return of 3% over 6-month, the APR = 6%
A 2-year bond gives return of 3%, APR=1.5%
This is the annual rate of return used by banks
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Risk and Return
What we talked about so far are “guaranteed” returns. What ifreturns are not “guaranteed”, i.e. is uncertain?
Expected rate of return
E (r) =∑
s
p(s)r(s)
where p(s) is probability of state s happens, r(s) is rate of return ifstate s occurs.
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Risk and Return
When evaluating risky assets, we must also consider the variance (orstandard deviation) of return
Variance (recall from stats) of return
σ2 =∑
s
p(s)[r(s)− E (r)]2
standard deviation, σ is the square root of variance
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Risk Premium
Risk premium: the expected return earned on a risky asset in excessof the risk-free rate
e.g. a risky asset has an expected return of 7% and T-bills (risk-free)are paying 2%, then risk premium is 5%
Excess return: the return earned on a risky asset in excess ofrisk-free rate in each state
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Sharpe Ratio
Sharpe ratio: measures “reward to volatility”, or risk-adjusted return
Sharpe Ratio =Risk Premium
SD of excess return
all else equal, a higher Sharpe ratio is “better”
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Other Measures of Risk
Sometimes we want to know more about the worst-case scenarios
Value at Risk (VaR): quantile of a distribution below which lies q%of the possible values of that distribution
e.g. the 5% VaR is the return at the 5th percentile when returns aresorted from high to low
Expected Shortfall (ES): expected return conditional on being in thebottom q% of scenarios
more conservation measure of downside risk than VaRVaR takes the highest return from the worst casesES takes an average return of the worst cases
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Chapter 6. Risk Aversion and Capital Allocation to RiskyAssets
Learning Objectives
Understand the ideas of risk and risk-aversion
Calculate utility of a portfolio given the function and parameters
Understand the concept of certainty equivalent and calculate it
Understand the concept of mean-variance criterion and draw itsgraphical representation
Calculate expected return and standard deviation of a portfolio
Understand what a capital allocation line is, calculate its slope,y-intercept, draw it
Calculate optimal capital allocation between risky and risk-free assets
Understand what a capital market line is, calculate its slope andy-intercept, draw it
Understand how to do a leverage, calculate expected return of aleveraged portfolio
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Allocation to Risky Assets
Investors will avoid risk unless there is a reward
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Risk Aversion
Most investors want to reduce variance in returns (risk) for givenexpected return
A risk-averse investor will
reject fair-games (or worse) when there is any uncertaintyrequire a higher expected return for higher variance
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Utility Function for Portfolios
Investor‘s utility for a risky portfolio depends on
Expected return, E(r)Variance, σ2
Investor‘s risk-aversion index, A (A > 0 for risk-averse, higher if morerisk-averse)
Utility from a portfolio
U = E (r)− 1
2Aσ2
So for risk-averse investors, utility increases with expected returnand decreases with variance
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Utility Function for Portfolios
For each investor, we can choose the best portfolio among givenportfolios using the given utility function
On the other hand, given investors‘ choice of portfolio, we can ranktheir risk-aversion
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Certainty Equivalent
The risk-free (thus “certainty”) rate of return needed for an investorto be indifferent between a portfolio of risk-free asset and a portfolioof risky assets
The certainty equivalent value varies by investor‘s risk-aversion
Example: an investor with risk-aversion index A = 2, a riskyportfolio of expected return and variance combination (0.7, 0.25).Calculate the certainty equivalent.
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Mean-Variance Criterion
For a risk-averse investor, the mean-variance criterion says thatportfolio A dominates portfolio B if
E (rA) ≥ E (rB )
andσA ≤ σB
with at least one strict inequality
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Graphical Representation of Mean-Variance Criterion
top-left quadrant is better than bottom-right quadrant
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Portfolio Choice Set
Investor chooses to invest in risk-free (f) and risky assets (p)
proportion in risky assets = y, so proportion in risk-free asset = 1-yusually, 0 ≤ y ≤ 1if borrow to invest in risky asset, y > 1, i.e. 1 − y < 0
For any value of y we can calculate expected return and standarddeviation of return for the portfolio
expected return: E(r) = yE(rp) + (1 − y)rf
variance: Var(r) = y 2Var(rp)i.e. the risk-free asset doesn‘t contribute to variancestandard deviation: σ = yσp
where E(rp) is expected return of the risky assets, E(r) is expectedreturn of portfolio, σp is standard deviation of risky assets
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Capital Allocation Line
We can put expected return and standard deviation of portfolios ona line.
the line: E(r) = rf + y(E(rp) − rf ) = rf +(E(rp )−rf )
σpσ
the slope is the reward-to-volatility ratio (Sharpe ratio)
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Capital Allocation Line: Example
Suppose we have risky asset P with E (rp) = 0.13 and σp = 0.25,rf = 0.05. Calculate the expected return and standard deviation of aportfolio when y = 0, 0.5, 1.
What is the y-intercept and slope of the capital allocation line?
Draw the capital allocation line
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Optimal Capital Allocation: Indifference Curve Analysis
optimal capital allocation: the capital allocation (i.e. portfoliochoice) that gives the highest utility and is attainable with the givenasset choices
points on capital allocation line are “attainable” portfolios; pointsabove CAL are not attainable; points below CAL are attainable bythrowing money away
points on the same indifference curve give the same utility, higherutility curve represents higher utility
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Optimal Capital Allocation: Indifference Curve Analysis
Using Capital Allocation Line and indifference curve we can illustrate theoptimal capital allocation between risk-free and risky assets
1 draw capital allocation line (CAL)
2 draw indifference curves for investor with risk-aversion index A
3 point on CAL and the highest utility indifference curve is optimal portfolio
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Optimal Capital Allocation: Derivative Analysis
Take derivative (not the asset class here) to find the optimal capitalallocation
1 given parameters of assets (rf ,E(rp), σp), and investor’s risk-aversionindex A, the utility of the portfolio as a function of y
U(y) = rf + y(E(rp) − rf ) − 1
2Ay 2σ2
p
2 take derivative with respect to y and set it to zero
U ′(y) = E(rp) − rf − Ayσ2p = 0
3 utility function is at a maximum when
y∗ =E(rp) − rf
Aσ2p
4 check for interior solution (make sure U(y = 0) and U(y = 1) areless than U(y∗))
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Optimal Capital Allocation: Example
Suppose an investor has risk aversion of A = 3. The investor hasaccess to a risk-free asset that pays 5% and a risky investmentwhich has an expected return of 8% and standard deviation of 25%.Preferences are modeled by the utility function U = E (r)− 0.5Aσ2.Find the optimal allocation to the risky and risk-free assets.
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Capital Market Line
The Capital Market Line (CML) is the capital allocation linebetween
1-month T-bill (risk-free asset)a broad index of stocks e.g. S&P 500 (“market portfolio”)
E (r) = rf +(E (rM )− rf )
σMσ
the slope is the Sharpe ratio of the market portfolio
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Leveraged Portfolios
An investor can borrow to increase investment in risky asset
proportion y > 1 of own wealth invested in risky assetsof which, (y − 1) is borrowedLender will demand a higher interest rate (rB ) than risk-free rate (rf )
Expected return of the leveraged portfolio
E (r) = yE (rp)− (y − 1)rB
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Leveraged Portfolios: Capital Allocation Line
Capital Allocation Line with leverage
E (r) = rB +(E (rp)− rB )
σpσ
Notice the kink at y = 1
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Leveraged Portfolios: Example
Suppose an investor borrows $5,000 at 6% to make a totalinvestment of $10,000 in an risky asset with an expected return of8% and standard deviation of 25%. Calculate the expected returnand standard deviation of the portfolio
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A Complete Example
Suppose an investor has risk aversion of A = 2. The investor hasaccess to a risk-free asset that pays 2% and a risky investmentwhich has an expected return of 8% and standard deviation of 25%.Preferences are modeled by the utility function U = E (r)− 0.5Aσ2.Find the optimal allocation to the risky and risk-free assets.
Draw the Capital Allocation LineFind the optimal allocation to risky and risk-free assetsCompute the expected return and standard deviation of the optimalportfolioWhat is the investor’s certainty equivalent of the risky investment?What is his certainty equivalent of the optimal portfolio?Sketch two indifference curves corresponding to the two certaintyequivalent above. Draw in the capital allocation line to illustrate theoptimal portfolio
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Part III: Optimal Risky Portfolios, Index Models
Chapter 7. Optimal Risky Portfolios
Chapter 8. Index Models
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Chapter 7. Optimal Risky PortfoliosLearning Objectives
Understand utility of wealth, calculate certainty equivalent and riskcompensation using utility of wealth
Understand the difference between the two portfolio risks: marketrisk and firm-specific risk
Calculate covariance and correlation of two risky securities
Understand how diversification works and how correlation affectsdiversification
Find the minimum-variance portfolio given two risky assets
Draw the portfolio opportunity set, minimum-variance portfolio andefficient frontier given two risky assets
Find the optimal risky portfolio given two risky assets and illustrateit in an expected return-standard deviation graph
Find the optimal combined (or complete) portfolio and illustrate it inan expected return-standard deviation graph
Understand the Markowitz Theorem and Separation Property
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The Investment Decision
On the first day of class, we briefly talked about top-down vs bottom upinvestment strategies. For specifically, Top-down process has 3 steps
1 Capital allocation between risky and risk-free assets (last week)
2 Asset allocation across broad asset classes
3 Security selection of individual assets within each asset class
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Utility of Wealth
Recall the utility for a portfolio U = E (r)− 0.5Aσ2
Utility of wealth measures utility from total wealth.
it could take different forms, e.g. U(w) = w 0.5, U(w) = ln(w)expected utility of wealth with a risky investment
E(U(w + r)) =∑
s
p(s)U(w + r(s))
Example: suppose investor has $1000 and faces a risky investmentthat pays $250 with probability 0.4 and $50 with probability 0.6.Investor’s utility from wealth is given by U(w) = w0.5.
find the certainty equivalent of the risky investmentfind the risk compensation for this investor.The risk compensation is the difference between expected value andthe certainty equivalent of a risky investment.
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Portfolio Risk
Portfolio is subject to market risk and firm-specific risk
market risk is systematic (i.e. non-diversifiable)firm-specific risk is diversifiable, i.e. can be reduced throughdiversification
Diversification
the process of reducing portfolio risk (measured by standard deviationof return on portfolio) by mixing a wide variety of investments
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Stats Review: Variance and Covariance
For one variable, X (a is a scalar)
Var(aX ) = a2Var(X )
With sum of two variables X and Y
Var(aX + bY ) = a2Var(X ) + b2Var(Y ) + 2abCov(X ,Y )
Covariance measures how much the two returns change together
Cov(X ,Y ) =∑
s
p(s)[X (s)− E (X )][Y (s)− E (Y )
if X and Y are independent, then Cov(X ,Y ) = 0Cov(X ,X ) = σ2
X
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Stats Review: Covariance and Correlation
Correlation ρXY is computed by normalizing or “scaling” covariance
ρXY =Cov(X ,Y )
σXσY
+1.0 ≥ ρXY ≥ −1.0ρXY = 1.0, X and Y are perfectly positively correlatedρXY = −1.0, X and Y are perfectly negatively correlated
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Stats Review: Exercise
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Two-Security Risky Portfolio
A portfolio with two risky securities: debt (D) and equity (E)
Expected return on portfolio
E (rp) = wDE (rD) + wEE (rE )
Risk of return on portfolio (measured by variance)
σ2p = w2
Dσ2D + w2
Eσ2E + 2wDwECov(rD , rE )
where rp is portfolio return; E (rp) is expected portfolio returnrD is debt return; E (rD) is expected debt returnrE is equity return; E (rE ) is expected equity returnwD is proportion in debt; wE is proportion in equity; wD + wE = 1σ2
D is variance of return on debt; σ2E is variance of return on equity
Cov(rD , rE ) is covariance of the two returns
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DiversificationThe amount of possible risk reduction through diversificationdepends on the correlation ρDE
When ρDE = 1, i.e. debt and equity are perfectly positivelycorrelated, there is no diversification
σ2p = w2
Dσ2D + w2
Eσ2E + 2wDwEσDσE
= (wDσD + wEσE )2
soσp = wDσD + wEσE
When ρDE = −1, i.e. debt and equity are perfectly negativelycorrelated, a perfect hedge (0 variance) is possible
σ2p = w2
Dσ2D + w2
Eσ2E − 2wDwEσDσE
= (wDσD − wEσE )2
soσp = |wDσD − wEσE |
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Minimum-Variance Portfolio
The portfolio mixture (wD ,wE ) that minimizes overall variance ofportfolio, for given asset variances and covariance
this is not necessarily the most efficient or optimal, just min variance
When ρDE = −1, σp = 0 minimum-variance portfolio is
wE =σD
σD + σE, wD =
σE
σD + σE
When ρDE > −1, minimum-variance portfolio is found by minimizingthe quadratic function of wD
σ2p = w2
Dσ2D + (1− wD)2σ2
E + 2wD(1− wD)Cov(rD , rE )
taking derivative and set to zero
wMVD =
σ2E − Cov(rD , rE )
σ2D + σ2
E − 2Cov(rD , rE )
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Portfolio Opportunity Set
different Portfolio Opportunity set (POS) curve is drawn for each correlation value, eachcurve shows all portfolios that can be built from the two assets
Minimum-Variance portfolio is the left-most point of each POS curve
Efficient Frontier is the portion of each POS curve above the Minimum-Variance portfolio
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Minimum-Variance Portfolio: Example
Suppose there are two risky assets: D has an expected return of 5%and a standard deviation of 15%; E has an expected return of 10%and a standard deviation of 20%. The correlation between D and Eis 0.2.
Find the allocation to D and E that minimizes the variance of therisky portfolio.
Find the minimum-variance portfolio If the correlation is -1.
Draw the portfolio opportunity set for both scenarios and illustratethe minimum variance portfolios.
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Asset Allocation with Diversified Portfolio
In Chapter 6, we studied the capital allocation decision betweenrisk-free asset (F) and some risky portfolio (P)
Now we also choose how to divide P into two risky assets D and E,and how much to put into F
all points on the POS are possible risky portfoliosfor each point in the Portfolio Opportunity Set (POS), draw aCapital Allocation Line (CAL) from the risk-free pointall points on this CAL are “attainable” combined portfolios (riskyportfolio + risk-free asset)a steeper CAL has higher expected return and/or lower risk (higherSharpe ratio)our objective is to choose the steepest CAL (highest Sharpe ratio)that goes through one of the points on POS
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Portfolio Opportunity Set and Capital Allocation Lines
portfolio opportunity set is the blue curve
CAL(B) is better than CAL(A) but we can do even better
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Optimal Risky Portfolio
optimal risky portfolio is P
CAL(P) is the steepest CAL, and P is also on the portfolio opportunity set
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Optimal Risky Portfolio
The portfolio mixture (wD ,wE ) that maximizes the slope of CAL(Sharpe ratio), for given assets
Recall
Sharpe ratio =E (rp)− rf
σp
where E (rp) = wDE (rD) + wEE (rE ) andσ2
p = w2Dσ
2D + (1− wD)2σ2
E + 2wD(1− wD)Cov(rD , rE )
Find optimal risky portfolio by maximizing Sharpe ratio subject towD + wE = 1. Taking derivative and set to zero
w∗D =
E (RD)σ2E − E (RE )Cov(RD ,RE )
E (RD)σ2E + E (RE )σ2
D − [E (RD) + E (RE )]Cov(RD ,RE )
where RD and RE are excess rates of return D and E
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Optimal Risky Portfolio: Example
Suppose there are two risky assets: D has an expected return of 5%and a standard deviation of 15%; E has an expected return of 10%and a standard deviation of 20%. The correlation between D and Eis 0.2. The risk-free asset has a return of 2%.
Find the optimal risky portfolio.
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Optimal Combined (or Complete) Portfolio
Once we have established the optimal asset allocation of D and E inthe risky portfolio P, we can find the optimal capital allocationbetween risk-free asset F and P, using the method from chapter 6.
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Optimal Combined (or Complete) Portfolio
Recall the optimal capital allocation between risky and risk-freeassets is given by
y∗ =E (rp)− rf
Aσ2p
this is proportion of combined portfolio in risky portfolio
By solving the optimal risky portfolio we get w∗D : this is proportion
of risky portfolio in asset D
Then the combined portfolio is given by
risk-free: 1 − y∗
risky asset D: y∗w∗Drisky asset E: y∗(1 − w∗D )
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Optimal Combined (or Complete) Portfolio: Example
Suppose there are two risky assets: D has an expected return of 5%and a standard deviation of 15%; E has an expected return of 10%and a standard deviation of 20%. The correlation between D and Eis 0.2. The risk-free asset has a return of 2%. with
An investor with risk-aversion index A = 3, utility functionU = E (r)− 0.5Aσ2
Find the optimal combined portfolio.
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Asset Allocation and Security Selection
Markowitz Theorem
every investor, regardless of risk aversion should choose the samerisky portfolio (same wD ). This gives the optimal CALinvestor with different risk-aversion will then choose differentmixtures of the risk-free asset and the optimal risky portfolio
more risk-averse investors put more in risk-free asset (smaller y)less risk-averse investors put more in the optimal risky portfolio(larger y)
Separation Property: asset allocation can be separated into twotasks
choice of optimal risky portfolio is technical and does not depend onindividual investorscapital allocation into risk-free and risky portfolio is personal,depends on risk preferences
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Chapter 8. Index Models
An important part of portfolio selection is calculating variances andcovariances
for 2 risky securities, we need 2 variances and 1 covariancefor n risky securities, we need n variances and n2 − n convariances
Index models simplify the description of portfolio risk. It captures
common “market forces”each security’s sensitivity to market forceseach firm’s value in excess of the marketrandomness
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Decompose Rate of Return
returns are considered random, because we cannot have perfectknowledge of future events and how they will affect each security
for an individual security, we can write return as
ri = E (ri ) + ei
where E (ri ) is the expected return and ei is the unexpectedcomponent (or “error term”)ei is a random variable with mean 0 and standard deviation σ(ei )
individual security returns are correlated, we can decompose ei into
uncertainty about the economy as a whole m, anduncertainty about the firm ei
ri = E (ri ) + βim + ei
βi measures i’s response to common factor
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Single-Index Model
Using excess returns and market return to proxy for common factorto get the Single-Index model:
Ri = E (Ri ) + βiRm + ei
security variance (total risk) = systematic risk + firm-specific risk
σ2i = β2
i σ2m + σ2(ei )
covarianceCov(ri , rj ) = βiβjσ
2m
correlation
Corr(ri , rj ) =βiβiσ
2m
σiσj
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Estimating Single-Index Model
Regression of single-index model using data over time
collect series of historical observations Ri (t) and Rm(t) for manydates t
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Estimating Single-Index Model
Regression of single-index model using data over time
collect series of historical observations Ri (t) and Rm(t) for manydates tregression equation
Ri (t) = αi + βi Rm(t) + ei (t)
alpha (αi ) is security’s expected excess return when market excessreturn is zerobeta (βi ) is security’s sensitivity to market returnei (t) is the residual (error term)
Using estimate αi and βi can calculate expected excess return
E (Ri ) = αi + βiE (RM )
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Regression Result: Excess return: HP vs. market
Security Characteristic Line (SCL)
RHP (t) = αHP + βHPRSP500(t) + eHP (t)
y-intercept αHP , slope βHP
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Regression Result: Analysis of Variance (ANOVA)
R2is a measure of “goodness of fit”: how much of the variation is explained bythe regression. Values range from 0 to 1, with 1 being a perfect linear fit.
The ANOVA gives more detailed information on how much of the variation isdue to variation in the explanatory variables (like market forces) and how muchis due to unexplained factors (error term)
The last panel gives the estimates for α and β as well as standard error, t-stat,and p-value which help us decide if the coefficients are significant.
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Index Model and Diversification
Excess return on portfolio: Rp = αp + βpRm + ep
With an equally-weighted portfolio of n stocks
Rp =1
n
n∑i=1
(αi + βiRm + ei ) =1
n
n∑i=1
αi︸ ︷︷ ︸αp
+ (1
n
n∑i=1
βi )︸ ︷︷ ︸βp
Rm +1
n
n∑i=1
ei︸ ︷︷ ︸ep
As number of stocks increases, nonmarket risk for an equallyweighted portfolio decreases
variance of return on portfolio
σ2p = β2
pσ2m + σ2(ep)
error terms are assumed to be independent (thus uncorrelated)
σ2(ep) =n∑
i=1
(1
n
)2
σ2(ei ) =1
nσ2(e)
this approaches zero as n gets larger
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Investment Portfolio Management
Passive portfolio invests in an index such as the S&P 500 (which hasbeta=1, alpha=0, and no firm-specific risk)
Active portfolio invests in a mix of individual stocks
Portfolio construction1 estimate risk premium (excess return) and variance of market index
by macroeconomics analysis. this is E(Rm) and σ2m
2 use regression to estimate βi and αi for each security, then findresidual variances σ2(ei )
3 establish expected excess return of each security βi E(Rm)4 use security analysis to adjust alphas αi
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Index Model and Optimal Risky Portfolio
Inputs for single-index model
risk premium on S&P 500, E(Rm)standard deviation of S&P 500, σm
for each of n stocks, estimates of
Alpha, αi
Beta, βi
Residual Variance, σ2(ei )
Parameters of portfolio
weights of the n component securities wi , adding up to 1passive portfolio (index fund) is the (n + 1)th securityAlpha, Beta, and Residual variance of portfolio are just weightedsums from the parameters of component securities
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Index Model and Optimal Risky PortfolioA portfolio with n assets each of proportion wi
Rp =n+1∑i=1
wi (αi + βiRm + ei ) =n+1∑i=1
wiαi︸ ︷︷ ︸αp
+ (n+1∑i=1
wiβi )︸ ︷︷ ︸βp
Rm +n+1∑i=1
wiei︸ ︷︷ ︸ep
Portfolio Alpha
αp =n+1∑i=1
wiαi
Portfolio Beta
βp =n+1∑i=1
wiβi
Portfolio Residual Variance
σ2(ep) =n∑
i=1
w 2i σ
2(ei )
market index has no residual variance i.e. σ2(en+1) = 0
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Forming an optimal Risky Portfolio Using Index Model
Choose portfolio weights: w1,w2, . . . ,wn+1
Maximize portfolio Sharpe ratio
Sp =E (Rp)
σp
E(Rp) = αp + βpE(RM )σp = [β2
pσ2M + σ2(ep)]1/2
αp, βp, σ2(ep) are as defined on previous slide
Subject to∑n+1
i=1 wi = 1
Combine an Active Portfolio (A) and a Passive Portfolio (M)
If the active portfolio has a beta of 1, the optimal weight of Adepends on αA/σ
2(eA), the ratio of alpha to residual varianceM contributes: E(RM )/σ2
M
We want to choose allocation into A and M, i.e. wA and wM
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Step 1: weights of stocks in A
Initial weights of stocks in active portolio
w0i =
αi
σ2(ei )
Scale these weights to make sure they add up to 1:
wi =w0
i∑w0
i
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Step 2: characterize optimal active portfolio
Alpha of active portfolio
αA =n∑
i=1
wiαi
Beta of active portfolio
βA =n∑
i=1
wiβi
Residual variance of active portfolio
σ2(eA) =n∑
i=1
w2i σ
2(ei )
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Step 3: weight of active portfolio
Initial position in the active portfolio (A) is
w0A =
αA/σ2(eA)
E (RM )/σ2M
When βA is close to 1, A correlates more with M, so the benefit ofdiversification through M is not as great
Update the position in A based on beta
w∗A =
w0A
1 + (1− βA)w0A
Example: if we had initially found w0A = 0.2 and A has a beta of 2,
then
w∗A =
0.2
1− (1− 2)(0.2)= 0.25
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Step 4: risk premium and variance of optimal riskyportfolio
Note that w∗M = 1− w∗
A, and w∗i = wiw
∗A
Risk premium:
E (Rp) = (w∗M + W ∗AβA)E (RM ) + αAw
∗A
Variance:σ2
p = (w∗M + w∗
AβA)2σ2M + [w∗
Aσ(eA)]2
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Part IV: The Capital Asset Pricing Model
Chapter 9. The Capital Asset Pricing Model
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Chapter 9. The Capital Asset Pricing Model
Learning Objectives
Know the assumptions of CAPM
Know the qualitative and quantitative results of CAPM
Calculate expected returns and risky premium of a security usingCAPM
Understand the meaning of Alpha and Beta
Calculate the implied Alpha given analyst forecast
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CAPM
The Capital Asset Pricing Model (CAPM) is an equilibrium pricingmodel
It predicts equilibrium rates of return (E (ri )) by taking into account
the asset’s sensitivity (β) to non-diversifiable risk (market risk)the expected return of the market (E(rM ))the risk-free asset (rf )
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CAPM: Assumption
investors are price-takers
single-period investment horizon
investments limited to traded financial assets
no taxes or transaction costs
information is free and perfect
investors are rational mean-variance optimizers
homogeneous expectations and information
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CAPM: Equilibrium Result
qualitative result
all investors hold the same portfolio for risky assets – the marketportfoliomarket portfolio contains all securities
quantitative result
market risk premiumindividual security risk premium
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Equilibrium Portfolio: Market Returns
The risk premium on the market portfolio is proportional to its riskand the degree of risk aversion of the investor
E (rM )− rf = Aσ2M
where σ2M is the variance of the market portfolio
A is the average degree of risk aversion across investors
The market price of risk is the ratio of the risk premium to varianceof market returns
E (rM )− rfσ2
M
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Equilibrium Portfolio: Individual Security Returns
In equilibrium, prices adjust so all investment have the samereward-risk ratio
E (rM )− rfσ2
M
=E (ri )− rfCov(ri , rM )
The risk premium on the security i
E (ri )− rf = βi [E (rM )− rf ]
where beta is given by
βi =Cov(ri , rM )
σ2M
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CAPM Example
Example 1: Suppose the risk-free rate is 5%, the expected return ofa market portfolio is 12%, the standard deviation of the marketportfolio is 30%. A given security has a covariance of 0.135 with themarket.
What is the beta of the security?What does CAPM predict for the expected return of the security?If the price forecast for the security in one year is $60, what would befair price today?
Example 2: Suppose that the expected returns for the market are12%. Security A has a beta of 0.75 and an expected return of 10%.Under CAPM, what must be the expected return for security B,which has a beta of 1.25?
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Adding in Alpha
Beta (βi ) is calculated from public data, is generally considered acharacteristic of the security and is stable in short-run.
Alpha (αi ) is something that gets added on “extra” when theanalyst thinks there is something special about the stock that otherpeople don’t see.
represents a deviation from the prediction of CAPM (roughly, returnsin excess of market)positive alpha arise occasionally from chance
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Reference
Bodie, Kane & Marcus, Investments, 10th edition.
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