Dimensions of Portfolio Performance€¦ · Introduction to Portfolio Analysis Benchmarking...
Transcript of Dimensions of Portfolio Performance€¦ · Introduction to Portfolio Analysis Benchmarking...
INTRODUCTION TO PORTFOLIO ANALYSIS
Dimensions of Portfolio Performance
Introduction to Portfolio Analysis
Interpretation of Portfolio Returns
Portfolio Return Analysis
Predictions About Future Performance
Conclusions About Past
Performance
Introduction to Portfolio Analysis
Risk vs. Reward
Risk
Rew
ard
Introduction to Portfolio Analysis
Need For Performance MeasurePortfolio Returns
Performance & Risk MeasuresReward portfolio mean return
Risk portfolio volatility
Interpretation
Introduction to Portfolio Analysis
Arithmetic Mean Return
● Reward Measurement: Arithmetic mean return is given:
● It shows how large the portfolio return is on average
● Assume a sample of T portfolio return observations:
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
Geometric mean
● Formula for Geometric Mean for a sample of T portfolio returnobservations R1, R2, …, RT :
Geometric mean
● Example: +50% & -50% return
Introduction to Portfolio Analysis
Application to the S&P 500
INTRODUCTION TO PORTFOLIO ANALYSIS
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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
Reward: measured by mean portfolio return
Risk: measured by volatility of the portfolio returns
Risk Free Asset E.g: US Treasury Bill
Reward: measured by risk free rate
Risk: No risk, volatility = 0return = risk free rate
Introduction to Portfolio Analysis
Risk-Return Trade-Off M
ean
Port
folio
Ret
urn
Excess Return of Risky Portfolio
Volatility of Portfolio
Risk Free Asset
Risk Free Rate
Risky Portfolio
Mean Return Risk
0
Introduction to Portfolio Analysis
Capital Allocation LineM
ean
Port
folio
Ret
urn
Volatility of Portfolio
50% in Risky Portfolio
50% in Risk Free
Leveraged Portfolios: Investor borrows capital to
invest more in the risky asset than she has
0
Risk Free Asset
Risky Portfolio
Introduction to Portfolio Analysis
The Sharpe RatioSlope
Volatility of Portfolio
Mea
n Po
rtfo
lio R
etur
n
0
Risk Free Asset
Risky Portfolio
Introduction to Portfolio Analysis
Performance Statistics In Action> library(PerformanceAnalytics) > sample_returns <- c( -0.02, 0.00, 0.00, 0.06, 0.02, 0.03, -0.01, 0.04)
returns -0.02, 0 , 0 , 0.06, 0.02, 0.03, -0.01, 0.04
arithmetric mean
geometric mean
volatility
sharpe ratio
> mean(sample_returns)
0.015
> mean.geometric(sample_returns)
0.01468148
0.02725541
> StdDev(sample_returns)
0.4035897
> (mean(sample_returns) - 0.004)/StdDev(sample_returns)
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 <- c( -0.02, 0.00, 0.00, 0.06, 0.02, 0.03, -0.01, 0.04)
monthly FACTOR annualized
arithmetric mean 0.015
geometric mean 0.01468148
volatility 0.02725541
sharpe ratio 0.4035897
> Return.annualized(sample_returns, scale = 12, geometric = FALSE)
12 0.18
0.1911235
> Return.annualized(sample_returns, scale = 12, geometric = TRUE)
sqrt(12) 0.0944155
> StdDev.annualized(sample_returns, scale = 12)
sqrt(12) 1.398076
> Return.annualized(sample_returns, scale = 12) / Std.Dev.annualized(sample_returns, scale = 12)
INTRODUCTION TO PORTFOLIO ANALYSIS
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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
High
Financial CrisisInternet Bubble
High Low Low
Introduction to Portfolio Analysis
Rolling Estimation Samples● Rolling samples of K observations:
● Discard the most distant and include the most recent
Rt-k+1 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
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INTRODUCTION TO PORTFOLIO ANALYSIS
Non-Normality of the Return Distribution
Introduction to Portfolio Analysis
Volatility Describes “Normal” RiskHigh 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% most extreme losses
5% VaR
5% ES is the average of the 5% most negative returns
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
● Distribution is symmetric
● Negative Skewness
● Large negative returns occur moreo#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
Fat-Tailed Distribution
Normal Distribution
INTRODUCTION TO PORTFOLIO ANALYSIS
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