INTRODUCTION TO PORTFOLIO ANALYSIS. Dimensions of Portfolio Performance

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1 INTRODUCTION TO PORTFOLIO ANALYSIS Dimensions of Portfolio Performance

2 Interpretation of Portfolio Returns Portfolio Return Analysis Conclusions About Past Performance Predictions About Future Performance

3 Risk vs. Reward Reward Risk

4 Need For Performance Measure Portfolio Returns Performance & Risk Measures Reward Risk portfolio mean return portfolio volatility Interpretation

5 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

6 Introduction to Portfolio Analysis Risk: Portfolio Volatility De-meaned return Variance of the portfolio Portfolio Volatility:

7 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 = ( ) / 2 = 0

8 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

9 Application to the S&P 500 Introduction to Portfolio Analysis

10 INTRODUCTION TO PORTFOLIO ANALYSIS Let s practice!

11 INTRODUCTION TO PORTFOLIO ANALYSIS The (Annualized) Sharpe Ratio

12 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

13 Risk-Return Trade-Off Risky Portfolio Mean Return Risk Mean Portfolio Return Risk Free Asset Excess Return of Risky Portfolio Risk Free Rate 0 Volatility of Portfolio

14 Capital Allocation Line Risky Portfolio Mean Portfolio Return 50% in Risky Portfolio 50% in Risk Free Risk Free Asset Leveraged Portfolios: Investor borrows capital to invest more in the risky asset than she has 0 Volatility of Portfolio

15 The Sharpe Ratio Slope Mean Portfolio Return Risk Free Asset Risky Portfolio 0 Volatility of Portfolio

16 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) > mean.geometric(sample_returns) StdDev(sample_returns) (mean(sample_returns) )/StdDev(sample_returns) returns -0.02, 0, 0, 0.06, 0.02, 0.03, -0.01, 0.04 arithmetric mean geometric mean volatility sharpe ratio

17 Annualize Monthly Performance Arithmetric mean: monthly mean * 12 Geometric mean, when Ri are monthly returns: Volatility: monthly volatility * sqrt(12)

18 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) > StdDev.annualized(sample_returns, Return.annualized(sample_returns, scale = 12, 12) geometric / = FALSE) TRUE) Std.Dev.annualized(sample_returns, scale = 12) monthly FACTOR annualized arithmetric mean geometric mean volatility sharpe ratio sqrt(12) sqrt(12)

19 INTRODUCTION TO PORTFOLIO ANALYSIS Let s practice!

20 INTRODUCTION TO PORTFOLIO ANALYSIS Time-Variation In Portfolio Performance

21 Bulls & Bears Business cycle, news, and swings in the market psychology affect the market

22 Clusters of High & Low Volatility Low High Low High Internet Bubble Financial Crisis

23 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

24 Rolling Performance Calculation Introduction to Portfolio Analysis

25 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

26 INTRODUCTION TO PORTFOLIO ANALYSIS Let s practice!

27 INTRODUCTION TO PORTFOLIO ANALYSIS Non-Normality of the Return Distribution

28 Volatility Describes Normal Risk High Volatility increases probability of large returns (positive & negative)

29 Non-Normality of Return Introduction to Portfolio Analysis

30 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

31 Value-at-Risk & Expected Shortfall 5% ES is the average of the 5% most negative returns 5% most extreme losses 5% VaR

32 Shape of the Distribution Is it symmetric? Check the skewness Are the tails fatter than those of the normal distribution? Check the excess kurtosis

33 Skewness Zero Skewness Distribution is symmetric Negative Skewness Large negative returns occur more often than large positive returns Positive Skewness Large positive returns occur more often than large negative returns

34 Kurtosis The distribution is fat-tailed when the excess kurtosis > 0 Fat-Tailed Distribution Normal Distribution Fat-Tailed Distribution

35 INTRODUCTION TO PORTFOLIO ANALYSIS Let s practice!