Econ133Notes Documentation · Econ133Notes Documentation, Release 0.2 1.8Properties of Expectation For random variables and and constants , ∈ , the expected value has the following

Econ133Notes DocumentationRelease 0.2

Eric M. Aldrich

June 07, 2014

Contents

1 Probability 31.1 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Probability Mass Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Probability Density Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Properties of Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 Cumulative Distribution Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.6 Expected Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.7 Expected Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.8 Properties of Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.9 Properties of Expectation - Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.10 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.11 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.12 Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.13 Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.14 Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.15 Properties of Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.16 Properties of Variance - Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.17 Properties of Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.18 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.19 Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.20 Normal Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.21 Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.22 Standard Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.23 Standard Normal Distribution - Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.24 Standard Normal Distribution - Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.25 Sum of Normals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.26 Sample Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.27 Sample Mean (Cont.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.28 Sample Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.29 Other Sample Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Markets 112.1 Asset Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Indexes and Funds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Trading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Risk and Returns 233.1 Rates of Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

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3.2 Risk and Risk Premiums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4 Portfolio Optimization 334.1 Asset Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2 Portfolio Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.3 Portfolio Optimization with Many Risky Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5 Index Models 495.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.3 Single-Factor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.4 Single-Factor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.5 Decomposing Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.6 Decomposing Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.7 Using an Index as a Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.8 Using an Index as a Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.9 Single-Index Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.10 Single-Index Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.11 Expected Return-Beta Relationship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.12 Why 𝛼𝑀𝑢𝑠𝑡𝐵𝑒𝑍𝑒𝑟𝑜 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.13 Single-Index Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.14 Security Characteristic Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.15 Security Characteristic Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.16 Advantages of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.17 Advantages of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.18 Advantages of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.19 Advantages of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.20 Cost of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.21 Index Model Portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.22 Index Model Portfolio Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.23 Index Model Portfolio Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.24 Index Model Portfolio Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.25 Index Model Diversification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.26 Index Model Diversification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.27 Index Model Diversification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.28 Index Model Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.29 Index Model Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.30 Index Model Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.31 Index Model Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

6 Bonds 576.1 Bond Prices and Yields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576.2 The Term Structure of Interest Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

7 Options 717.1 Call Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717.2 Call Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717.3 Call Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717.4 Call Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717.5 Call Option Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727.6 Call Option Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727.7 Put Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727.8 Put Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727.9 Put Option Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727.10 Put Option Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

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7.11 Moneyness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737.12 American vs. European . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737.13 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737.14 Call Option Payoff (Buyer) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747.15 Call Option Payoff (Buyer) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747.16 Call Option Payoff (Buyer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747.17 Call Option Payoff (Seller) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747.18 Call Option Payoff (Seller) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747.19 Put Option Payoff (Buyer) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747.20 Put Option Payoff (Buyer) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757.21 Put Option Payoff (Buyer) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757.22 Speculation and Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757.23 Speculation and Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757.24 Speculation and Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757.25 Speculation and Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767.26 Speculation and Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767.27 Speculation and Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767.28 Speculation and Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777.29 Protective Put . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777.30 Protective Put . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777.31 Protective Put . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777.32 Protective Put . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777.33 Covered Call . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777.34 Covered Call . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787.35 Straddle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787.36 Straddle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787.37 Straddle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797.38 Spread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797.39 Bullish Spread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797.40 Bullish Spread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797.41 Collar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797.42 Protective Put Alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797.43 Put Call Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807.44 Put Call Parity Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807.45 Put Call Parity Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807.46 Put Call Parity Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

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Contents:

Contents 1


2 Contents

CHAPTER 1

Probability

1.1 Random Variables

Suppose 𝑋 is a random variable which can take values 𝑥 ∈ 𝒳 .

• 𝑋 is a discrete r.v. if 𝒳 is countable.

– 𝑝(𝑥) is the probability of a value 𝑥 and is called the probability mass function.

• 𝑋 is a continuous r.v. if 𝒳 is uncountable.

– 𝑓(𝑥) is called the probability density function and can be thought of as the probability of a value𝑥.

1.2 Probability Mass Function

For a discrete random variable the probability mass function (PMF) is

𝑝(𝑎) = 𝑃 (𝑋 = 𝑎),

where 𝑎 ∈ 𝑅.

1.3 Probability Density Function

If 𝐵 = (𝑎, 𝑏)

𝑃 (𝑋 ∈ 𝐵) = 𝑃 (𝑎 ≤ 𝑋 ≤ 𝑏) =

∫︁ 𝑏

𝑎

𝑓(𝑥)𝑑𝑥.

Strictly speaking

𝑃 (𝑋 = 𝑎) =

∫︁ 𝑎

𝑎

𝑓(𝑥)𝑑𝑥 = 0,

but we may (intuitively) think of 𝑓(𝑎) = 𝑃 (𝑋 = 𝑎).

3


1.4 Properties of Distributions

For discrete random variables

• 𝑝(𝑥) ≥ 0, ∀𝑥 ∈ 𝒳 .

•∑︀

𝑥∈𝒳 𝑝(𝑥) = 1.

For continuous random variables

• 𝑓(𝑥) ≥ 0, ∀𝑥 ∈ 𝒳 .

•∫︀𝑥∈𝒳 𝑓(𝑥)𝑑𝑥 = 1.

1.5 Cumulative Distribution Function

For discrete random variables the cumulative distribution function (CDF) is

• 𝐹 (𝑎) = 𝑃 (𝑋 ≤ 𝑎) =∑︀

𝑥≤𝑎 𝑝(𝑥).

For continuous random variables the CDF is

• 𝐹 (𝑎) = 𝑃 (𝑋 ≤ 𝑎) =∫︀ 𝑎

−∞ 𝑓(𝑥)𝑑𝑥.

1.6 Expected Value

For a discrete r.v. 𝑋 , the expected value is

𝐸[𝑋] =∑︁𝑥∈𝒳

𝑥𝑝(𝑥).

For a continuous r.v. 𝑋 , the expected value is

𝐸[𝑋] =

∫︁𝑥∈𝒳

𝑥 𝑓(𝑥)𝑑𝑥.

1.7 Expected Value

If 𝑌 = 𝑔(𝑋), then

• For discrete r.v. 𝑋

𝐸[𝑌 ] = 𝐸[𝑔(𝑋)] =∑︁𝑥∈𝒳

𝑔(𝑥)𝑝(𝑥).

• For continuous r.v. 𝑋

𝐸[𝑌 ] = 𝐸[𝑔(𝑋)] =

∫︁𝑥∈𝒳

𝑔(𝑥)𝑓(𝑥)𝑑𝑥.

4 Chapter 1. Probability


1.8 Properties of Expectation

For random variables 𝑋 and 𝑌 and constants 𝑎, 𝑏 ∈ 𝑅, the expected value has the following properties (for bothdiscrete and continuous r.v.’s):

• 𝐸[𝑎𝑋 + 𝑏] = 𝑎𝐸[𝑋] + 𝑏.

• 𝐸[𝑋 + 𝑌 ] = 𝐸[𝑋] + 𝐸[𝑌 ].

Realizations of 𝑋 , denoted by 𝑥, may be larger or smaller than 𝐸[𝑋].

• If you observed many realizations of 𝑋 , 𝐸[𝑋] is roughly an average of the values you would observe.

1.9 Properties of Expectation - Proof

𝐸[𝑎𝑋 + 𝑏] =

∫︁ ∞

−∞(𝑎𝑥 + 𝑏)𝑓(𝑥)𝑑𝑥

=

∫︁ ∞

−∞𝑎𝑥𝑓(𝑥)𝑑𝑥 +

∫︁ ∞

−∞𝑏𝑓(𝑥)𝑑𝑥

= 𝑎

∫︁ ∞

−∞𝑥𝑓(𝑥)𝑑𝑥 + 𝑏

∫︁ ∞

−∞𝑓(𝑥)𝑑𝑥

= 𝑎𝐸[𝑋] + 𝑏.

1.10 Variance

Generally speaking, variance is defined as

𝑉 𝑎𝑟(𝑋) = 𝐸[︀(𝑋 − 𝐸[𝑋])2

]︀.

If 𝑋 is discrete:

𝑉 𝑎𝑟(𝑋) =∑︁𝑥∈𝒳

(𝑥− 𝐸[𝑋])2𝑝(𝑥).

If 𝑋 is continuous:

𝑉 𝑎𝑟(𝑋) =

∫︁𝑥∈𝒳

(𝑥− 𝐸[𝑋])2𝑓(𝑥)𝑑𝑥

1.11 Variance

Using the properties of expectations, we can show 𝑉 𝑎𝑟(𝑋) = 𝐸[𝑋2] − 𝐸[𝑋]2:

𝑉 𝑎𝑟(𝑋) = 𝐸[︀(𝑋 − 𝐸[𝑋])2

]︀= 𝐸

[︀𝑋2 − 2𝑋𝐸[𝑋] + 𝐸[𝑋]2

]︀= 𝐸[𝑋2] − 2𝐸[𝑋]𝐸[𝑋] + 𝐸[𝑋]2

= 𝐸[𝑋2] − 𝐸[𝑋]2.

1.8. Properties of Expectation 5


1.12 Standard Deviation

The standard deviation is simply

𝑆𝑡𝑑(𝑋) =√︀

𝑉 𝑎𝑟(𝑋).

• 𝑆𝑡𝑑(𝑋) is in the same units as 𝑋 .

• 𝑉 𝑎𝑟(𝑋) is in units squared.

1.13 Covariance

For two random variables 𝑋 and 𝑌 , the covariance is generally defined as

𝐶𝑜𝑣(𝑋,𝑌 ) = 𝐸 [(𝑋 − 𝐸[𝑋])(𝑌 − 𝐸[𝑌 ])]

Note that 𝐶𝑜𝑣(𝑋,𝑋) = 𝑉 𝑎𝑟(𝑋).

1.14 Covariance

Using the properties of expectations, we can show

𝐶𝑜𝑣(𝑋,𝑌 ) = 𝐸[𝑋𝑌 ] − 𝐸[𝑋]𝐸[𝑌 ].

This can be proven in the exact way that we proved

𝑉 𝑎𝑟(𝑋) = 𝐸[𝑋2] − 𝐸[𝑋]2.

In fact, note that

𝐶𝑜𝑣(𝑋,𝑋) = 𝐸[𝑋𝑌 ] − 𝐸[𝑋]𝐸[𝑌 ]

= 𝐸[𝑋2] − 𝐸[𝑋]2 = 𝑉 𝑎𝑟(𝑋).

1.15 Properties of Variance

Given random variables 𝑋 and 𝑌 and constants 𝑎, 𝑏 ∈ 𝑅,

𝑉 𝑎𝑟(𝑎𝑋 + 𝑏) = 𝑎2𝑉 𝑎𝑟(𝑋).

𝑉 𝑎𝑟(𝑎𝑋 + 𝑏𝑌 ) = 𝑎2𝑉 𝑎𝑟(𝑋) + 𝑏2𝑉 𝑎𝑟(𝑌 )

+ 2𝑎𝑏𝐶𝑜𝑣(𝑋,𝑌 ).

The latter property can be generalized to

𝑉 𝑎𝑟

(︃𝑛∑︁

𝑖=1

𝑎𝑖𝑋𝑖

)︃=

𝑛∑︁𝑖=1

𝑎2𝑖𝑉 𝑎𝑟(𝑋𝑖)

+ 2

𝑛−1∑︁𝑖=1

𝑛∑︁𝑗=𝑖+1

𝑎𝑖𝑎𝑗𝐶𝑜𝑣(𝑋𝑖, 𝑋𝑗).



1.16 Properties of Variance - Proof

𝑉 𝑎𝑟(𝑎𝑋 + 𝑏𝑌 ) = 𝐸[︀(𝑎𝑋 + 𝑏𝑌 )2

]︀− 𝐸 [𝑎𝑋 + 𝑏𝑌 ]

2

= 𝐸[𝑎2𝑋2 + 𝑏2𝑌 2 + 2𝑎𝑏𝑋𝑌 ] − (𝑎𝐸[𝑋] + 𝑏𝐸[𝑌 ])2

= 𝑎2𝐸[𝑋2] + 𝑏2𝐸[𝑌 2] + 2𝑎𝑏𝐸[𝑋𝑌 ]

− 𝑎2𝐸[𝑋]2 − 𝑏2𝐸[𝑌 ]2 − 2𝑎𝑏𝐸[𝑋]𝐸[𝑌 ]

= 𝑎2(︀𝐸[𝑋2] − 𝐸[𝑋]2

)︀+ 𝑏2

(︀𝐸[𝑌 2] − 𝐸[𝑌 ]2

)︀+ 2𝑎𝑏 (𝐸[𝑋𝑌 ] − 𝐸[𝑋]𝐸[𝑌 ])

= 𝑎2𝑉 𝑎𝑟(𝑋) + 𝑏2𝑉 𝑎𝑟(𝑌 ) + 2𝑎𝑏𝐶𝑜𝑣(𝑋,𝑌 ).

1.17 Properties of Covariance

Given random variables 𝑊 , 𝑋 , 𝑌 and 𝑍 and constants 𝑎, 𝑏 ∈ 𝑅,

𝐶𝑜𝑣(𝑋, 𝑎) = 0.

𝐶𝑜𝑣(𝑎𝑋, 𝑏𝑌 ) = 𝑎𝑏𝐶𝑜𝑣(𝑋,𝑌 ).

𝐶𝑜𝑣(𝑊 + 𝑋,𝑌 + 𝑍) = 𝐶𝑜𝑣(𝑊,𝑌 ) + 𝐶𝑜𝑣(𝑊,𝑍)

+ 𝐶𝑜𝑣(𝑋,𝑌 ) + 𝐶𝑜𝑣(𝑋,𝑍).

The latter two can be generalized to

𝐶𝑜𝑣

⎛⎝ 𝑛∑︁𝑖=1

𝑎𝑖𝑋𝑖,

𝑚∑︁𝑗=1

𝑏𝑗𝑌𝑗

⎞⎠ =

𝑛∑︁𝑖=1

𝑚∑︁𝑗=1

𝑎𝑖𝑏𝑗𝐶𝑜𝑣(𝑋𝑖, 𝑌𝑗).

1.18 Correlation

Correlation is defined as

𝐶𝑜𝑟𝑟(𝑋,𝑌 ) =𝐶𝑜𝑣(𝑋,𝑌 )

𝑆𝑡𝑑(𝑋)𝑆𝑡𝑑(𝑌 ).

• It is fairly easy to show that −1 ≤ 𝐶𝑜𝑟𝑟(𝑋,𝑌 ) ≤ 1.

• The properties of correlations of sums of random variables follow from those of covariance and standard devia-tions above.

1.19 Normal Distribution

The normal distribution is often used to approximate the probability distribution of returns.

• It is a continuous distribution.

• It is symmetric.

• It is fully characterized by 𝜇 (mean) and 𝜎 (standard deviation) – i.e. if you only tell me 𝜇 and 𝜎, Ican draw every point in the distribution.

1.16. Properties of Variance - Proof 7


1.20 Normal Density

If 𝑋 is normally distributed with mean 𝜇 and standard deviation 𝜎, we write

𝑋 ∼ 𝒩 (𝜇, 𝜎).

The probability density function is

𝑓(𝑥) =1√2𝜋𝜎

exp

{︂1

2𝜎2(𝑥− 𝜇)2

}︂.

1.21 Normal Distribution

From Wikipedia:

1.22 Standard Normal Distribution

Suppose 𝑋 ∼ 𝒩 (𝜇, 𝜎).

Then

𝑍 =𝑋 − 𝜇

𝜎

is a standard normal random variable: 𝑍 ∼ 𝒩 (0, 1).

• That is, 𝑍 has zero mean and unit standard deviation.

We can reverse the process by defining

𝑋 = 𝜇 + 𝜎𝑍.

1.23 Standard Normal Distribution - Proof

𝐸[𝑍] = 𝐸

[︂𝑋 − 𝜇

𝜎

]︂=

1

𝜎𝐸[𝑋 − 𝜇]

=1

𝜎(𝐸[𝑋] − 𝜇)

=1

𝜎(𝜇− 𝜇)

= 0.


http://en.wikipedia.org/wiki/Normal_distribution


1.24 Standard Normal Distribution - Proof

𝑉 𝑎𝑟(𝑍) = 𝑉 𝑎𝑟

(︂𝑋 − 𝜇

𝜎

)︂= 𝑉 𝑎𝑟

(︂𝑋

𝜎− 𝜇

𝜎

)︂=

1

𝜎2𝑉 𝑎𝑟(𝑋)

=𝜎2

𝜎2

= 1.

1.25 Sum of Normals

Suppose 𝑋𝑖 ∼ 𝒩 (𝜇𝑖, 𝜎𝑖) for 𝑖 = 1, . . . , 𝑛.

Then if we denote 𝑊 =∑︀𝑛

𝑖=1 𝑋𝑖

𝑊 ∼ 𝒩

⎛⎝ 𝑛∑︁𝑖=1

𝜇𝑖,

⎯⎸⎸⎷ 𝑛∑︁𝑖=1

𝜎2𝑖 + 2

𝑗∑︁𝑖=1

𝑛∑︁𝑗=1

𝐶𝑜𝑣(𝑋𝑖, 𝑋𝑗)

⎞⎠ .

How does this simplify if 𝐶𝑜𝑣(𝑋𝑖, 𝑋𝑗) = 0 for 𝑖 ̸= 𝑗?

1.26 Sample Mean

Suppose we don’t know the true probabilities of a distribution, but would like to estimate the mean.

• Given a sample of observations, {𝑥𝑖}𝑛𝑖=1, of random variable 𝑋 , we can estimate the mean by

�̂� =1

𝑛

𝑛∑︁𝑖=1

𝑥𝑖.

• This is just a simple arithmetic average, or a probability weighted average with equal probabilities: 1𝑛 .

• But the true mean is a weighted average using actual (most likely, unequal) probabilities. How do we reconcilethis?

1.27 Sample Mean (Cont.)

Given that the sample {𝑥𝑖}𝑛𝑖=1 was drawn from the distribution of 𝑋 , the observed values are inherently weighted bythe true probabilities (for large samples).

• More values in the sample will be drawn from the higher probability regions of the distribution.

• So weighting all of the values equally will naturally give more weight to the higher probabilityoutcomes.

1.24. Standard Normal Distribution - Proof 9


1.28 Sample Variance

Similarly, the sample variance can be defined as

�̂�2 =1

𝑛− 1

𝑛∑︁𝑖=1

(𝑥𝑖 − �̂�)2.

Notice that we use 1𝑛−1 instead of 1

𝑛 for the sample average.

• This is because a simple average using 1𝑛 underestimates the variability of the data because it doesn’t account

for extra error involved in estimating �̂�.

1.29 Other Sample Moments

Sample standard deviations, covariances and correlations are computed in a similar fashion.

• Use the definitions above, replacing expectations with simple averages.


CHAPTER 2

Markets

Contents:

2.1 Asset Classes

2.1.1 Introduction

What is an investment?

• A commitment of resources with the expectation of a future payoff.

• Financial investments: stocks, bonds, options, futures.

• Other investments: education, physical fitness, relationships.

Decisions faces by investors:

• Risk-return trade-off.

• Efficient pricing of financial assets.

2.1.2 Real vs. Financial Assets

Real assets are goods (generally tangible) that are used to produce other goods or services: buildings, machines, land,knowledge.

• Productivity of economy determined by real assets.

Financial assets are claims to income generated by real assets.

• Firms use the money raised through financial assets to invest in plants, equipment, labor, etc.

• Holder of financial asset receives a portion of the resulting returns from real assets.

While real assets determine wealth, financial assets determine distribution of wealth.

2.1.3 Equity Assets

Equity (stock) is an ownership claim in a particular firm.

• Equity holder owns a prorated share of the real assets of a firm and is entitled to a portion of the profits that isnot reinvested (dividends).

11


• The return to an equity asset is not guaranteed: it is tied to the financial well-being of the firm.

2.1.4 Equity Example

Consider The Coca-Cola Company.

• On 14 Jan 2014, there were 4.42 billion shares of KO outstanding.

• On 28 Nov 2013, KO decided to pay $1.24 billion in dividends to shareholders.

• This amounted to $0.28 paid to each share.

• If the company were liquidated, each share would entitle the holder to 1/4,420,000,000 of all KO assets.

2.1.5 Equity Prices

There are several ways to think about the price of an equity share.

• Add the values of all firm assets and divide by the number of shares outstanding.

• Alternatively, it is the present value of all future payments (dividends).

𝑃 =

∞∑︁𝑡=1

𝐷𝑡

(1 + 𝑟)𝑡

• 𝐷𝑡 represents the dividend payment at future date 𝑡, and 𝑟 is the competitive interest rate that you could otherwiseearn on your money.

2.1.6 Present Value

Recall that present value is a way to value future payments in current terms.

• Money paid to you in the future should be less than money paid now.

• Why?

2.1.7 Present Value

Suppose you can earn 10% interest on every dollar you save tonight.

• Then if you save $0.9091 tonight, it will be worth $0.9091 × 1.1 = $1 tomorrow.

• If I offer you either $0.92 today or $1 tomorrow, which would you pick?

• If I offer you either $0.90 today or $1 tomorrow, which would you pick?

2.1.8 Fixed Income Assets

Fixed income securities are assets with payments determined by a formula (e.g. bonds).

• U.S. Treasury bills/bonds, commercial paper, CDs.

• Although fixed income payments are determined mathematically there is still risk (interest rate risk, defaultrisk).

• Longer maturity fixed income assets tend to be more risky.

• Fixed income is typically less risky than equity because the payments are guaranteed.

12 Chapter 2. Markets

http://bit.ly/1ckSWAC

http://en.wikipedia.org/wiki/Present_value


2.1.9 Fixed Income Assets

• Fixed income assets are typically a way for firms or governments to borrow money.

• It is possible to hold both equity and debt (fixed income) assets for the same firm.

2.1.10 Fixed Income Prices

Fixed income assets typically promise a stream of payments at future dates.

• Coupon payments at regular intervals.

• A final principal payment (face value) at the end.

• The price of a fixed income asset is the present value of these payments.

𝑃 =

𝑇∑︁𝑡=1

𝐶𝑡

(1 + 𝑟)𝑡+

𝐹

(1 + 𝑟)𝑇.

2.1.11 Derivatives

Derivatives are assets whose payoffs are determined by another (underlying) asset.

• Options are assets which allow the holder the option to buy or sell an asset at a pre-specified price at a futuredate.

• Futures are contracts to buy and sell assets (real or financial) for a pre-specified price at a future date.

• These assets are called derivatives because their value derives from the value of the underlying asset.

• Derivatives are commonly used for hedging (insurance) but are sometimes used for speculation, too.

2.2 Indexes and Funds

2.2.1 Indexes

Indexes are weighted averages of asset characteristics.

• For example, it might be a weighted average of stock prices, stock returns, or bond yields.

• The two most widely cited indexes are:

– The Dow Jones Industrial Average (DJIA).

– The Standard and Poor’s Composite 500 (S&P 500).

• These are reported every day in common news outlets.

2.2.2 Dow Jones

The Dow Jones Industrial Average (DJIA) is the oldest U.S. index, dating to 1896.

• Since 1926 it has included 30 large stocks.

• Originally a simple average of the prices.

2.2. Indexes and Funds 13

http://en.wikipedia.org/wiki/Djia

http://en.wikipedia.org/wiki/S%26P_500

http://wsj.com


• Percentage change in the Dow was originally the return on a portfolio consisting of one share invested in eachof the stocks in the index.

2.2.3 Dow Jones

• The DJIA is a price-weighted average: the amount of money invested in each asset of the portfolio is proportionalto the share price.

• Due to splits and changes in the composition of the index, the DJIA is no longer a simple weighted average ofprices.

2.2.4 Price-Weighted Indexes

Consider a price-weighted index of two stocks, 𝑋 and 𝑌 .

• The price of 𝑋 is originally $25 and increases to $30.

• The price of 𝑌 is originally $100 and decreases to $90.

Then

• Initial index value = $25+$1002 = $62.5.

• Final index value = $30+$902 = $60.

• Percentage change = −$2.5$62.5 = −0.04 = −4%.


Note that price-weighted indexes give higher priced stocks more weight.

• The percentage change in stock 𝑋 is:

$30 − $25

$25= 0.2 = 20%

• The percentage change in stock 𝑌 is:

$90 − $100

$100= −0.1 = −10%


• The overall percentage change in the index is

%change in index =𝑝0𝑋

𝑝0𝑋 + 𝑝0𝑌∆𝑋 +

𝑝0𝑌𝑝0𝑋 + 𝑝0𝑌

∆𝑌

= 0.2 * 0.2 + 0.8 * (−0.1) = −0.04.

• 𝑝0𝑖 is the initial price of stock 𝑖.

• ∆𝑖 is the percentage change in the price of stock 𝑖.



2.2.7 Splits and Price-Weighted Averages

Suppose that stock 𝑌 split, causing its price to fall to $50.

• This would cause a large fall in the value of the index, unless an adjustment is made to the divisor.

• That is, the index value before the split is

$25 + $100

2= $62.5.

• The post-split divisor, 𝑑, should be the value such that

$25 + $50

𝑑= $62.5.

2.2.8 Splits and Price-Weighted Averages

• Hence, 𝑑 falls from 2 to 1.2.

• Notice that since the split causes the price of 𝑌 to fall, it’s relative weight in the portfolio will fall.

• Movements in the price of 𝑌 will have a smaller impact on the index.

2.2.9 Standard and Poor’s Composite 500

The S&P 500 stock index has two advantages over the Dow:

• It is comprised of 500 large stocks, and hence is more broadly based and a better indicator of the market as awhole.

• It is a value-weighted, rather than price-weighted, index.

• The market value or market capitalization of a firm is simply its total value on the market: price per share timesthe number of shares outstanding.

• A value-weighted index weights each stock in the index according to its market cap.

2.2.10 Value-Weighted Indexes

If stock 𝑋 currently has 20 shares trading in the market and stock 𝑌 only has 1 share, the market caps for 𝑋 and 𝑌are

𝑀𝐶0𝑋 = 20 * $25 = $500

𝑀𝐶0𝑌 = 1 * $100 = $100.

• A value-weighted index of the two stocks would give 𝑋 five times the weight as 𝑌 .

• Compare to the price-weighted index which gives 𝑌 four times the weight.

• Initially, the total stock on the market is equal to $500 + $100 = $600.

2.2. Indexes and Funds 15



After the price changes, market caps become

𝑀𝐶1𝑋 = 20 * $30 = $600

𝑀𝐶1𝑌 = 1 * $90 = $90.

• The total value of stock is now $690.

• If the initial value of the value weighted index was $100, after the price changes it would be $100* $690$600 = $115.


• In this case the value of the index rises since it gives a relatively higher weight to 𝑋 .

%change in index =𝑀𝐶0

𝑋

𝑀𝐶0𝑋 + 𝑀𝐶0

𝑌

∆𝑋 +𝑀𝐶0

𝑌

𝑀𝐶0𝑋 + 𝑀𝐶0

𝑌

∆𝑌

=5

6* 0.2 +

1

6* (−0.1) = 0.15.

2.2.13 Equally-Weighted Indexes

One of the advantages of price-weighted and value-weighted indexes is that they correspond to buy-and-hold portfoliostrategies:

• A price-weighted index is equivalent to buying and holding one share (or an equal number of shares) of eachstock in the index.

• A value-weighted index is equivalent to buying and holding each share of the index in proportion to its marketcap.

2.2.14 Equally-Weighted Indexes (Cont.)

In contrast, one could form an equally-weighted index, where all stocks receive the exact same weight.

• This does not correspond to a buy-and-hold strategy.

• Consider starting with with equal amounts of money invested in stocks 𝑋 and 𝑌 .

• If the price of 𝑋 increases by 20% and the price of 𝑌 falls by 10%, the dollar amount invested in each stock isno longer equal.

• To keep the investment equally weighted, you would have to sell some shares of 𝑋 and buy shares of 𝑌 .

2.2.15 Other Indexes

There are a wide number of published indexes:

• Nasdaq Composite.

• NYSE Composite.

• Wilshire 5000.


http://en.wikipedia.org/wiki/Nasdaq_Composite

http://en.wikipedia.org/wiki/NYSE_Composite

http://en.wikipedia.org/wiki/Wilshire_5000


• Sub-indexes of the S&P 500 and others above.

To hold these as part of a portfolio one could

• Buy shares of a mutual fund that tracks the index.

• Buy shares of an Exchange Traded Fund (ETF) which is a portfolio of the shares in the index.

2.3 Trading

2.3.1 Exchanges

Most financial assets today are traded on exchanges.

• The first stock exchange, the Antwerp Bourse, was established in Antwerp, Belgium in 1460.

• Others were established throughout Europe in the 1500s.

• The oldest existing stock exchange is the Amsterdam Stock Exchange, which was established in 1602.

• The Amsterdam Stock Exchange merged with the Paris and Brussels exchanges in 2000 to form Euronext.

• Euronext merged with the New York Stock Exchange (NYSE) in 2007.

2.3.2 Early Trading

For centuries, trading of financial assets consisted of exchanging physical certificates.

• The New York Stock Exchange was Conceived in 1792 under a buttonwood tree on Wall Street.

• 24 brokers signed an agreement to exclusively deal with each other.

• They frequently met under the tree on Wall Street to conduct business.

2.3. Trading 17

http://en.wikipedia.org/wiki/Mutual_fund

http://en.wikipedia.org/wiki/Exchange-traded_fund

http://lat.ms/1jqdmAQ

http://en.wikipedia.org/wiki/Amsterdam_Stock_Exchange

http://en.wikipedia.org/wiki/Euronext

http://en.wikipedia.org/wiki/New_York_Stock_Exchange


2.3.3 Early Trading

2.3.4 Early Trading

2.3.5 Electronic Trading

In the 1990s, several electronic exchanges were developed.

• These are often referred to as Electronic Communication Networks or ECNs.

• Island was one of the first ECNs, later purchased by Nasdaq.

• Archipelago was developed at roughly the same time and was later purchased by NYSE (hence the name, NYSEArca).

2.3.6 High-Frequency Trading

Originally electronic systems were simply used for quoting and displaying prices.

• Trades were still executed by hand on the trading floor.

• Eventually, traders started placing orders electronically.


http://www.investopedia.com/terms/e/ecn.asp

http://en.wikipedia.org/wiki/Island_Exchange

http://en.wikipedia.org/wiki/NASDAQ

http://en.wikipedia.org/wiki/Archipelago_Exchange


• This led to algorithmic trading: traders not only submitted orders electronically, but wrote software algorithmsthat executed trades intelligently, depending on market conditions.

• Algorithmic trading has resulted in very fast execution, commonly referred to as high-frequency trading.

2.3.7 Exchanges Today

Exchanges today no longer consist of open floors populated by traders yelling at each other.

• They consist of large warehouses, populated by computer servers that electronically match orders.

• These are referred to as matching engines.

• Since order execution speed is so important, many traders co-locate their trading computers with the matchingengines.


• The NYSE Arca matching engine is located in Secaucus, NY.

• The Chicago Board Option Exchange (CBOE) has co-located with NYSE Arca in Secaucus.

• The Nasdaq matching engine is located in Carteret, NJ.

• The Chicago Mercantile Exchange (CME) matching engine is located in Aurora, IL.




2.3.12 Market Makers

Participants on exchanges include makers and takers.

• Makers are required to post both quotes to buy and sell assets on the exchange.

• Quotes to buy are called ‘bids’.

• Quotes to sell are called ‘offers’.

• Typically there are many bids and offers posted on the exchange at different prices.

• Market makers are compensated for providing quotes (also referred to as providing liquidity).

2.3.13 Takers

Takers actively take orders that have been passively posted by market makers.

• When a taker wants to buy the asset, the buy at the lowest posted offer quote.

• When a take wants to sell an asset, they sell at the highest posted bid quote.

• The difference between the highest bid quote and lowest offer quote is known as the spread.

2.3. Trading 19

http://en.wikipedia.org/wiki/Chicago_Board_Options_Exchange

http://en.wikipedia.org/wiki/CME_Group


2.3.14 Order Book Example: time 1





2.3.19 Buying on Margin

When investors borrow money from a broker to purchase an asset, they are buying on margin.

• The margin is the proportion of the purchase price provided by the investor.

• The purchased securities are maintained in an account by the broker and are monitored.

• The Board of Governors has limited initial margins to be 50% of the initial purchase (i.e. investors must provideat least half of the funds for asset purchase).

2.3.20 Margin Example

Suppose an investor has $6000 and would like to buy 100 shares of asset 𝑊 , currently selling for $100.

• Since $10,000 is required for the purchase, the investor must borrow $4000 from a broker.

• The initial balance sheet would be

Assets Liabilities and Owners’ EquityValue of Stock: $10000 Loan from broker: $4000

Equity: $6000

Thus, the initial margin is $6000$10000 = 0.6.

2.3.21 Margin Example (Cont.)

If the stock price falls to $70 per share, the balance sheet becomes

Assets Liabilities and Owners’ EquityValue of Stock: $7000 Loan from broker: $4000

Equity: $3000

The resulting margin is $3000$7000 = 0.43.

2.3.22 Maintenance Margin

Brokers can set minimal margin requirements, known as maintenance margins.

• If the asset value in the account falls too low, relative to the loan from the broker, the broker can issue a margincall.

• This requires the investor to add cash to the account or liquidate some the stock.

• If the investor doesn’t act accordingly, the broker then has the right to sell some of the stock to pay off enoughof the loan to restore the margin to an acceptable level.


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2.3.23 Maintenance Margin

For example, if a broker has a maintenance margin of 30%, the lowest the price could fall before a margin call is

100𝑃 − 4000

100𝑃= 0.3,

or P = $57.14.

2.3.24 Why Margins?

Margins have the ability to magnify upside returns.

• Suppose asset 𝑊 is selling for $100 and an investor believes its price will increase to $120 next period.

• If the investor buys $10,000 worth of stock, she will realize a return of 20% on her money (after one period hernet worth will be $12,000).

2.3.25 Why Margins?

• However, if the investor borrows an additional $10,000 at 5% interest, then she will earn a gross return of $4000on the $20,000 invested.

• After repaying $10,500 (loan plus interest), her net return will be

($24, 000 − $10, 500) − $10, 000

$10, 000= 0.35.

• So the investor was able to increase her return from 20% to 35% by earning a return on borrowed money.

2.3.26 Why Margins?

While margins provide the ability to magnify returns, they also increase downside risk exposure.

• Suppose that after borrowing $10,000 from the broker, the price of asset 𝑊 fell from $100 to $80.

• The investor’s gross return would be -$4000 on the $20,000 invested.

2.3.27 Why Margins?

• After repayment of the loan, the investor’s net return would be

($16, 000 − $10, 500) − $10, 000

$10, 000= −0.45.

• If the investor hadn’t borrowed money, she would have realized a net return of -0.2. So, by borrowing on marginshe has magnified her loss from -20% to -45%.

2.3. Trading 21


2.3.28 Short Sales

Short sales allow investors to profit from declines in stock prices.

• To engage in a short sale, an investor borrows a share of a stock from a broker and sells it on the market.

• The investor is then obligated to replace the share at a later date.

• If the stock price falls, the investor benefits from selling at a high price and repurchasing at a low price.

• In between, the investor is responsible for paying the broker any dividends that are realized (since the brokerwould have received these if it hadn’t lent the stock).

2.3.29 Short Sales

• When an investor engages in a short sale, we say that they are ‘short’ the stock, or have taken the ‘short position’.

• Conversely, if an investor buys a stock (the usual transaction), we say they are ‘long’, or have taken the ‘longposition’.

2.3.30 Short Sale Example

Suppose asset 𝑊 is selling for $100 and you think its price will fall.

• You decide to borrow 1000 shares, selling them for $100 a piece, yielding a total sale value of $100,000.

• If the stock price falls to $80 per share next period, you can repurchase the 1000 shares for $80,000 and returnthem to the lender.

• You earn a total of $20,000 in the process.


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CHAPTER 3

Risk and Returns

Contents:

3.1 Rates of Return

3.1.1 Holding Period Return

Consider a stock with beginning price 𝑃0, ending price 𝑃1 and a dividend payment of 𝑑.

• The holding period return is

𝐻𝑃𝑅 =𝑃1 − 𝑃0 + 𝑑

𝑃0

=𝑃1 − 𝑃0

𝑃0⏟ ⏞ capital gains yield

+𝑑

𝑃0⏟ ⏞ dividend yield

.

This definition can be used for assets other than stocks (e.g. a bond with a coupon payment).

3.1.2 Holding Period Return Example

• On Nov 9th 2012, Apple stock closed at 𝑃0 = $547.06.

• On Nov 12th, Apple payed a dividend of 𝑑 = $2.65 per share and the price closed at 𝑃1 = $542.83.

• What was the HPR?

𝐻𝑃𝑅 =$542.83 − $547.06

$547.06+

$2.65

$547.06

=−$4.23

$547.06+

$2.65

$547.06

=−$1.58

$547.06= −0.00289.

23


3.1.3 Gross and Net Returns

Forget dividends or cash payouts for a moment.

• The capital gains yield is

𝑃1 − 𝑃0

𝑃0⏟ ⏞ net return

=𝑃1

𝑃0⏟ ⏞ gross return

− 1.

3.1.4 Gross and Net Returns

What’s the difference between net and gross returns?

• The net return is the fraction of your invested money that you gain by holding the asset, excluding the originalmoney.

• The gross return is the total gain, including your original money. It is the factor by which you multiply youroriginal invested amount to determine the final invested amount.

3.1.5 Multi-period Returns

Suppose an asset has net returns {𝑟𝑡}𝑇𝑡=0. Consider two forms of average returns:

Arithmetic Average =1

𝑇

𝑇∑︁𝑡=0

𝑟𝑡

and

Geometric Average =

(︃𝑇∏︁

𝑡=0

(1 + 𝑟𝑡)

)︃ 1𝑇

.

The geometric average is the constant return that would have to be earned each period to yield the same final value ofthe asset.

3.1.6 Annualized Returns - EAR

Suppose you enter into a contract to pay or receive a net rate of return 𝑟 on an asset for each of 𝑛 periods in a year.

• 𝑛 = 12 is a monthly contract.

• 𝑛 = 4 is a quarterly contract.

• The Effective Annual Rate (EAR) is

1 + EAR = (1 + 𝑟)𝑛.

3.1.7 Annualized Returns - APR

Suppose you enter into a contract to pay or receive a net rate of return 𝑟 on an asset for each of 𝑛 periods in a year.

• The Annual Percentage Rate (APR) is

APR = 𝑛× 𝑟.

The APR ignores compounding (as seen in the following example).

24 Chapter 3. Risk and Returns


3.1.8 Annualized Returns - Example

You invest $100 in an asset that pays 5% return each quarter for one year.

𝑄1 : $100 × 1.05 = $105

𝑄2 : $105 × 1.05 = $110.25

𝑄3 : $110.25 × 1.05 = $115.76

𝑄4 : $115.76 × 1.05 = $121.55

𝐸𝐴𝑅 : (1.05)4 − 1 = 0.2155

𝐴𝑃𝑅 : 0.05 × 4 = 0.2

𝐻𝑃𝑅 :$121.55 − $100

$100= 0.2155.

3.1.9 EAR and APR

What is the relationship between EAR and APR?

• Since

𝑟 =APR𝑛

we have

1 + EAR =

(︂1 +

𝐴𝑃𝑅

𝑛

)︂𝑛

.

We can rearrange the equation above to get

APR =[︁(1 + EAR)

1𝑛 − 1

]︁× 𝑛.

3.1.10 Continuous Compounding

Continuous compounding is what occurs when we allow the number of periods in the year, 𝑛, to become large.

• For daily returns, 𝑛 = 365.

• For hourly returns, 𝑛 = 8760.

• For returns each minute, 𝑛 = 525, 000.

• For returns each second, 𝑛 = 31, 536, 000.

3.1. Rates of Return 25


3.1.11 Continuous Compounding

Continuous compounding is the limit, when 𝑛 = ∞. In this case

lim𝑛→∞

(︂1 +

APR𝑛

)︂𝑛

= 𝑒APR.

So, under continuous compounding

1 + EAR = 𝑒APR

or

APR = ln(1 + EAR).

3.1.12 Inflation

Inflation is the increase of the general price level over time.

• Inflation erodes the purchasing power of a given amount of money over time.

• In the presence of inflation, an asset that yields a return of 𝑟 doesn’t actually generate 𝑟 units of additional realpurchasing power for each dollar invested.

3.1.13 Nominal vs. Real Returns

In the previous slides we computed nominal returns.

• Let us momentarily change notation and refer to the nominal return of an asset as 𝑅.

• Then the real return of the asset is the nominal return discounted by inflation:

1 + 𝑟 =1 + 𝑅

1 + 𝜋.

• 𝑟 is the net real return and 𝜋 is net inflation.

3.1.14 Nominal vs. Real Returns

• This relationship is approximated by

𝑟 ≈ 𝑅− 𝜋.

See the proof on the next slide.

3.1.15 Nominal vs. Real Returns - Proof

The proof requires an approximation. For some small number 𝜖 > 0,

ln(1 + 𝜖) ≈ 𝜖.

Thus,

1 + 𝑟 =1 + 𝑅

1 + 𝜋



⇒ ln(1 + 𝑟) = ln

(︂1 + 𝑅

1 + 𝜋

)︂

⇒ ln(1 + 𝑟) = ln(1 + 𝑅) − ln(1 + 𝜋)

⇒ 𝑟 ≈ 𝑅− 𝜋.

3.1.16 Nominal vs. Real Returns - Example

Suppose you can invest in a CD that pays 8% return over the next year and that inflation is 5% during the same period.

• 𝑅 = 0.08.

• 𝜋 = 0.05.

• 𝑟 ≈ 0.08 − 0.05 = 0.03.

The actual real rate of return is

𝑟 =1.08

1.05− 1 = 0.0286.

3.1.17 Expected Inflation

In practice, future inflation is not known, even though the nominal rate of return may be known with certainty.

• Think of a fixed-income asset.

• In this case

𝑅 = 𝑟 + 𝐸[𝜋].

• 𝐸[𝜋] is expected inflation.

3.1.18 Expected Inflation

• The returns to typical government bonds are nominal.

• In 1997, the U.S. Treasury introduced “Treasury Inflation-Protected Securities” (TIPS).

• These have coupon and principle payments that are corrected for observed inflation over time.

• The difference between these rates of return on these two instruments can be treated as a measure of expectedinflation.

3.2 Risk and Risk Premiums

3.2.1 Probabilistic Returns

Since we don’t know future returns, we will treat them as random variables.

• We can model them as discrete random variables, taking one of a finite set of possible values in the future: 𝑟(𝑠),𝑠 = 1, . . . , 𝑆.

3.2. Risk and Risk Premiums 27


– In this case the probability of each value is 𝑝(𝑠), 𝑠 = 1, . . . , 𝑆.

• We can model them as continuous random variables, taking one of an infinite set of possible values in the future:𝑟(𝑠), 𝑠 ∈ 𝒮 (e.g. 𝒮 = (−∞,∞)).

– In this case the probability of each value (kind of) is 𝑓(𝑠), 𝑠 ∈ 𝒮.

3.2.2 Expected Returns

Our best guess for the future return is the expected value:

𝐸[𝑟] ≡ 𝜇 =

𝑆∑︁𝑠=1

𝑟(𝑠)𝑝(𝑠),

or

𝐸[𝑟] ≡ 𝜇 =

∫︁𝑠∈𝒮

𝑟(𝑠)𝑓(𝑠)𝑑𝑟(𝑠).

3.2.3 Return Volatility

The amount of uncertainty in potential returns can be measured by the variance or standard deviation.

• Volatility of returns specifically refers to standard deviation, NOT VARIANCE.

𝑆𝑡𝑑(𝑟) ≡ 𝜎 =

⎯⎸⎸⎷ 𝑆∑︁𝑠=1

(𝑟(𝑠) − 𝜇)2𝑝(𝑠),

or

𝑆𝑡𝑑(𝑟) ≡ 𝜎 =

√︃∫︁𝑠∈𝒮

(𝑟(𝑠) − 𝜇𝑟)2𝑓(𝑠)𝑑𝑟(𝑠).

3.2.4 Expectation and Variance Example

State Probability ReturnSevere Recession 0.05 -0.37Mild Recession 0.25 -0.11Normal Growth 0.40 0.14Boom 0.30 0.30

What are 𝜇 and 𝜎?

𝜇 = 0.05 * (−0.37) + 0.25 * (−0.11)

+ 0.40 * 0.14 + 0.30 * 0.30 = 0.10

𝐸[𝑟2] = 0.05 * (−0.37)2 + 0.25 * (−0.11)2

+ 0.40 * (0.14)2 + 0.30 * (0.30)2 = 0.04471

𝜎 =√︀𝐸[𝑟2] − 𝜇2 = 0.04471 − 0.102 = 0.03471



3.2.5 Assumption of Normality

It will often be convenient to assume asset returns are normally distributed.

• In this case, we will treat returns as continuous random variables.

• We can use the normal density function to compute probabilities of possible events.

• We will not assume that returns of different assets come from the same normal, but instead FROM DIFFERENTnormal distributions.

3.2.6 Differing Normal Distributions

As an example, suppose that

• Amazon stock (AMZN) has an expected monthly return of 3% and a volatility (standard deviation) of 8%.

• Coca-Cola stock (KO) has an expected monthly return of 1% and a volatility (standard deviation) of 4%.

What do their probability distributions look like?

3.2.7 Amazon Distribution

3.2.8 Coca-Cola Distribution

3.2.9 Implications of Normality

The assumption of normality is convenient because

• If we form a portfolio of assets that are normally distributed, then the distribution of the portfolio return is alsonormally distributed.

– Recall that if 𝑋𝑖 ∼ 𝒩 (𝜇𝑖, 𝜎𝑖), 𝑖 = 1, . . . , 𝑁 , then 𝑊 =∑︀𝑁

𝑖=1 𝑤𝑖𝑋𝑖 is also normally distributed (where𝑤𝑖 are constant weights).

• The mean and the variance (or standard deviation) fully characterize the distribution of returns.

• The variance or standard deviation alone is an appropriate measure of risk (no other measure is needed).

3.2.10 Estimating Means and Volatilities

Typically we don’t know the true mean and standard deviation of Amazon and Coca-Cola. What do we do?

• Use historical data to estimate them.

• Collect 𝑁 + 1 past prices of each asset for a particular interval of time (daily, monthly, quarterly, annually).

• Compute 𝑁 returns using the formula

𝑟𝑡 =𝑃𝑡 − 𝑃𝑡−1

𝑃𝑡−1.

We don’t include dividends in the return calculation above, because we use ADJUSTED closing prices, which accountfor dividend payments directly in the prices.



3.2.11 Estimating Means and Volatilities

Compute the sample mean of returns

�̂� =1

𝑁

𝑁∑︁𝑡=1

𝑟𝑡.

Compute the sample standard deviation of returns

�̂�2 =1

𝑁 − 1

𝑁∑︁𝑡=1

(𝑟𝑡 − �̂�)2.

The “hats” indicate that we have estimated 𝜇 and 𝜎: these are not the true, unknown values.

3.2.12 Estimating Means and Volatilities - Example

Let’s collect the 𝑁 = 13 closing prices for Amazon and Coca-Cola between 3 Jan 2012 and 2 Jan 2013.

• We will only keep the first closing price on the first trading day of each month.

• We can then compute 12 monthly returns by computing the difference in month prices at the beginning of eachmonth, dividing by the price of the previous month.

• This will give us 12 returns that we can use to estimate the means and standard deviations.

3.2.13 Amazon Monthly Prices

3.2.14 Coca-Cola Monthly Prices

3.2.15 Computing Returns and Moments

3.2.16 Risk-Free Returns

We will typically assume that a risk-free asset is available for purchase.

• We will denote the risk-free return as 𝑟𝑓 .

• If an asset is risk free, its return is certain and has no variability:

𝐸[𝑟𝑓 ] = 𝑟𝑓

𝑉 𝑎𝑟(𝑟𝑓 ) = 0.

3.2.17 T-Bills as Risk-Free Assets

The return on a short-term government t-bill is usually considered risk free:

• Although the price changes over time, the risk of default is extremely low.

• Also, the holding period return can be determined at the beginning of the holding period (unlike other riskyassets).



3.2.18 Compensation for Risk

If you can invest in a risk-free asset, why would you purchase a risky asset instead?

• Risky assets compensate for risk through higher expected return.

• If risky assets didn’t offer higher expected return, everyone would sell them, leading to a price decline todayand a higher expected return:

↑ 𝐸[𝑟𝑡] =𝐸[𝑃𝑡]− ↓ 𝑃𝑡−1

↓ 𝑃𝑡−1

• There is no guarantee that the actual return will be higher – only its expected value.

3.2.19 Risk Premium & Excess Returns

The amount by which the expected return of some risky asset 𝐴 exceeds the risk-free return is known as the riskpremium:

rp𝐴,𝑡 = 𝐸[𝑟𝐴,𝑡] − 𝑟𝑓,𝑡.

The excess return measures the difference between a previously observed holding period return of 𝐴 and the risk-free:

er𝐴,𝑡−1 = 𝑟𝐴,𝑡−1 − 𝑟𝑓,𝑡−1.

3.2.20 Risk Premium & Excess Returns

• Note that excess returns can only be computed with past returns.

• We estimate risk premia with the sample mean of historical excess returns.

3.2.21 Sharpe Ratio

The Sharpe Ratio is a measure of how much risk premium investors require, per unit of risk:

SR𝐴,𝑡 =𝜇𝐴,𝑡 − 𝑟𝑓,𝑡

𝜎𝐴,𝑡

• The Sharpe Ratio is a measure of risk aversion.

• It is often referred to as the price of risk.

• The Sharpe Ratio for a broad market index of assets (like the S&P 500) is referred to as the market price of risk.

• The true Sharpe Ratio is unknown, since we don’t know 𝜇𝐴,𝑡 and 𝜎𝐴,𝑡, but we can estimate these with historicalreturns.

3.2.22 Risk Premium Example

Suppose the monthly risk-free rate is 0.2%. What is the estimated risk premium and Sharpe Ratio for Amazon stock?

𝑟𝑝𝐴𝑀𝑍𝑁 = 0.03 − 0.002 = 0.028

𝑆𝑅𝐴𝑀𝑍𝑁 =𝑟𝑝𝐴𝑀𝑍𝑁

0.08= 0.35




CHAPTER 4

Portfolio Optimization

Contents:

4.1 Asset Allocation

4.1.1 Utility

Investors usually care about maximizing utility.

• Suppose all investors have utility function

𝑈(𝜇, 𝜎) = 𝜇− 𝛾𝜎2.

• 𝜇 and 𝜎 are the mean and standard deviation of asset returns.

• What is the utility of holding a risk-free asset 𝑈(𝜇𝑓 , 𝜎𝑓 )?

𝑈(𝜇𝑓 , 𝜎𝑓 ) = 𝑟𝑓 .

4.1.2 Certainty Equivalent

For a risky portfolio, 𝑈(𝜇𝑓 , 𝜎𝑓 ) can be thought of as a certainty equivalent return.

• The return that a risk-free asset would have to offer to provide the same utility level as a risky asset.

4.1.3 Risk Aversion

The parameter 𝛾 is a measure of risk preference.

• If 𝛾 > 0 individuals are risk averse - volatility detracts from utility.

• If 𝛾 = 0 individuals are risk neutral - volatility doesn’t enter into the utility function.

– In this case, investors rank portfolios by their expected return and don’t care about portfolio riskiness.

33


4.1.4 Risk Aversion

• If 𝛾 < 0 individuals are risk lovers - volatility is rewarded in the utility function.

– In this case, investors enjoy and get utility by taking on risk.

• We will generally assume investors are risk averse, with the magnitude of 𝛾 dictating the amount of risk aversion.

4.1.5 Mean-Variance Criterion

Under this utility model, investors prefer higher expected returns and lower volatility.

• Let portfolio 𝐴 have mean and volatility 𝜇𝐴 and 𝜎𝐴.

• Let portfolio 𝐵 have mean and volatility 𝜇𝐵 and 𝜎𝐵 .

• If 𝜇𝐴 ≥ 𝜇𝐵 and 𝜎𝐴 ≤ 𝜎𝐵 , then 𝐴 is preferred to 𝐵.

4.1.6 Mean-Variance Criterion

4.1.7 Indifference Curves

Portfolios in quadrant I are preferred to 𝑃 , which is preferred to portfolios in quadrant IV.

• What about quadrants II and III?

• If a portfolio 𝑄 has a mean and volatility that differ from 𝑃 but yields the same utility level, then either

𝜇𝑄 > 𝜇𝑝 and 𝜎𝑄 > 𝜎𝑝

or

𝜇𝑄 < 𝜇𝑝 and 𝜎𝑄 < 𝜎𝑝.

• That is, Q must be in quadrants II or III.

4.1.8 Indifference Curves

• The portfolios that yield the same utility as 𝑃 constitute an indifference curve.

• We conclude that the indifference curve must cut through quadrants II and III.

4.1.9 Indifference Curve Plot

34 Chapter 4. Portfolio Optimization


4.1.10 Portfolios of Assets

Suppose an individual can invest in two assets: a risky portfolio 𝑃 and a risk-free asset 𝐹 .

• 𝜔 will be the fraction wealth invested in 𝑃 .

• 1 − 𝜔 will be the fraction wealth invested in 𝐹 .

• We will typically assume that portfolio weights sum to 1.

• 𝑟𝑝 will denote the return on asset 𝑃 , with 𝜇𝑝 = 𝐸[𝑟𝑝] and 𝜎2𝑝 = 𝑉 𝑎𝑟(𝑟𝑝).

• 𝑟𝑓 will denote the return on asset 𝐹 , with 𝜇𝑓 = 𝑟𝑓 and 𝜎2𝑓 = 0.

4.1.11 Portfolio Return

Let 𝐶 denote the portfolio that combines the two assets.

• 𝐶 is a weighted average of 𝑃 and 𝐹 :

𝐶 = 𝜔𝑃 + (1 − 𝜔)𝐹.

• The return to 𝐶, is

𝑟𝑐 = 𝜔𝑟𝑝 + (1 − 𝜔)𝑟𝑓 .

4.1.12 Portfolio Return

By the linearity of expectations:

𝜇𝑐 = 𝐸[𝑟𝑐]

= 𝜔𝐸[𝑟𝑝] + (1 − 𝜔)𝐸[𝑟𝑓 ]

= 𝜔𝜇𝑝 + (1 − 𝜔)𝑟𝑓

= 𝑟𝑓 + 𝜔(𝜇𝑝 − 𝑟𝑓 ).

• The term in the parentheses is the risk premium of 𝑃 .

4.1.13 Portfolio Volatility

According to the properties of variance,

𝜎2𝑐 = 𝑉 𝑎𝑟(𝑟𝑐)

= 𝜔2𝑉 𝑎𝑟(𝑟𝑝) + (1 − 𝜔)2𝑉 𝑎𝑟(𝑟𝑓 ) + 2𝜔(1 − 𝜔)𝐶𝑜𝑣(𝑟𝑝, 𝑟𝑓 )

= 𝜔2𝑉 𝑎𝑟(𝑟𝑝)

= 𝜔2𝜎2𝑝.

• The third equality follows because 𝑟𝑓 is a constant.

• Thus,

𝜎𝑐 = 𝜔𝜎𝑝.

4.1. Asset Allocation 35


4.1.14 Portfolios and Risk Aversion

Portfolio 𝐶 earns a base return of 𝑟𝑓 plus the risk premium associated with 𝑃 , weighted by the amount of wealth theinvestor allocates to 𝑃 .

• More risk averse investors (small 𝜔) expect a rate of return closer to 𝑟𝑓 .

• Less risk averse investors (high 𝜔) expect a rate closer to 𝜇𝑝 − 𝑟𝑓 .

• More risk averse investors have smaller portfolio volatilities.

• Less risk averse investors have higher portfolio volatilities.

4.1.15 Portfolio Weight

Since 𝜎𝑐 = 𝜔𝜎𝑝,

𝜔 =𝜎𝑐

𝜎𝑝.

4.1.16 Sharpe Ratio

Thus

𝜇𝑐 = 𝑟𝑓 + 𝜔(𝜇𝑝 − 𝑟𝑓 )

= 𝑟𝑓 +𝜎𝑐

𝜎𝑝(𝜇𝑝 − 𝑟𝑓 )

= 𝑟𝑓 +𝜇𝑝 − 𝑟𝑓

𝜎𝑝𝜎𝑐

= 𝑟𝑓 + SR𝑝𝜎𝑐.

• SR𝑝 is the Sharpe Ratio of portfolio 𝑃 .

4.1.17 Capital Allocation Line

The Capital Allocation Line (CAL) depicts the set of portfolios available to an investor (a budget constraint).

• It plots pairs of 𝜎𝑐 and 𝜇𝑐 that the investor can choose by selecting 𝜔.

• This is simply a plot of the equation

𝜇𝑐 = 𝑟𝑓 + SR𝑝𝜎𝑐.

• Clearly, the intercept will be 𝑟𝑓 and the slope will be SR𝑝.

4.1.18 Capital Allocation Line Example

Suppose

• 𝑟𝑓 = 0.07.

• 𝜇𝑝 = 0.15.

• 𝜎𝑝 = 0.22.

What is the CAL?



4.1.19 Capital Allocation Line Plot

4.1.20 Simple Portfolio Choice

Individuals seek to maximize utility subject to the available choice set.

max𝜇𝑐,𝜎𝑐

𝑈(𝜇𝑐, 𝜎𝑐) = max𝜇𝑐,𝜎𝑐

𝜇𝑐 −1

2𝛾𝜎2

𝑐 ,

subject to

𝜇𝑐 = 𝑟𝑓 + 𝜔(𝜇𝑝 − 𝑟𝑓 )

𝜎𝑐 = 𝜔𝜎𝑝.

4.1.21 Simple Portfolio Choice (Cont.)

Substituting the constraints, the maximization problem becomes

max𝜔

{︂𝑟𝑓 + 𝜔(𝜇𝑝 − 𝑟𝑓 ) − 1

2𝛾𝜔2𝜎2

𝑝

}︂.

Taking the derivative of this equation w.r.t. 𝜔,

𝜇𝑝 − 𝑟𝑓 = 𝛾𝜔*𝜎2𝑝

⇒ 𝜔* =𝜇𝑝 − 𝑟𝑓𝛾𝜎2

𝑝

=SR𝑝

𝛾𝜎𝑝.


Since the CAL is a choice set (budget constraint), we can find the optimal portfolio by observing where an indifferencecurve is tangent to the CAL.

• Fix the utility value at �̄� = 𝑟𝑓 .

• Use the relation �̄� = 𝜇𝑐 − 12𝛾𝜎

2𝑐 to solve for 𝜇𝑐:

𝜇𝑐 = �̄� +1

2𝛾𝜎2

𝑐 .


• Using this equation, we find the pairs of 𝜇𝑐 and 𝜎𝑐 that corresponds to utility �̄� , which we plot.

• Repeat this process, increasing the values of �̄� until a tangent indifference curve is found. The tangency corre-sponds to the optimal portfolio.

4.1.24 Spreadsheet Optimization

Given 𝛾 = 2, 𝜇 = 0.15, 𝜎 = 0.22 and 𝑟𝑓 = 0.07.

4.1. Asset Allocation 37


4.1.25 Graphical Optimization

4.1.26 Capital Market Line

Consider a value-weighted portfolio of all assets in the market.

• We will call this the market portfolio and denote it by 𝑀 .

• The CAL which connects 𝑟𝑓 with 𝑀 is called the Capital Market Line (CML).

• Because the true market portfolio is unobserved, we use a proxy - a well diversified portfolio that provides agood representation of the entire market.

• Typically we use the S&P 500.

4.1.27 Passive vs. Active Strategies

Holding the market portfolio or a market proxy is known as a passive strategy.

• It requires no security analysis.

• An active strategy is one that requires individual security analysis.

4.2 Portfolio Optimization

4.2.1 Types of Risk

We classify risk using two broad categories.

• Systematic risk (also called market or non-diversifiable risk) which is common to all assets in the market.

• Idiosyncratic risk (also called firm-specific, non-systematic or diversifiable risk) which is unique to a particularasset.

4.2.2 Benefits of Diversification

Idiosyncratic risk can be eliminated through portfolio diversification.

• Imagine holding one stock in your portfolio. The risk of your portfolio will be equal to the sum of the stock’ssystematic and idiosyncratic risks.

• Now add another stock to the portfolio. If the two stocks are perfectly correlated, then they will always movetogether and your portfolio will have the same risk properties as before.


• What if the idiosyncratic components of the two stocks are uncorrelated?

– Systematic shocks will cause them to move together.

– Idiosyncratic shocks will occur at different times and magnitudes, so the stocks will not move together asmuch.



– Sometimes when one falls in price another will rise price, counteracting each other and reducing risk.


• What if the idiosyncratic components of the two stocks are negatively correlated?

– Systematic shocks will cause them to move together.

– Idiosyncratic shocks will cause them to move in opposite directions, reducing risk.

4.2.5 Limits to Diversification

Adding more and more assets to a portfolio compounds the diversification effect.

• In theory, it should be possible to eliminate all idiosyncratic risk by increasing the number of assets in a portfolio.

• Systematic risk, however, cannot be eliminated since it is common to all assets.

4.2.6 Limits to Diversification

4.2.7 Portfolios of Two Risky Assets

Suppose you can invest in two risky assets.

• A debt asset (bonds) denoted by 𝐷.

• An equity asset (stocks) denoted by 𝐸.

• 𝜔𝑑 is the fraction of wealth invested in 𝐷.

• 𝜔𝑒 is the fraction of wealth invested in 𝐸.

• We require 𝜔𝑑 + 𝜔𝑒 = 1, so that 𝜔𝑒 = 1 − 𝜔𝑑.

4.2.8 Returns to Portfolios of Risky Assets

Suppose the returns to 𝐷 and 𝐸 are 𝑟𝑑 and 𝑟𝑒, respectively.

• The portfolio return is the weighted average of returns

𝑟𝑝 = 𝜔𝑑𝑟𝑑 + 𝜔𝑒𝑟𝑒.

• By the properties of expectations, the portfolio expected return is

𝜇𝑝 = 𝜔𝑑𝜇𝑑 + 𝜔𝑒𝜇𝑒.

• By the properties of variance, the portfolio variance is

𝜎2𝑝 = 𝜔2

𝑑𝜎2𝑑 + 𝜔2

𝑒𝜎2𝑒 + 2𝜔𝑑𝜔𝑒𝐶𝑜𝑣(𝑟𝑑, 𝑟𝑒).

4.2. Portfolio Optimization 39


4.2.9 Covariance Matrix

It is often useful to express the variance and covariance information in matrix form

Σ =

[︂𝜎2𝑑 𝜎𝑑,𝑒

𝜎𝑒,𝑑 𝜎2𝑒

]︂,

where 𝜎𝑑,𝑒 = 𝜎𝑒,𝑑 = 𝐶𝑜𝑣(𝑟𝑑, 𝑟𝑒).

• Σ is referred to as a covariance matrix.

• It can be generalized to more than two assets.

• The diagonal terms always represent variances.

• The off-diagonal terms represent covariances.

4.2.10 Correlation

Define 𝜌𝑥𝑦 = 𝐶𝑜𝑟𝑟(𝑋,𝑌 ). Then recall that

𝜌𝑥𝑦 =𝐶𝑜𝑣(𝑋,𝑌 )

𝜎𝑥𝜎𝑦.

• Hence, 𝐶𝑜𝑣(𝑋,𝑌 ) = 𝜌𝑥𝑦𝜎𝑥𝜎𝑦 .

• Remember that 𝜌𝑥𝑦 ∈ [−1, 1].

4.2.11 Portfolio Variance

The previous relationship allows us to write portfolio variance as

𝜎2𝑝 = 𝜔2

𝑑𝜎2𝑑 + 𝜔2

𝑒𝜎2𝑒 + 2𝜔𝑑𝜔𝑒𝜌𝑑𝑒𝜎𝑑𝜎𝑒.

• Since 𝜌𝑑𝑒 ∈ [−1, 1], 𝜎2𝑝 is greatest when 𝜌𝑑𝑒 = 1. Why?

• Note that when 𝜌𝑑𝑒 = 1,

𝜎2𝑝 = (𝜔𝑑𝜎𝑑 + 𝜔𝑒𝜎𝑒)

2

⇒ 𝜎𝑝 = 𝜔𝑑𝜎𝑑 + 𝜔𝑒𝜎𝑒.

4.2.12 Portfolio Standard Deviation

• In this special case, the portfolio standard deviation is the weighted average of asset standard deviations.

• So the maximal possible portfolio standard deviation is the simple weighted average of component standarddeviations.

4.2.13 Portfolio Variance

Note that

• 𝜇𝑝 is the weighted average of asset means.

• 𝜎𝑝 is less than the weighted average of asset volatilities when 𝜌𝑑𝑒 ̸= 1.

• So the risk-return properties of the portfolio are better than those of the component assets.



4.2.14 Portfolio Variance Lower Bound

Portfolio variance is made smaller with smaller values of 𝜌𝑑𝑒.

• When 𝜌𝑑𝑒 = −1,

𝜎2𝑝 = (𝜔𝑑𝜎𝑑 − 𝜔𝑒𝜎𝑒)

2

⇒ 𝜎𝑝 = |𝜔𝑑𝜎𝑑 − 𝜔𝑒𝜎𝑒|.

• If we choose 𝜔𝑑 = 𝜎𝑒

𝜎𝑒+𝜎𝑑(and 𝜔𝑒 = 1 − 𝜔𝑑) then 𝜎𝑝 = 0.

• This means that if assets are perfectly negatively correlated, the portfolio has zero variance.

4.2.15 Portfolio Variance Lower Bound

4.2.16 Portfolio Frontier

The lines in the figure above are known as portfolio opportunity sets or portfolio frontiers.

• Since individuals prefer less risk and greater return, frontiers that are bowed to the northwest are better.

• Clearly, the frontiers are better for smaller values of 𝜌𝑑𝑒.

4.2.17 Portfolios of Three Assets

Assume you can invest your money in three assets:

• Two risky assets.

• A risk-free asset.

The portfolio frontier now consists of

• The curved frontier between the two risky assets. We will call this the risky frontier.

• Every capital allocation line (CAL) connecting the risk-free to a portfolio on the risky frontier.

• Steeper CALs are better since they offer portfolios of equivalent expected return with lower standard deviation.

4.2.18 Efficient Frontier

4.2.19 Optimal CAL

The optimal CAL is the steepest possible line that intersects both the risk-free asset and a portfolio on the risky frontier.

• The slope of the CAL is the Sharpe Ratio of the risky portfolio it intersects.

• So the investor’s optimization problem is

max𝜔

𝑆𝑅𝑝 =𝜇𝑝 − 𝑟𝑓

𝜎𝑝subject to

𝜇𝑝 = 𝜔𝑑𝜇𝑑 + (1 − 𝜔𝑑)𝜇𝑒

𝜎𝑝 =√︁𝜔2𝑑𝜎

2𝑑 + (1 − 𝜔𝑑)2𝜎2

𝑒 + 2𝜔𝑑(1 − 𝜔𝑑)𝜌𝑑𝑒𝜎𝑑𝜎𝑒.

4.2. Portfolio Optimization 41


4.2.20 Optimal CAL

To solve the problem we

• Substitute the constraints.

• Take the derivative with respect to 𝜔𝑑 and set the derivative equal to zero.

• Perform some tedious algebra.

4.2.21 Optimal CAL

The solution is

𝜔*𝑑 =

𝑟𝑝𝑑𝜎2𝑒 − 𝑟𝑝𝑒𝜌𝑑𝑒𝜎𝑑𝜎𝑒

𝑟𝑝𝑒𝜎2𝑑 + 𝑟𝑝𝑑𝜎2

𝑒 − (𝑟𝑝𝑒 + 𝑟𝑝𝑑)𝜌𝑑𝑒𝜎𝑑𝜎𝑒.

• 𝑟𝑝𝑑 = 𝜇𝑑 − 𝑟𝑓 and 𝑟𝑝𝑒 = 𝜇𝑒 − 𝑟𝑓 .

• The optimal portfolio 𝑇 is known as the tangency portfolio.

4.2.22 Tangeny Portfolio

4.2.23 Optimal CAL

The optimal CAL is the investor’s efficient choice set, since it offers minimal variance portfolios for every choice ofexpected return.

• We have said nothing about which portfolio the investor will actually consume; we have only identified the setfrom which the investor will choose.

• We must combine the optimal CAL with a specification of preferences to determine the investor’s actual choice.

• The portfolio choice problem with three assets now reduces to the simple problem of choosing between onerisky asset (the tangency portfolio) and the risk-free asset.

4.2.24 Investor Choice

Let’s assume again that agents’ utility is given by

𝑈(𝜇, 𝜎) = 𝜇− 1

2𝛾𝜎2.

Recall that when allocating wealth between a risky asset 𝑃 and a risk-free asset, the optimal portfolio weight, 𝜏 , is

𝜏 =𝜇𝑝 − 𝑟𝑓𝛾𝜎2

𝑝

.

4.2.25 Investor Choice

If the agent invests in the tangency portfolio 𝑇 and the risk-free,

𝜏 =𝜇𝑇 − 𝑟𝑓𝛾𝜎2

𝑇

.

• Given that the tangency portfolio is a weighted average of assets 𝐷 and 𝐸, with optimal weight 𝜔*𝑑



𝑤𝑑 = 𝜏𝜔*𝑑

𝑤𝑒 = 𝜏(1 − 𝜔*𝑑).

4.3 Portfolio Optimization with Many Risky Assets

4.3.1 Portfolio Returns

Suppose you can now invest in an arbitrary number (𝑁 ) of risky assets.

• Index the assets by 𝑖 = 1, . . . , 𝑁 .

• Let 𝜔𝑖 be the fraction of income invested in asset 𝑖.

• We will always assume that∑︀𝑁

𝑖=1 𝜔𝑖 = 1.

• We will denote the return to asset 𝑖 by 𝑟𝑖.

• The portfolio return is expressed as

𝑟𝑝 =

𝑁∑︁𝑖=1

𝜔𝑖𝑟𝑖.

4.3.2 Portfolio Moments

From the properties of expectation and variance, we can compute the mean and variance of the portfolio return.

• Recognize that the 𝑁 asset returns, 𝑟𝑖, are random variables.

• Denote the means of 𝑟𝑖 as 𝜇𝑖.


• The 𝑁 ×𝑁 covariance matrix of the returns contains the variances, 𝜎2𝑖 , and covariances, 𝐶𝑜𝑣(𝑟𝑖, 𝑟𝑗) = 𝜎𝑖𝑗 :

Σ𝑃 =

⎡⎢⎢⎢⎣𝜎21 𝜎12 · · · 𝜎1𝑁

𝜎21 𝜎22 · · · 𝜎2𝑁

......

. . ....

𝜎𝑁1 𝜎𝑁2 · · · 𝜎2𝑁

⎤⎥⎥⎥⎦


Thus resulting moments of the portfolio are

𝜇𝑝 =

𝑁∑︁𝑖=1

𝜔𝑖𝜇𝑖

𝜎2𝑝 =

𝑁∑︁𝑖=1

𝜔2𝑖 𝜎

2𝑖 + 2

𝑁−1∑︁𝑖=1

𝑁∑︁𝑗=𝑖+1

𝜔𝑖𝜔𝑗𝜎𝑖𝑗 .

What are other ways to express 𝜎2𝑝?

4.3. Portfolio Optimization with Many Risky Assets 43


4.3.5 Optimization: Risky MV Frontier

To determine the set of efficient risky portfolios (the risky frontier), the investor solves

min{𝜔𝑖}𝑁−1

𝑖=1

𝜎2𝑃 =

𝑁∑︁𝑖=1

𝜔2𝑖 𝜎

2𝑖 + 2

𝑁−1∑︁𝑖=1

𝑁∑︁𝑗=𝑖+1

𝜔𝑖𝜔𝑗𝜎𝑖𝑗

subject to

𝜇𝑝 =

𝑁∑︁𝑖=1

𝜔𝑖𝜇𝑖

where 𝜇𝑝 is some prespecified value of the portfolio mean return.

4.3.6 Optimization: Risky MV Frontier

Note that

• The optimization problem has 𝑁 − 1 choice variables: {𝜔𝑖}𝑁−1𝑖=1 .

• 𝜔𝑁 is not a choice variable because it is found from the constraint: 𝜔𝑁 = 1 −∑︀𝑁−1

𝑖=1 𝜔𝑖.

• This is a challenging problem that is only tractable with linear algebra (we won’t solve it).

4.3.7 Risky Minimum-Variance Frontier

4.3.8 Risky Minimum-Variance Frontier

The frontier generated by multiple risky assets is known as the risky minimum-variance (MV) frontier.

• The lower portion of the frontier is inefficient since a higher mean portfolio exists with the same volatility onthe upper portion of the frontier.

• The efficient MV frontier is generated by allowing investment in a risk-free asset and finding the CAL which istangent to the risky efficient MV frontier.

4.3.9 Efficient Minimum-Variance Frontier

4.3.10 Optimization: Efficient MV Frontier

To determine the tangency portfolio, the investor solves the same problem as before

max𝜇𝑝,𝜎𝑝

𝑆𝑅𝑝 =𝜇𝑝 − 𝑟𝑓

𝜎𝑝

subject to

𝜇𝑝 =

𝑁∑︁𝑖=1

𝜔𝑖𝜇𝑖



𝜎𝑝 =

⎯⎸⎸⎷ 𝑁∑︁𝑖=1

𝜔2𝑖 𝜎

2𝑖 + 2

𝑁−1∑︁𝑖=1

𝑁∑︁𝑗=𝑖+1


4.3.11 Optimization: Investor Choice

So far we have specified two optimization problems:

1. To determine the risky minimum-variance frontier by minimizing variance subject to a particular expected re-turn.

2. To determine the tangency portfolio, by maximizing the Sharpe Ratio subject to constraints on the mean andstandard deviation.

Neither of these made use of preferences. A final optimization problem would be the same as before:

3. Maximize utility, 𝑈(𝜇𝑝, 𝜎𝑝), subject to investing in the tangency portfolio and a risk-free asset.

4.3.12 Estimation

In practice we must estimate 𝜇𝑖, 𝜎2𝑖 and 𝜎𝑖𝑗 for 𝑖 = 1, . . . , 𝑁 and 𝑗 = 𝑖 + 1, . . . , 𝑁 .

• A total of 𝑁 estimates of means.

• How many variances and covariances must we estimate?

• A total of 𝑁 elements on the diagonal (variances).

• All of the elements above or below the diagonal (not both because of symmetry).

4.3.13 Estimation

• The resulting number of variance and covariance estimates is

𝑁 + (𝑁 − 1) + (𝑁 − 2) + . . . + 2 + 1 =

𝑁∑︁𝑖=1

𝑖 =𝑁(𝑁 + 1)

2.

4.3.14 Estimation

The total number of estimates is

𝑁 +𝑁(𝑁 + 1)

2=

𝑁(𝑁 + 3)

2.

• As an example, a portfolio of 50 stocks requires 50×532 = 1325 estimates.

• The models of subsequent lectures will reduce this estimation burden.

4.3.15 Portfolio Optimization Recipe

For an arbitrary number, 𝑁 , of risky assets:

1. Specify (estimate) the return characteristics of all securities (means, variances and covariances).

2. Establish the optimal risky portfolio.



• Calculate the weights for the tangency portfolio.

• Compute mean and std. deviation of the tangency portfolio.

4.3.16 Portfolio Optimization Recipe

3. Allocate funds between the optimal risky portfolio and the risk-free asset.

• Calculate the fraction of the complete portfolio allocated to the tangency portfolio and to the risk-free asset.

• Calculate the share of the complete portfolio invested in each asset of the tangency portfolio.

4.3.17 Separation Property

All investors hold some combination of the same two assets: the risk-free asset and the tangency portfolio.

• The optimal risky (tangency portfolio) is the same for all investors, regardless of preferences.

• The tangency portfolio is simply determined by estimation and a mathematical formula.

• Individual preferences determine the exact proportions of wealth each investor will allocate to the two assets.

• This is known as The Separation Property (or Two Fund Separation).


The separation property implies that portfolio choice can be separated into two independent steps:

• Determining the optimal risky portfolio (preference independent).

• Deciding what proportion of wealth to invest in the risk-free asset and the tangency portfolio (preference depen-dent).


The separation property will not hold if

• Individuals produce different estimates of asset return characteristics (since differing estimates will result indifferent tangency portfolios).

• Individuals face different constraints (short-sale, tax, etc.).

4.3.20 The Power of Diversification

Let’s formalize the benefits of diversification. The variance of a portfolio of 𝑁 risky assets is

𝜎2𝑝 =

𝑁∑︁𝑖=1

𝑁∑︁𝑗=1

𝜔𝑖𝜔𝑗𝜎𝑖𝑗 =

𝑁∑︁𝑖=1

𝜔2𝑖 𝜎

2𝑖 + 2

𝑁−1∑︁𝑖=1

𝑁∑︁𝑗=𝑖+1


In the case of an equally weighted portfolio,

𝜎2𝑝 =

1

𝑁2

𝑁∑︁𝑖=1

𝜎2𝑖 +

2

𝑁2

𝑁−1∑︁𝑖=1

𝑁∑︁𝑗=𝑖+1

𝜎𝑖𝑗

=1

𝑁𝑉 𝑎𝑟 +

𝑁 − 1

𝑁𝐶𝑜𝑣.




Where

𝑉 𝑎𝑟 =1

𝑁

𝑁∑︁𝑖=1

𝜎2𝑖

and

𝐶𝑜𝑣 =2

𝑁(𝑁 − 1)

𝑁−1∑︁𝑖=1

𝑁∑︁𝑗=𝑖+1

𝜎𝑖𝑗 .

These are the average variance and covariance.


The limit of portfolio variance is

lim𝑁→∞

𝜎2𝑝 = lim

𝑁→∞

1

𝑁𝑉 𝑎𝑟 + lim

𝑁→∞

𝑁 − 1

𝑁𝐶𝑜𝑣 = 𝐶𝑜𝑣.

• If the assets in the portfolio are uncorrelated or not correlated on average (𝐶𝑜𝑣 = 0), there is no limit todiversification: 𝜎2

𝑝 = 0.

• If there are systemic sources of risk that affect all assets (𝐶𝑜𝑣 > 0) there will be a lower bound on ability todiversify: 𝜎2

𝑝 > 0.




CHAPTER 5

Index Models

5.1 Motivation

Recall that for portfolio optimization with 𝑁 assets we must estimate all means, variances and covariances:

# of estimates =𝑁(𝑁 + 3)

2.

Suppose you have a dataset consisting of 5 years of monthly returns for each asset (i.e. 60 observations per asset):

𝑁 50 100 1000 3000# Estimates 1325 5150 501,500 4,504,500Data 3000 6000 60,000 180,000

Note: it is impossible to estimate 𝑃 parameters with 𝑁 data observations if 𝑁 < 𝑃 .

5.2 Motivation

The return to an asset can always be decomposed into two parts:

𝑟𝑖 = 𝐸[𝑟𝑖] + 𝜖𝑖 = 𝜇𝑖 + 𝜖𝑖,

where 𝜖𝑖 has zero mean (𝜇𝜖𝑖 = 0) and standard deviation 𝜎𝜖𝑖 .

• The first term of the equation above is the expected return.

• The second term is the unexpected, or unanticipated return (also referred to as a shock).

• This decomposition is not dependent on a model or special assumptions.

5.3 Single-Factor Model

Suppose there is a market factor 𝑚 that influences the returns to all firms.

• Assume we can further decompose the shock, 𝜖𝑖, into two parts:

𝜖𝑖 = 𝛽𝑖𝑚 + 𝜀𝑖.

49


5.4 Single-Factor Model

In this case the return can be written as a single-factor model:

𝑟𝑖 = 𝜇𝑖 + 𝛽𝑖𝑚 + 𝜀𝑖. (5.1)

• 𝑚 and 𝜀𝑖 have means 𝜇𝑚 = 𝜇𝜀𝑖 = 0, standard deviations 𝜎𝑚 and 𝜎𝜀𝑖 , and are uncorrelated (𝐶𝑜𝑣(𝑚, 𝜀𝑖) = 0).

• 𝜖𝑖 is still unanticipated since 𝜇𝜖𝑖 = 𝜇𝑚 + 𝜇𝜀𝑖 = 0.

• 𝛽𝑖 is a measure of the sensitivity of 𝑟𝑖 to 𝑚.

5.5 Decomposing Risk

We can now compute variances using the model:

𝜎2𝑖 ≡ 𝑉 𝑎𝑟(𝑟𝑖) = 𝑉 𝑎𝑟(𝜇𝑖 + 𝛽𝑖𝑚 + 𝜀𝑖)

= 𝛽2𝑖 𝑉 𝑎𝑟(𝑚) + 𝑉 𝑎𝑟(𝜀𝑖) + 2𝛽𝑖𝐶𝑜𝑣(𝑚, 𝜀𝑖)

= 𝛽2𝑖 𝜎

2𝑚 + 𝜎2

𝜀𝑖 .

• We made use of 𝑉 𝑎𝑟(𝜇𝑖) = 0 and 𝐶𝑜𝑣(𝑚, 𝜀𝑖) = 0.

• 𝑉 𝑎𝑟(𝑟𝑖) arises from two separate sources.

– 𝛽2𝑖 𝜎

2𝑚: risk due to 𝑚. Since this is common to all assets, it is the systematic component.

– 𝜎2𝜀𝑖 : idiosyncratic component specific to each asset.

5.6 Decomposing Risk

We can use the model to compute covariances between assets:

𝐶𝑜𝑣(𝑟𝑖, 𝑟𝑗) = 𝐶𝑜𝑣(𝜇𝑖 + 𝛽𝑖𝑚 + 𝜀𝑖, 𝜇𝑗 + 𝛽𝑗𝑚 + 𝜀𝑗)

= 𝛽𝑖𝛽𝑗𝐶𝑜𝑣(𝑚,𝑚)

= 𝛽𝑖𝛽𝑗𝜎2𝑚.

• 𝑉 𝑎𝑟(𝜇𝑖) = 𝑉 𝑎𝑟(𝜇𝑗) = 𝐶𝑜𝑣(𝜇𝑖, 𝜇𝑗) = 0, since 𝜇𝑖 and 𝜇𝑗 are constants.

• We assumed 𝐶𝑜𝑣(𝜀𝑖, 𝜀𝑗) = 0.

• Intuitively, unanticipated shocks to different assets shouldn’t be correlated.

5.7 Using an Index as a Factor

So what is 𝑚?

• We would like to find a macroeconomic variable correlated with all assets.

• It is common to use a market index portfolio, such as the S&P 500 (we expect the return of a broad index to becorrelated with individual assets).

50 Chapter 5. Index Models


5.8 Using an Index as a Factor

In particular, let’s use 𝑚 = 𝑟𝑚 − 𝜇𝑚, where 𝑟𝑚 is the return to the S&P 500 (𝑚 stands for market).

• In this case

𝐸[𝑚] = 𝐸[𝑟𝑚 − 𝜇𝑚]

= 𝐸[𝑟𝑚] − 𝜇𝑚

= 𝜇𝑚 − 𝜇𝑚

= 0.

• So our assumption of 𝐸[𝑚] = 0 is satisfied.

5.9 Single-Index Model

Substituting this factor into the single-factor model of Equation (5.1):

𝑟𝑖 = 𝜇𝑖 + 𝛽𝑖(𝑟𝑚 − 𝜇𝑚) + 𝜀𝑖.

We then manipulate the equation:

𝑟𝑖 = 𝑟𝑓 − 𝑟𝑓 + 𝜇𝑖 + 𝛽𝑖(𝑟𝑚 − 𝑟𝑓 + 𝑟𝑓 − 𝜇𝑚) + 𝜀𝑖

= 𝑟𝑓 + (𝜇𝑖 − 𝑟𝑓 ) − 𝛽𝑖(𝜇𝑚 − 𝑟𝑓 )⏟ ⏞ 𝛼𝑖

+𝛽𝑖(𝑟𝑚 − 𝑟𝑓 ) + 𝜀𝑖

⇒ 𝑟𝑖,𝑒 = 𝛼𝑖 + 𝛽𝑖𝑟𝑚,𝑖 + 𝜀𝑖. (5.2)

5.10 Single-Index Model

Equation (5.2) is the single-index model.

• Note: 𝛼𝑖 = 𝑟𝑝𝑖 − 𝛽𝑖𝑟𝑝𝑚.

5.11 Expected Return-Beta Relationship

Taking expectations of Equation (5.2), we find

𝑟𝑝𝑖 = 𝛼𝑖 + 𝛽𝑖𝑟𝑝𝑚,

since 𝐸[𝜀𝑖] = 0.

• 𝛽𝑖 is known as the security Beta and is a measure of the sensitivity of asset 𝑖 to the market index.

• 𝛽𝑖𝑟𝑝𝑚 is the systematic risk premium, since it is the premium one could expect for taking on systematic risk.

• 𝛼𝑖 is the non-market premium. It is the risk premium expected above that provided by the market.

In equilibrium we expect 𝛼𝑖 = 0.

5.8. Using an Index as a Factor 51


5.12 Why 𝛼 Must Be Zero

Why do we expect 𝛼𝑖 = 0?

• Suppose 𝛼𝑖 > 0.

• We expect individuals to buy more of asset 𝑖, putting more weight on it in their individual portfolios relative tothe market portfolio.

• If everyone did this, the market portfolio would put higher weight on asset 𝑖.

– Everyone deviates from the market portfolio by holding more of asset 𝑖.

– But since everyone constitutes the market, the market portfolio shifts by the exact amount that they wantto hold asset 𝑖.

5.13 Single-Index Regression

We express the single-index model as a regression:

𝑟𝑖,𝑒(𝑡) = 𝛼𝑖 + 𝛽𝑖𝑟𝑚,𝑒(𝑡) + 𝜀𝑖(𝑡),

where the 𝑡 denotes that the relationship must hold for all observations through time.

• We estimate the model by collecting historical observations for 𝑟𝑚, 𝑟𝑖 and 𝑟𝑓 and then computing the regressionestimates for 𝛼𝑖 and 𝛽𝑖.

5.14 Security Characteristic Line

The regression estimates of 𝛼𝑖 and 𝛽𝑖 are denoted by �̂�𝑖 and 𝛽𝑖.

• The fitted values of the regression are

𝑟𝑖,𝑒(𝑡) = �̂�𝑖 + 𝛽𝑖𝑟𝑚,𝑒(𝑡). (5.3)

• 𝑟𝑖,𝑒(𝑡) are the values of the regression line.

• They are the values we expect 𝑟𝑖,𝑒 to take for given values of 𝑟𝑚,𝑒.

5.15 Security Characteristic Line

• Residuals are not included since they are unexpected (deviations from the line).

• Equation (5.3) is known as the Security Characteristic Line or SCL.

5.16 Advantages of the Model

Suppose we want to use the Index Model to produce the estimates required for portfolio optimization.

• Assume there are 𝑁 assets in the portfolio.

• From the model,



𝜇𝑖,𝑒 = 𝛼𝑖 + 𝛽𝑖𝜇𝑚,𝑒

𝜎2𝑖 = 𝛽2

𝑖 𝜎2𝑚 + 𝜎2

𝜀𝑖

and

𝜎𝑖𝑗 = 𝛽𝑖𝛽𝑗𝜎2𝑚.


Let’s count the number of parameters we must estimate.

• 𝑁 estimates of 𝛼𝑖.

• 𝑁 estimates of 𝛽𝑖.

• 𝑁 estimates of 𝜎2𝜀𝑖 .

• One estimate of 𝜇𝑚.

• One estimate of 𝜎𝑚.


That’s a total of 3𝑁 + 2 estimates.

• We’ll see this is much better than 𝑁(𝑁+3)2 estimates without the model.


Suppose you have a dataset consisting of 5 years of monthly returns for each asset (i.e. 60 observations per asset):

𝑁 50 100 1000 3000# Estimates No Model 1325 5150 501,500 4,504,500# Estimates Index Model 152 302 3002 9002Data 3000 6000 60,000 180,000

Clearly estimation is much more reasonable with the single-index model for large 𝑁 .

5.20 Cost of the Model

The index model restricts the relationship among the asset variances and covariances to be of a specific form.

• It is precisely this imposed structure that relieves us of the estimation burden.

• However, it oversimplifies the true nature of the world.

• For example, it dichotomizes security risk into two components: market and asset specific.

• But neglects to account for industry specific risk, etc.

• In this sense, the model may fail to capture important aspects of market.

5.17. Advantages of the Model 53


5.21 Index Model Portfolios

Suppose you hold a portfolio of 𝑁 assets with weights 𝜔𝑖, 𝑖 = 1, . . . , 𝑁 .

• Then

𝑟𝑝,𝑒 =

𝑁∑︁𝑖=1

𝜔𝑖𝑟𝑖,𝑒 =

𝑁∑︁𝑖=1

𝜔𝑖(𝛼𝑖 + 𝛽𝑖𝑟𝑚,𝑒 + 𝜀𝑖)

=

𝑁∑︁𝑖=1

𝜔𝑖𝛼𝑖⏟ ⏞ 𝛼𝑝

+

(︃𝑁∑︁𝑖=1

𝜔𝑖𝛽𝑖

)︃⏟ ⏞

𝛽𝑝

𝑟𝑚,𝑒 +

𝑁∑︁𝑖=1

𝜔𝑖𝜀𝑖⏟ ⏞ 𝜀𝑝

= 𝛼𝑝 + 𝛽𝑝𝑟𝑚,𝑒 + 𝜀𝑝.

5.22 Index Model Portfolio Coefficients

The mathematics highlight some very nice results.

• 𝛼𝑝: The non-market return of the portfolio is a weighted average of the non-market returns of individual assets.

• 𝛽𝑝: The sensitivity of the portfolio to the market excess return is a weighted average of the sensitivities ofindividual assets.

• 𝜀𝑝: The portfolio shock is a weighted average of individual shocks.

5.23 Index Model Portfolio Risk

Since the portfolio is described by a single index model, its risk can be decomposed as

𝜎2𝑝 = 𝛽2

𝑝𝜎2𝑚 + 𝜎2

𝜀𝑝 .

Note that,

𝛽2𝑝 =

(︃𝑁∑︁𝑖=1

𝜔𝑖𝛽𝑖

)︃2

not

𝛽2𝑝 =

𝑁∑︁𝑖=1

𝜔2𝑖 𝛽

2𝑖 .

5.24 Index Model Portfolio Risk

The variance of 𝜀𝑝 can be written as

𝜎2𝜀𝑝 ≡ 𝑉 𝑎𝑟(𝜀𝑝) = 𝑉 𝑎𝑟

(︃𝑁∑︁𝑖=1

𝜔𝑖𝜀𝑖

)︃(5.4)



=

𝑁∑︁𝑖=1

𝜔2𝑖 𝑉 𝑎𝑟(𝜀𝑖) (5.5)

=

𝑁∑︁𝑖=1

𝜔2𝑖 𝜎

2𝜀𝑖

(5.6)

Equation (5.5) follows from Equation (5.4) because we assume that 𝐶𝑜𝑣(𝜀𝑖, 𝜀𝑗) = 0 for 𝑖 ̸= 𝑗.

5.25 Index Model Diversification

In the special case of an equally-weighted portfolio

𝜎2𝜀𝑝 =

1

𝑁2

𝑁∑︁𝑖=1

𝜎2𝜀𝑖 =

1

𝑁𝜎2𝜀𝑝 ,

where

𝜎2𝜀𝑝 =

1

𝑁

𝑁∑︁𝑖=1

𝜎2𝜀𝑖 .

Clearly,

lim𝑁→∞

𝜎2𝜀𝑝 = 0.


Recall

𝜎2𝑝 = 𝛽2

𝑝𝜎2𝑚 + 𝜎2

𝜀𝑝 .

Thus,

lim𝑁→∞

𝜎2𝑝 = lim

𝑁→∞𝛽2𝑝𝜎

2𝑚 + lim

𝑁→∞𝜎2𝜀𝑝

= 𝛽2𝑝𝜎

2𝑚.


5.25. Index Model Diversification 55


5.28 Index Model Example

Let’s estimate an index model.

• Use SPY (S&P 500 SPDR) as a surrogate for market returns.

• Estimate the model for SWY (Safeway).

• Download 5 years of monthly data from Yahoo Finance, between 1 Jan 2007 and 31 Dec 2012.

• Use adjusted closing prices to compute returns.

• Estimate the regression.



Parameter Estimate Standard Error P-Value𝛼 0.003903 0.007387 0.599𝛽 0.7620 0.1319 1.92e-07

The Adjusted 𝑅2 is 0.3133.



CHAPTER 6

Bonds

Contents:

6.1 Bond Prices and Yields

6.1.1 Bond Basics

A bond is a financial asset used to facilitate borrowing and lending.

• A borrower has an obligation to make pre-specified payments to the lender on specific dates.

• Bonds are generally referred to as fixed-income securities.

– Historically, their payments have been fixed.

– In general, they can have variable payments, but those payments are determined by a formula.

• At the end of the bond’s maturity, the borrower pays the face value, or par value of the bond to the lender.

6.1.2 Coupons

Typical coupon bonds require the borrower to make semiannual coupon payments to the lender.

• The coupon rate is the total annual amount paid in coupons divided by the face value.

• Zero-coupon bonds pay no coupons - they simply pay the face value at maturity.

• The only way for such assets to be marketable is for their sale value to be below the face value.

6.1.3 Bond Example

Suppose a semiannual coupon bond has a face value of $1000 and a coupon rate of 8%.

• The total amount of coupons paid annually is

0.08 * $1000 = $80.

• Since coupons are paid semiannually, this amount is divided by two.

• So a $40 coupon is paid every six months and $1000 is paid at maturity.

57


6.1.4 Government Bonds

U.S. Government debt assets fall into three categories.

• Treasury bills (T-bills). These pay no coupons and mature in one year or less from the time of issue.

• Treasury notes. These typically pay semiannual coupons and mature in 2 to 10 years from the time of issue(typical maturities are 2, 5 and 10 years).

• Treasury bonds. These typically pay semiannual coupons and mature between 10 to 30 years from the time ofissue.

6.1.5 Government Bonds

6.1.6 Corporate Bonds

U.S. corporations also issue bonds in order to borrow money. These debt instruments are generally quite similar totheir government counterparts.

6.1.7 Treasury Inflation Protected Securities

Typical bonds offer nominal returns that don’t account for inflation.

• TIPS are bonds whose face values (and hence their coupons) are indexed to the general level of prices.

• If inflation is equal to 2% over the course of a year, the face value of the bond will also rise by 2%.

• The result is a bond with no inflation risk and which should offer a rate of return that is equal to the real, risk-freerate.

6.1.8 TIPS Example

Consider the following bond.

• Face value = $1000.

• Coupon rate = 4%.

• Annual coupon, with three years to maturity.

• Inflation will be 2%, 3% and 1% for the next three years.

6.1.9 TIPS Example

58 Chapter 6. Bonds


6.1.10 TIPS Example

We can compute the nominal and real rates of return.

𝑟𝑛𝑜𝑚 =Interest + Price Appreciation

Initial Price

=40.80 + 20

1000

= 0.0608

6.1.11 TIPS Example

𝑟𝑟𝑒𝑎𝑙 =1 + Nominal Return

1 + Inflation− 1

=1.0608

1.02− 1

= 0.04.

6.1.12 TIPS Example

The real rate of return is simply equal to the coupon rate.

• Inflation has been eliminated since the bond’s face value has implicitly incorporated it into the value by tying itsface value to increases in prices.

6.1.13 STRIPS

STRIPS stands for “Separate Trading of Registered Interest and Principle of Securities”.

• It is a U.S. Treasury program which identifies bond payments as a separate securities.

• A coupon paying bond can be “stripped” into multiple assets - each coupon and the face value being marketedas individual zero-coupon bonds.

– A 10 year, semiannual bond is comprised of 20 coupon payments and a final payment of the face value.

– It could be stripped into 21 separate, zero-coupon assets with varying maturities.

6.1.14 Discounting Cash Flows

A bond’s value should be equal to the net present value of its cash flows.

• Each payment should be discounted by the product of interest rates over the period of interest.

• For a coupon payment in four years

𝑃𝑉 =𝐶

(1 + 𝑟1)(1 + 𝑟2)(1 + 𝑟3)(1 + 𝑟4).

• 𝑟𝑖 is the interest rate over year 𝑖.

6.1. Bond Prices and Yields 59


6.1.15 Discounting Cash Flows

Assuming the interest rate is constant over all periods (𝑟𝑖 = 𝑟, ∀𝑖)

𝑃𝑉 =𝐶

(1 + 𝑟)4.

6.1.16 Geometric Series

The following mathematical result will be useful for deriving the value of a bond:

• For |𝑥| < 1,

∞∑︁𝑖=0

𝑎 𝑥𝑖 =𝑎

1 − 𝑥.

6.1.17 Bond Pricing

Let 𝐹 = Face Value, 𝐶 = Coupon Payment and 𝑉 = Bond Value.

• Then,

𝑉 =𝐶

1 + 𝑟+

𝐶

(1 + 𝑟)2+

𝐶

(1 + 𝑟)3+ . . . +

𝐶

(1 + 𝑟)𝑇+

𝐹

(1 + 𝑟)𝑇

=

𝑇∑︁𝑖=1

𝐶

(1 + 𝑟)𝑖+

𝐹

(1 + 𝑟)𝑇

=

∞∑︁𝑖=1

𝐶

(1 + 𝑟)𝑖−

∞∑︁𝑖=𝑇+1

𝐶

(1 + 𝑟)𝑖+

𝐹

(1 + 𝑟)𝑇

6.1.18 Bond Pricing

=

∞∑︁𝑖=1

𝐶

(1 + 𝑟)𝑖− 1

(1 + 𝑟)𝑇

∞∑︁𝑖=1

𝐶

(1 + 𝑟)𝑖+

𝐹

(1 + 𝑟)𝑇

=

(︃ ∞∑︁𝑖=1

𝐶

(1 + 𝑟)𝑖

)︃(︂1 − 1

(1 + 𝑟)𝑇

)︂+

𝐹

(1 + 𝑟)𝑇

=𝐶

1 + 𝑟

(︃ ∞∑︁𝑖=0

1

(1 + 𝑟)𝑖

)︃(︂1 − 1

(1 + 𝑟)𝑇

)︂+

𝐹

(1 + 𝑟)𝑇

=𝐶

1 + 𝑟

(︃1

1 − 11+𝑟

)︃(︂1 − 1

(1 + 𝑟)𝑇

)︂+

𝐹

(1 + 𝑟)𝑇

60 Chapter 6. Bonds


6.1.19 Bond Pricing

=𝐶

1 + 𝑟

(︃1

1+𝑟−11+𝑟

)︃(︂1 − 1

(1 + 𝑟)𝑇

)︂+

𝐹

(1 + 𝑟)𝑇

=𝐶

1 + 𝑟

1 + 𝑟

𝑟

(︂1 − 1

(1 + 𝑟)𝑇

)︂+

𝐹

(1 + 𝑟)𝑇

=𝐶

𝑟

(︂1 − 1

(1 + 𝑟)𝑇

)︂+

𝐹

(1 + 𝑟)𝑇.

We made use of the geometric series result, recognizing 0 < 𝑥 = 11+𝑟 < 1.

6.1.20 Bond Value Formula

The bond price formula can be decomposed as

𝑉 =𝐶

𝑟

(︂1 − 1

(1 + 𝑟)𝑇

)︂+

𝐹

(1 + 𝑟)𝑇(6.1)

= 𝐶 ×[︂

1

𝑟

(︂1 − 1

(1 + 𝑟)𝑇

)︂]︂⏟ ⏞

Annuity factor(𝑟,𝑇 )

+𝐹 ×[︂

1

(1 + 𝑟)𝑇

]︂⏟ ⏞ PV factor(𝑟,𝑇 )

.

• Annuity factor(𝑟, 𝑇 ) is the present value of a $1 annuity that lasts for 𝑇 periods with interest rate 𝑟.

• PV factor(𝑟, 𝑇 ) is the present value of $1 paid in 𝑇 periods.

6.1.21 Value vs. Price

So far we have only computed the present value of the bond.

• What is its price?

• The bond price should be equal to the present value of payments.

• If the price was lower, you could buy the bond for 𝑃 < 𝑉 and receive the promised cash flow: your net gainwould be 𝑉 − 𝑃 .

• If the price was higher, you could short the bond for 𝑃 > 𝑉 and pay the promised cash flow: your net gainwould be 𝑃 − 𝑉 .

6.1.22 Bond Pricing Example

Consider a bond with the following characteristics.

• 𝐹 = $1000.

• 30 years to maturity.

• 8% coupon rate.

• Semiannual coupons.



Suppose the nominal interest rate is constant at 8% for the next 30 years. Note that

• The coupon payment is $40 every six months.

• There are 60, six-month time periods.

6.1.23 Bond Pricing Example

The bond price is

𝑃 =$40

0.04

(︂1 − 1

1.0460

)︂+

$1000

1.0460= $1000.

• If the annual interest rate equals the coupon rate, 𝑃 = 𝐹 .

If 𝑟 = 0.10, then

𝑃 =$40

0.05

(︂1 − 1

1.0560

)︂+

$1000

1.0560= $810.71.

6.1.24 Prices and Yields

The return for holding a bond is typically referred to as a yield.

• A central feature of bonds is that prices and yields are negatively related.

• Since a bond promises a fixed payment at maturity, you want to purchase for a very low price.


Think of a zero-coupon bond that costs 𝑃 and pays 𝐹 at maturity.

• The yield, 𝑦, is the value such that 𝑃 × (1 + 𝑦) = 𝐹 :

𝑦 =𝐹

𝑃− 1.

• Note that 𝑦 is a net return (not gross).

• Since the bond price is in the denominator, 𝑦 and 𝑃 have an inverse relationship: higher prices mean loweryields.

• This relationship holds for coupon bonds as well (as seen in the bond pricing formula in Equation (6.1)).


Consider a zero-coupon bond with a face value of $1000 and one period until maturity.

• If 𝑃 = $990, the yield is

𝑦 =$1000

$990− 1 = 0.01010.

• If 𝑃 = $995, the yield is

𝑦 =$1000

$995− 1 = 0.005025.

62 Chapter 6. Bonds


6.1.27 Coupon Rate and Yield

We saw above that if the market interest rate 𝑟 is equal to the coupon rate, 𝑃 = 𝐹 .

• Consider two competing investments: put your money in a savings account that pays 𝑟 each period, or buy abond.

• If the coupon rate is exactly equal to 𝑟, then the bond exactly compensates you for market return and does notneed to provide price appreciation.


• If the coupon rate is less than 𝑟, the bond does not compensate you enough - to be competitive, it must sell at adiscount (less than the face value).

• If the coupon rate is greater than 𝑟, the bond overcompensates you - to be competitive, it can sell at a premium(more than the face value).


Why would a borrower sell a bond at a discount when 𝑟 is greater than the coupon rate?

• Because nobody would buy the bond unless the price formula holds.


Later, the bond price will fluctuate on the secondary market.

• If the market rate rises, investors will sell the bond (because of its low coupon).

– The price will fall until the overall return is commensurate with the market return.

• If the market rate falls, investors will buy the bond (because of its high coupon).

– The price will rise until the overall return is commensurate with the market return.

6.1.31 Interest Rate Risk

Unfortunately, interest rates don’t remain constant.

• Suppose you purchase a bond for the face value when the coupon rate and market interest rates are equal.

• If the market rate suddenly rises, you will have your money tied up in an investment paying a rate that is too low(opportunity cost).

• If you sell the bond on the secondary market, the price will be lower than the face (your purchase price) because𝑟 is greater than the coupon rate.

6.1.32 Interest Rate Risk

The opposite would be true if the market rate falls.

• As a result, bond holders are subject to interest rate fluctuations, or interest rate risk.



6.1.33 Maturity Risk

Interest rate risk is exacerbated over longer maturities.

• Suppose the market interest rate moves after you purchase two bonds with 10 and 30 years to maturity.

• If the market rate rises you will suffer a greater loss on the 30 year bond since your money is tied to it for alonger period.

• As a result, prices of longer maturity bonds are more sensitive to interest rate movements.

6.1.34 Maturity Risk

6.1.35 Bond Prices Over Time

As a bond approaches maturity, its price will approach the face value.

• Fewer payments are subject to interest rate fluctuations.

• This is true whether the bond sells at a premium or discount.

6.1.36 Bond Prices Over Time

Consider buying a bond the day before maturity.

• If a coupon is paid at maturity, you only receive a prorated share of the coupon (one day’s worth).

• Net of the prorated coupon, you should pay only the slightest discount to face value.

• The slight discount arises because you can invest your money overnight and earn a small amount ofinterest.

6.2 The Term Structure of Interest Rates

6.2.1 Yield Curve

Bonds of different maturities often have different yields to maturity.

• The relationship between yield and maturity is summarized graphically in the yield curve.

• Consider several examples below.

6.2.2 Yield Curve Slope

An upward sloping yield curve is evidence that short-term interest rates are going to rise.

• Consider two investment strategies.

– Buy and hold a two-year zero-coupon bond, offering 6% return each year.

– Buy a one-year bond today, offering a 5% return over the coming year, and roll the investment into anotherone-year zero-coupon bond a year from now, offering an interest rate of 𝑟2.

64 Chapter 6. Bonds


– These investments should be equivalent. Why?

6.2.3 Yield Curve Slope

Suppose you begin with $890 to invest (the price of a two-year zero-coupon bond with 6% YTM.

• Equating the returns to each strategy gives:

$890 × (1.06)2 = $890 × (1.05) × (1 + 𝑟2)

⇒ 1 + 𝑟2 =1.062

1.05= 1.0701

⇒ 𝑟2 = 0.0701.

6.2.4 Two Investment Strategies

6.2.5 Spot Rates and Short Rates

We distinguish between two types of interest rates.

• Spot rate: the rate offered today on zero-coupon bonds of different maturities.

– In the previous example, the one-year spot rate is 5% and the two year spot rate is 6%.

• Short rate: the rate for given time interval (one year) offered at different points in time.

– In the previous example, the first-year short rate is 5% (same as the spot!) and the second-year short rateis 7.01%.


The spot rate for a given period should be the geometric average of short rates over that interval.

• Let 𝑦2 be the two-year spot rate.

• Let 𝑟1 and 𝑟2 be the first-year and second-year short rates.

• Don’t forget that 𝑦1 = 𝑟1.

(1 + 𝑦2)2 = (1 + 𝑟1) × (1 + 𝑟2)

⇒ 1 + 𝑦2 =√︀

(1 + 𝑟1) × (1 + 𝑟2).


So, if the yield curve slopes up (𝑦2 > 𝑦1 = 𝑟1), we conclude that short-term rates will rise (𝑟2 > 𝑟1).

• Reverse reasoning holds for a downward sloping yield curve.

6.2. The Term Structure of Interest Rates 65


6.2.8 Spot Rate and Short Rate Example

Assume the following spot rates and short rates:

• Spots: 𝑦1 = 0.05, 𝑦2 = 0.06 and 𝑦3 = 0.07.

• Shorts: 𝑟1 = 𝑦1 and 𝑟2 = 0.0701.

• What is the three-year short rate, 𝑟3?


Buying a three-year zero-coupon bond should be identical to buying a two-year zero and rolling into a one-year zero.

(1 + 𝑦3)3 = (1 + 𝑦2)2 × (1 + 𝑟3)

⇒ 1.073 = 1.062 × (1 + 𝑟3)

⇒ 𝑟3 =1.073

1.062− 1 = 0.09025.


We know

(1 + 𝑦2)2 = (1 + 𝑟1) × (1 + 𝑟2).

So the full decomposition is

(1 + 𝑦3)3 = (1 + 𝑦2)2 × (1 + 𝑟3)

= (1 + 𝑟1) × (1 + 𝑟2) × (1 + 𝑟3)

⇒ 1.073 = 1.05 × 1.0701 × 1.09025.


6.2.12 General Short Rates

We can generalize the previous results.

• Investing in an 𝑛 period zero-coupon bond should be the same as investing in an 𝑛 − 1 zero and rolling into aone-period zero at time 𝑛− 1.

(1 + 𝑦𝑛)𝑛 = (1 + 𝑦𝑛−1)𝑛−1 × (1 + 𝑟𝑛)

⇒ 1 + 𝑟𝑛 =(1 + 𝑦𝑛)𝑛

(1 + 𝑦𝑛−1)𝑛−1.

66 Chapter 6. Bonds


6.2.13 Forward Rates

In the development above, we assumed no uncertainty.

• All future rates were known at time zero.

• In reality, we don’t have perfect knowledge of time 𝑛 short rates at time zero.

6.2.14 Forward Rates

To distinguish between actual short rates that occur in the future, we define the forward rate to be

⇒ 1 + 𝑓𝑛 =(1 + 𝑦𝑛)𝑛

(1 + 𝑦𝑛−1)𝑛−1.

• The time 𝑡 = 𝑛 forward rate is the break-even interest rate that equates the returns of an n-period zero-couponbond with an (𝑛− 1) -period zero rolled into a one-period zero.

• It may not be equal to the expected future short rate.

6.2.15 Expectations Hypothesis

The Expectations Hypothesis of the yield curve says that expected short rates equal forward rates:

𝐸[𝑟𝑛] = 𝑓𝑛

⇒ (1 + 𝑦𝑛)𝑛 = (1 + 𝑦𝑛−1)𝑛−1(1 + 𝐸[𝑟𝑛]).

• If the yield curve slopes upward, short rates are expected to rise: 𝐸[𝑟𝑛] > 𝐸[𝑟𝑛−1] > 𝑟1 = 𝑦1.

• If the yield curve slopes downward, short rates are expected to fall: 𝐸[𝑟𝑛] < 𝐸[𝑟𝑛−1] < 𝑟1 = 𝑦1.

6.2.16 Liquidity Preference Theory

According to the Liquidity Preference Theory of the yield curve, investors must be compensated for holding longer-term bonds.

• Longer-term bonds are subject to greater risk, and so investors should demand a premium for holding them.

• In reality, a premium means that investors will only buy them for a lower price (which means greater yield).

6.2.17 Liquidity Preference Theory

The Liquidity Preference Theory can be expressed as forward rates being equal to expected short rates plus a premium,𝜑:

𝑓𝑛 = 𝐸[𝑟𝑛] + 𝜑

⇒ (1 + 𝑦𝑛)𝑛 = (1 + 𝑦𝑛−1)𝑛−1(1 + 𝐸[𝑟𝑛] + 𝜑).

• According to this theory, expected short rates can be constant if the yield curve is upward sloping.

• If the yield curve is downward sloping, expected short rates must be falling. Why?



6.2.18 Liquidity Preference Example

Suppose you buy a two-year bond and that

• Short rates for the next two years are constant at 8%: 𝑟1 = 𝐸[𝑟2] = 0.08.

• The liquidity premium for year two is 1%: 𝜑 = 0.01.

6.2.19 Liquidity Preference Example

What is the yield to maturity of the two year bond?

(1 + 𝑦2)2 = (1 + 𝑟1)(1 + 𝑓2)

= (1 + 𝑟1)(1 + 𝐸[𝑟2] + 𝜑)

= (1.08)(1.09)

⇒ 1 + 𝑦2 =√

1.08 × 1.09

= 1.085.

• So the yield curve slopes up (𝑦2 > 𝑦1) even though expected short rates are constant.

6.2.20 Expectations Hypothesis Example

However, if there is no liquidity premium

(1 + 𝑦2)2 = (1 + 𝑟1)(1 + 𝑓2)

= (1 + 𝑟1)(1 + 𝐸[𝑟2])

= (1.08)(1.08)

⇒ 1 + 𝑦2 =√

1.08 × 1.08

= 1.08.

• Now the yield curve is flat.

6.2.21 Implications of the Theories

The slope of the yield curve always determines whether forward rates are rising or falling.

• 𝑦2 > 𝑦1 means 𝑓2 > 𝑓1 (by the definition of forward rates!).

• If the Expectations Hypothesis holds, 𝐸[𝑟2] = 𝑓2, so 𝑦2 > 𝑦1 means 𝐸[𝑟2] > 𝑟1 = 𝑓1 = 𝑦1.

68 Chapter 6. Bonds


6.2.22 Implications of the Theories

• If the Liquidity Preference Theory holds, we have no guarantee that 𝐸[𝑟2] > 𝑟1 if 𝑦2 > 𝑦1.

– Short rates could be constant with a moderate liquidity premium.

– Short rates could be rising some, with a small liquidity premium.

– Short rates could be falling, with a large liquidity premium.

– What about if 𝑦2 < 𝑦1?

6.2.23 Exp. Hypothesis vs. Liquidity Pref Theory

6.2.24 Historical Term Spread



70 Chapter 6. Bonds

CHAPTER 7

Options

7.1 Call Options

A call option is a contract that allows the buyer the option to purchase an asset for a pre-specified price on or before aparticular date. It has the following ingredients:

• Underlying asset: The asset that may be bought or sold when the option is exercised.

• Maturity (exercise) date: The date at which the contract expires.

• Strike price: The pre-specified price at which the underlying can be purchased.

7.2 Call Options

The buyer of the option is not obligated to buy the underlying asset at the strike price.

• The buyer has the option to buy.

• The seller of the call option is obligated to sell the underlying if the buyer wants to exercise the option.

• If the price of the underlying asset is above the strike price on the maturity date, the buyer will exercise. Why?

• If the price of the underlying asset is below the strike price on the maturity date, the buyer will not exercise.Why?

7.3 Call Options

The call option has no downside risk for the buyer.

• The buyer is better off if the underlying asset price rises.

• If the underlying asset price falls, the buyer doesn’t lose anything.

7.4 Call Options

However, the seller of the option only faces downside risk.

• The seller is worse off if the underlying asset price rises.

• If the underlying asset price falls, the seller doesn’t gain anything.

71


The seller must be compensated for taking the risk of having to sell the underlying for a low price.

• The buyer pays a premium to purchase the option contract.

7.5 Call Option Example

On March 8th 2013, stock for Chipotle Mexican Grill (CMG) sold for $321.84 and an option contract with a strikeprice of $320.00 and maturity date of March 15th 2013 cost $5.30.

• If the price of Chipotle is less than $320.00 on March 15th, the option will not be exercised.

• If the price is $325.00 on March 15th, the option holder (buyer) will exercise the contract.

• The gain to the buyer will be $5.00.

7.6 Call Option Example

• Remember that the contract cost $5.30, so the buyer has a net loss of $0.30.

• If the price on March 15th is greater than $325.30, the buyer will have a net gain.

7.7 Put Options

A put option is a contract that allows the buyer the option to sell an asset (the underlying) for a pre-specified price (thestrike) on or before a particular date (the maturity date).

• The buyer of the put benefits when the price of the underlying asset falls below the strike.

• The buyer of the option can buy the asset at the market price and sell it at the higher strike price (to the writer ofthe put option).

• If the price of the asset rises above the strike, the buyer will not exercise the option and has no downside loss.

7.8 Put Options

• The put is an option (not an obligation) for the buyer to sell the asset at the strike price.

• The writer of the put is under obligation to buy the asset whenever the buyer chooses to exercise the option.

7.9 Put Option Example

Consider again Chipotle stock which sold for $321.84 on March 8th 2013.

• A put option with a strike price of $320.00 and a maturity date of March 15th 2013 costs $3.30.

• If the price of the stock is above $320.00 on March 15th, the option will not be exercised.

72 Chapter 7. Options


7.10 Put Option Example

• Suppose the price is below $320.00 on March 15th: $315.00.

• The buyer of the put will exercise the contract, buying the stock for $315.00 on the market and selling to the putwriter for $320.00.

• The gross profit would be $320.00 - $315.00 = $5.00.

• The net profit would be: $5.00 - $3.30.

7.11 Moneyness

An option is

• In the money when its strike price would produce profits for the holder.

• Out of the money when exercise would be unprofitable.

• At the money when the strike price is equal to the asset price.

The moneyness can be determined at any time, as if the option were exercised at that instant.

7.12 American vs. European

• An American option allows the buyer to exercise the contract on or before the maturity date.

• A European option only allows exercise on the maturity date.

• Since an American option encompasses all of the possibilities of a European option, it should always be morevaluable and cost more.

• As the name denotes, virtually all options traded in the U.S. are of the American flavor.

7.13 Notation

We use the following notation:

𝑇 = Maturity date

𝑆𝑡 = Underlying asset price at time 𝑡

𝑋 = Strike Price

𝐶𝑡 = Value of a call option at time 𝑡

𝑃𝑡 = Value of a put option at time 𝑡

7.10. Put Option Example 73


7.14 Call Option Payoff (Buyer)

The payoff to a call option holder (buyer) at expiration is

𝐶𝑇 =

{︃𝑆𝑇 −𝑋, if 𝑆𝑇 > 𝑋

0, if 𝑆𝑇 ≤ 𝑋.

• If the asset price is above the strike, the holder can buy the underlying for 𝑋 and immediately sell it for 𝑆𝑇 ,yielding a profit of 𝑆𝑇 −𝑋 .

• If the asset price is below the strike, the option is worthless.

7.15 Call Option Payoff (Buyer)

The payoffs above did not account for the cost of the option.

• If the option is purchased at time 𝑡 for a price of 𝐶𝑡, the net payoff to the holder at expiration is

𝐶𝑇 =

{︃𝑆𝑇 −𝑋 − 𝐶𝑡, if 𝑆𝑇 > 𝑋

−𝐶𝑡, if 𝑆𝑇 ≤ 𝑋.

7.16 Call Option Payoff (Buyer

7.17 Call Option Payoff (Seller)

On the flip side, the gross payoff to the call option writer at expiration is

𝐶𝑇 =

{︃𝑋 − 𝑆𝑇 , if 𝑆𝑇 > 𝑋

0, if 𝑆𝑇 ≤ 𝑋.

The net payoff is

𝐶𝑇 =

{︃𝑋 − 𝑆𝑇 + 𝐶𝑡, if 𝑆𝑇 > 𝑋

𝐶𝑡, if 𝑆𝑇 ≤ 𝑋.

7.18 Call Option Payoff (Seller)

7.19 Put Option Payoff (Buyer)

The gross payoff to put option holders at expiration is

𝑃𝑇 =

{︃0, if 𝑆𝑇 > 𝑋

𝑋 − 𝑆𝑇 , if 𝑆𝑇 ≤ 𝑋.



• If the underlying asset price is below the strike, the holder can purchase it for 𝑆𝑇 and immediately resell for 𝑋 ,yielding a profit of 𝑋 − 𝑆𝑇 .

• If the asset price is above the strike at expiration, the option is worthless.


The net payoff to put option holders is

𝑃𝑇 =

{︃−𝑃𝑡, if 𝑆𝑇 > 𝑋

𝑋 − 𝑆𝑇 − 𝑃𝑡, if 𝑆𝑇 ≤ 𝑋.


7.22 Speculation and Hedging

Options can be used for both speculation and hedging.

• Suppose you have $10,000 available for investment.

• A share of stock costs $100.

• An option with a strike price of $100 and six-month maturity costs $10.

• You can lend money (purchase the risk-free asset) at a rate of 3% for the next six months.


Consider three strategies.

• Strategy A: Invest entirely in stock, buying 100 shares at the current price of $100.

• Strategy B: Invest entirely in at-the-money options, buying 10 call contracts (each for 100 shares) selling for$1000 a piece.

• Strategy C: Purchase 100 call options (1 contract) for $1,000 and invest the remaining $9,000 in the risk-freeasset, which will yield a total of $9, 000 × 1.03 = $9, 270 at the end of the six months.


The values of the three strategies are:

7.20. Put Option Payoff (Buyer) 75



The returns to the three strategies are:


From these tables we see two features of options.

• Options offer leverage.

– For the all-option portfolio, the return plummets to -100% when the stock price is below the strike.

– The return rockets to numbers that are much greater than simply holding the stock when the stock priceincreases above the strike.


• Options offer insurance.

– The mixed portfolio has limited downside loss: the investor can’t lose more than -7.3%.

– It also has limited upside gains: if the stock price is above the strike, its returns are always below theportfolio comprised of only stock.




7.29 Protective Put

A protective put strategy consists of simultaneously purchasing a share of stock and a put option on that stock.

• This limits the potential downside loss of the stock while leaving the potential gains intact.

7.30 Protective Put

The put acts as insurance against loss.

• Comparing the net payoff of the protective put with the strategy of holding stock alone shows that the formercomes at a cost.

• This is the insurance premium.

7.31 Protective Put

7.32 Protective Put

7.33 Covered Call

A covered call strategy consists of simultaneously purchasing a share of stock and writing a call option on that stock.

• It doesn’t eliminate downside loss (like the protective put).

• It covers the obligation to deliver the stock for less than its market value if the stock price is above the strike.

7.28. Speculation and Hedging 77


• The call writer is charging a premium (the call price) in order to forsake the upside gain of holding the stock.

7.34 Covered Call

7.35 Straddle

A straddle consists of purchasing call and put options for the same asset and strike price.

• It is a bet on volatility.

• The straddle holder will earn (gross) positive returns if the stock price moves up or down, but nothing if itremains at the strike.

7.36 Straddle

So why doesn’t everyone hold straddles?

• Because the investor must pay for both contracts.

• The investor doesn’t earn a net return unless the stock price moves enough to compensate for the initial outlay.



7.37 Straddle

7.38 Spread

A spread is a combination of two or more options (both calls or both puts) on the same stock with different strikes.

• Some of the options are purchased while others are sold.

• A money spread is the simultaneous purchase and sale of options with different strikes.

• A time spread is the simultaneous purchase and sale of options with different maturities.

7.39 Bullish Spread

A bullish spread:

7.40 Bullish Spread

7.41 Collar

An example of a collar is the purchase of a protective put for one strike price and the sale of a call option, on the samestock, for a higher strike.

• This strategy eliminates downside losses below the strike of the put and also upside gains beyond the strike ofthe call.

• In this case, the investor constrains gains and losses within a region close to the current price of the stock.

7.42 Protective Put Alternative

A protective put eliminates the downside loss of holding stock. We could achieve this with an alternative strategy.

• Purchase a call option with strike price 𝑋 .

7.37. Straddle 79


• Purchase a T-bill (lend at the risk-free rate) with a face value equal to the call strike price, 𝑋 , and the samematurity date as the call.

7.43 Put Call Parity

The payoffs in the previous table are identical to those for the protective put.

• Hence, the cost of the protective put strategy should be equal to the cost of the call plus bonds strategy (why?!!!).

• This fact is known as the Put-Call Parity Relationship.

• Mathematically, it can be expressed as:

𝐶0 +𝑋

1 + 𝑟𝑓= 𝑆0 + 𝑃0.

• This relationship is very useful because it allows us to compute the value of a call option if we know the priceof the corresponding put, and vice versa.

7.44 Put Call Parity Example

Assume

• An asset currently sells for $110.

• A call option with strike 𝑋 = $105 and 1-year maturity sells for $17.

• A put option with strike 𝑋 = $105 and 1-year maturity sells for $5.

• The risk-free interest rate is 5% per year.

• Does the parity relationship hold?


𝐶0 +𝑋

1 + 𝑟𝑓

?= 𝑆0 + 𝑃0.

$117 = $17 +$105

1.05̸= $110 + $5 = $115.



• The relationship doesn’t hold.

• How might an investor take advantage of the situation?


7.46. Put Call Parity Example 81

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Econ133Notes Documentation · Econ133Notes Documentation, Release 0.2 1.8Properties of Expectation For random variables and and constants , ∈ , the expected value has the following