Mean–variance & information
Where quadratic methods meet exact information loss
Mean–variance analysis describes a portfolio through its expected return and a quadratic measure of risk. The information-theoretic perspective describes its whole payoff through an induced distribution. The connection becomes precise once we specify which measure supplies the quadratic penalty, and where the approximation is taken.
Use \(G=V(T)/V(0)\) for a gross payoff and \(Z=G-1\) for its excess return, all in units of \(Y\). Fair pricing means \(\mathbb E^Y[G]=1\), hence \(\mathbb E^Y[Z]=0\).
Start with classical mean–variance
Let \(r\) be the vector of asset excess returns. With a reference asset available, risky weights \(w\) produce \(Z=w^\top r\); the remaining budget goes into the reference asset. Define
\[ \mu=\mathbb E^M[r], \]
\[ \Sigma_M=\operatorname{Cov}^M(r). \]
Classical mean–variance with risk penalty \(\lambda>0\) maximizes
\[ w^\top\mu-\frac\lambda2w^\top\Sigma_Mw. \]
Both the mean and covariance are evaluated under the investor’s probability model. When \(\Sigma_M\) is positive definite and there are no constraints on risky positions,
\[ w^*=\frac1\lambda\Sigma_M^{-1}\mu. \]
This formula does not check that terminal wealth stays nonnegative. Budget, leverage, short-sale and solvency constraints must be imposed separately. Under constraints, the solution is a quadratic program rather than this inverse-matrix formula.
The ordinary Taylor approximation
For \(Z\) near zero,
\[ \log(1+Z)=Z-\frac12Z^2+O(Z^3). \]
Taking expectation under the investor’s belief gives
\[ T_M(Z)=\mathbb E^M[Z]-\frac12\mathbb E^M[Z^2]. \]
The penalty is a second moment, not just a variance:
\[ \mathbb E^M[Z^2] =\operatorname{Var}^M(Z)+(\mathbb E^M[Z])^2. \]
Thus the literal second-order approximation to expected log growth differs from classical mean–variance with \(\lambda=1\). It becomes unreliable near \(Z=-1\), where log utility diverges but the quadratic remains finite.
The book’s mixed-measure quadratic
The likelihood construction leads to another objective,
\[ J(Z)=\mathbb E^M[Z]-\frac12\mathbb E^Y[Z^2]. \]
Beliefs determine the expected reward; prices determine the quadratic geometry. Because \(\mathbb E^Y[Z]=0\), the penalty here also equals \(\tfrac12\operatorname{Var}^Y(Z)\).
This objective has an exact information interpretation. Set
\[ L_M=\frac{d\mathbb P^M}{d\mathbb P^Y}, \qquad \chi^2=\mathbb E^Y[(L_M-1)^2]. \]
Completing the square gives
\[ J(Z)=\frac12\chi^2 -\frac12\mathbb E^Y[(G-L_M)^2]. \]
Write \(Q(G)=\mathbb E^Y[(G-L_M)^2]\) for the ordinary squared-error loss, without a factor of \(1/2\). Then \(J=\chi^2/2-Q(G)/2\): maximizing \(J\) is exactly equivalent to minimizing \(Q\). The factor of \(1/2\) belongs to the reward objective \(J\), while the loss \(Q\) uses the same convention as the comparison page.
The first term is fixed. Maximizing \(J\) chooses the attainable payoff closest to the likelihood ratio \(L_M\) in squared error under the pricing measure. In a complete market, \(G=L_M\) is attainable and achieves the maximum \(\chi^2/2\) exactly. We reserve \(R\) for a traded asset’s gross return.
In a complete market, both log growth and this mixed quadratic select \(G=L_M\). Their objective values are different: the log optimum is \(D(\mathbb P^M\Vert\mathbb P^Y)\), whereas the quadratic optimum is \(\chi^2/2\). In a restricted asset menu, their best attainable portfolios can also differ.
Since \(\mathbb E^Y[Z]=0\),
\[ \mathbb E^M[Z]=\mathbb E^Y[(L_M-1)Z]. \]
For real numbers \(a,z\), the identity \(az-z^2/2=a^2/2-(z-a)^2/2\) applies state by state with \(a=L_M-1\). Taking \(\mathbb E^Y\) proves the formula. The continuous-state version requires \(L_M-1\) and \(Z\) to be square integrable.
A different expansion: close to the desired payoff
The near-cash Taylor expansion uses \(G\approx1\). A target-centered expansion instead uses \(G\approx L_M\). For a strictly positive target \(L_M\) and an equal-cost positive approximation \(G\), define
\[ \delta=\frac{G-L_M}{L_M}. \]
Because both payoffs cost one, \(\mathbb E^M[\delta]=\mathbb E^Y[G-L_M]=0\). Expanding \(-\log(1+\delta)\) gives
\[ \begin{aligned} D_{\mathrm{KL}}(\mathbb P^M\Vert\mathbb P^G) &=-\mathbb E^M[\log(1+\delta)]\\ &\approx\frac12\mathbb E^M[\delta^2]\\ &=\frac12\mathbb E^Y\left[\frac{(G-L_M)^2}{L_M}\right]. \end{aligned} \]
This criterion is \(\operatorname{KL}_2=\tfrac12\chi^2(\mathbb P^G\Vert\mathbb P^M)\). Its weight \(1/L_M\) emphasizes errors relative to the target size. It differs from ordinary quadratic hedging, which minimizes \(\mathbb E^Y[(G-L_M)^2]\).
For \(G=1+w(R-1)\), its unconstrained regression coefficient is
\[ w_{\mathrm{KL}_2} =\frac{\mathbb E^Y[(R-1)(L_M-1)/L_M]} {\mathbb E^Y[(R-1)^2/L_M]}, \]
provided these moments are finite and the denominator is positive. Feasibility still needs to be checked.
Small absolute payoff error is not enough for the target-centered expansion when \(L_M\) can approach zero. A sufficient condition is uniformly small relative error, with integrability controlling the Taylor remainder. In the geometric-shock example, KL2 selects \(w=0.8125\), ordinary quadratic hedging selects \(0.7\), and exact KL selects approximately \(0.745331\). The local approximation does not rank those portfolios by exact growth.
Ordinary quadratic hedging emerges in an additional near-cash limit. If \(L_M=1+\varepsilon h\) and \(G=1+\varepsilon a\), with bounded \(h,a\) of zero \(\mathbb P^Y\) mean, then
\[ D_{\mathrm{KL}}(\mathbb P^M\Vert\mathbb P^G) =\frac{\varepsilon^2}{2}\mathbb E^Y[(h-a)^2]+O(\varepsilon^3). \]
Here both beliefs relative to prices and the chosen payoff approach the reference asset. A target far from cash need not have this unweighted local geometry.
Experiment with the approximation
The tool uses three scenarios and a portfolio of the reference asset and one risky asset. Its illustrative inputs are \(q=(0.30,0.40,0.30)\) and \(\mathbb P^X=(0.60,0.30,0.10)\). Belief strength moves the investor from \(q\) toward \((0.50,0.35,0.15)\). These are an asymmetric teaching example, chosen so the optimizers can differ.
Change belief strength, move the risky allocation \(w\) between zero and one, and compare exact growth with the ordinary Taylor and mixed quadratic objectives. Use each optimizer to see whether it chooses the same position.
Interactive comparison. It needs JavaScript.
The approximation depends on the size of statewise returns, not just on an apparently small average return. A small mean can coexist with a substantial probability of a nearly zero payoff.
Their difference is
\[ T_M(Z)-J(Z) =-\frac12\mathbb E^Y[(L_M-1)Z^2]. \]
If both \(L_M-1\) and \(Z\) are uniformly of order \(\varepsilon\), this difference is of order \(\varepsilon^3\). In the same regime, both approximate expected log growth to second order. Small returns alone justify the ordinary Taylor expansion; using the pricing measure in the quadratic penalty also needs control of the difference between beliefs and prices, or the separate Pearson projection interpretation.
From payoffs to a portfolio calculation
For excess-return vector \(r\) with \(\mathbb E^Y[r]=0\), define
\[ \Sigma_Y=\mathbb E^Y[rr^\top]. \]
The mixed objective becomes
\[ J(w)=w^\top\mu-\frac12w^\top\Sigma_Yw. \]
With invertible \(\Sigma_Y\) and unconstrained risky weights, its solution is
\[ w^*=\Sigma_Y^{-1}\mu. \]
The matrix resembles the classical covariance formula, but its entries come from \(\mathbb P^Y\). A singular matrix can reflect redundant assets; removing redundancy or solving the constrained program avoids treating its inverse as defined. Even when the matrix is invertible, an unrestricted projection can produce negative wealth. The portfolio laboratory keeps long-only cost weights and nonnegative payoffs.