• Utility Values and risk aversion

risk aversion investor gain the same amount of utility spending more on the risk. Because risk can be quantified as the sum of the variance of the returns over time, it is possible to assign a utility score (aka utility value, utility function) to any portfolio by subtracting its variance from its expected return to yield a number that would be commensurate with an investor's tolerance for risk, or a measure of their satisfaction with the investment. \[Utility Score = Expected Return – 0.005\sigma^2 × Risk Aversion Coefficient\]

• Risk-indifference curve

The set of all portfolios with the same utility score plots as a risk-indifference curve. An investor will accept any portfolio with a utility score on her risk-indifference curve as being equally acceptable.

The weight vector that maximizes the expected portfolio growth rate E log(1 + Rtp) (subject to 1T w = 1, w ≥ 0) is called the Kelly optimal portfolio or log-optimal portfolio. Using the quadratic approximation of the logarithm log(1 + a) ≈ a − (1/2)a2 we obtain \[E[log(1 + Rtp)] ≈ E[Rtp − (1/2)(Rtp)2] = µT w − (1/2)wT (Σ + µµT )w\],

where µ = Ert and Σ = E(rt−µ)(rt−µ)T are the mean and covariance of the return rt.

Alpha is used to determine by how much the realized return of the portfolio varies from the required return, as determined by CAPM. \[\alpha = R_{portfolio} - [R_f + (R_m - R_f)\beta]\]

In algebra, a quadratic function, a quadratic polynomial, a polynomial of degree 2, or simply a quadratic, is a polynomial function in one or more variables in which the highest-degree term is of the second degree. For example, a quadratic function in three variables x, y, and z contains exclusively terms x2, y2, z2, xy, xz, yz, x, y, z, and a constant:

\[ {\displaystyle f(x,y,z)=ax^{2}+by^{2}+cz^{2}+dxy+exz+fyz+gx+hy+iz+j,} f(x,y,z)=ax^{2}+by^{2}+cz^{2}+dxy+exz+fyz+gx+hy+iz+j,\] with at least one of the coefficients a, b, c, d, e, or f of the second-degree terms being non-zero.

• Quadratic programming (QP) is the process of solving a special type of mathematical optimization problem—specifically, a (linearly constrained) quadratic optimization problem, that is, the problem of optimizing (minimizing or maximizing) a quadratic function of several variables subject to linear constraints on these variables. Quadratic programming is a particular type of nonlinear programming.

the process of solving an optimization problem defined by a system of equalities and inequalities, collectively termed constraints, over a set of unknown real variables, along with an objective function to be maximized or minimized, where some of the constraints or the objective function are nonlinear.

Quadratic programming is particularly simple when Q is positive definite and there are only equality constraints; specifically, the solution process is linear. By using Lagrange multipliers and seeking the extremum of the Lagrangian, it may be readily shown that the solution to the equality constrained problem \[{\text{Minimize}}\quad {\tfrac {1}{2}}\mathbf {x} ^{\mathrm {T} }Q\mathbf {x} +\mathbf {c} ^{\mathrm {T} }\mathbf {x} {\text{subject to}}\quad E\mathbf {x} =\mathbf {d}\]

is given by the linear system

\[{\begin{bmatrix}Q&E^{T}\\E&0\end{bmatrix}}{\begin{bmatrix}\mathbf {x} \\\lambda \end{bmatrix}}={\begin{bmatrix}-\mathbf {c} \\\mathbf {d} \end{bmatrix}}\] where \({\displaystyle \lambda }\) is a set of Lagrange multipliers which come out of the solution alongside \({\displaystyle \mathbf {x} }\) .

OLS assumptions:

• Linear in Parameters
• Random Sample of n Observations
• Zero Conditional Mean
• No Perfect Collinearity
• Homoskedasticity

Shortfall of Modern Portfolio Theory(MPT):

• It is based on the historical return performance. If the political visk shows up or the fund changes policy or manager, the corporate or the fund performance may change, investor views can't be integrated into the problem.

Modern portfolio theory looks at total risk and total return.

The institutional investor cares about active risk and active return. For that reason, we will concentrate on the more general problem of managing relative to a benchmark. This focus on active management arises for several reasons:

• the current portfolio(with certainty)
• alphas(often unreasonable and subject to hidden biases)
• covariance estimates(noisy estimates)
• transactions cost estimates(noisy estimates)
• an active risk aversion(self-biased)

We can replace any portfolio construction process, regardless of its sophistication, by a process that first refines the alphas and then uses a simple unconstrained mean/variance optimization to determine the active positions.

• Rank the stocks by alpha.
• Choose the first 50 stocks (for example).
• Equal-weight (or capitalization-weight) the stocks.
  • rebalancing:
    • divide the stocks into three categores, top 40, next 60, remaining 100.
    • buy any stocks on the top 40 not in the portfolio.
    • sell any stocks on the bottom 100 in the portfolio.
    • holding any stocks on the middle in the portfolio.
  • pros:
    • The screen enhances alphas by concentrating the portfolio in the high-alpha stocks.
    • It strives for risk control by including a sufficient number of stocks (50 in the example) and by weighting them to avoid concentration in any single stock.
    • Transactions costs are limited by controlling turnover through judicious choice of the size of the buy, sell, and hold lists.
  • cons:
    • They ignore all information in the alphas apart from the rankings.
    • They do not protect against biases in the alphas.

• splitting the list of followed stocks into categories.
• classify the stocks in each sector by size: big, medium, and small.
• industry neutral.

The result is that portfolios constructed using returns-based analysis are very close to mean/variance portfolios, although they require much more effort to construct.

• placing limits on active stock positions.
• limiting turnover.
• constraining holdings in certain categories of stocks to match the benchmark holdings.

• stocks
• bonds
• cash

• forecasting returns
• building portfolios
• analyzing out-of-sample performance.

Markowitz mean-variance optimization: \[max: \omega\mu - \omega\Sigma\omega'/2\]

Lagrange multiplier solution: \(\omega=(\lambda\Sigma)^{-1}\mu\)

Two most important elements: Expected return \(\mu\), Covariance matrix \(\Sigma\).

• Extreme long or short positions

It could bring extreme long or short positions without constraint(excluding risk exposure constraint, weight constraint).

• Heavy weight on a small number of assets

When subject to a long only constraint, portfolios will concentrate in a relatively small number of assets.

• Importance of expected return

A small increase in the expected return of one of the portfolio's assets can force half of the assets from the portfolio.

• Lack of investor views

Expected return estimate is based on the historical return performance.

If the political risk shows up, macro economy forecast changes, or the fund changes policy or manager, the corporate or the fund performance may change. Investor views can't be integrated into the problem.

The Black-Litterman model uses a Bayesian approach to combine the subjective views of an investor regarding the expected returns of one or more assets with the market equilibrium vector of expected returns (the prior distribution) to form a new, mixed estimate of expected returns.

prior return distribution -> posterior expected return estimated distribution.

Black–Litterman model is a mathematical model for portfolio allocation developed in 1990 at Goldman Sachs by Fischer Black and Robert Litterman, and published in 1992. It seeks to overcome problems that institutional investors have encountered in applying modern portfolio theory in practice, although the covariances of a few assets can be adequately estimated, it is difficult to come up with reasonable estimates of expected returns. The model starts with the equilibrium assumption that the asset allocation of a representative agent should be proportional to the market values of the available assets, and then modifies that to take into account the 'views' (i.e., the specific opinions about asset returns) of the investor in question to arrive at a bespoke asset allocation.

Black–Litterman overcame this problem by not requiring the user to input estimates of expected return; instead it assumes that the initial expected returns are whatever is required so that the equilibrium asset allocation is equal to what we observe in the markets. The user is only required to state how his assumptions about expected returns differ from the market's and to state his degree of confidence in the alternative assumptions. From this, the Black–Litterman method computes the desired (mean-variance efficient) asset allocation.

In general, when there are portfolio constraints - for example, when short sales are not allowed.

• the easiest way to find the optimal portfolio is to use the Black–Litterman model to generate the expected returns for the assets,
• and then use a mean-variance optimizer to solve the constrained optimization problem.

\(\Omega\), which representing the uncertainty of the views, and \(\tau\) are the most abstract mathematical parameters of this model.

how to specify a probability density function for each view?

criteria for specifying asset classes are:

• Assets within an Asset Class should be homogenous.
• Asset Classes should be mutually exclusive.
• Asset Classes should be diversifying.
• All Asset Classes as a group should make up a significant fraction of all investor's wealth.
• The Asset Class should have the capacity to absorb a significant fraction of the investor's wealth.

• Domestic Equity (can be further sub-divided by value vs growth, or large cap vs small cap).
• International Equity (can be further divided by developed, emerging, or frontier, or large cap vs small cap, or value vs growth).
• Domestic Fixed Income (can be further divided by sovereign vs corporate, nominal vs inflation protected, short vs long term, furher sub-divided by issuer or credit rating).
• International Fixed Income (can be further divided by sovereign vs corporate, nominal vs inflation protected, developed vs emerging, or other distinctions.
• Real Estate (can be further divided by public vs private, type of real estate holding, loans vs properties, domestic vs international).
• Commodities (can be further divided by public vs private, by type, e.g. energy vs crops vs metals).
• Private Equity (can be divided by domestic vs international)
• Cash or cash equivalents

• 非正态分布市场

由于 B-L 模型里假设的资产收益都是服从正态分布，这对绝大多数市场来说并不现实， Attilio Meucci 提出了在非正态分布的条件下如何去使用 B-L 模型，并且在对风险的处理中，不再采用传统B-L 模型里的方式，也就是说不再用方差来表示风险，改用 CVAR 等来描述风险，以便能够捕捉到非对称和尾部的风险。

• 多因子扩展模型

在传统的 B-L模型中，投资者给出的主观观点是对资产的预期收益，而很多时候投资者可能是一些间接的观点，例如对一些指标的观点（股息率、 EPS、 ROE 等），这些因子可能驱动股价波动。Wing Cheung 提出了 ABL 模型（Augmented B-L 模型），首次将因子模型融入到传统 B-L 模型的框架中，大幅度拓宽了 B-L 模型的适用面。

In order for investors to seek higher returns, they must take on higher risk. The question is how to take that risk. The traditional approach is to concentrate in riskier assets, in particular equities. In contrast, the Risk Parity approach is to start with a diversied lower-risk portfolio and then use leverage to raise the expected return. (The use of leverage introduces its own risks, of course, particularly when investments are illiquid. To mitigate this, Risk Parity portfolios tend to invest in liquid instruments such as fnancial futures contracts.) Risk Parity investors believe that some leveraging of a more diversied liquid portfolio is a fundamentally better way to achieve higher returns than the traditional approach of concentrating in the riskiest assets.

• Breadth of instruments used.

While the Simple Risk Parity Strategy invests in only three asset classes, actual Risk Parity portfolios can incorporate additional asset classes. Since these instruments are not perfectly correlated with each other, this further enhances the level of risk diversication and the efficiency of the total portfolio.

• Correlation and volatility forecasting.

While the Simple Risk Parity Strategy targets an equal amount of risk in each asset class, a real-life implementation would incorporate correlations across different asset classes in order to equalize risk contributions. In addition, proprietary risk models can be utilized to improve volatility forecasts, which can help maintain the risk balance across asset classes and more consistent portfolio level volatility over time.

• Tactical over/underweights.

The Simple Risk Parity approach described so far was based on an equal allocation of risk to each of three major risk categories. Indeed, some practical implementations use this “passive” approach to budgeting risk across these categories. However, it is also possible to use the “equal risk across categories” portfolio as the neutral allocation, and then over- or under-weight the risk allocations based on the manager’s tactical views.

• Different volatility targets.

In the exposition above, we used an annualized volatility target of about 10%, which corresponds to the approximate average volatility of a 60/40 portfolio. However, by changing the amount of leverage used, it is easy to construct portfolios of an arbitrary level of volatility — e.g. that of a 70/30 portfolio, 80/20 portfolio, or even a 100% equity portfolio. We believe that Risk Parity is a far superior approach to building “target risk” portfolios than those conventionally used.

• Trading systems and risk control.

Managers can also use proprietary algorithmic trading systems in order to trade passively and reduce trading costs or market impact while adjusting position sizes. Finally, systematic portfolio level drawdown control systems can be used in order to minimize portfolio losses during challenging periods for the strategy.

• Core/Satellite Approach.

This core portfolio can also be supplemented by other uncorrelated strategies, such as alternative investments. Some large institutions have moved in the direction of this “core/ satellite” approach to building their portfolios.

• Alternative investments.

Risk Parity portfolios, which have approximately a 0.5 correlation to equities, would be classi ed as a “directional” alternative rather than a zero beta, non-directional alternative strategy.

• Opportunistic or Flexible Allocation.
• Global Tactical Asset Allocation (GTAA).

• reduced equity concentration and reduced tail risk, • more meaningful diversi cation than traditional approaches, • a portfolio that is more robust in different economic environments, and • an opportunity to improve the risk/return characteristics of an overall portfolio, by either enhancing return, reducing risk, or a combination of both.