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FINS2624: PORTFOLIO MANAGEMENT NOTES

Daniel Quinn UNIVERSITY OF NEW SOUTH WALES

FinS2624: Portfolio Management Notes

Daniel Quinn

TABLE OF CONTENTS Bond Pricing _____________________________________________________________________ 3 Bonds _______________________________________________________________________________ 3 Arbitrage Pricing ______________________________________________________________________ 3 YTM and Bond prices___________________________________________________________________ 4 Realized Compound Yield _______________________________________________________________ 4

Term Structure of Interest Rates ______________________________________________________ 5 What is the Term Structure? _____________________________________________________________ 5 Inferring the Term Structure _____________________________________________________________ 5 Exploiting Mispricing with Three Bonds (Example) ____________________________________________ 5 Reinvestment Risk _____________________________________________________________________ 6 Forward Rates ________________________________________________________________________ 6 Liquidity Risk _________________________________________________________________________ 6 Market Expectations ___________________________________________________________________ 7

Duration ________________________________________________________________________ 8 Macaulay’s Duration: Measure of Maturity (D) ______________________________________________ 8 Modified Duration: Measure of Yield Sensitivity (D*)__________________________________________ 8 Portfolio Duration ____________________________________________________________________ 10 Immunization________________________________________________________________________ 10 Rebalancing _________________________________________________________________________ 10

Markowitz Portfolio Theory ________________________________________________________11 Utility ______________________________________________________________________________ 11 Statistics____________________________________________________________________________ 11 Portfolio Variance ____________________________________________________________________ 11 Diversification _______________________________________________________________________ 12

The Optimal Portfolio _____________________________________________________________13 Complete Portfolio ___________________________________________________________________ 13 Separation Theorem __________________________________________________________________ 13 Capital Asset Pricing Model _____________________________________________________________ 15

CAPM__________________________________________________________________________ 16 The Security Market Line_______________________________________________________________ 16

Chapter: Table of Contents

Portfolio Selection ____________________________________________________________________ 12

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Unsystematic Risk ____________________________________________________________________ 16 Systematic Risk ______________________________________________________________________ 17 The SML and the CAL __________________________________________________________________ 17 Portfolio Beta________________________________________________________________________ 17 Assumptions of CAPM _________________________________________________________________ 17 Using the CAPM ______________________________________________________________________ 17

Portfolio Management in Practice ___________________________________________________18 Criticisms of CAPM ___________________________________________________________________ 18 The Single Index Model ________________________________________________________________ 18 Exploiting Mispricing __________________________________________________________________ 18 Factor Models _______________________________________________________________________ 19

Behavioural finance and Market Efficiency ____________________________________________20 Efficient Market Hypothesis ____________________________________________________________ 20 Behavioural Finance __________________________________________________________________ 20

Performance Measures ____________________________________________________________ 21 Active Investments: Risk Adjusted Performance Measures ____________________________________ 21 Passive Investments __________________________________________________________________ 22 Practical Considerations _______________________________________________________________ 22 Sources of Performance _______________________________________________________________ 22 Performance attribution _______________________________________________________________ 23

Options Strategies ________________________________________________________________ 24 Value of Options _____________________________________________________________________ 24 Options Strategies ____________________________________________________________________ 24

Black-Scholes Formula ____________________________________________________________27 Assumptions ________________________________________________________________________ 27 Greeks _____________________________________________________________________________ 27

Chapter:

Delta Hedging _______________________________________________________________________ 27

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BOND PRICING BONDS x x x x

A claim on fixed future cash flows Typically a “large” cash flow (face value) at maturity (FV) May be series of smaller cash flows before maturity (coupons) Sum of annual coupons are expressed as fraction of FV (coupon rate)

x

Bond’s current yield =

x

Assumptions in the pricing model: o No default risk o No transaction costs o Constant interest rates o Complete markets

ARBITRAGE PRICING

x x

x x

Arbitrage: set of trades that generate zero cash flows in the future, but a positive, risk free cash flow today Arbitrage pricing: constructing replicating portfolios using assets with known prices to exactly mimic the cash flows of some other asset For example price a bond with coupon rate 5%, FV $100, 2 year maturity, when interest rate is 8% for both lending and borrowing

Exploiting mispricing: Buy Æ riskless profit o T = 0 will result in riskless profit, every other period will have 0 net cash flow Arbitraging increases demand for bond, increases price until no further arbitrage is possible – arbitrage free price

Arbitrage free - all the prices in the market are same Buy low, Sell high

Chapter: Bond Pricing

x

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PRICING FORMULA x

Replicate: o One coupon stream of c from 1 to T o One large payment of FV at T

x

PV(FC) =

x x x

PV(Coupon stream) = PV(perpetuity starting at time 1) – PV(perpetuity starting at time T+1) In practice, interest rates are not constant We take P as given and define Y as a yield to maturity (YTM)

x

Holding period return =

( )

(

)

YTM AND BOND PRICES x x x

Bond price decreases with YTM Price is less sensitive to changes in YTM when YTM is high When YTM = C, P = FV o C = YTM Æ P = FV Æ bond trades at par o C < YTM Æ P < FV Æ bond trades at a discount o C > YTM Æ P > FV Æ bond trades at a premium

REALIZED COMPOUND YIELD

x

If bond A and B have the same YTM, t2 cash flows will differ However if coupon in B can be reinvested at an interest rate that equals YTM, time two cash flows will be equal Realized compound yield solves for the annualized return Æ useful when reinvestment rate is different from YTM o Collect all cash flows at maturity of bond o Divide by price and solve for annualised return o ( )

Chapter: Bond Pricing

x x

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TERM STRUCTURE OF INTEREST RATES WHAT IS THE TERM STRUCTURE? x x x x

t spot rate (0yt): fixed interest rate for an investment starting today and ending at time t Spot rates together make up the term structure of interest rates or the yield curve Interest rate refers to the price of a future cash flow Æ set in equilibrium Typically upwards sloping

INFERRING THE TERM STRUCTURE x

Consider the pricing equation of a two year bond at time 0:

x

Bootstrapping technique: find y1 from a one year coupon bond

x

Then substitute into the equation to find y2

x x x

Buy the under-priced Bond C and sell the overpriced synthetic bond Sell 0.1A Æ P = $0.09 with a CF of $10 at T1 to match the CF of C Sell 1.1B Æ P = $87.69 with a CF of $110 at T2 to match the CF of C

Chapter: Term Structure of Interest Rates

EXPLOITING MISPRICING WITH THREE BONDS (EXAMPLE)

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Riskless profit of t = 0 Scaling these trades up will increase demand and push bond prices to their arbitrage free values

REINVESTMENT RISK x x x

If we must reinvest coupons, the interest rate of reinvestment is unknown This risk arises when cash flows do not match our investment horizon Therefore, our final cash flow is unknown

FORWARD RATES x x x

Interest rates for investments we agree on today but exercised in the future sft = interest rate determined today that starts at time s and ends at time t In absence of arbitrage opportunities, the term structure determines all forward rates

LIQUIDITY RISK x x x

If one’s investment horizon is less than the bond maturity, they must sell the bond However, the yield at the time is unknown and hence, the price and holding period return will be also unknown As with reinvestment risk, liquidity risk arises when cash flows do not match our investment horizon

LIQUIDITY PREMIUM x x x x

Investors can theoretically pick a bond with maturity that matches their horizon However, issuers and investors may have different horizons Because markets must be clear, someone must carry some risk Æ risk will be priced Liquidity premium offered to induce investors to hold bonds whose maturity does not match their horizon

Chapter: Term Structure of Interest Rates

DETERMINING FORWARD RATES

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If issuers prefer a longer investment horizon than investors Æ pushes slope of term structure upwards

MARKET EXPECTATIONS x

Assume future interest rates are known

x x x

In reality, future interest rates are unknown but expectations work the same way Effect of interest rate expectations on the shape of term structure is the expectations hypothesis Since the term structure implies all forward rates, a common way to phrase EH is:

x

Incorporating the liquidity premium theory:

Chapter: Term Structure of Interest Rates

EXPECTATIONS HYPOTHESIS

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DURATION MACAULAY’S DURATION: MEASURE OF MATURITY (D) x

“Average time to cash flows”

x

Useful formula when calculating duration:

x x

Higher duration Æ price is more sensitive to yield changes Durations of various assets can be compared by simply estimating them in any of the following ways: o Three year bond with coupons Æ D < 3 o A coupon bond with a long maturity Æ calculate the weight x time period of last cash flow and estimate duration to be larger than that

MODIFIED DURATION: MEASURE OF YIELD SENSITIVITY (D*) Interested in how much price of a bond changes when interest changes

Chapter: Duration

x

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For very small changes in yield, the following approximation holds:

CONVEXITY

x

Convexity gives rise to a favourable asymmetry in price effects of yield changes With greater convexity, the price increases more than it would fall with equal magnitudes of yield change This is priced in as investor like such asymmetry

Chapter: Duration

x x

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PORTFOLIO DURATION x

Duration of a portfolio is the weighted average of the duration of its components

IMMUNIZATION

x x x x

Buying a bond Æ postponing cash flows (saving) Æ possibly for a future liability Buying a zero coupon bond whose maturity matches that of liability, there are no reinvestment or liquidity risks However, if there is no such bond, we are exposed to changes in yield Reinvestment risk: higher yields are beneficial since we can reinvest our coupons at better interests Liquidity risk: lower yields are beneficial since we can sell our bonds at a higher price Immunization: constructing a bond portfolio with a duration equal to the duration of our liabilities so price changes resulting from parallel yield movements will cancel out o Because duration is a measure of sensitivity of our bonds to changes in yield

REBALANCING x

Portfolios must be consistently rebalanced to keep it immune as yield and maturity parameters will constantly change

Chapter: Duration

x x

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MARKOWITZ PORTFOLIO THEORY UTILITY x x x x x

Assume wealth is consumed rationally to maximise utility We want to model insatiation and risk aversion Insatiation: increasing wealth will always increase utility (continuous, increasing function) Risk aversion: We prefer certain outcomes to uncertain ones Risk premium: the incentive for risk averse investors to take on risk

x x

Concave down function: a decrease in wealth results in a larger decrease in utility than the increase in utility derived from the equal increase in wealth Utility compares investment portfolios on basis of expected return and risk of those portfolios

x

Common utility function:

x

Utility score = certainty equivalent rate of return o The rate risk free investments would need to offer to provide the same utility score as the risky portfolio Mean-variance criterion: PA dominates PB if: o ( ) () o

x

()

(

()

)

STATISTICS ∑

x

( ) ∑ () ( ) () () ( ) () ( ) ( ])( ] )] () ( ]) ] () ( ) ( ) ( ) ( ) ( ) () ( ) ( ) ( ) ( ) ( ) () () ( )

( )

PORTFOLIO VARIANCE x x

)

( o

(

) (

)

( )

(

)

Chapter: Markowitz Portfolio Theory

x x x x x x x x x x x x

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DIVERSIFICATION x

Combination of two or more less than perfectly correlated assets in one portfolio is diversification

x x x

Higher Æ higher risk When = 1, the potential combinations of risk/return given by different weights can be plotted as a straight line and the weight of each asset determines where the portfolio lies on the line

x

As decreases, the portfolio standard deviation must be lower for each level of expected return

x

Efficient Frontier: plotting a set of portfolios after removing dominated ones

x

Efficient frontier is constructed by finding the sets of weights that minimize s.d. for all expected returns Indifference curve: a plot of equally preferred portfolios depending on an investor’s risk aversion P and Q are equally desirable as the increase in risk is, to him, sufficiently compensated by the increase in return

x x

Chapter: Markowitz Portfolio Theory

PORTFOLIO SELECTION

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THE OPTIMAL PORTFOLIO COMPLETE PORTFOLIO

x x

Combination of risky assets and risk free assets Let rf = return of risk free asset o Var(rf) = 0 o E(rf) = rf ( ) ( ) ( ( ) or ( ) (

)

[( (

)

x

x x x

]

]

) (

) (

x

)

)

(

(

) (

)

)

Different value for y determines the position of the portfolio on the line on the expected returnvariance space

Some portfolios dominate others (lower s.d. and higher E(r) The optimal risky portfolio, P*: o Must be on the efficient frontier o Must be on the highest sloping CAL (capital allocation line)  E.g. for every s.d. in CAL1, it would be dominated by a portfolio in CAL2 with same s.d.

SEPARATION THEOREM x x x

All efficient portfolios is the presence of a risk free asset are on the CAL We pick the portfolio on the CAL that offers the amount of risk we want to take, expressed in y value Two steps in portfolio choice: o Find optimal risky portfolio, P*

Chapter: The Optimal Portfolio

x x

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Choose how much risk we want by choosing the fraction of our wealth we invest in that portfolio Finding optimal risky portfolio P* (and CAL as a result) o

o o o x

Choose P* by choosing portfolio weights wp that maximises Sp, the slope of the CAL Sp = Sharpe ratio

Choosing the risky share, y

o

o o o

o o o o o o o o

y* = the solution for y when the first derivative is 0

y can be > 1 Æ a negative position in risk-free asset Æ a short position Æ borrowing money Æ taking a leveraged position in P* Rational investors increase risk by increasing leverage All investors will end up holding the same risky optimal portfolio Since prices adjust to set supply = demand for stocks, P* is the market portfolio, M Attractiveness of a stock is determined by its risk and return effects on this portfolio Expected return increase of a stock on a portfolio is linear and that risk effect depends on its covariance with other stocks in the portfolio Expected return is essentially required return of a stock to make it as attractive as all other stocks  If price is lower, expected return of investors is higher

Chapter: The Optimal Portfolio

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CAPITAL ASSET PRICING MODEL x x x x

Assume a portfolio consisting only of market portfolio ( ) Suppose we borrow a very small amount of money with weight of and invest it in arbitrary asset i

x

()

x x

()

[( )

] (

)

(

[( ) (

x x o

→
...