Nonlinear Econometrics for Finance Lecture 2
. Econometrics for Finance Lecture 2 1 / 26
Asset Pricing
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Last class: Asset Pricing
Contrary to Lecture 1, the risk-less rate Rf is not zero
In general, all asset pricing models imply the following equivalence:
pt = Et[mt+1xt+1] = 1 EQt [xt+1], 1 + Rf
pt = price of the asset at time t
xt+1 = asset cash flow at time t + 1 (e.g., pt+1 plus dividends, if there are dividends)
mt+1 = stochastic discount factor
Et = expectation computed at time t (given information at time t)
1 Prices are discounted expectations of future cash flows which take risk into account.
2 In (1) the risk adjustment is in the discounting.
3 In (2) the risk adjustment is in the probabilities.
4 Different asset pricing models imply a different risk adjustment mt+1 (or Q).
. Econometrics for Finance Lecture 2 2 / 26
Asset Pricing
Let us consider a specific model: the Consumption-CAPM
. Econometrics for Finance Lecture 2 3 / 26
Asset Pricing
The representative agent
Consider a “representative” agent.
The agent lives two periods: today and tomorrow.
The agent has utility over consumption today and tomorrow.
U(ct, ct+1) = u(ct) + βu(ct+1).
β is a subjective (time) discount factor (β < 1).
We will assume an increasing u(.) reflecting non-satiation.
We will also assume a concave u(.) reflecting decreasing marginal utility from consumption.
For example, u(ct) = 1 c1−γ, a constant relative risk aversion, or 1−γ t
CRRA, utility.
. Econometrics for Finance Lecture 2 4 / 26
Asset Pricing
The CRRA utility function
u(c)= 1 c1−γ withγ=0.7 1−γ
Figure: The figure represents the CRRA utility function for γ = 0.7. The function is concave.
. Econometrics for Finance Lecture 2 5 / 26
Asset Pricing
Choosing between consumption and investment
The agent has some endowed wealth in both periods: (et,et+1)
In the first period, the agent has to allocate his/her wealth et: he/she can consume (ct) or he/she can invest in a risky asset (and buy θ shares). Given et, the less he/she consumes now (the lower ct), the more he/she can invest (the larger θ) and the more he/she can hope to consume later (if the price of the risky asset goes up).
In the second period, everything is consumed (et+1 and the cash flow from the risky asset, namely xt+1 = pt+1 + dt+1, i.e., price plus potential dividends).
The agent will choose θ, i.e., the number of shares, to maximize his/her two-period utility:
max[U(ct(θ), ct+1(θ))] = max [u(ct(θ)) + βEt(u(ct+1(θ)))] , θθ
subject to
et = ct +ptθ,
ct+1 = et+1 + xt+1θ.
. Econometrics for Finance Lecture 2 6 / 26
Asset Pricing
Once more ...
The agent will choose θ, i.e., the number of shares, to maximize his/her two-period utility:
max[U(ct(θ), ct+1(θ))] = max [u(ct(θ)) + βEt(u(ct+1(θ)))] , θθ
subject to
et = ct +ptθ,
ct+1 = et+1 + xt+1θ.
Recall, how do we maximize a function f(θ) with respect to θ? We find θ so that ∂f(θ) = 0 (we should also check that the second derivative is negative).
Now, let us plug in ct = et −ptθ and ct+1 = et+1 +xt+1θ, we have max[U(ct(θ), ct+1(θ))] = max [u(et − ptθ) + βEt(u(et+1 + xt+1θ))] .
Setting the first derivative equal to zero, we have:
∂U(ct(θ),ct+1(θ)) ∂u ∂u
= −pt (ct) + βEt (ct+1) (pt+1 + dt+1) = 0.
∂θ ∂c ∂c xt+1
. Econometrics for Finance Lecture 2 7 / 26
Asset Pricing
The first-order condition
Recall the first-order condition (writing ∂u (c) = u′ (c)): ∂c
Interpretation of Eq. (1): The agent invests to the point where the loss of utility from consuming one less unit today is equal to the expected (discounted) utility gain from that unit tomorrow.
The agent continues to buy (or sell) until the marginal loss equals the marginal gain.
−ptu′ (ct) + βEt(u′ (ct+1)(pt+1 + dt+1)) = 0 ′′
ptu (ct) = βEt(u (ct+1)(pt+1 + dt+1)), (1)
. Econometrics for Finance Lecture 2 8 / 26
Interpretation
If we divide by u (ct), we can rewrite Eq. (1) as
u(ct+1) pt = Et β ′ (pt+1 + dt+1) =
Asset Pricing
u(ct) 1+R
f Et [xt+1]. (2)
mt+1 Interpretation of Eq. (2):
price today = expected discounted future cash flows. ′
The discount factor mt+1 = β u′ (ct ) is stochastic.
Risk is accounted for by mt+1. The riskier the asset, the more we discount future cash flows, the lower the price. (Notice that the stochastic discount factor accounts for risk, but it also accounts for the time value of money.)
Ignoring dividends and interest rates, prices are martingales under the new probabilities induced by mt+1.
. Econometrics for Finance Lecture 2 9 / 26
Asset Pricing
Interpretation, continued
Notice that, tautologically, the risky return on any asset can be written as
Rt+1 = pt+1 +dt+1 −pt ⇒1+Rt+1 = pt+1 +dt+1. pt pt
Without uncertainty (i.e., without risk) any return should be the same as the risk-free return, i.e., Rt+1 = Rf . But this implies that
1+Rf = pt+1 +dt+1 ⇒pt = 1 (pt+1 +dt+1). pt 1+Rf
Thus, if there is no uncertainty about future cash flows (i.e., if there is no risk), then
Prices would be future cash flows discounted at the risk-free rate.
Since there is uncertainty, and investors are risk-averse, some risk-correction ought to take place. Let’s see ...
pt = 1 (pt+1 + dt+1). 1+Rf
. Econometrics for Finance Lecture 2 10 / 26
Asset Pricing
Interpretation, again
By the “alternative” covariance formula (from the last class), we obtain pt = Et(mt+1(pt+1 + dt+1))
= Ct(mt+1, pt+1 + dt+1) + Et(mt+1)Et(pt+1 + dt+1). (3) In terms of returns (i.e., dividing by pt):
pt+1 + dt+1
1 = Et(mt+1(1 + Rf)) ⇒ 1 = Et(mt+1)(1 + Rf) ⇒ Et(mt+1) = 1
and, substituting in Eq. (3), we obtain
1 = Et mt+1
= Et(mt+1(1 + Rt+1)). (4) By applying Eq. (4) to the risk-free asset (Eq. (4) applies to all assets, right?), we obtain
pt = f Et(pt+1 + dt+1) + Ct(mt+1, pt+1 + dt+1) .
1+R risk adjustment
Eq. (5) clarifies the nature of the risk-adjustment. Assets which pay-off (high xt = pt+1 + dt+1) when the marginal utility of consumption is high or, equivalently, when consumption is low (high
m = β u (ct+1 ) ) are assets which sell at a premium (their price is relatively higher). Why? t+1 ′
Because investors want to hedge fluctuations in their consumption streams.
. Econometrics for Finance Lecture 2 11 / 26
Asset Pricing
From prices to returns: a β representation
We are used to asset pricing models written in terms of excess (with respect to the risk-free rate) returns (think about the CAPM).
Let us derive one in this case!
Assume existence of N assets.
Denote a generic asset by the superscript i, with i = 1, ..., N . Using Eq. (4) - and the alternative covariance formula - write
1=E(m (1+Ri ))=C(m ,1+Ri )+E(m )E(1+Ri ). t t+1 t+1 t t+1 t+1 t t+1 t t+1
. Econometrics for Finance Lecture 2 12 / 26
Asset Pricing
A β representation, continued
Let’s repeat the expression:
1 = Ct(mt+1, 1 + Ri ) + Et(mt+1)Et(1 + Ri ).
Thus, dividing by Et(mt+1) and re-arranging:
1 Ct(mt+1,1+Ri ) Et(1+Ri )= − t+1.
Et (mt+1 ) Et (mt+1 )
1 ⇒1+Rf = 1 .
Recall that
Then, we can write
Et(mt+1) Ct(mt+1, 1 + Ri )
Et(1+Ri )=1+Rf − t+1
t+1 . Et (mt+1 )
Therefore, in terms of excess returns, we have
Ct(mt+1, 1 + Ri ) Et(Ri )−Rf =− t+1 .
t+1 Et (mt+1 )
. Econometrics for Finance Lecture 2 13 / 26
A β representation, continued
Now, multiply by Vt (mt+1 ) (which is, clearly, 1):
Ct(mt+1,1+Ri )−Rf =− t+1
Vt (mt+1 )
) Vt(mt+1) Et (mt+1 )
Vt (mt+1 )
Asset Pricing
whereβi,m =
Interpretation of Eq. (6): Assets which pay-off (high Ri
marginal utility of consumption is high or, equivalently, when consumption is
low (high mt+1 = β u′ (ct ) ) are good for you. People require relatively lower
returns (notice the − sign) to hold these stocks. This is a consumption-CAPM (C-CAPM) model.
People care about fluctuations in consumption rather than about fluctuations in market returns (as in the traditional CAPM).
,1+Ri ) t+1 t+1
V (m ) t t+1 .
Vt (mt+1 )
Et (mt+1 )
Et(Ri )−Rf =−βi,mλm, t+1
) when the
. Econometrics for Finance Lecture 2 14 / 26
Testing asset pricing models: Generalized Method of Moments (GMM)
. Econometrics for Finance Lecture 2 15 / 26
From Hansen’s web site
In 2013, Hansen was a recipient of the Prize in Economic Sciences in Memory of for his work advancing understanding of asset prices through empirical analysis. Hansen developed statistical tools and methods that deepen analysis of the connections between financial markets and the macroeconomy. “Economists build models to try to understand risk aversion and how it affects prices. My work tries to understand ways those models work and the ways they do not”, explained Hansen.
In the 1980s, Hansen became the leading contributor to the development and application of rigorous estimation and testing methods for financial data. His 1982 Econometrica paper, “Large Sample Properties of Generalized-Methods of Moments Estimators,” outlined time series statistical methods that made it possible to investigate one part of an economic model without having to fully specify and estimate all of the model ingredients. Known as GMM, this method fundamentally altered empirical research in finance and macroeconomics.
“When you do analysis of these models, it’s nice to be able to study some things without having to study a full, fleshed out view of all parts of the economy”, Hansen said.
. Econometrics for Finance Lecture 2 16 / 26
Estimating and testing asset pricing models: GMM
Let us consider how to estimate and test asset pricing models. We start again with the “usual” equation
E(m (1+Ri ))=1. t t+1 t+1
Take expectations of both sides:
EE(m (1+Ri ))=E(1).
By the law of iterated expectations, we have
E(m (1+Ri ))=1. t+1 t+1
Taking, now, the 1 to the other side, we have: E(m (1+Ri )−1)=0.
This is, now, an unconditional “moment condition” that needs to
hold for each asset i with i = 1,...,N.
. Econometrics for Finance Lecture 2 17 / 26
Estimating and testing asset pricing models: GMM
Consider, now, N assets.
Stack the moment conditions for all N assets one on top of the other to obtain
E(mt+1(1 + Rt+1) − 1) = 0,
Alternatively, even more explicitly, write
]⊤ and 1 = [1,...,1]⊤
Rt+1 = [R1 ,...,RN t+1 t+1
1 1+Rt+1 1
Emt+1 ...
− ... = 0
1+Rt+1 1
N vector N vector
Notice that mt+1 depends on parameters. Different asset pricing models will,
therefore, lead to a different stochastic discount factor mt+1 and different moment conditions. Here is one example.
. Econometrics for Finance Lecture 2 18 / 26
Estimating and testing asset pricing models: GMM
An example: The Consumption CAPM with CARA utility
In the C-CAPM model with CRRA utility: u(ct)= 1 c1−γ.
1−γ t u′(ct) = c−γ.
Finally, the moment conditions become
Thus, And,
ct+1 −γ mt+1=β′ =β .
1 −γ 1+Rt+1 1
Eβ ct+1 ...
ct N 1+Rt+1 1
− ... =0.
N vector N vector
There are two parameters to estimate: the subjective discount factor β and the
coefficient of relative risk aversion γ. We could write mt+1(θ) with θ = (β, γ). . Econometrics for Finance Lecture 2 19 / 26
Estimating and testing asset pricing models: GMM
The moment conditions depend on an expectation. We do not know the expectation. We can, however, compute empirical means. Sample means converge to expectations by the law of large numbers.
Estimation: GMM estimates θ by setting the difference between the sample mean of mt+1(θ)(1 + Rt+1) and 1 as close as possible to 0:
− ... =
1
mt+1(θ) ...
gT (θ) ≈ 0
Tt=1 1+RN 1
t+1 more compact notation
N vector N vector
Testing: Given an estimate for θ, denoted by θT , GMM evaluates the size of
the pricing errors (Hansen, 1982): how close to 0 is the difference between the
sample mean of mt+1(θT )(1 + Rt+1) and 1? The larger the pricing errors, the worse the pricing model.
. Econometrics for Finance Lecture 2 20 / 26
Estimating and testing asset pricing models: GMM
Estimation of θ:
arg min gT (θ)⊤ WT gT (θ) = θ
arg min QT (θ)
Thus,wechooseθT sothat ∂Q (θ ) ≈0.
1×N N×N N×1
Assume the dimension of the vector θ is d with N ≥ d. (The number of
parameters is not larger than the number of assets.)
Typically, we cannot estimate θ to make the pricing errors exactly zero.
However, we want to make the pricing errors as small as possible.
In order to do so, we minimize a quadratic criterion: arg min QT (θ)
θ 1×1
Note: the weight matrix WT tells you how much emphasis you are putting on specific moments (i.e., on specific assets).
If WT = IN , i.e., the identity matrix, then you are effectively treating all assets in the same way. In this case, the criterion minimizes the sum of the squared pricing errors. 2 asset example
. Econometrics for Finance Lecture 2 21 / 26
Next week we will ...
Understand the GMM criterion better
Discuss consistency and asymptotic normality of the GMM estimates
See GMM at work using Matlab
. Econometrics for Finance Lecture 2 22 / 26
Proving the equivalence
Proof: We have
this is just the definition of expectation
= 1 EQ[X] 1+Rf
now, let us just multiply and divide by the true probs
re-arranging, we have ...
= 1 xj pQ pj
. Econometrics for Finance Lecture 2 23 / 26
1+Rf j j=1
= J1pQjxjpj
1+Rf pj j=1
= mjxjpj j=1
where the SDF m is a random variable with outcomes mj = 1 pQj for j = 1, ..., J.
The expectation of the stochastic discount factor
putting in the definition of m ...
= 1 J pQj pj
1+Rf pj j=1
1+Rf j j=1
E(m) = mjpj
... since the risk-adjusted probabilities also sum up to 1 Conclusion: the SDF takes both risk and time into account. back
= 1 . 1+Rf
. Econometrics for Finance Lecture 2
Estimating and testing asset pricing models: GMM
Example with 2 assets
The model:
) − 1 ) − 1
(θ)(1 + R1 t+1
(θ)(1 + R2 t+1
g1(X g2(X
, θ) , θ)
= E(g(X , θ)) = 0, t+1
where 2 is the number of assets (1 moment condition per asset).
Empirically (after replacing “expectations” with “sample means”):
T−1 1 T−1 T−1
1 mt+1(θ)(1+Rt+1)−1 = 1 g1(Xt+1,θ) = 1 g(X ,θ)=g (θ)≈0.
T t=1 mt+1(θ)(1+Rt+1)−1 Estimation criterion:
T t=1 g (Xt+1,θ)
T t=1 2×1
1 T−1 1
1 T−1 2
t=1 g (Xt+1,θ)
t=1 g (Xt+1,θ) θT = argmin T t=1 g (Xt+1,θ) T t=1 g (Xt+1,θ) WT 1 T−1 2
= argming (θ)⊤ W g (θ) = argminQ (θ). T TT T
θ θ 1×2 2×2 2×1 1×1
. Econometrics for Finance Lecture 2 25 / 26
Estimating and testing asset pricing models: GMM
Example with 2 assets
Recall the estimation criterion:
θT = argmin θ
T t=1 g (Xt+1,θ) 2×2
θT = argmin θ
t=1 g (Xt+1,θ)
g (Xt+1,θ) T
argminw1
g (Xt+1,θ)
T t=1 g (Xt+1,θ) T
1 T−1 T t=1
g (Xt+1,θ)
g (Xt+1,θ)
t=1 g (Xt+1,θ) WT 1 T−1 2 .
If WT = I2, then we minimize the sum of the squared pricing errors:
T t=1 g (Xt+1,θ) 1 T−1 2
T t=1 g (Xt+1,θ)
If WT is a generic symmetric matrix, then we minimize a “weighted” sum of the squared pricing errors:
2
= argmin 1 1 g1(Xt+1,θ) +1 1 g2(Xt+1,θ) .
T−1 θ T t=1 T t=1
1 T−1 T t=1
w3 T t=1 g (Xt+1,θ)
T−1 2
g (Xt+1,θ) +w2
11 12 back
1T−1 1
1T−11
1T−1 1
w 1 T−1 2
3 2 T t=1g(Xt+1,θ)
g (Xt+1,θ) +
g (Xt+1,θ) g (Xt+1,θ). T t=1
. Econometrics for Finance Lecture 2
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