fin5122 - Outlines of Lecture 2 Objectives 1. Discuss if...

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Unformatted text preview: Outlines of Lecture 2 Objectives 1. Discuss if asset returns are normally distributed 2. Measure the expected return and risk 3. Discuss if time diversification works References Quant_Review.pdf; Chapter 5 & 7.5, BKM; Handouts Hsiu-lang Chen Portfolio Analysis 1 Are assets returns normally distributed? Individual stock returns may not be normally distributed. The normal return distribution allows for any outcome, including a negative stock price. An alternative assumption: the continuously compounded annual rate of return (r) is normally distributed. [Chapter 5, BKM] [Quant_Review.pdf] Hsiu-lang Chen Portfolio Analysis 2 Are assets returns normally distributed? If re is the effective annual rate, then re=er-1 and re will be lognormal. Because er cannot be negative, the smallest possible value for re is –100%. x ~ ~ N ( µ , σ 2 ) ⇒ ~ = e ~ is lognormal. Note : x z ~ ) = e µ + 1 σ 2 and Var ( ~ ) = e 2 µ +σ 2 ( eσ 2 − 1) 2 Its E ( z z Hsiu-lang Chen Portfolio Analysis 3 Further Discussion on Lognormal Suppose a stock’s price at the beginning of the year is P0=$10. With continuous compounding, if r turns out to be .23, then the end-of-year price will be P1=P0 (1+re)= P0e.23 =$12.59 representing an observed effective annual rate of return of re=(P1-P0)/P0=er-1=.259. That is why BlackScholes Option Pricing Model assumes that the price of the asset underlying the option is lognormally distributed. Recall: ln(P̃1/P0 )=ln(1+r̃e)=ln(er̃)=r̃ is normally distributed! If a stock’s continuously compounded return is normally distributed, then the future stock price is necessary lognormally distributed. Hsiu-lang Chen Portfolio Analysis 4 Why are continuously compounded rates extensively used in Finance? Since information flows into capital markets continuously, rates of return are best modeled as evolving and compounding continuously. Pricing models for derivative assets often are based on continuously rebalanced portfolios. The need to model stock prices as trading continuously calls for continuously compounded rates of return. Hsiu-lang Chen Portfolio Analysis 5 Are assets returns normally distributed? Call re (t) the effective rate over an investment period of length t. With rt normally distributed, the effective holding return over short time period t (a fraction of a year) may be taken as approximately normally distributed. re(t) = ert-1 ≈ rt as t─>0 Suppose continuously compounded annual rate of return (r) is normally distributed with µ and σ2. The monthly continuously compounded return on the stock has: µ(monthly)= µ(annual)/12; σ2(monthly )= σ2(annual)/12 Hsiu-lang Chen Portfolio Analysis 6 Will the distribution of returns of a large portfolio resemble a normal distribution? NORMINV in Excel Hsiu-lang Chen Portfolio Analysis 7 Measuring Mean Returns Arithmetic mean T rA = Geometric mean ∑r t =1 t T T rG = [∏ (1 + rt )] 1/ T −1 t =1 When returns are normally distributed, rG = rA-0.5σ2 Hsiu-lang Chen Portfolio Analysis 8 When do we use rG and rA? Consider a stock that will either double in value with probability of 0.5, or halve in value with probability 0.5. t-2 t-1 +100% What is rG? Hsiu-lang Chen Now -50% What is rA? Portfolio Analysis 9 When do we use rG and rA? Which one describes the average past performance per year? Why? Which one approximates the expected return per year in the future? Why? Hsiu-lang Chen Portfolio Analysis 10 Measuring Risk An ideal risk measure can be used to describe the likelihood and magnitudes of “surprises” (deviations from the mean). A typical measure is the standard deviation (SD) of returns, the square root of the variance. When the returns are normally distributed, the probability of having a return that is within one, two, three SD of the mean is approximately 0.68, 0.95, and 0.997, respectively. Hsiu-lang Chen Portfolio Analysis 11 Asymmetric Return Distribution Consider the two probability distributions for returns of a portfolio. [BKM, p. 132 (8th) or 136 (9th)] [Quant_Review.pdf] A Pr(r) Pr(r) B small losses more likely; extreme gains less likely E(rA) small gains more likely; extreme losses less likely E(rB) By construction, both have identical expected values and variances. Which one is more likely preferred by a risk-averse investor? Hsiu-lang Chen Portfolio Analysis 12 The asymmetry of the distribution is called skewness, which we measure by the third central moment, given by n M3 = ∑ Pr( s )[ r ( s ) − E ( r )] 3 s =1 It gives greater weight to larger deviations, so the “long tail” of the distribution dominates the measure. Thus the skewness of the distribution will be positive for a distribution skewed to the right such as A and negative for a distribution skewed to the left such as B. The σ overestimates risk when the distribution is positively skewed while the σ underestimates risk when the distribution is negatively skewed. Why? The use of options or nonlinear trading rules in the portfolio may cause portfolio returns be non-normally distributed because such strategies reduce variance asymmetrically. Hsiu-lang Chen Portfolio Analysis 13 Figure 5.5A Normal and Skewed Distributions 5-14 Fat-Tail Risk n M4 = ∑ Pr( s )[r ( s ) − E ( r )]4 s =1 Kurtosis A measure of the thickness of the tails. High frequency of extreme returns leads to fat tails! SD underestimates the likelihood of extreme events. Portfolio Analysis Hsiu-lang Chen 15 Other risk measures: Tracking error = σ(∆r) ≡ σ(r-rB) where rB is the benchmark return. It measures how closely the investment tracks the benchmark. Probability of shortfall = Probability (r<r*). It gives the probability that undesirable event might occur but gives no hint as to how sever it might be. Expected shortfall = E[r-r*] over the range where r-r*<0. It represents the magnitude of shortfall times the probability of occurring. A major problem is that it treats a large probability of a small shortfall as equivalent to a small probability of a large shortfall. However, the last two measures about downside risk cannot be built up so easily from individual securities to portfolios. Q: Using usdata.xls to construct Slide 18 & 20. Hsiu-lang Chen Portfolio Analysis 16 Usdata.xls (1926-2009) Rates of Return 1926-2009 World Portfolios World Equity Return in US Dollars Year US Markets World Bond Return in US Dollars Small Stocks Large Stock Long-Term T-Bonds T-Bills Real Tbill Inflation Rates Diversified 1926 25.24 8.10 -8.91 12.21 4.54 3.19 -1.12 4.36 12.20 1927 23.15 9.62 35.48 35.99 8.11 3.13 -2.26 5.51 21.10 1928 28.62 2.44 51.31 39.29 -0.93 3.54 -1.16 4.76 24.29 1929 -12.56 3.45 -48.35 -7.66 4.41 4.74 0.59 4.13 -14.63 1930 -22.6 6.04 -48.36 -25.90 6.22 2.43 -6.40 9.43 -19.11 1931 -39.94 -12.32 -53.17 -45.56 -5.31 1.09 -9.32 11.48 -32.20 1932 1.46 18.26 8.58 -9.14 11.89 0.95 -10.27 12.50 6.01 1933 70.81 29.26 153.18 54.56 1.03 0.30 0.76 -0.46 66.35 1934 0.15 3.87 34.78 -2.32 10.15 0.18 1.52 -1.32 10.08 1935 22.44 -1.41 72.87 45.67 4.98 0.14 2.99 -2.77 27.29 1936 18.84 -0.49 77.01 33.55 6.52 0.18 1.45 -1.25 26.78 1937 -17.7 -0.96 -55.05 -36.03 0.43 0.29 2.86 -2.50 -19.73 1938 6.21 0.65 15.46 29.42 5.25 -0.04 -2.78 2.82 7.77 1939 -5.6 -5.11 -6.79 -1.06 5.90 0.01 0.00 0.01 -2.39 1940 7.97 11.32 -14.15 -9.65 6.54 -0.02 0.71 -0.72 3.12 1941 13.26 5.61 -13.42 -11.20 0.99 0.04 9.93 -9.00 4.24 1942 -0.56 -3.69 39.66 20.80 5.39 0.28 9.03 -8.03 9.27 Hsiu-lang Chen Portfolio Analysis 17 Figure 5.6 Histograms of Rates of Return for 1926-2009 Hsiu-lang Chen Portfolio Analysis 18 Inferred Normal Distribution Hsiu-lang Chen Portfolio Analysis 19 Wealth Indexes of Investment Hsiu-lang Chen Portfolio Analysis 20 Investment horizon: Is it relevant? It is commonly assumed that investors with longer horizons should allocate a larger fraction of their savings to risky assets than investors with shorter horizons. [Chapter 7.5, BKM] Hsiu-lang Chen Portfolio Analysis 21 Consider a one-year investment versus a fiveyear investment. Suppose that a risky asset yield an annual rate of return with E(rP)=15% & σP=30%, and returns are independent over time. What is the return standard deviation of the fiveyear investment if your primary concern is on the averaged annual returns? What is the return standard deviation of the fiveyear investment if your primary concern is on the total returns? Hsiu-lang Chen Portfolio Analysis 22 Hsiu-lang Chen Portfolio Analysis 23 The Relevant Metrics Paul Samuelson argues that the notion of time diversification is specious, because the relevant metrics is terminal wealth, not annualized return. As the investment horizon increases, the dispersion of terminal wealth diverges from the expected terminal wealth. Thus, the total return becomes more uncertain the longer the investment horizon. Hsiu-lang Chen Portfolio Analysis 24 Hsiu-lang Chen Portfolio Analysis 25 Consider a risky investment of $100 that has a 50 percent chance of a one-third gain and a 50 percent chance of a one-fifth loss under two cases. Case1: Returns are independent from one period to the next. Case 2: The risky asset has a 60 percent chance of reversing direction and only a 40 percent chance of repeating its prior return. Hsiu-lang Chen Portfolio Analysis 26 Case 1: Random Returns Initial Wealth Distribution of Wealth after One Period Two Periods Three Periods 1/8 x 237.04 ¼ x177.78 1/8 x 142.22 ½ x133.33 1/8 x 142.22 ¼ x106.67 1/8 x 85.33 100 1/8 x 142.22 ¼ x106.67 1/8 x 85.33 ½ x 80.00 1/8 x 85.33 ¼ x 64.00 1/8 x 51.20 Expected Wealth? Expected Utility if U= -1/W? Hsiu-lang Chen Portfolio Analysis 27 Case 2: Mean-Reverting Returns Initial Wealth Distribution of Wealth after One Period Two Periods Three Periods 0.08x237.04 0.2x177.78 0.12x142.22 ½ x133.33 0.18x142.22 0.3x106.67 0.12x 85.33 100 0.12x142.22 0.3x106.67 0.18x 85.33 ½ x 80.00 0.12x 85.33 0.2x 64.00 0.08x 51.20 Expected Wealth? Expected Utility if U= -1/W? Hsiu-lang Chen Portfolio Analysis 28 Conclusion of Time Diversification If investors are more risk-averse than log wealth investors, time diversification might work when investment returns meanrevert rather than follow a random walk. Questions: Will the implication of time diversification be changed if investors’ degree of risk aversion changes, the gain/loss ratio changes, or the probability of mean-reverting changes? Hsiu-lang Chen Portfolio Analysis 29 ...
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