Financial Markets Spring 2012 Final 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 ). 2 The we have the following formulas for means (µX , µY ), variance (σX ), 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 2 σX = var(X) = E (X − µX )2 = p1 (X1 − µX )2 + p2 (X2 − µX )2 + · · · + pn (Xn − µX )2 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 EAR = APR 1+ m 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 C r
years to double
(f) The price of a perpetuity that pays C forever (if the discount rate is r) is: P (Perpetuity) =
(g) The price of an annuity that pays C for T periods is: P (Annuity) = C 1 C C − × = 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: P (Growing Annuity) = C (1 + g)T C C − × = r − g (1 + r)T r−g r−g 1− (1 + g)T (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(˜i , rM ) r ˜ 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 ˜ ri,t = αi + βi rM,t + εi,t ˜e where ri = ri − rf is the excess return of security i ˜e ˜ (e) The total risk (variance) of asset i can be decomposed var(˜i ) r total risk
βi2 var(˜M ) r
var(˜i ) ε
(f) The R-squared of the...