FairPortfolio ~ a simple and stable optimal portfolio

[email@1m1.io]

Usage

Choose n assets which you believe are going to increase in value over the long term. For each asset, have a price vector. The price vectors for each asset should have equal length. The prices are denominated in a common ccy.

using FairPortfolio
optimize(price_asset_1, price_asset_2, price_asset_3, ...)

The standard output displays the optimal weights your portfolio should use per asset. E.g.

weights based on cross-risk:
0.33130302872447653,0.3320587287004601,0.33663824257506336

weights based on own-risk to get 3.906250000000001e-5 variance per time period (scale accordingly):
0.2303491927483238,0.2437090738769763,0.18799540576581383

weights based on combined risks: 0.0763153852217579,0.08092572524435526,0.06328644300918951

shares based on both risks and prices: 2.721552378761487e-6,4.513655979086473e-5,0.8497999937377919

The first line describes the optimal weights considering only cross-risks (covariances) of the assets.
The second line are the weights to have homogeneous own-risks (variances). If the input prices were e.g. per day, then these weights are to get a yearly standard deviation of 0.1.
The third line is a multiplication of the first two lines. This is the final weighting.
The fourth line is the number of shares you should hold per 1 ccy of funds.

The above output is from the example/crypto.jl and tells us to hold approx 0.0000027 BTC, 0.000045 ETH, 0.85 DOGE in our portfolio to minimize our risk and have 10% annualized standard deviation (volatility) per 1 USD investment, given current data.

Abstract

This whitepaper describes a method of creating an stable, optimal portfolio based on arbitrary list of assets. The model was practically tested vs other existing models and resulted in significantly better results (the test data is to be provided later or left as an exercise to the interested). The result is achieved via simplications of multiple types.

I. Introduction

We assume that we have chosen a list of $n$ assets that we want to invest into. Us choosing these assets implies that we believe that each of these assets has a long term positive return.

The distribution of future returns of any asset consists of moments of increasing degrees. The first degree moment is the expected value and is the hardest to predict. An investor should only choose assets that are believed to have positive first moment.

The second degree moments are the covariances between the assets, $\sigma_{ij}$, including the variances of each asset $\sigma_{ii}$. These covariances should be estimated by employing shrinkage.

Third degree moments are referred to as skewness and fourth degree moments as kurtosis. However, moments of degree 3 and higher have very large errors in measurement and hence are completely ignored in this method. This is because financial data is highly noisy.

II. Homogeneous Variance

To get a stable portfolio, we first homogenize the variances across all assets and time

$$\sigma_{ii} = \sigma$$

An investor could choose any model to predict the variance of an asset for the following time period and pre-scale the asset such that all assets have approx. the same variance (or volatility) over any time period. One simple model to achieve that is to take the rolling window average variance.

III. Minimal Variance Portfolio

Now that we have well estimated covariances, equal variances and we ignore moments of degree 1 and degrees 3 and higher, we find the optimal portfolio by minimizing the total portfolio variance.

With the symmetric covariance matrix $C$

$$C = \begin{pmatrix} \sigma & \ldots &\sigma_{1n} \\ \vdots & \ddots & \vdots \\ \sigma_{1n} & \ldots &\sigma \end{pmatrix}$$

the total portfolio variance $V$ is

$$V = w^T\cdot C \cdot w = (w_1 \ldots w_n) C \begin{pmatrix} w_1 \\ \vdots \\ w_n \end{pmatrix}$$

where $w$ are the weights of assets that we want to calculate.

We simplify algebraicly

$$V = w^T\cdot C \cdot w = (w_1 \ldots w_n) \begin{pmatrix} \sigma_{11} & \ldots &\sigma_{n1} \\ \vdots & \ddots & \vdots \\ \sigma_{1n} & \ldots &\sigma_{nn} \end{pmatrix} \begin{pmatrix} w_1 \\ \vdots \\ w_n \end{pmatrix}$$

$$= (\sum_{i < n} w_i \sigma_{1i} + w_n \sigma_{1n} \ldots \sum_{i < n} w_i \sigma_{ni} + w_n \sigma_{nn}) \begin{pmatrix} w_1 \\ \vdots \\ w_n \end{pmatrix}$$

$$= \sum_{j < n} w_j \sum_{i < n} w_i \sigma_{ji} + w_n \sum_{i < n} w_i \sigma_{n1} + \sum_j w_j w_n \sigma_{jn}$$

$$= \underbrace{\sum_{i, j < n} w_i w_j \sigma_{ij}}A + \underbrace{2 \sum{j < n} w_j w_n \sigma_{jn}}B + \underbrace{w_n^2 \sigma{nn}}_C$$

To find the $w_k$ that minimizes $V$, we will solve for

$$\frac{\partial{V}}{\partial{w_k}} = 0$$

for all $k < n$.

This is equivalent (and slightly simpler) to solving

$$0 = \frac{1}{2} \cdot \frac{\partial{V}}{\partial{w_k}}= \frac{1}{2} \cdot \frac{\partial{(A+B+C)}}{\partial{w_k}}$$

Also, we know that all the asset weights sum to 100%, i.e. we can reduce the problem by 1 dimension by realising

$$ w_n = 1 - \sum_{i < n} w_i $$

which also means

$$\frac{\partial{w_n}}{\partial{w_k}} = -1$$

For $A$, we get

$$\frac{1}{2} \cdot \frac{\partial{A}}{\partial{w_k}} = \sum_{i < n} w_i \sigma_{ki}$$

For $B$, we get

$$\frac{1}{2} \cdot \frac{\partial{B}}{\partial{w_k}} = \sigma_{kn}\frac{\partial{}}{\partial{w_k}}(w_kw_n) + \sum_{k \ne j < n} w_j \sigma_{jn} \frac{\partial{w_n}}{\partial{w_k}}$$

$$= \sigma_{kn} (w_n-w_k) - \sum_{k \ne j < n} w_j \sigma_{jn}$$

$$= \sigma_{kn} - \sigma_{kn} w_k - \sigma_{kn}\sum_{i < n}w_i - \sum_{k \ne j < n} w_j \sigma_{jn}$$

$$= \sigma_{kn} - \sigma_{kn}\sum_{i < n}w_i - \sum_{j < n} w_j \sigma_{jn}$$

$$= \sigma_{kn} - \sum_{i < n} w_i (\sigma_{in}+\sigma_{kn})$$

For $C$, we get

$$\frac{1}{2} \cdot \frac{\partial{C}}{\partial{w_k}} = -w_n\sigma_{nn} = -\sigma_{nn} + \sigma_{nn}\sum_{i < n} w_i$$

Putting it together, we get

$$\frac{1}{2} \cdot \frac{\partial{V}}{\partial{w_k}} =\frac{1}{2} \cdot \frac{\partial{(A+B+C)}}{\partial{w_k}}$$

$$ = \sigma_{kn}-\sigma_{nn} + \sum_{i < n} w_i(\sigma_{nn}-\sigma_{in}-\sigma_{kn}+\sigma_{ki})$$

We can put previous formula into matrix form as follows

$$0 = \frac{\partial{V}}{\partial{w_k}} \Leftrightarrow \hat{S} \hat{w} = \hat{b}$$ with

$$\hat{w} = \begin{pmatrix} w_1 \\ \vdots \\ w_{n-1} \end{pmatrix}$$

$$\hat{b} = \begin{pmatrix} \sigma-\sigma_{1n} \\ \vdots \\ \sigma-\sigma_{n-1,n} \end{pmatrix}$$

and $\hat{S}$ the matrix containing $s_{ik}$ with

$$s_{ik} = \sigma-\sigma_{in}-\sigma_{kn}+\sigma_{ki} = s_{ki}$$

$$s_{ii} = 2(\sigma - \sigma_{in})$$

We can break up $\hat{S}$ as follows

$$\hat{S} = \tilde{C} + \tilde{C}^T + \sigma \cdot (I + \mathbb{1}) - (C_n + C_n^T)$$

with $\tilde{C}$ the upper triangular covariance matrix

$$\tilde{C} = \begin{pmatrix} 0 && \sigma_{12} && \sigma_{13} && \ldots && \sigma_{1,n-1} \\ && 0 && \sigma_{23} && \ldots && \sigma_{2,n-1} \\ && && \ddots && \ldots && \vdots \\ && \text{\huge0} && && 0 && \sigma_{n-1,n-1} \\ && && && && 0 \end{pmatrix}$$

$C_n$ the row repeating matrix

$$C_n = \begin{pmatrix} \sigma_{1n} && \ldots && \sigma_{1n} \\ \vdots && \ddots && \vdots \\ \sigma_{n-1,n} && \dots && \sigma_{n-1,n} \\ \sigma && \ldots && \sigma\end{pmatrix}$$

$I$ the identity matrix

$$I = \begin{pmatrix} 1 && && \text{\large0}\\ && \ddots \\ \text{\large0} && && 1\end{pmatrix}$$

and $\mathbb{1}$ the constant 1 matrix

$$\mathbb{1} = \begin{pmatrix} 1 && \ldots && 1 \\ \vdots && \ddots && \vdots \\ 1 && \ldots && 1\end{pmatrix}$$

Now we are left with solving a linear equation system

$$\hat{S} \hat{w} = \hat{b} \Leftrightarrow \hat{w}=\hat{S} \backslash \hat{b}$$

which gives us our optimal weights $w$.

IV. Conslusion

Separating the task of optimising the portfolio from homogenizing the variance of each asset across time and the different assets enables a significantly more stable portfolio. Using shrinkage to estimate the covariances also plays helps this stability.

Reducing the optimisation dimension helps further.

The described model was found to perform significantly better than other tested models in 2012. This data will be provided at some point and is left as an excercise for the interested reader until then.