Tài liệu tham khảo về Phân tích rủi ro

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Tài liệu học Phân tích về Rủi ro

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Real vs. Nominal Rates Fisher effect: Approximation R = r + i or r = R i Example: r = 3%, i = 6% R = 9% = 3%+6% or r = 3% = 9%6% Fisher effect: Exact 1 +R R −i 1 +r = or r = ; 1 +i 1 +i 0 . 09 − 0 . 06 Numerically: r = 2 . 83 % = 1 + 0 . 06
Rates of Return: Single Period HPR = P 1 −P0 +D1 P 0 HPR = Holding Period Return 1
Rates of Return: Single Period Example Ending Price = 48 Beginning Price = 40 Dividend = 2 48 − 40 + 2 HPR = = 25 % 40
Characteristics of Probability Distributions 1) Mean: most likely value 2) Variance or standard deviation 3) Skewness * If a distribution is approximately normal, the distribution is described by characteristics 1 and 2
Normal Distribution s.d. s.d. r Symmetric distribution
Measuring Mean: Scenario or Subjective Returns Subjective returns s E( r ) = ∑p i =1 i ⋅ ri i ‘s’ = number of scenarios considered
Numerical example: Scenario Distributions Scenario Probability Return 1 0.1 5% 2 0.2 5% 3 0.4 15% 4 0.2 25% 5 0.1 35% E(r) = (.1)(.05)+(.2)(.05)...+(.1)(.35) E(r) = .15 = 15%
Measuring Variance or Dispersion of Returns Subjective or Scenario Distributions s 2 Variance = σ = 2 ∑ p(i) ⋅ [ r (i) − E(r )] i =1 Using Our Example: 1/2
Risk Uncertain Outcomes p = .6 W = 100 1p = .4 2 1 2 2 2
Risky Investments with RiskFree Investment p = .6 Risky Investment 1p = .4 100 Risk Free Tbills Profit = 5 Risk Premium = 225 = 17
Risk Aversion & Utility  Investor’s view of risk  Risk Averse  Risk Neutral  Risk Seeking  Utility  Utility Function U = E ( r ) – .005 A σ 2  A measures the degree of risk aversion
Risk Aversion and Value: The Sample Investment U = E ( r ) .005 A σ 2 = 22% .005 A (34%) 2 Risk Aversion A Utility High 5 6.90 Tbill = 5% 3 4.66 Low 1 16.22
Dominance Principle Expected Return 4 2 3 1 Variance or Standard Deviation • 2 dominates 1; has a higher return • 2 dominates 3; has a lower risk • 4 dominates 3; has a higher return
Utility and Indifference Curves  Represent an investor’s willingness to tradeoff return and risk Example (for an investor with A=4): Exp Return St Deviation U=E(r).005Aσ 2 (%) (%) 10 20.0 2 15 25.5 2 20 30.0 2 25 33.9 2
Indifference Curves Expected Return Increasing Utility Standard Deviation
Portfolio Mathematics: Assets’ Expected Return Rule 1 : The return for an asset is the probability weighted average return in all scenarios. s E( r ) = ∑p i =1 i ⋅ ri
Portfolio Mathematics: Assets’ Variance of Return Rule 2: The variance of an asset’s return is the expected value of the squared deviations from the expected return. s 2 2 σ = ∑p i =1 i ⋅ [ ri − E( r )]
Portfolio Mathematics: Return on a Portfolio Rule 3: The rate of return on a portfolio is a weighted average of the rates of return of each asset comprising the portfolio, with the portfolio proportions as weights. rp = w1r1 + w2r2
Portfolio Mathematics: Risk with RiskFree Asset Rule 4: When a risky asset is combined with a riskfree asset, the portfolio standard deviation equals the risky asset’s standard deviation multiplied by the portfolio proportion invested in the risky asset. σ p = w risky asset ×σrisky asset
Portfolio Mathematics: Risk with two Risky Assets Rule 5: When two risky assets with variances σ 12 and σ 22 respectively, are combined into a portfolio with portfolio weights w1 and w2, respectively, the portfolio variance is given by: σ p 2 = w 1 2 σ1 2 + w 2 2 σ2 2 + 2 w 1w 2Cov ( r 1, r 2)