Download Formula page Beta and covariance cov(x,y) = E(xy) − E(x)E(y) = E[(x

Formula page
Beta and covariance
cov(x, y) = E(xy) − E(x)E(y) = E[(x − E(x))(y − E(y))] = E[x(y − E(y))]
β y onx = cov(x, y)/var(x)
Prediction and present value
e
Forecast regression Rt+1
= a + bxt + εt+1
If xt = φxt−1 + εt then Et (xt+j ) = φj xt
Gordon growth (const r, g) :
pt − d t
rt − Et−1 rt
≈ Et
∞
X
j=1
ρj−1 ∆dt+j − Et
⎡
≈ (Et − Et−1 ) ⎣∆dt +
∞
X
P
1
=
D
r−g
ρj−1 rt+j ; ρ =
j=1
∞
X
j=1
ρj−1 ∆dt+j −
∞
X
j=1
1
≈ 0.96
1 + D/P
⎤
ρj−1 rt+j ⎦
rt+1 ≈ ρ(pt+1 − dt+1 ) + ∆dt+1 − (pt − dt )
∞
X
zj =
j=0
1
if kzk < 1
1−z
Discount factors, consumption and models
µ 0
¶
u (ct+1 )
pt = Et (mt+1 xt+1 ) = Et β 0
xt+1
u (ct )
mt+1 = β
µ
ct+1
ct
¶−γ
≈ 1 − δ − γ∆ct+1
e
0 = Et (mt+1 Rt+1
); 1 = Et (mt+1 Rt+1 )
Rf = 1/E(mt+1 ) ≈ 1 + δ + γEt (∆ct+1 )
e
e
e
) = −Rtf cov(mt+1 , Rt+1
) ≈ γcov(∆ct+1 , Rt+1
)
E(Rt+1
APT:
max SR2
GRS:
e
= α + βft+1 + εt+1
Rt+1
0
= E(f ) cov(f, f 0 )−1 E(f ) + α0 cov(ε, ε0 )−1 α
h
i−1
¯
α̂0 cov(α̂)−1 α̂ = T 1 + f¯0 Σ−1
α̂0 Σ−1 α̂˜χ2N
f
f
i−1
T −N −K h
¯
α̂0 Σ̂−1 α̂˜FN,T −N −K
1 + f¯0 Σ−1
f
f
N
I promise you do not need cross sectional regression standard errors and cov(α)
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Term structure
(n)
pt
(n)
yt
=
log price at t of bond that comes due at t + n, e.g. -0.20
1 (n)
(n)
(n−1)
(n)
(n)
(n−1)
(n)
≡ − pt ; ft ≡ pt
− pt ; rt+1 ≡ pt+1 − pt
n
Expectations:
(n)
y0
(n)
h
ft
(n)
Et rt+1
i
rtU S
1
E (r0 + r1 + r2 + ...rN −1 ) + (risk premium)
n
(1)
= Et (yt+n−1 ) + (risk premium)
=
(1)
= yt
=
FB regression
+ st − Et st+1 + (risk premium)
³
´
(n)
(1)
(n)
(1)
= rt+1 − yt = a + b ft − yt
+ εt+1
³
´
(n)
(1)
= a + b ft − yt
+ εt+1
(n)
rxt+1
(1)
+ (risk premium)
rtEuro
(1)
yt+n−1 − yt
CP regression
rxt+1
(n)
rxt+1
Eigenvalues
5
1 X (n)
rx
= γ 0 ft + εt+1
4 n=2 t+1
¡
¢
(n)
= bn γ > ft + εt+1
=
Σ = QΛQ0 ; Q0 Q = QQ0 = I
yt = Qxt ; xt = Q0 yt ;
cov(y, y 0 ) = Σ, cov(x, x0 ) = Λ
Lognormal trick
1
E (ex ) = eE(x)+ 2 σ
2
(x)
if x is normal
Portfolios
Quadratic utility, or mean-variance objective,
1
w0 = Σ−1 E(Re ); Σ = cov(Re )
γ
With a factor model
p
em
Rt+1
= Rf + w0 Rt+1
+ wα0 (α + ε).
w0 =
1 E(Rem )
1
; wα = Σ−1 α; Σ ≡ E(εt+1 ε0t+1 )
2
em
γ σ (R )
γ
Bayesian portfolios
f (R) =
Z
α̂ =
f (R|μ)f (μ)dμ.
¡ −1
¢
¢−1 ¡ −1
Σα + Σ−1
Σα α + Σ−1
p
p 0
R˜N (μ, σ 2 ),μ˜N (μ̄, σ 2μ ) thenf (R)˜N (μ̄, σ 2 + σ 2μ )
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List of papers
Cochrane, John H., Asset Pricing
Fama Eugene F. and Kenneth R. French 1996 "Multifactor Explanations of Asset Pricing Anomalies," Journal
of Finance 51, 55-84.
Fama, Eugene F., and Kenneth R. French 2006, “Dissecting Anomalies” Journal of Finance 51, 55-84.
Carhart, Mark M., 1997, “On Persistence in Mutual Fund Performance,” Journal of Finance 52, 57-82.
Fama, Eugene F., and Kenneth R. French, 2008, “Mutual Fund Performance”
Berk, Jonathan, “Five Myths of Active Portfolio Management”
Mitchell, Mark and Todd Pulvino, 2001, “The characteristics of risk and return in risk arbitrage” Journal of
Finance 56, 2135-2176
Malkiel, Burton, and Atanu Saha, 2005, “Hedge Funds: Risk and Return,” Financial Analysts Journal 61 (6)
80-89.
Asness, Cifford, Robert Krail and John Liew, 2001, Do hedge funds hedge? Journal of Portfolio Management,
28 (Fall) 6-19.
Agarwal, Vikas and Narayan Naik, 2004, “Risk and Portfolio Decisions Involving Hedge Funds” Review of
Financial Studies. 17: 63 - 98.
Lamont Owen, and Richard Thaler 2003, “Can the Market Add and Subtract?: Mispricing in Tech-Stock
Carve-Outs” Journal of Political Economy 111: 227-268
Cochrane, John H., “Stock as Money: Convenience Yield and the Tech-Stock Bubble” Manuscript. (Eventually
published in William C. Hunter, George G. Kaufman and Michael Pomerleano, Eds., Asset Price Bubbles
Cambridge: MIT Press 2003
Lamont, Owen, (2004) “Go Down Fighting: Short Sellers vs. Firms” Manuscript, Yale University.
Lamont, Owen, “Go Down Fighting: Short Sellers vs. Firms”
Brandt Michael and Kenneth A. Kavajecz, 2004, “Price Discovery in the U.S. Treasury Market: The impact of
Orderflow and Liquidity on the Yield Curve” Journal of Finance 59, (Dec) 2623-2654.
Mitchell, Mark, Lasse Heje Pedersen, and Todd Pulvino, 2007, “Slow-Moving Capital” Manuscript
Cochrane, John H. and Monika Piazzesi, “Bond Risk Premia” American Economic Review March 2005.
Cochrane, John H. “Portfolio Advice for a Multifactor World”
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