returns
INTRODUCTION TO PORTFOLIO ANALYSIS
Dimensions of Portfolio Performance
Introduction to Portfolio Analysis
Interpretation of Portfolio Returns Portfolio Return Analysis
Conclusions About Past Performance
Predictions About Future Performance
Introduction to Portfolio Analysis
Reward
Risk vs. Reward
Risk
Introduction to Portfolio Analysis
Need For Performance Measure Portfolio Returns
Performance & Risk Measures Reward portfolio mean return Risk portfolio volatility
Interpretation
Introduction to Portfolio Analysis
Arithmetic Mean Return ●
Assume a sample of T portfolio return observations:
●
Reward Measurement: Arithmetic mean return is given:
●
It shows how large the portfolio return is on average
Introduction to Portfolio Analysis
Risk: Portfolio Volatility De-meaned return
●
●
Variance of the portfolio
●
Portfolio Volatility:
Introduction to Portfolio Analysis
No Linear Compensation In Return ●
Mismatch between average return and effective return
final value=
initial value * (1 +0.5)*(1-0.5)= 0.75 * initial value
Average Return = (0.5 - 0.5) / 2 = 0
Introduction to Portfolio Analysis
Geometric Mean Return ●
Formula for Geometric Mean for a sample of T portfolio return observations R1, R2, …, RT : Geometric mean
●
Example: +50% & -50% return Geometric mean
Introduction to Portfolio Analysis
Application to the S&P 500
INTRODUCTION TO PORTFOLIO ANALYSIS
Let’s practice!
INTRODUCTION TO PORTFOLIO ANALYSIS
The (Annualized) Sharpe Ratio
Introduction to Portfolio Analysis
Benchmarking Performance Risky Portfolio
E.g: portfolio invested in stocks, bonds, real estate, and commodities
Risk Free Asset E.g: US Treasury Bill
Reward: measured by mean portfolio return
Reward: measured by risk free rate
Risk: measured by volatility of the portfolio returns
Risk: No risk, volatility = 0 return = risk free rate
Introduction to Portfolio Analysis
Risk-Return Trade-Off Risky Portfolio
Mean Portfolio Return
Mean Return Risk
0
Risk Free Asset
Excess Return of Risky Portfolio Risk Free Rate
Volatility of Portfolio
Introduction to Portfolio Analysis
Capital Allocation Line Mean Portfolio Return
Risky Portfolio
0
50% in Risky Portfolio 50% in Risk Free Risk Free Asset
Volatility of Portfolio
Leveraged Portfolios: Investor borrows capital to invest more in the risky asset than she has
Introduction to Portfolio Analysis
The Sharpe Ratio Mean Portfolio Return
Slope
0
Risky Portfolio Risk Free Asset
Volatility of Portfolio
Introduction to Portfolio Analysis
Performance Statistics In Action > library(PerformanceAnalytics) > sample_returns mean.geometric(sample_returns) StdDev(sample_returns) (mean(sample_returns) - 0.004)/StdDev(sample_returns)
returns
-0.02, 0 , 0 , 0.06, 0.02, 0.03, -0.01, 0.04
arithmetric mean
0.015
geometric mean
0.01468148
volatility
0.02725541
sharpe ratio
0.4035897
Introduction to Portfolio Analysis
Annualize Monthly Performance
Arithmetric mean: monthly mean * 12 Geometric mean, when Ri are monthly returns:
Volatility: monthly volatility * sqrt(12)
Introduction to Portfolio Analysis
Performance Statistics In Action > library(PerformanceAnalytics) > sample_returns Return.annualized(sample_returns, 12, / geometric = TRUE) StdDev.annualized(sample_returns, scale = 12) FALSE) Std.Dev.annualized(sample_returns, scale = 12)
monthly
FACTOR
annualized
arithmetric mean
0.015
12
0.18
geometric mean
0.01468148
volatility
0.02725541
sqrt(12)
0.0944155
sharpe ratio
0.4035897
sqrt(12)
1.398076
0.1911235
INTRODUCTION TO PORTFOLIO ANALYSIS
Let’s practice!
INTRODUCTION TO PORTFOLIO ANALYSIS
Time-Variation In Portfolio Performance
Introduction to Portfolio Analysis
Bulls & Bears ●
Business cycle, news, and swings in the market psychology affect the market
Introduction to Portfolio Analysis
Clusters of High & Low Volatility
Low
High
Low High
Internet Bubble Financial Crisis
Introduction to Portfolio Analysis
Rolling Estimation Samples ●
Rolling samples of K observations: ●
Rt-k+1
Discard the most distant and include the most recent
Rt-k+2
Rt-k+3
…
Rt
Rt+1
Rt+2
Rt+3
Introduction to Portfolio Analysis
Rolling Performance Calculation
Introduction to Portfolio Analysis
Choosing Window Length ●
Need to balance noise (long samples) with recency (shorter samples)
●
Longer sub-periods smooth highs and lows
●
Shorter sub-periods provide more information on recent observations
INTRODUCTION TO PORTFOLIO ANALYSIS
Let’s practice!
INTRODUCTION TO PORTFOLIO ANALYSIS
Non-Normality of the Return Distribution
Introduction to Portfolio Analysis
Volatility Describes “Normal” Risk High Volatility increases probability of large returns (positive & negative)
Introduction to Portfolio Analysis
Non-Normality of Return
Introduction to Portfolio Analysis
Portfolio Return Semi-Deviation ●
Standard Deviation of Portfolio Returns: ●
●
Take the full sample of returns
Semi-Deviation of Portfolio Returns: ●
Take the subset of returns below the mean
Introduction to Portfolio Analysis
Value-at-Risk & Expected Shortfall 5% ES is the average of the 5% most negative returns 5% most extreme losses 5% VaR
Introduction to Portfolio Analysis
Shape of the Distribution ●
Is it symmetric? ●
●
Check the skewness
Are the tails fa"er than those of the normal distribution? ●
Check the excess kurtosis
Introduction to Portfolio Analysis
Skewness ●
Zero Skewness ●
●
Negative Skewness ●
●
Distribution is symmetric
Large negative returns occur more o#en than large positive returns
Positive Skewness ●
Large positive returns occur more o#en than large negative returns
Introduction to Portfolio Analysis
Kurtosis ●
The distribution is fat-tailed when the excess kurtosis > 0 Fat-Tailed Distribution Normal Distribution
Fat-Tailed Distribution
INTRODUCTION TO PORTFOLIO ANALYSIS
Let’s practice!
Dimensions of Portfolio Performance
Introduction to Portfolio Analysis
Interpretation of Portfolio Returns Portfolio Return Analysis
Conclusions About Past Performance
Predictions About Future Performance
Introduction to Portfolio Analysis
Reward
Risk vs. Reward
Risk
Introduction to Portfolio Analysis
Need For Performance Measure Portfolio Returns
Performance & Risk Measures Reward portfolio mean return Risk portfolio volatility
Interpretation
Introduction to Portfolio Analysis
Arithmetic Mean Return ●
Assume a sample of T portfolio return observations:
●
Reward Measurement: Arithmetic mean return is given:
●
It shows how large the portfolio return is on average
Introduction to Portfolio Analysis
Risk: Portfolio Volatility De-meaned return
●
●
Variance of the portfolio
●
Portfolio Volatility:
Introduction to Portfolio Analysis
No Linear Compensation In Return ●
Mismatch between average return and effective return
final value=
initial value * (1 +0.5)*(1-0.5)= 0.75 * initial value
Average Return = (0.5 - 0.5) / 2 = 0
Introduction to Portfolio Analysis
Geometric Mean Return ●
Formula for Geometric Mean for a sample of T portfolio return observations R1, R2, …, RT : Geometric mean
●
Example: +50% & -50% return Geometric mean
Introduction to Portfolio Analysis
Application to the S&P 500
INTRODUCTION TO PORTFOLIO ANALYSIS
Let’s practice!
INTRODUCTION TO PORTFOLIO ANALYSIS
The (Annualized) Sharpe Ratio
Introduction to Portfolio Analysis
Benchmarking Performance Risky Portfolio
E.g: portfolio invested in stocks, bonds, real estate, and commodities
Risk Free Asset E.g: US Treasury Bill
Reward: measured by mean portfolio return
Reward: measured by risk free rate
Risk: measured by volatility of the portfolio returns
Risk: No risk, volatility = 0 return = risk free rate
Introduction to Portfolio Analysis
Risk-Return Trade-Off Risky Portfolio
Mean Portfolio Return
Mean Return Risk
0
Risk Free Asset
Excess Return of Risky Portfolio Risk Free Rate
Volatility of Portfolio
Introduction to Portfolio Analysis
Capital Allocation Line Mean Portfolio Return
Risky Portfolio
0
50% in Risky Portfolio 50% in Risk Free Risk Free Asset
Volatility of Portfolio
Leveraged Portfolios: Investor borrows capital to invest more in the risky asset than she has
Introduction to Portfolio Analysis
The Sharpe Ratio Mean Portfolio Return
Slope
0
Risky Portfolio Risk Free Asset
Volatility of Portfolio
Introduction to Portfolio Analysis
Performance Statistics In Action > library(PerformanceAnalytics) > sample_returns mean.geometric(sample_returns) StdDev(sample_returns) (mean(sample_returns) - 0.004)/StdDev(sample_returns)
returns
-0.02, 0 , 0 , 0.06, 0.02, 0.03, -0.01, 0.04
arithmetric mean
0.015
geometric mean
0.01468148
volatility
0.02725541
sharpe ratio
0.4035897
Introduction to Portfolio Analysis
Annualize Monthly Performance
Arithmetric mean: monthly mean * 12 Geometric mean, when Ri are monthly returns:
Volatility: monthly volatility * sqrt(12)
Introduction to Portfolio Analysis
Performance Statistics In Action > library(PerformanceAnalytics) > sample_returns Return.annualized(sample_returns, 12, / geometric = TRUE) StdDev.annualized(sample_returns, scale = 12) FALSE) Std.Dev.annualized(sample_returns, scale = 12)
monthly
FACTOR
annualized
arithmetric mean
0.015
12
0.18
geometric mean
0.01468148
volatility
0.02725541
sqrt(12)
0.0944155
sharpe ratio
0.4035897
sqrt(12)
1.398076
0.1911235
INTRODUCTION TO PORTFOLIO ANALYSIS
Let’s practice!
INTRODUCTION TO PORTFOLIO ANALYSIS
Time-Variation In Portfolio Performance
Introduction to Portfolio Analysis
Bulls & Bears ●
Business cycle, news, and swings in the market psychology affect the market
Introduction to Portfolio Analysis
Clusters of High & Low Volatility
Low
High
Low High
Internet Bubble Financial Crisis
Introduction to Portfolio Analysis
Rolling Estimation Samples ●
Rolling samples of K observations: ●
Rt-k+1
Discard the most distant and include the most recent
Rt-k+2
Rt-k+3
…
Rt
Rt+1
Rt+2
Rt+3
Introduction to Portfolio Analysis
Rolling Performance Calculation
Introduction to Portfolio Analysis
Choosing Window Length ●
Need to balance noise (long samples) with recency (shorter samples)
●
Longer sub-periods smooth highs and lows
●
Shorter sub-periods provide more information on recent observations
INTRODUCTION TO PORTFOLIO ANALYSIS
Let’s practice!
INTRODUCTION TO PORTFOLIO ANALYSIS
Non-Normality of the Return Distribution
Introduction to Portfolio Analysis
Volatility Describes “Normal” Risk High Volatility increases probability of large returns (positive & negative)
Introduction to Portfolio Analysis
Non-Normality of Return
Introduction to Portfolio Analysis
Portfolio Return Semi-Deviation ●
Standard Deviation of Portfolio Returns: ●
●
Take the full sample of returns
Semi-Deviation of Portfolio Returns: ●
Take the subset of returns below the mean
Introduction to Portfolio Analysis
Value-at-Risk & Expected Shortfall 5% ES is the average of the 5% most negative returns 5% most extreme losses 5% VaR
Introduction to Portfolio Analysis
Shape of the Distribution ●
Is it symmetric? ●
●
Check the skewness
Are the tails fa"er than those of the normal distribution? ●
Check the excess kurtosis
Introduction to Portfolio Analysis
Skewness ●
Zero Skewness ●
●
Negative Skewness ●
●
Distribution is symmetric
Large negative returns occur more o#en than large positive returns
Positive Skewness ●
Large positive returns occur more o#en than large negative returns
Introduction to Portfolio Analysis
Kurtosis ●
The distribution is fat-tailed when the excess kurtosis > 0 Fat-Tailed Distribution Normal Distribution
Fat-Tailed Distribution
INTRODUCTION TO PORTFOLIO ANALYSIS
Let’s practice!
