econ 4751 financial economics instructor: zoe xie...agency problem 10/137 chapter 1chapter 2chapter...

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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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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econ 4751 financial economics instructor: zoe xie...agency problem 10/137 chapter 1chapter 2chapter...

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