Another great contribution by Lars Hansen (one of the 2013 Nobel Prize in Economics recipi-ents) was the Generalized Method of Moments and its applications in the context of the canonical consumption-based asset pricing model. We know that under CRRA utility we have
Et
" ✓ Ct+1
Ct
◆ ⇣
Rt+1 Rft⌘#
= 0
That is, the expected discounted excess return should always be zero. How do we take this to data?
How do we find parameters beta and gamma that best fit the data? How do we check this over many di↵erent times and returns, to see if those two parameters can actually explain empirical facts? What do we do about that conditional expectationEt, conditional on information in people’s heads? How do we bring in all the variables that seem to forecast returns over time (e.g. by the dividend-price ratio) and across assets (value, size, etc.)? How do we handle the fact that return variance changes over time, and consumption growth may be autocorrelated?
When Hansen wrote his paper, this was a big headache. He suggested to just multiply by any variable z that we think forecasts returns or consumption, and take the unconditional average of this conditional average: the model predicts that the unconditional average obeys
E
" ✓ Ct+1
Ct
◆ ⇣
Rt+1 Rft⌘
⇥ zt
#
= 0
So we can just take this average in the data, and do this for lots of di↵erent assets R and lots of di↵erent instruments z. Finaly, we pick the and that make some of the averages as close to zero as possible, and then look at the other averages and see how close they are to zero. (Hansen worked out the statistics of this procedure - how close should the other averages be to zero, and what is a good measure of the sample uncertainty in and estimates - taking in to account a wide variety of statistical problems that may arise).
The results were not favorable to the consumption model: a huge is needed to fit the di↵erence between stocks and bonds, questioning the validity of the utility function used, the measurements of consumption, and even of the whole consumption-based asset pricing framework.
Mean-Variance Analysis and CAPM
In section 3.4 we discussed the State-Price Beta model, which essentially states that for an asset j E⇥
Rj⇤
Rf = j⇣
E [R⇤] Rf⌘
Where j = Cov(R⇤,Rj)
V ar(R⇤) and R⇤ is the return of an asset whose payo↵ is equal to the discount factor m⇤ derived from the pricing kernel q⇤ 2 hXi. This is a very general statement about returns, but it does not help us identify what assets in reality have a return of R⇤.
Suppose all agents have the same quadratic utility u (c0, c1) = v (c0) (c1 ↵)2. Recall that the expected utility is E [U (c)] = PS
s=0
⇡su (c0, cs) and therefore m = E[@@1u
0u]: in the quadratic utility case we have
@1u =h
2 (c1 ↵) · · · 2 (cS ↵) i
So the excess return can be written as E⇥
This holds for any asset xj 2 hXi, and can be generalized to any portfolio h:
Eh Rhi
Rf = Rf ⇥2Cov c1, Rh E [@0u]
Now consider the market portfolio defined as xmkt= P
j:xj2hXi
for each (linearly independent) asset in the asset span.
87
for it too, and therefore dividing side by side we get
Where Rmkt is the return of the market portfolio. If moreover agents are homogeneous and live in an exchange economy, then we know that c1 corresponds to the aggregate endowment and this is perfectly correlated with Rmkt:
E⇥
Which we call the Market Security Line (MSL). Note that in order for the above result to hold, R⇤ must be a linear function of Rmktof the type
R⇤ = a + bRmkt a + bRf
With b < 0 since we know that the stochastic discount factor m is high (hence high R⇤) in states in which the economy is doing poorly, so in which the market portfolio xmkt is low (hence low Rmkt).
6.1 The Traditional Derivation of CAPM
We define the mean return of the portfolio h as
µh ⌘ E [rh] =E of Rh is still 1. The variance of the portfolio is given by
2h⌘ V ar (rh) = w0V w
And h⌘p
V ar (rh) is the standard deviation of portfolio h, where w2 RJ is the vector of weights introduced above and V is the covariance matrix of the assets in portfolio h.
2Note that we can always write the excess return in gross or net terms: E⇥ Rj⇤
Rf =E [rj] rf.
Definition: We say that portfolio A mean-variance dominates portfolio B if µA µB and A< B, or µA> µB and A B.
Note that in the ( h, µh) space, for a given value of µh and h we can immediately find the subset of portfolios which are not dominated by the given µh and h:
Definition: For given µ and , the Efficient Frontier is the locus of all non-dominated payo↵s in the ( h, µh) space.
It follows directly from the definition of efficient frontier that no rational investor with mean-variance preferences would choose to hold a portfolio outside of the efficient frontier.
In the J = 2 case, for portfolio h we have w2 = 1 w1, so the mean return is µh= w1µ1+(1 w2) µ2 and the variance is h2 = w12 12+ (1 w1)2 22+ 2w1(1 w1) 1 2⇢1,2. Note that in general, we can specify the vector µ of returns and covariance matrix V and back out the weights vector w: in the 2 assets example, for ⇢1,2= 1 we get
w1 = ± h 2
1 2
So that plugging back in the expressions for µh we get µh= µ1+µ2 µ1
2 1
(± h 1)
Which looks like:
For ⇢1,2 = 1 we get
w1 = ± h+ 2
1+ 2
And therefore
µh = 2
1+ 2
µ1+ 1
1+ 2
µ2± µ2 µ1
1+ 2 h
Which looks like:
While for ⇢1,22 ( 1, 1) we have
The same concept generalizes to the J assets case: a frontier portfolio has minimum variance among all feasible portfolios with the same expected portfolio return, so the problem is
maxw
1 2w0V w s. t. w0µ = E, w0I = 1
Where I = (1, . . . , 1) and E 2 R is a given (fixed) expected return. The idea is that we fix an expected return E and then look for the minimum variance possible given that expected return:
The FOCs are:
@L
@w = V w µ I = 0
@L
@ = E w0µ = 0
@L
@ = I w0I = 0
Where and are the Lagrange multipliers for the two constraints. Pre-multiplying the first FOC by µ0V 1 we get
µ0w = µ0V 1µ + µ0V 1I ⌘ B + A And since µ0w = w0µ = E by the third FOC,
E = B + A
While pre-multiplying the same FOC by I0V 1 we get
1 = I0w = I0V 1µ + I0V 1I ⌘ A + C
Since µ0V 1I = I0V 1µ. Solving for and we get
= E⇥ C A D
= B E⇥ A D
Where D ⌘ B ⇥ C A2. Therefore, pre-multiplying the first FOC by V 1 and plugging in and we have
Suppose now that for some portfolio with return r we set E = E [r]. Similarly to the 2 assets problem, the solution portfolio weights are linear in the (expected) portfolio returns:
w⇤ = g + hE [r]
Note that for E [r] = 0 and E [r] = 1 we get g and g + h respectively, which are therefore frontier portfolios as well. We established the following facts:
Proposition 1: The entire set of frontier portfolios can be generated by g and g + h:
any portfolio in the frontier is a linear combination of these two portfolios.
Proposition 2: Any linear combination of frontier portfolios is also a frontier port-folio: the portfolio frontier can be described as linear combinations of any two frontier portfolios (not just g and g + h).
It is possible to show that
2(r) = C
So that we know that 1) the expected return of the minimum variance portfolio is AC, 2) the variance of the minimum variance portfolio is C1, 3) is the equation of a parabola with vertex at C1,CA in the expected return/variance space and of a hyperbola in the expected return/standard deviation space.
That is, in the expected return-variance space (V ar(r),E [r]) space the set of frontier portfolios looks like:
Where A, B, C and D are constants. In the expected return-standard deviation space (SD(r),E [r]) space the set of frontier portfolios is:
Given two portfolios A and B in the frontier:
We can see that points above B and below A correspond to portfolios in which we short-sell assets.
How does all this change if we have a risk-free asset (i..e an asset whose variance is zero)? We can adapt the problem considered above as follows:
maxw
1 2w0V w
s. t. w0µ + (1 w0· I)rf =E [r]
Where rf is the net risk-free rate.3 Note that in this problem weights automatically sum to one since w0I + (1 w0I) = 1, so we don’t have the second constraint in the problem without the risk-free asset. First order conditions yield
w⇤ = V 1(µ I· rf)
So premultiplying both sides by (µ I· rf)0 yields
= E [r] rf
(µ I· rf)0V 1(µ I· rf)
Since (µ I· rf)0w⇤ = µ0w⇤ I0w⇤· rf = µ (w⇤)0 I (w⇤)0· rf =E [r] rf from the constraint in the problem above. Plugging this back in w⇤ we finally get
w⇤ = V 1(µ I· rf)
(µ I· rf)0V 1(µ I· rf) ⇥ (E [r] rf)
Where the denominator is the square of the sharpe ratio H. We have two important results at this
3That is, rf = Rf 1.
point:
Result 1: For any two frontier portfolios p and q, we have E [rq] rf = q,p(E [rp] rf).
To see this, note that Cov (rq, rp) = w0qV wp = wq0 (µ I· rf)E[rHp] r2 f = (E[rq] rf)(E[rp] rf)
H2 and
V ar (rp) = (E[rp] rf)2
H2 , and dividing side by side yields the result. Importantly, this holds for any two frontier portfolio p, thus in particular it also holds for the market portfolio.
Result 2: The frontier is linear in the (SD(r),E [r]) space.
This follows immediately from V ar (rp) = (E[rp] rf)2
H2 by taking the square root and rearranging: we get E [rp] = rf + H ⇥ SD (rp). Therefore the efficient frontier with a risk-free asset looks like the following:
6.1.1 Two Fund Separation
What is the result of individual optimization on the aggregate supply and demand of assets? That is, what can we say about the equilibrium state given what we have seen so far? We approach this problem in two steps: first we solve for the efficient frontier of the J risky assets, then we solve for the tangency point with the agents’ indi↵erence curves in the (SD(r),E [r]) space. The advantage of this approach is that we have the same portfolio of J risky assets for di↵erent agents with di↵ering risk aversion, and this makes it easier to apply equilibrium aguments. Represented in the graph
below are the optimal portfolios of two investors with di↵erent degrees of risk aversion (the black and red indi↵rerence curves).
In this setup, mean-variance preferences represented by a utility function U µ, 2 simply satisfy
@U
@µ > 0 and @@U2 < 0: a simple example is U µ, 2 = µ ⇢2 2. As we have already seen, these preferences are equivalent to those of a Von Neumann-Morgenstern quadratic utility: if u (X) = a + bX + cX2 then E [u (X)] = a + bµ + c 2 + cµ2 = U µ, 2 . Again as already seen, if asset returns are gaussian then also any portfolio is gaussian and therfore preferences can only depend on the first two moments. Moreover if agents have CARA utility function we know that the certainty equivalent for a lottery with mean µ and variance 2 is going to be µ ⇢2 2 where ⇢ is the absolute risk aversion of agents.
6.1.2 Equilibrium leads to CAPM
The theory examined so far only tells us about the demand of assets: prices are taken as given, and the composition of the optimal (risky) portfolio is the same for all investors. Setting the aggregate demand for assets equal to the supply of assets, that is, the market portfolio. Through CAPM we can find assets’ equilibrium prices and agents’ risk premium. As we have seen, the market portfolio is efficient (since it lies on the efficient frontier); moreover, all individual optimal portfolios are located on the half line originating at the point (0, rf). This half-line is called capital market line (CML) and can be written as
E [rh] = rf+E [rmkt] rf (rmkt) h
Note that the slope is the E[rmkt(r ] rf
mkt) sharpe ratio of the market portfolio.
Similarly, since (rh
mkt) = h,mkt we can write
E [rh] = rf + h,mkt(E [rmkt] rf) Which defines the security market line (SML)in the ( ,E [r]) space: