14
Quick summary
Basics:
1. Value? To an investor.
∙ 0
¸
u (ct+1 )
xt+1
pt = Et β 0
u (ct )
p = E(mx)
u0 (ct+1 )
m = β 0
u (ct )
2. Alternative expressions and special cases
basic p = E(mx)
riskfree Rf = 1/E(m)
E(x)
risk corrections, price discount p =
+ cov(m, x)
Rf
expected returns E(R) = Rf − Rf cov(m, R)
beta models = Rf + βλ; λ = −Rf var(m)
µ
¶
dΛ
f
continuous time Et (dRt ) = rt dt − Et dRt
Λ
3. Economics
rf =
1
− γ (γ + 1) σ 2
| 2 {z
}
intertemporal substitution. precautionary saving
µ
¶
dct
e
E(R ) = γcovt dRt ,
ct
δ + γμ
| {z }c
Risk (measured by covariance), risk aversion
4. Equity premium/risk free rate puzzle.
kEt (dRte )k
= γσ t
σ t (dRte )
µ
dct
ct
¶
ρ
(a) SR = 0.5 needs huge (50) risk aversion
(b) Worse (250) if ρ (∆c, R) = 0.2.
(c) High γ means rf high (γμc dominates initially), or unbelievable balance around
γ = 200.
(d) High γ means rf is very sensitive to ∆c.
(e) At a minimum, need high risk aversion, Epstein-Zin or habits to separate risk
aversion from intertemporal substitution.
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Representation
1. States s; Contingent claims, risk neutral probabilities. Introduction to state space
geometry.
2. LOOP and Arbitrage
(a) Loop (linearity) ⇔ ∃ a unique x∗ ∈ X s.t. p = E(x∗ x)∀x ∈ X.
(b) Loop + No arbitrage ⇔ ∃m > 0 s.t. p = E(mx).
(c) Formulas for x∗
x∗ = p0 E(xx0 )−1 x
¤ −1
1
1 £
f
x∗ =
−
E(R)
−
R
Σ [R − E(R)]
Rf
Rf
£
¤ −1
dΛ∗
f
f
=
−r
dt
−
E(dR)
−
r
dt
Σ dR (σdz)
Λ∗
3. Mean variance frontier
(a) m is minimum second moment return
(b) mvf implied by p = E(mx).
kE(Rte )k
σ(Rte )
<
σ(m)
E(m)
Cone.
(c) Hansen-Richard.
i. R∗ = x∗ /p(x∗ ); Re∗ = proj(1|Re )
ii. all returns R = R∗ + wRe∗ + n
iii. mvf: Rmv = R∗ + wRe∗
4. MVF ⇔ p = E(mx) ⇔ E(Re ) = βλ. All three representations are equivalent.
(a) m, x∗ , R∗ , Rmv = R∗ + wRe∗ all are reference variables for ER − β.
(b) x∗ = proj(m|X), R∗ = E(x∗ )/E(x∗2 ) on mvf
(c) Rmv on mvf → can construct m (m = a + bRmv )
(d) f : E(Re ) = β f λ then m = b0 f
(e) f = Rmv satisfies beta representation
5. Conditioning information
(a) Condition down:
pt = Et (mt+1 xt+1 ) ⇒ E(p) = E(mx)
(b) Add instruments=managed portfolios.; In principle this is sufficient for all conditioning information
pt = Et (mt+1 xt+1 ) ⇔ E(zt pt ) = E [mt+1 (xt+1 zt )]
(c) Models with time t parameters do not necessarily condition down.
R
mv
mt+1 = a − bft+1 ⇒ mt+1 = at − bt ft+1
on unconditional mvf ⇒ Rmv on conditional mvf
Not ⇐. Don’t forget scaled portfolios in the definition of unconditional mvf.
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(d) Scaled factors model
mt+1 = a(zt ) − b(zt )ft+1 ⇔ mt+1 = a ⊗ zt − b (ft+1 ⊗ zt )
Factor Pricing Models
1. Basic idea:
β
u0 (ct+1 )
≈ a + b0 ft+1
u0 (ct )
f: Market. (CAPM). News (ICAPM) Macro variables. Mimicking portfolios. APT:
other big portfolios.
2. CAPM: 2 period quadratic, quadratic iid, exponential normal, log utility...
3. ICAPM: Other state variables
4. APT: use f ∗ to price portfolios ”near” f ? Works with Sharpe ratio limits. Then
E(Re ) = α + βf + ε, extra sharpe ratio = α0 Σα, so small ε ⇒ small α.
Estimation and testing
1. General: procedure for estimating free parameters (α, β, b, γ,etc.), standard errors, test
for 0 pricing errors α̂0 V −1 α̂
2. GMM for consumption-based models and in general.
(a)
agT (b) = 0
d S gT (b) = 0
0
−1
formulas for σ(b̂), cov(gT ). (Don’t have to memorize!)
(b) Linear combinations from minimization
min gT (b)0 W gT (b)
∂gT
W gT (b) = 0
∂b0
[d0 W ] gT (b) = 0
(Solve by numerical minimization. a = d0 W for distribution theory)
(c) Two step procedure for consumption model
(d) Interpretation: gT = E(mx − p) = Rf α = pricing errors. J = α0 V −1 a = test
3. GMM corrections for OLS standard errors
gT (β) = E(xt (yt − βxt )) = 0
4. Time series regressions for E(Re ) = βλ
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(a) Idea: Rtei = αi + β i ft + εit t = 1, 2, ...T ∀i., OLS estimates for α, β. λ̂ = ET (ft )
(b) OLS and GMM distributions.
(c) GRS statistic for α0 V −1 α and its GMM counterpart.
5. Cross sectional regressions for E(Re ) = βλ
(a) Idea. TS for βs. OLS and GLS E(Rei ) = (γ) + β i λ + αi i = 1, 2, ...N
(b) Difference between CS and TS
(c) Fixed β standard errors
(d) GMM standard errors
(e) Shanken as a special case of GMM, correct for β̂
(f) Note: If errors are not iid, (β 0 Σ−1 β)−1 β 0 Σ−1 ET (Re ) is not d0 S −1 gT (β, λ). Efficient
GMM is no longer GLS cross sectional regression.
6. Fama MacBeth
(a) Idea, and procedure.
(b) Point: correct OLS std. errors for cross sectional correlation
(c) Equivalence to pooled, XS when betas are fixed
7. No need to assume iid homoskedastic any more! Do α0 V −1 α even if FMB
8. GMM for linear factor models m = a − b0 ft ; gT = E(mRe ) = 0 or E(mR) = 1.
(a) E(p) on E(fx0 )
(b) E(Re ) on E(Re f 0 )(a = 1)
³ ´
e
e 0
(c) E(R ) on cov(R f ) (E f˜ = 0); but harder s.e. since Ef is estimated.
9. All work about the same
10. λ vs. b. λ is not a test for “do we need this factor to price other assets.”
Options and bonds
1. Options without intermediate trading, p = E(mx) treatment
(a) Call, put definitions and payoff diagrams
(b) LOOP, ∃m : put-call parity
(c) NA, m > 0. Arbitrage bounds. Payoff approach (A > B ⇒ P (A) > P (B)). m
approach and linear program
(d) ( σ 2 (m) or other restrictions can narrow arbitrage bounds. Like APT)
min(max){m} C = E(mxc ) s.t.S = E(mST ); 1 = E(mRf ), m > 0, σ 2 (m) < A,...
2. Options with dynamic trading, Black Scholes
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(a) Use Λ∗ to price xc .
Λ∗
(b) Solve Λ∗ forward, C = E( ΛT0 xcT )
(c) Guess C(S, t), use E(dR) − rf = −E
3. Bonds
¡ dΛ
Λ
¢
dR , to pde for C. Solve PDE
(a) y,p,f, definitions. Expectations hypothesis
(b) Discrete time term structure model. Example,
mt+1 + δ = ρ (mt + δ) + εt+1
(n)
pt
= ln Et (emt,t+n )
i. Solve m forward
ii. Solve p backward
(c) Character of result: one factor model with constant risk premium
(d) Continuous time term structure models. Form
dΛ
= −r(xt )dt − σ Λ (xt )dz
Λ
dxt = μx (xt )dt + σ x (xt )dz
(e) Bond PDE derivation
(f) Solution for CIR, Vasicek cases.
(g) Example: setting up differential equation for a term structure option.
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4. Big picture for APT, options, bonds, relative pricing
X, prices you know
m>0, arbitrage bounds
Hedge portfolio b’f
Small or 0 error
Payoff you want to price
f*, discount factor for f
m with Sharpe ratio, other limts
(a) If the payoff xc is in X (Black Scholes), you can price it exactly (given prices in
X) by using f ∗ (Λ∗ ), p = E(f ∗ xc )
(b) If the payoff is near X — small error — it’s often priced with f ∗ anyway, assuming
market price of extra risk is zero (Term structure models.)
(c) Other m give other answers. For options m > 0 gives arbitrage bounds. For APT,
σ(m) < h gives Sharpe ratio bounds. Small ε can lead to usefully tight bounds.
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Important things you’re missing
1. Ch 20, 21 empirical work!
(a) Size, B/M factors, momentum, the latest alphas, (Fama).
(b) Predictability of stocks (D/P) bonds (long-short yields) and foreign exchange
(if oreign − idomestic ).
(c) Present
P value formulas and volatility. (Campbell Shiller approximation pt − dt =
Et ρj+1 (∆dt+j − rt+j ))
2. Economic models that capture Equity premium, predictability, size and B/M.
(a) Habits
(b) Nonrecursive Epstein-Zin etc. utility
(c) Heterogeneity and incomplete markets. (Heaton!)
3. Portfolio theory!
(a) Classic mean-variance
(b) Merton, ICAPM hedge portfolios
(c) Dynamic and multifactor portfolio theory — How to incorporate stock, bond predictability; how to incorporate size, value, momentum, etc.
(d) Bayesian portfolio theory — how parameter uncertainty affects portfolios
(e) Portfolio facts — the crazy things people hold.
4. Much more fixed income, risk management, options, futures, swaps, etc.
5. New things: Liquidity, short sales constraints, etc. (My 35150 reading list)
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