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# Exam 1 Cheatsheet MGMT 411

Purdue

GPA 3.76

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This 3 page Class Notes was uploaded by Zijing Tang on Saturday October 1, 2016. The Class Notes belongs to MGMT 411 at Purdue University taught by Lulu Zeng in Fall 2016. Since its upload, it has received 4 views.

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Date Created: 10/01/16

Empirical Facts on Stock Returns Expected Return s= State r(s)= Return if a r state occurs Nominal interest rate ( nominal: Growth rate of money p(s)= Probability of a state Real interest rate ( r ): Growth rate of purchasing real power Rate of inflation: i The relationship: rnominal rreal+i Variance and Standard Deviation Exact relationship: Variance: The equilibrium nominal rate of return: 2 Real after-tax rate: σ = 0.25(0.31 − 0.0976)^2 + 0.45(0.14 − . rnominal(1−t)−i = ( rreal +i)(1−t)−i = rreal (1−t) 0976)^2 + 0.25(−0.0675 − 0.0976)^2 + 0.05(−0.52 − 0.0976)^2 = 0.038 −i ×t Holding Period Return Zero coupon bond: Par = $100, Standard deviation: σ = √ 0.038 = 0.1949 = 19.49% Maturity = T, Price = P The holding period rate: 100−P rf(T = P Returns Using Arithmetic and Geometric Averaging Effective Annual Rate (EAR) － Arithmetic Average － Geometric (Time- Percentage increase in funds invested over a 1-year Weighted) Average horizon Annual Percentage Rate (APR) Where terminal value of the investment: TV = (1+r1) stated /quoted rate, no compounding (1+r2)...(1+rn) The relationship between EAR and APR Variance and Standard Deviation 1+EAR = [1+r (T )] 1/T = [1+T∗APR] 1/T － Estimated Variance －Unbiased(better) f Estimation Continuous compounding: The shorter compounding period is, the bigger EAR is. And EAR has a limit. When T →0, 1 + EAR = eAPR －Estimated Standard Deviation The APR here is called “limiting case continuously compounded interest rate”, rcc. 1 + EAR = e rcc EAR’s limit (instantly The Reward-to-Volatility (Sharpe) Ratio compounding) Excess Return: The difference in any particular period between the actual rate of return on a risky asset and the actual risk-free rate Risk Premium: The difference between the expected HPR on a risky asset and the risk-free rate Normality and Risk Measures Investment management is easier when returns are normal Reason: 1. Standard deviation is a good measure of Anticipating Inflation risk when returns are symmetric 2. If security returns are symmetric, portfolio returns will be as well 3. The Fisher equation: rnominal= r real + E[i] Future scenarios can be estimated using only the (1) The expected real rate is stable and realized mean and the standard deviation 4. The dependence inflation matches initial expectations, then of returns across securities can be summarized using only the pairwise correlation coefficients corr ( rnominal,i) = 1, corr (rreal ,i) = 0 (2) If investors ignored or were very poor at predicting What if excess returns are not normally distributed? 1. inflation, Standard deviation is no longer a complete measure of risk 2. Sharpe ratio is not a complete measure of corr ( rnominal,i) = 0, corr (rreal,i) = −1 portfolio performance 3. Need to consider skewness Holding Period Return and kurtosis Rates of return: Single period HPR= Holding period return P0 = Beginning price P1 = Ending price D1 = Dividend during period one － The simplest way to control risk is to manipulate the fraction of the portfolio invested in risk-free assets versus the portion invested in the risky assets Alternative Measures of Risk y = Weight of the risky portfolio, P, in the complete Value at Risk (VaR): Loss corresponding to a very low portfolio percentile of the entire return distribution, such as the (1−y) = Weight of risk-free assets fifth or first percentile return Expected Shortfall (ES) Also called conditional tail expectation (CTE): focuses on the expected loss in the worst-case scenario (left tail of the distribution) Lower Partial Standard Deviation (LPSD): Similar to usual standard deviation, Risky portfolio (denoted as P) weights: but uses only negative deviations from the risk-free return, thus, addressing the asymmetry in returns The Risk-Free Asset issue Sortino Ratio (replaces Sharpe Ratio): The ratio of average excess returns to LPSD Frequency of Only the government can issue default-free securities. extreme (3-sigma) returns: A good way to measure T-bills are viewed as “the” risk-free asset. Money risk; more extreme returns, or jumps, means more risk market funds also considered risk-free in practice. One Risky Asset and a Risk-Free Asset: Example 5 Risky Portfolios All U.S. equity portfolio － Big/Value －y= Portion allocated to the risky portfolio, P － Big/Growth Small/Value Small/Growth B/M= Book Value/Market Value E( r P )=15% σ P =22% －(1−y)=Portion to be invested in risk-free asset, F rf =7% －The expected return on the complete portfolio: E ( r ) = y E( r ) + (1−y) r = r + y[E ( C P f f rP )− r f ] = 7+y (15−7) Capital Allocation to Risky Assets －The risk of the complete portfolio: σ C =y σ P =22y Risk Aversion and Utility Values Capital Allocation Line 1/2 = scaling factor U= Utility E(r)= Expected return on the asset or portfolio － A= Coefficient of risk aversion σ2 = Variance of returns Speculation: Taking considerable risk for a commensurate gain Gamble: Bet on an uncertain outcome for enjoyment Trade-off between risk and return of a potential portfolio The expected return on the complete portfolio: The risk of the complete portfolio: The equation for CAL: Mean-Variance (M-V) Criterion The slope of the CAL is Sharpe Ratio Portfolio A dominates portfolio B if: E ( r ) ≥ E ( r ) and σ ≤ σ And at least Optimal Portfolio: Indifference Curve Meets CAL A B A B one inequality is strict Indifference Curve The investor must choose one optimal portfolio, C, from the set of feasible choices. The Utility Maximization Problem Asset Allocation The choice among broad asset classes that represents a very important part of Optimal position in the risky asset: portfolio construction. The opportunity set with differential borrowing and lending rates B What if rf<r f ? resulting in a reward-to-volatility ratio of 0.40 Implied Risk Aversion It is estimated that approximately 65.6% of net worth Passive Strategy The passive strategy avoids any is invested in a broad array of risky assets. direct or indirect security analysis. A natural candidate for a passively held risky asset would be a well- This implies a coefficient of risk aversion of diversified portfolio of common stocks such as the S&P 500. Capital Market Line The Capital Market Line (CML) is a capital allocation line formed investment in two passive portfolios: (1) Virtually risk-free short-term T- bills (or a money market fund) (2) Fund of common stocks that mimics a broad market index － From 1926 to 2012, the passive risky portfolio offered: － * an average risk premium of 8.1% － * a standard deviation of 20.48%,

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