HEC Paris Financial Markets Fall 2013 Midterm Exam “Cheat Sheet” 0. Basic Statistics (a) Consider an n-outcome probability space with probabilities p1 , p2 , . . . , pn . Consider two discrete random variables X and Y with outcomes (X1 , X2 , . . . , Xn ) and (Y1 , Y2 , . . . , Yn ). The we have the following formulas for means (µX , µY ), variance (σX2 ), standard deviation (σX ), covariance (σX,Y ), and correlation (ρX,Y ) µX = EX = E(X) = p1 X1 + p2 X2 + · · · + pn Xn µY = EY = E(Y ) = p1 Y1 + p2 Y2 + · · · + pn Yn σX2 = var(X) = E (X − µX )2 = p1 (X1 − µX )2 + p2 (X2 − µX )2 + · · · + pn (Xn − µX )2 p var(X) σX = σ(X) = σX,Y = cov(X, Y ) = E (X − µX )(Y − µY ) = p1 (X1 − µX )(Y1 − µY ) + p2 (X2 − µX )(Y2 − µY ) + · · · + pn (Xn − µX )(Yn − µY ) cov(X, Y ) ρX,Y = corr(X, Y ) = σX σY (b) Some formulas relating covariances, correlations, standard deviations and variances cov(X, Y ) cov(a1 X1 + a2 X2 , Y ) var(X1 + X2 ) var(aX + b) = = = = corr(X, Y ) σX σY a1 cov(X1 , Y ) + a2 cov(X2 , Y ) var(X1 ) + var(X2 ) + 2 cov(X1 , X2 ) a2 var(X) (c) Univariate regression: By regressing the dependent variable Y on the independent (or explanatory) variable X, one gets the regression line: Y t = α + β X t + εt , where α is the intercept, β is the slope, and εt is the residual (or the error term). One typically assumes E(εt ) = 0, and cov(Xt , εt ) = 0. The slope β is given by β = cov(X, Y )/ var(X). The variance of Y decomposes as var(Y ) = β 2 var(X) + var(ε). The goodness of fit of the regression is measured by R2 = β 2 var(X)/ var(Y ). 1. Present Value (a) Consider an asset with cash flows Ct+1 , Ct+2 , Ct+3 , . . . If the discount rate r is constant, the price of the asset is given by the present value formula Pt = E (Ct+1 ) E (Ct+2 ) E (Ct+3 ) + + + ... 2 1+r (1 + r) (1 + r)3 The discount rate r is the same as the expected return of the asset, and is given by a model such as CAPM or APT (b) Similarly, consider a project involving a series of (net) cash flows C0 , C1 , C2 , . . . , CT occurring in 0, 1, 2, . . . , T periods. The NPV of this project is NPV = C0 + C2 CT C1 + + ··· + 2 (1 + r) (1 + r) (1 + r)T (c) The future value of a cash flow of C, invested over T periods at a rate of return r is: FV (C) = C × (1 + r)T (d) If we know the Annual Percentage Rate (APR), the Effective Annual Rate (EAR), when interest is compounded each of the m subdivisions of a year, is given by m APR EAR = 1 + −1 m (e) If r is the annual rate of return of an investment, it takes T = 72 the investment. The rule of 72 approximates this by T ≈ 100r ln(2) ln(1+r) years to double (f) The price of a perpetuity that pays C forever (if the discount rate is r) is: P (Perpetuity) = C r (g) The price of an annuity that pays C for T periods is: C 1 C C P (Annuity) = − × = T r (1 + r) r r 1 1− (1 + r)T (h) The price of a growing perpetuity that pays initially C and then grows at a rate g per period forever is: C P (Growing Perpetuity) = r−g For stocks this is also called the “Gordon dividend growth formula.” (i) The price of a growing annuity over T periods that pays initially C and then grows at a rate g is: C (1 + g)T C C (1 + g)T P (Growing Annuity) = − × = 1− r − g (1 + r)T r−g r−g (1 + r)T 2. Capital Asset Pricing Model (CAPM) (a) The tangency portfolio T is the market portfolio, with weights given by the market capitalization of each asset. (b) The only thing that matters for equilibrium returns is market risk, measured by beta βi = cov(˜ ri , r˜M ) 2 σM (c) The relationship between expected returns and beta is the Security Market Line (SML) Ei = rf + βi (EM − rf ) (d) Beta can be estimated as the regression coefficient in the CAPM regression e e r˜i,t = αi + βi r˜M,t + ε˜i,t where r˜ie = r˜i − rf is the excess return of security i (e) The total risk (variance) of asset i can be decomposed var(˜ r) | {z i} = total risk βi2 var(˜ r ) | {z M } + systematic risk var(˜ ε) | {z i} idiosyncratic risk (f) The R-squared of the regression is the ratio of systematic risk to the total risk R2 = rM ) systematic risk β 2 var(˜ = i total risk var(˜ ri ) 3. APT (a) Assume asset returns depend on k common factors (systematic sources of risk) r˜i,t = ai + β1i |{z} first factor loading × F1t + · · · + |{z} first factor βki |{z} k-th factor loading × Fkt + |{z} k-th factor ε˜i,t |{z} idiosyncratic shock (b) Then Arbitrage Pricing Theory says that E i − rf | {z } asset i’s risk premium = β1i |{z} E T P − rf | 1{z } + ··· + 1st factor 1st factor loading risk premium βki |{z} ET Pk − rf | {z } k-th factor k-th factor loading risk premium where T P1 , . . . , T Pk are tracking portfolios: they move one-to-one with the factors (c) The Fama–French 3-factor model (version of APT) Ei − rf = βi,MKT EMKT + βi,SMB ESMB + βi,HML EHML • MKT = Market minus rf , • SMB = Small minus Big • HML = Value minus Growth (High minus Low Book/Market ratio)

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