Unformatted text preview: Risk and Return Risk Measures of Return Measures
s s s s s s Holding Period Return (HPR) = [P(1)P(0)+D]/P(0). This Holding can be generalized. can Riskless Rate of Return = Return on ShortTerm US Riskless Treasury Bills. Treasury Expected Return = Probability weighted average return Excess Return = Return to nondiversifiable risk = Return Excess in excess of the riskless rate of return for a fully diversified portfolio. Also called the Risk Premium. portfolio. Abnormal Return = Return to diversifiable risk = Return in Abnormal excess of the Excess Return (Positive or Negative). excess Expected Return and Standard Deviation (Normal Expected Distributions). Distributions). Returns & Statistical Moments Returns
s s s Expected Rate of Return: E(r) = Σsp(s)r(s) Expected Standard Deviation of Expected return = σ2 = Standard Σsp(s)[r(s)E(r)]2 Skew = E[r(s)E(r)]3 / σ3 Kurtosis = [E[r(s)E(r)]4 / σ4 ] – 3 – Measures asymmetry of deviation about the mean (i.e. Measures expected rate of return). expected – Measures degree of fat tails in the return distribution – Subtract 3 because the ratio for a normal distribution Subtract would be 3, and this transformation normalizes the measure. measure. s Measuring Returns Measuring
s Time Series v. Scenario Analysis
– Forward looking scenarios v. History (Time Forward Series) Series) Arithmetic Average: E(r) = (1/n) Σsr(s) Arithmetic r(s) where each r(s) is one observation, Not Necessarily a Distinct Value. Necessarily s Geometric Average: The value of g such Geometric that Πs=1,n(1+r(s)) = (1+g)n.
s Measures of Risk & RiskAdjusted Returns RiskAdjusted
Variance & Standard Deviation: Measures of Variance Dispersion (i.e. Volatility) Dispersion s Reward to Variability (Sharpe Ratio) = Risk Reward Premium / Standard Deviation of Returns. Premium
s – The Sharpe Ratio measures the return per unit of The risk. risk. – Ignores effects of skew & kurtosis.
» Large Cap stocks exhibit significant negative skew » Small Cap stocks exhibit much less negative skew » Kurtosis is relatively insignificant NonNormal Measures NonNormal of Risk1
s Value at Risk (VaR): Value
– Measures the potential loss from extreme Measures negative returns. negative – Measured using a “quantile” of the distribution Measured of returns (usually the 5% quantile). of – The quantile gives the probability of incurring a The loss greater than or equal to the VaR (i.e. the quantile value of returns). quantile NonNormal Measures NonNormal of Risk2
s Conditional Tail Expectation (CTE):
– Given a VaR quantile, the CTE is the Expected Given Return of the possible returns in the bottom quantile. quantile. – Improves on VaR because it calculates the Improves expected value of the Bad Case scenarios captured by the VaR quantile, providing a better measure of worst case return outcomes. better NonNormal Measures NonNormal of Risk3
s Lower Partial Standard Deviation (LPSD):
– Provides a measure of “Downside Risk,” i.e. the Provides risk of a return below the Expected Return. risk – Defined to be the standard deviation of returns Defined that are less than the expected return. that – Does not provide information that materially Does improves on the Sharpe Ratio because the effects of skew & kurtosis of the conditional distribution are similar to those of the complete distribution. distribution. Risk in a Portfolio Context Risk
Standard deviation of an asset's rate of Standard return is a useful measure of its standalone risk risk s It is not an appropriate measure of the It asset's risk when it is part of a portfolio. asset's
s Example Example
You own a home you purchased for You $100,000. $100,000. s The home is likely to be worth $120,000 by The next year. next s During this period there is a 0.5% During probability of the home being completely destroyed in a fire destroyed s Suppose you are able to buy an insurance Suppose policy policy
s Payoff across two states Payoff
Fire Fire No Fire House Policy Total Payoff 0 120,000 120,000 120,000 0 120,000 Examine Portfolio payoff Examine
Portfolio’s payoff is riskfree Portfolio’s s Which characteristic of the payoffs on the Which two assets is responsible for this result? two s How much would you be willing to pay for How the insurance policy? the
s Risk preferences Risk
Willingness to pay for insurance depends Willingness on risk aversion on s Consider the following utility function: U = E(rp)  0.5Aσ p2
s s Characterizes a meanvariance optimizer No Preference for Skewness No Calculate utility from owning the asset
s The expected rate of return on the home itself is The given by: given
E(r) = {[0.005(0) + 0.995(120,000)]  100,000}/100,000 E(r) = 0.194 s The variance of the rate of return on the The home = .005(1  0.194)2 + 0.995(0.2  0.194)2 .005(1 0.995(0.2 = 0.007164 Calculate utility Calculate
s Suppose that your risk aversion parameter Suppose is equal to 2.0. is Then your utility from the home is given Then by, by, U = 0.194  0.5(2)(0.007164) = 0.186836 s Acceptable cost of insurance Acceptable
Calculate utility from a portfolio consisting Calculate of the home and the policy of s U = E(rportfolio)  0.5Aσ 2portfolio
s = {(120,000) (100,000+C)}/(100,000+C) {(120,000)
s Set this equal to 0.186836 and solve for Set "C". "C". Answer: C = $1,109.17 s Another choice problem Another
Consider a riskaverse investor with Consider coefficient of relative risk aversion ‘A’ = 2. Suppose she was offered the opportunity to invest in one of two opportunities: invest s Option A: CD with a 5% sure rate of return Option s Option B: A stock with expected return = Option 20% and variance (σ2) = 15% 20% s Which would she choose to invest in?
s Choice Problem Choice
s Utility from investing in the stock:
– Ustock = E(r )  0.5 A σ 2 = 0.20  (0.5)(2) (0.15) stock – Ustock = 0.05 s Utility from investing in the CD:
– UCD = E(r )  0.5 A σ 2 = 0.05  (0.5)(2) (0) – UCD = 0.05 The investor is indifferent between the stock The investment and a certain return of 5% investment s Hence, the stock’s certainty equivalent rate Hence, certainty of return for the investor is 5% of
s Another example Another
Since 1945, the S&P 500 has earned an Since average return of 12.28% per year with a standard deviation of 14.62%. What is the certainty equivalent rate of return of a S&P500 index fund for our investor? S&P500 s Answer : 10.14%
s Inferring the coefficient of relative risk aversion ‘A’ relative
Suppose a meanvariance optimizing Suppose investor is indifferent between investing in indifferent an S&P 500 index fund and riskfree Tbills which have yielded 3.75% per year. which s What can you infer about her relative risk What aversion?
s Portfolio expected return and Standard Deviation Standard
E (rp ) = W A E ( rA ) + W B E ( rB )
2 2 2 W A σ A + W B2σ B + 2W Aσ AWBσ B ρ A, B σp = ...
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 Spring '09
 TongYao
 Management, Standard Deviation, Variance

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