Quantitative Methods in Finance

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FIN617-chapter11AFQ2020.pptx

Quantitative Methods in Finance

FIN 617

Professor Edwalds

Chapter 11 Topics 1 - 3

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Review Questions

What is seasonality and how do you detect it?

Answer:

Seasonality in a data series is a consistent pattern of movement of the data series within each year

Seasonality is often evident from viewing a graph of the data series over several years. It can be detected by performing an AR(1) regression on the series and observing the autocorrelation of the residuals at the year-ago lag. For quarterly data, the year-ago lag is lag 4. For monthly data, the year-ago lag is lag 12

Review Questions

What is ARCH and how do you test for it?

Answer:

ARCH is Autoregressive Conditional Heteroscedasticity in a data series. This violates the covariance stationary assumptions for the data series, since the error variance is not the same for every observation

Test for ARCH by performing an AR(1) regression on the data series, then perform an AR(1) regression on the squared residuals from the first regression. If the Lag 1 slope coefficient in the second regression is statistically significant (based on the t-statistic), the data series has ARCH

Chapter 11 – Multifactor Models

Brief background of portfolio theory

1952 – Markowitz – Modern Portfolio Theory

Investors risk-averse (for same return, prefer less volatility)

Describe assets by mean, variance, and covariance of expected returns

1964 – Sharpe – Capital Asset Pricing Model

Based on Markowitz assumptions

Asset returns are sensitivity factor (Beta) times market returns

Market risk is only source of priced risk

1976 – Ross – Arbitrage Pricing Theory

Various sources of systemic risk

Each priced separately in the market

Arbitrage Pricing Theory

Assumptions:

Model describes asset returns based on sensitivity to risk factors priced in market

Many assets: diversified portfolios eliminate asset-specific risk

No arbitrage opportunities

Resulting model: , where

expected return to portfolio

risk-free rate

expected reward for bearing the risk of factor

Factor risk premium or price

sensitivity of portfolio to factor

number of factors

Example 1: One-Factor APT Model Parameters

Portfolio Expected Return Factor Sensitivity
A 0.075 0.5
B 0.150 2.0
C 0.070 0.4

Model:

Portfolio equations:

Solving:

Example 1: Check Model Parameters on C

Portfolio Expected Return Factor Sensitivity
A 0.075 0.5
B 0.150 2.0
C 0.070 0.4

Model:

Portfolio C equation:

Example 2: Finding Arbitrage Opportunities

Portfolio Expected Return Factor Sensitivity
A 0.075 0.50
B 0.150 2.00
C 0.070 0.40
D 0.080 0.45

Model:

Portfolio equation:

So there is arbitrage opportunity

Example 2: Constructing an Arbitrage

Portfolio Expected Return Factor Sensitivity
A 0.0750 0.50
B 0.1500 2.00
C 0.0700 0.40
D 0.0800 0.45
(A+C)/2 0.0725 0.45

Risk premium fluctuates with risk factor results

Match risk sensitivity with consistent portfolios:

Example 2: Constructing an Arbitrage

Portfolio Expected Return Factor Sensitivity
A 0.0750 0.50
B 0.1500 2.00
C 0.0700 0.40
D 0.0800 0.45
(A+C)/2 0.0725 0.45

Purchase one with larger return, short one with smaller return

Buy D for some amount, say $10,000

Short sell $5,000 of A and $5,000 of C

Net initial investment is $0

Will return profit with certainty

Gain 8% ($800) on D, plus .45 times risk return

Lose 7.25% ($725) on A & C, + .45 times risk return

Net gain is $75

Homework

Workbook Chapter 11 problems:

#1 and #3

Review Question

What are the assumptions of Arbitrage Pricing Theory?