My first list of wealth management terminology for software developers is mostly related to portfolio and asset aspects. In this post, I want to focus on the most popular mathematical models for portfolio optimization.
Modern Portfolio Theory (MPT)
MPT is a mathematical model for asset allocation, which makes it possible to maximize expected return for a given variance (level of risk). When choosing from two portfolios with the same expected return, MPT highlights the less risky one; increased risk may then be taken only if it will be compensated by higher expected return.
The return of the whole portfolio may be defined as follows:
where:
μ = E[R] = expected return of a portfolio with n different assets;
μi = return of the particular asset;
Ri = weight of portfolio allocation to a particular asset (specifies how much money is invested in a particular asset related to the total amount invested in the entire portfolio). This should satisfy the following:
MPT analyzes the efficiency of the particular portfolio based on the means and the variance of returns of the assets in the portfolio. The variance of the portfolio is defined as follows:
where:
σ2 = Var[R] = variance of the portfolio;
σi,j = covariance between Ri and Rj.
For different choices of R1, …, Rn you different combinations of μ and σ2 emerge. The set of all possible (σ2, μ) combinations is called the attainable set. By choosing (σ2, μ) combinations with a minimum σ2 for a given μ, and maximum μ for a given σ2, it is possible to build an efficient frontier, which facilitates building an optimal portfolio that offers the highest expected return for a defined level of risk, or the lowest risk for a given level of expected return.
In the following figure, the attainable set is the interior of the ellipse, and the efficient frontier is the highlighted part of its boundary:
Although MPT facilitates optimal portfolio creation, it uses a huge amount of data and is based on expected values that often do not take into account new circumstances. Nonetheless, MPT laid the foundation for other models of portfolio optimization.
Black–Litterman Model (BLM)
BLM is a mathematical model for asset allocation that tries to overcome typical problems of MPT. It is not necessary to estimate the expected return; instead, the equilibrium asset allocation is assumed to be equal to what is observed in the markets. The investor only needs to stipulate the degree to which his or her view of expected returns matches market projections for the method to estimate the returns. The idea for the model belongs to Goldman Sachs associates who tried to get the best of both worlds by integrating MPT and the capital asset pricing model (CAPM), with Harry Markowitz’s mean-variance optimization theory on top. BLM was first introduced in 1990.
Here is the BLM formula to calculate portfolio return:
where:
E[R] = new (posterior) combined return vector (n x 1 column vector)
τ = scalar
Σ = covariance matrix of returns (n x n matrix)
P = assets involved in the views (k x n matrix or 1 x n row vector in the special case of 1 view)
Ω = diagonal covariance matrix of error terms in expressed views representing the level of confidence in each view (k x k matrix)
П = implied equilibrium return vector (n x 1 column vector)
Q = view vector (k x 1 column vector)
(Note: ‘ indicates the transpose and -1 indicates the inverse)
In the figure below, we can see the efficient frontier for the portfolio that would bear the most fruit:
The main advantage of BLT is that within a mean-variance optimization framework, it offers managers a toolkit by which to present a set of expected returns. The method is instrumental when a portfolio manager is not planning to concentrate portfolio assets in a single handful of the assets under optimization.
According to experts, compared to other models used in the portfolio allocation process, BLM has two main points that take it ahead of the curve. First, it offers a tentative base, the equilibrium market portfolio, from which to calculate the asset returns. Second, it offers a specific way to determine investors’ views on returns, since it outputs estimates of expected excess returns and the estimates’ corresponding precision.
Fama–French Model (FFM)
FFM is another popular model used to understand portfolio performance. Expanding on CAPM, FFM adds three more factors that contribute to required return on equity. Along with the risk-free return present in CAPM, the FFM takes into account market risk premium, size premium, as well as value premium, thereby representing a more advanced tool for predicting portfolio performance. While CAPM can explain about 70% of returns from a 5% increase in a single portfolio, FFM brings the number close to 95%.
The formula to calculate portfolio performance is as follows:
where:
RF = risk-free rate;
βmkt = correlation coefficient for the market;
βsize = sensitivity of the i-security to movements in small stocks;
βvalue = sensitivity of i-security to movements in value stocks.
The benchmark factors are as follows:
RMRF = the excess return on the market portfolio; the value-weighted return on all NYSE, AMEX, and NASDAQ stocks (from CRSP) minus the one-month treasury bill rate;
SMB (“small minus big”) = return to small stocks minus return to large stocks;
HML (“high minus low”) = return to value stocks minus return to growth stocks;
Taking into account outperformance tendency, the benefit of FFM over a more simplistic CAPM is that it includes size and value premiums, which are not captured in CAPM.
Conclusion
As can be seen from the above discussion, newer models evolved as an expansion of previous ones, with more factors added in an effort to more precisely quantify the expected ROIs with a certain amount of risk.
This is not an exhaustive list of all pricing models; however, understanding the most popular ones and the differences between them is crucial for developers who create solutions to help capital markets and asset and wealth management firms harness the full potential technology has to offer.
