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CONTENTS

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FINS2624 Portfolio Management Contents 1 Introduction to Bond Pricing 1.1 Bond Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Bond Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Bond Yields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Default Risk and Other Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Approaches to Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Term Structure of Interest Rates 2.1 The Yield Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Yield Curve and Future Interest Rates . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Theories of the Term Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Duration 3.1 Interest Rate Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Immunization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 1 2 2 3 4 4 5

6 7 7

4 Markowitz Portfolio Theory 8 4.1 Utility Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.2 Return of Portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.3 Variance and Covariance of Portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.4 Diversif ication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5 Optimal Portfolios 11 5.1 Complete Portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.2 Optimal Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.3 Borrowing Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 6 Capital Asset Pricing Model 13 6.1 Optimal Risky Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 7 SIM 7.1 7.2 7.3 7.4

and Pricing Models 14 Single Index Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Exploiting Mispricing: One Asset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Exploiting Mispricing: Multiple Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Factor Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

8 EMH and Behavioural Finance 8.1 Efficient Market Hypothesis 8.2 Behavioural Finance . . . . 8.2.1 Limits to Arbitrage . 8.2.2 Behavioural Biases .

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17 17 17 18 18

9 Performance Evaluation 19 9.1 Sharpe Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 9.2 The M 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 9.3 Treynor Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 9.4 Information Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 9.5 Performance Attribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 9.6 Practical Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

FINS2624: Portfolio Management

Yichen Han

CONTENTS 10 Option Strategies 10.1 Derivatives . . . . . . . . 10.2 Options . . . . . . . . . . 10.3 Options Strategies . . . . 10.4 Put Call Parity Theorem

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22 22 22 23 23

11 Option Valuation 24 11.1 Binomial Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 11.2 Traditional Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

FINS2624: Portfolio Management

Yichen Han

1 INTRODUCTION TO BOND PRICING

Introduction to Bond Pricing

1 1.1

1

Bond Characteristics A bond is a security that is issued in connection with a borrowing arrangement. The borrower issues the bond to the lender for some amount of cash, pays specified payments called coupon payments on specified dates, and when the bond matures, repays the debt by paying the bond’s par value (or face value). The coupon rate of the bond determines the interest payment: the annual payment is the coupon rate times the bond’s par value. The coupon rate, maturity date, and par value of the bond are part of the bond indenture, which is the contract between the issuer and the bondholder. Bonds usually are issued with coupon rates set just high enough to induce investors to pay par value to buy the bond. Sometimes, however, zero-coupon bonds are issued that make no coupon payments, that is, the coupon rate is zero.

1.2

Bond Pricing The value of the bond is equal to the present value of its coupons plus the present value of its par value. Bond Value Bond Value =

T X Coupon t=1

(1 + r)t

+

Par Value (1 + r)T

for maturity date T and constant interest rate r . This can be further simplified to   1 1 1 Price = Coupon × 1− + Par Value × T (1 + r)T r (1 + r)   1 1 1 is the PV factor. where 1− is called the T -period annuity factor and T (1 + r)T r (1 + r) There is an inverse relationship between prices and yields, and this property is called convexity due to the convex shape of the bond price curve. Corporate bonds typically are issued at par value. After the bonds are issued, bondholders may buy or sell bonds in secondary markets, in which prices will fluctuate inversely with the market interest rate. Interest rate fluctuations represent the main source of risk in the fixed-income market. As a general rule, keeping all other factors the same, the longer the maturity of the bond, the greater the sensitivity of price to fluctuations in the interest rate.

FINS2624: Portfolio Management

Yichen Han

1 INTRODUCTION TO BOND PRICING

1.3

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Bond Yields The yield to maturity (YTM) is defined as the interest rate that makes the present value of the bond’s payments equal to its price. This interest rate is often interpreted as a measure of the average rate of return that will be earned on a bond if it is bought now and held until maturity. The bond’s yield to maturity is the internal rate of return on an investment in the bond. The yield to maturity can be interpreted as the compound rate of return over the life of the bond under the assumption that all bond coupons can be reinvested at that yield. Yield to maturity differs from the current yield of a bond, which is the bond’s annual coupon payment divided by the bond price. Premium bonds are bonds selling above par value, and discount bonds are bonds selling below par value. Coupon rate is greater than current yield for premium bonds, and the vice versa for discount bonds. The yield to call, where the bond is retired prior to the maturity date, is calculated just like the yield to maturity except that the time until call replaces time until maturity, and the call price replaces the par value. Yield to maturity will equal the rate of return realized over the life of the bond (the realized compound return) if all coupons are reinvested at an interest rate equal to the bond’s yield to maturity. If the reinvestment rate is greater than the the yield to maturity, the realized compound return will also be greater, and vice versa. Unlike yield to maturity, reinvestment rates of interim cash flows (coupons) will be determined by the market interest rates available at the time of each coupon and realized in the future (and are thus uncertain). Forecasting the realized compound yield over various holding periods or investment horizons is called horizon analysis. Realized Return without Interim Cash Flows Rt+1 =

Pt+1 + ct+1 −1 Pt

where Pt is the price at time t and ct+1 is the cash flow at time t + 1. With interim cash flows, reinvest them to the end of the holding period (at the market interest rates available at different horizons). Calculate the gross return over the holding period (1 + R) by dividing aggregate cash flows by the price. Annualized Return 1

(1 + R) T − 1 where T is the holding period in years.

1.4

Default Risk and Other Assumptions Bond default risk, usually called credit risk is the risk that the bond does not pay the promised fixed flow of income. Other frequent assumptions are:

FINS2624: Portfolio Management

Yichen Han

1 INTRODUCTION TO BOND PRICING

3

– No transaction costs – Constant interest rates – Complete markets

1.5

Approaches to Pricing Two approaches to pricing are fundamental pricing (under a supply-demand equilibrium) and arbitrage pricing (replicating portfolios). Arbitrage is a set of trades that generate zero cash flows in the future, but a positive cash flow today (that is risk free by default). This is also equivalent to zero cash flows today, but a positive and risk-free cash flow in the future. All arbitrage pricing is based on the same no-arbitrage principle, that is, securities with identical cash flows should have the same price in the equilibrium (the law of one price). To price an asset based on the no-arbitrage principle, construct a portfolio of other assets that exactly mimic the cash flows of the asset to price (the replicating portfolio or synthetic asset). This will usually involve short selling (or shorting) strategies (selling securities you don’t hold). The bond pricing formula in 1.2 is the arbitrage-free price.

FINS2624: Portfolio Management

Yichen Han

2 TERM STRUCTURE OF INTEREST RATES

2 2.1

4

Term Structure of Interest Rates The Yield Curve The yield curve is a plot of yield to maturity as a function of time to maturity. The pure yield curve refers to the curve for stripped, or zero-coupon, Treasuries. In contrast, the on-the-run yield curve refers to the plot of yield as a function of maturity for recently issued coupon bonds selling at or near par value.

2.2

The Yield Curve and Future Interest Rates To distinguish between yields on long-term bonds versus short-term rates that will be available in the future, practitioners use the following terminology. They call the yield to maturity on zero-coupon bonds (ZCB) the spot rate, meaning the rate that prevails today for a time period corresponding to the zero’s maturity. In contrast, the short rate for a given time interval (e.g. 1 year) refers to the interest rate for that interval available at different points in time. Conceptually, t-period spot rate (yt ) clearly differs from yield to maturity ( y) of a t-period bond. For zero-coupon bonds, these two values are the same. Therefore, spot rate is also informally called pure yield. Price of a t-year ZCB FV P = (1 + yt )t

⇔ yt =



FV P

1 t

−1

Price of a T -year Coupon-payment Bond P =

T X

t=1

FV CF t + (1 + yt )t (1 + yT )T

To infer the term structure from 1-year zero and 2-year coupon-payment bonds, the 1-year spot rate y1 is easily inferred, and sub the result into the price equation for the second bond to find y2. To infer the term structure from two coupon-payment bonds, simultaneously solve the two price equations for the two bonds. The forward rate can be defined as the break-even interest rate that equates the return on an n -period zero-coupon bond to that of an (n − 1)-period zero-coupon bond rolled over into a 1-year bond in year n. This is denoted by n−1 fn . One-period Forward Rate (1 + n−1 fn ) =

(1 + yn )n (1 + yn−1 )n−1

Multi-period Forward Rate (t > s)

(1 + s ft ) =

FINS2624: Portfolio Management

"

(1 + yt )t (1 + ys )s

1 # t−s

Yichen Han

2 TERM STRUCTURE OF INTEREST RATES

2.3

5

Theories of the Term Structure The simplest theory of the term structure is the expectations hypothesis . A common version of this hypothesis states that the forward rate equals the market consensus expectation of the future short interest rate. That is, s ft = E(s yt ), and liquidity premiums are zero. However, the term structure is typically upward sloping i.e. 1 f2 > y2 > y1 and thus the market expects the interest rate to be higher in the future, which is not always the case. The liquidity preference theory (or preferred habitat theory) takes into account the different horizons, or preferred habitats bond issuers and investors may have. Maturity mismatches between issuers and investors result in liquidity and reinvestment risks. – If investor’s investment horizon < issuer’s horizon, liquidity risk occurs. – If investor’s investment horizon > issuer’s horizon, reinvestment risk occurs. Since short-term investors carry the liquidity risk of holding long-term bonds, a liquidity premium is offered to induce these investors to hold the long-term bonds. This results in higher yields for long-term bonds, and thus a upward-sloping term structure. Since long-term investors carry the reinvestment risk of holding short-term bonds, a liquidity premium is offered to induce these investors to hold the short-term bonds. This results in higher yields for short-term bonds, and thus a downward-sloping term structure.

FINS2624: Portfolio Management

Yichen Han

3 DURATION

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3 3.1

Duration Interest Rate Risk Bond prices and yields are inversely related: as yields increase, bond prices fall; as yields fall, bond prices rise. An increase in a bond’s yield to maturity results in a smaller price change in a decrease in yield of equal magnitude. Prices of long-term bonds tend to be more sensitive to interest rate changes than prices of short-term bonds. The sensitivity of bond prices to changes in yields increases at a decreasing rate as maturity increases. In other words, interest rate risk is less than proportional to bond maturity. Interest rate risk is inversely related to the bond’s coupon rate. Prices of low-coupon bonds are more sensitive to changes in interest rates than prices of high-coupon bonds. The sensitivity of a bond’s price to a change in its yield is inversely related to the yield to maturity at which the bond currently is selling. Macaulay’s duration equals the weighted average of the times to each coupon or principal payment. The weight associated with each payment time clearly should be related to the “importance” of that payment to the value of the bond. Define the weight wt associated with the cash flow made at time t (denoted CFt ) as:

wt =

CFt (1+y)t

Bond Price

where y is the bond’s yield to maturity. Macaulay’s duration formula is as follows: Macaulay’s Duration D= =

T X

t × wt

t=1 T X

PV(CFt ) t × PT t=1 PV(CFt ) t=1

Duration can also be expressed as part of the change in price relative to the change in yield.

D=−

∂P P ∂y (1+y )

∂y ∂P = −D × (1 + y) P Modified Duration (MD) ∂P D × ∂y = −MD × ∂y =− 1+y P

FINS2624: Portfolio Management

Yichen Han

3 DURATION

7

Duration can be determined in three ways: 1. as an interest risk measure: the sensitivity of market value change to change in interest rate, 2. as a maturity/life measure: the average economic life of a bond, 3. as an ‘economic’ payback period: how long on average does it take to get back the cost of the investment in present value. Rules for duration: 1. The duration of a zero-coupon bond equals its time to maturity. 2. Holding maturity constant, a bond’s duration is lower when the coupon rate is higher. 3. Holding the coupon rate constant, a bond’s duration generally increases with its time to maturity. Duration always increases with maturity for bonds selling at par or at a premium to par. 4. Holding other factors constant, the duration of a coupon bond is higher when the bond’s yield to maturity is lower. 1+y . 5. The duration of a perpetuity is y

3.2

Convexity Convexity is the curvature of the price yield curve. Convexity 1 ∆P = −MD × ∆y + × C × (∆y)2 2 P

3.3

Immunization Match the duration of assets and liabilities such that the two cancel each other out, i.e. DA = DL. Using two bonds A and B, DP = XA × DA + (1 − XA )DB = DL where XA is the weight of bond A. XA =

DL − DB DA − DB

Immunization is a dynamic process as parameters change constantly.

FINS2624: Portfolio Management

Yichen Han

4 MARKOWITZ PORTFOLIO THEORY

Markowitz Portfolio Theory

4 4.1

8

Utility Function A utility function assigns a value to each outcome so that preferred outcomes get higher values. A perfect utility function should be able to used to assign values to every possible outcome, but it is difficult in practice. In finance and especially investment setting, risk usually refers to possibility that realized outcomes differ (better or worse) than what is expected. Investors tend to be risk averse, i.e. prefer certain outcomes. A risk premium provides risk averse investors an incentive to take investments with risk (e.g. liquidity premium). The risk-free rate rf is positive as there should be a positive return despite there being no risk. In this course’s context, the utility function of a portfolio is based on a quadratic function. Utility Function of a Portfolio 1 U = E(r) − Aσ 2 2 where U is the utility, E(r) is the expected return, A being the degree of risk aversion, and σ 2 being the variance of the portfolio’s return. For a risk-free portfolio σ 2 = 0. We want higher expected return, and lower risk (standard deviation). Whenever one portfolio has higher E(r) and lower σ 2 than some other portfolio, the former is dominating the latter. We can illustrate our preferences by indifference curves, i.e. curves in risk-return space that connect points giving equal utility. We want to achieve the optimal investment outcome that maximises our utility.

4.2

Return of Portfolios Return of a Portfolio of Assets rP = ...