HEC Paris Financial Markets Fall 2013 Midterm Exam “Cheat Sheet”

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)