Quantitative Methods in Finance
137071407Quantitative 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?