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# Comprehensive Notes Sheet FIN 500

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This 2 page Study Guide was uploaded by D S on Tuesday October 20, 2015. The Study Guide belongs to FIN 500 at University of Illinois at Urbana-Champaign taught by Adam Clark-Joseph in Summer 2015. Since its upload, it has received 44 views. For similar materials see Introduction to Finance in Finance at University of Illinois at Urbana-Champaign.

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Date Created: 10/20/15

DON T PANIC CARRY A BLANKET Foundation Probability space Qsample space 7 sigma eld lerobability measure A rv X is a measurable function X 9 gt S that maps from the sample space Q to the state space S g R Return 122 1ln1 7 2 Excess Return difference on return and reference US Treasury Bill zit Tit mt Dividends are payments made by a company to its shareholders currently use 0 Stats Expectation X EX 2 221 snip Ea bX cZ a bEX cEZ 1 2 3 2 Variance varX a EX EX2 EX2 EX vara bX b2varX VarX1 X2 varX1 varX2 2covX1X2 12 12 Sd stdX 2 am 2 xva X assume prices behave like Gaussian random walk the volatility increases with square root of time as time increases one reason is that uctuations are re ected by canceling each other out Covariance 012 COUX1X2 EXl EXllX2 EX2ll cova bX c dY bdcovX1 X2 when bd not zero 2 countries 39rf 03p5c1Er 12 sd 2p 333 02E39r 18 sd 3 va39rrm 33222 66232 2 333 66 5 2 3 covr2rm 33 5 2 3 66 32 Erm 33 12 66 18 Correlation corX1X2 2 p12 2 12 cor39ra bX1c l ng corrX1X2 0102 Gaussian Generalized CLT states that the sum of a number of rvs with a power law tail Paretian tail distributions decreasing as x l where 0 i a i 2 and therefore having in nite variance will tend to a stable distribution fx 0c0 as the number of summands grows facu a2 exp2 Does a conditional density function always exist No Stable distributions often don t have closed forms in all forms and thus need characteristic functions Limiting distribution for the averages of most things Linear combinations of Gaussian RVs are also Gaussian so return on portfolio of stocks Fully characterized by two parameters ua Multivariate Gaussian distributions fully characterized by two analogous objects theoretically sensible Return Assumptions Standard deviations of stock returns tend to be large and grow approximately with the square root of time Stock returns tend to have low correlations with previous returns of itself Stock returns tend to be reasonably highly correlated both with each other and the indexes positively correlated factors like expected consumption and oil prices effect all rms Volatility standard deviation of the continuously compounded return It is typically denoted by a To extend to Tperiod volatility multiply one period a by xT Volatility Clustering periods of high and low volatility reason not gaussian The mixture of two Gaussian distributions with different variances has a larger kurtosis than a Gaussian distribution ie not a sum Daily Stock Returns Is approximately symmetric Has fat tails Has a high peak kurtosis equal to three In the data values of k often exceed six for daily returns but are nearer to three for monthly returns CLT RV are said to be independent if information about the outcomes of some of the rvs does not provide any info about the conditional distributions of the other variables independence and zero correlation are not identical properties uncorrelated rv with a Gaussian joint distribution are independent Suppose that X1 X2 is a sequence of independent and identically distributed iid rvs with EXz u and Var Xi a2 less than 00 Then as n gt oo nazn1X u gtd N0 a2 weaken to not identically distributed when have some finite moment strictly greater than 2 and this moment doesnt grow too quickly as i gt oo iid allows for MGH and continuous pdf and therefore bounded cdf mixing process central limit theorem just say rv almost independent for large values nite variance requirement matters Do daily returns have infinite variance cause problems for the CLT Same with Perpetual dependence volatility clustering Since portfolio optimization is based on minimizing variance we cannot perform portfolio optimization at least in the conventional sense for the cases where 0 i a i 2 MV Holding variability constant we prefer higher expected returns Holding expected returns constant we prefer lower variability monotonically increasing utility function return of portfolio 7quot 2 221 win wTr i 7 expected return of portfolio Erp 2 221 wi variance of portfolio va39rrp 2 221 2311 wiwjaij wTZw 1 minEwTZwst worf 111T 77p 2 Owo le 1 0 reducewo 1 lerf wT F 39rf 7 0 1 LaGrangian Lw A EwTZw Tf wT7 7fl 730 3 5 2wA7 rf1 0 39rf F rf1Tw Fp 0 2 12w A7 771 2 10 wAZ1rf1 f 39rf 77 39rflTZ1rf1 77 7 0 7710 I f A 77 rf1TE17 rrfl 2 D1vers1 cation systematic risk w111 remain same mean Will be a straight line but left most point is minimum point same sd just shifts either left or right dependent of which one chosen triangle for risk free asset and risky asset Fun Funds Two Fund Theorem Investors seeking minimum variance portfolios need only invest in combinations of two minimum variance portfolios Assumptions Short Selling risk and same means variances and covariance One Fund Theorem Theorem there is a single fund F of risky assets such that any ef cient portfolio can be constructed as a combination of the fund F and the risk free asset Assumption shortselling risk free same means variances and covariances CAPM 7 rf 8rn 39rf BetaIB Sharpe I39 r j rf SML m 1 SMLzTiszIB i Tm rf Um rh rf Um E39r2 39rf 8Erm 39rf Sharpe slope of line in portfolio with risk free is sharpe so higher sharpe better MV portfolio returns to market risk only when diversified neta is the exposure of the asset to 7 2 Fan 71 w my CML r p 39rf 7p slope CAPM implied expected return d d d 710 w2ag 2w1 wpaMaa 1 w2aI dgi d Fa TMUM Fa TM TmTr Fa m paMaa a l Pad UM 0M pUa UM r7 rrf TA5quotf paaaM 39rf 8rjr 39rf HW he weighted sum standard deviation is ignoring the covariance between the assets whereas the properly calculated standard deviation is not Explain why the portfolio returns would be different from the expected returns of the current holdings Trivial answer past returns are realizations of random variablesthey will inevitably differ from expected returns Less trivial answer there is also something deeper going on namely that the composition of the portfolio can and in fact did change over time My portfolio holdings during the 21 month period were not the same as my current holdings Performance of CAPM a indicates how over or under performed with respect to market and re ect exposure to systematic risk factors that you failed to take into your benchmark model superior skill or stat noise Systematic 203 Idiosyncratic 5 CAPM only uses market risk to determine risk premium which can be argued that n variane terms and n2 n unique covariance terms so for large portfolio covariance dominate the sum and thus no need to have idiosyncratic a 203 va39rei aij ninja Consider two assets A and B and suppose that the standard deviation of As returns is twice as large as the standard deviation of Bs return Asset A must have a larger expected return than asset B False Larger standard deviation does not automatically mean that Asset A would have a larger return too Asset A might be below the mean variance ef cient frontier in which case it might even have a lower return than the asset B Empirical 7 3970 7181 5 CAPM predicts that 3970 not be different from zero if deviate then missing risk factor beta is only facto 2 small minus big HML Thigh now is diff high value and low book to market ratios growth divided stocks into 6 portfolios small value big value top 70 small neutral big neutral 30 70 small growth big growth bottom 30 split big and small by median market equity SMB SV SN SG BV BN BG HML SV BV SG BG differences of returns on portfolios return on zeroinvestment portfolio product of the coef cient 5 and the factor return SMB that is the product 5SM B is the return on 5 in small stocks and si in big stocks excess return returns are explained by 3 factors common to all stocks plus an idiosyncratic component 5 which can be interpreted as the returns on zero investment portfolios FF3 implies that returns on the smallest stocks would have a positive expected Jensens alpha when alpha is calculated using the CAPM T since CAPM ignore two risk factors which tend to be factor for small rms and for rms with high B M small rms positive loadings for SMB Grossman Stiglitz Paradox If markets are ef cient then there is no incentive to gather and or analyze information since prices fully re ect all available information if nobody is gathering or analyzing any info how can prices fully re ect all available information Rational Expectation Equilibrium fully revealing information function that maps joint signal to equilibrium price when each agent solves the consumption portfolio choice using conditional probability where investor j can infer investor i from looking at price noise was supposed to solve this paradox nvestors only gather information until the marginal bene t equals the marginal cost EMH de nitions Informationally Ef cient perfect information goes into price and immediate unbiased reaction nance 7 s EMH rationality risk preferences frictions Yeah if EMH true then questions ability of everyone in nance and corporate nance ie no free lunch no arbitrage no mispricing active asset management useless Quiz If fewer people are buying and selling shares you expect the market to become Unclear signal to noise ratio informed to uninformed increasing or decreasing Semi Strong Generally supported by data if prices re ect all publicly available information such as published imp not possible generate excess returns by poring over the press and company reports if true then investors would be able to make better returns than the market by studying earnings reports etc Strong Disproved in data case money managers consistently outperform risk adjusted benchmarkmarket prices re ect all information whether public or private insider trading doesn t work if true then people with inside information will be able to make larger returns than the market Weak form No evidence that technical trading rules are consistently pro table NO trading rule has an expected riskadjusted net return greater than that provided by riskfree investment if it fully incorporates the inf in past stock prices ie can t generate excess returns by simply studying the history of the share price if true then it would be possible to spot patterns in prices statement on expectation where expectation probability weighted average of all possible outcomes Practical Issues about EMH Transactions costs Regulatory restrictions Missing risk factors Liquidity Taxes ex trading rule Buy when the stock market opens Sell at the close Repeat RWH associates steps with identical means and uncorrelated returns so that returns cannot be predicted from past values mean is stationary E I t u Returns are uncor corrrtrtT 0 Implications on RWH and EMH If the RWH hypothesis is true The latest return and all previous returns are irrelevant if we attempt to predict future returns using linear predictors There is no reason to believe that non linear predictors are usefulmartingale difference0 The expected returns do not depend on the history of time series information Any tests we develop of the RWH can provide insights into market ef ciency Can t disprove EMH weak with rejecting RWH Net returns are gross returns minus all trading costs always make our statement of the EMH hold by de ning an investors opportunity cost to equal Net returns have to be adjusted for risk perhaps by estimating the systematic risk b or bsh of the trading rule b c costs risk preferences vary then market ef ciency not universal EMH refers to expected returns Actual returns may be high by chance so that a hypothesis test is required Expected returns may vary through time as interest rates and risk premia change without contradicting ef ciency Likewise if a trading rule is informative it does not necessarily imply rejection of EMH informative trading rule function that translates price history into investment decision has expected returns not always equal to the same number meaning condition on investment quantity useful Possible to nd ITR with evidence against market ef cient but de nitely not for EMH trades can be made at recorded prices without changing the price path F even though there is asset price patterns still not pro table amount of easy predictability is too small relative to the costs and or risks of trading on it Marginal vs Average Performance indifferent to being in or out of the market the marginal trader will not earn any excess risk adjusted expected net returns if arbitrageurs are heterogeneous average L marginal Yale a6 8 Yale 13 Er Yale CAPM 8 Yale 13 m 118 Conservative mark to market valuation of illiquid holdings might exaggerate bias upwards estimates of beta etc Yales historical real estate portfolios beta appeared to be x 7 8 looking back from the standpoint of 2008 Yale holds lots of illiquid assets Yale typically pays its fund managers something like 2 amp 20 2 of AUM 20 of performance above benchmark If a Yale 0 net of fees we can conclude that on average a Yale return fund managers L 0 assume relative to a perfect benchmark here If RWH or perfect market ef ciency holds then mark to market valuation is ideal Since current market value is our best estimate of future value mark to market for highly liquid assetsUS debt commodities SampP 500 futures etc For illiquid assets mark to market valuation is less accurate Fire sale market price unlikely to re ect fundamental value of illiquid assets Eg imagine specialized industrial equipment technology Roll s Critique impossible to create or observe a truly diversi ed market portfolio one of the key variables of the capital asset pricing model CAPM According to this view a true market portfolio would include every investment in every market including commodities collectibles and virtually anything with marketable value Stocks only a faction of all assets How to Exploit construct superior MV portfolio including some of the omitted assets and while not true market still outperform the current MV portfolio generalized to apply to linear factor asset pricing models such as FF3 by getting closer to true market we can generate a with respect to models CAPM FF3 etc that treat the portfolio of marketable equities as the True market portfolio Verbatim Does this mean that we should expect to get a positive a on an individual non easily marketable asset NO The ostensible a arises from the additional opportunities for diversi cation Ie you can reduce the amount of uncompensated idiosyncratic risk that you face systematic will always remain Asset Classes divide the universe of assets into classes on the basis of assets characteristics and function in portfolio Treat structural and legal characteristics as second order considerations typically why since universe of available assets way too large to analyze Ex Debt vs equity Public vs private Liquid vs Illiquid Quiz Would xed income work Why why not For debt instruments in ation sensitive except TIPS publicly traded somew

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