Portfolio dynamics 101
Portfolio Dynamics 101

June 21, 2022

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A portfolio is a collection of financial investments like stocks, bonds, commodities, cash, and cash equivalents, including closed-end funds and exchange-traded funds (ETFs). People generally believe that stocks, bonds, and cash comprise the core of a portfolio. Though this is often the case, it does not need to be the rule. A portfolio may contain a wide range of assets including real estate, art, and private investments. Usually, these types of investments are popular with insurance companies and pension funds, since these kinds of companies want stable returns over a very long period. However, we will not focus on the type of assets in a portfolio, but on the general dynamics in a portfolio consisting of mainly stocks and bonds.

A Portfolio

You may think of an investment portfolio as a pie that has been divided into pieces of varying wedge-shaped sizes, each piece representing a different asset class and/or type of investment. Investors aim to construct a well-diversified portfolio to achieve a risk-return portfolio allocation appropriate for their risk tolerance level. A very famous mathematical framework to achieve such a portfolio is the Markowitz model. This model maximizes the expected return E[R_p] = \sumw_iE[R_i], where R_p is the return of the portfolio, w_i the weight of asset i and R_i the return of asset i. We maximize this expected return subject to a certain level of variance in the portfolio. So in this model, the variance is used as a proxy for risk. In addition, from this model, we find the well-known efficient frontier depicted in the figure below. A portfolio lying on the efficient frontier represents the combination offering the best possible expected return for a given risk level.

Efficient frontier. The hyperbola is sometimes referred to as the ‘Markowitz Bullet’ and is the efficient frontier if no risk-free asset is available. With a risk-free asset, the straight line is the efficient frontier.

The straight line in the figure is called the capital allocation line and can be compared to the famous CAPM. However, the CAL is calculated as E[R_c] = R_f + \sigma_c * \dfrac{E[R_p] - R_f}{\sigma_p}, where p depicts a sub-portfolio of risky assets, f is a risk-free asset and c is a combination of c and p in one new portfolio.

Although the Markowitz model in modern portfolio theory is of great theoretical importance and gives a lot of intuition in finding the optimal portfolio strategy, it has some major drawbacks. First, the risk, return, and correlation measures used by this model are based on expected values, which means that they are statistical statements about the future. Such measures often cannot capture the true statistical features of the risk and return which often follow highly skewed distributions and can give rise to inflated growth of return. One could use a scenario-based approach to overcome this flaw but this is out of the scope of this article. Another problem is that it relies on the efficient-market hypothesis and uses fluctuations in share price as a substitute for risk. It is well known that the efficient-market hypothesis does not hold in the financial markets in our world.

Dynamics of assets

In order to continue with the portfolio dynamics that will be shown in the next section, a fundamental concept first needs to be explained. It is called stochastic differential equations (SDE) and are of great importance to capture the dynamics of a stock, bond or a portfolio. An SDE is usually dependent on a Wiener process W. The only thing that you have to know is that a Wiener process follows a normal distribution with mean 0 and variance dt, where dt is a very small time interval. Then a general SDE used for modelling stocks is often a Geometric Brownian Motion (GBM) and has the following form:

(1)   \begin{equation*} <!-- /wp:paragraph --> <!-- wp:paragraph --> dS_t = \muS_tdt + \sigmaS_tdW_t <!-- /wp:paragraph --> <!-- wp:paragraph --> \end{equation*}

where \mu is called the drift term and \sigma the diffusion term. We can find an explicit solution to this equation, but this is not really interesting. However, SDE’s can help us solve certain problems that model the price dynamics of options really well (like the Black-Scholes model). If you would like to more about this model, read my previous article on differential equations. We will use these SDE’s for our portfolio dynamics.

Portfolio dynamics

Suppose we have two assets, S_1 and S_2, that follow a GBM as described before. Now we will combine these two ‘assets’ into a portfolio with value V. Then V = h_1 S_1 + h_2 S_2, where h_i is the amount of shares that we have of asset i. The amount we invest in each asset changes over time, as share prices fluctuate. Therefore, we have dynamic portfolio strategies. We need to describe how we mathematically represent such strategies. Moreover, it turns out that we should limit ourselves to so-called self-financing portfolios. A self-financing portfolio is defined as:

(2)   \begin{equation*} <!-- /wp:paragraph --> <!-- wp:paragraph --> dV_t = h_1dS_1(t) + h_2dS_2(t). <!-- /wp:paragraph --> <!-- wp:paragraph --> \end{equation*}

Instead of using h_i, we can also define portfolio weights, which was also described in the beginning of the article. The portfolio weight of an asset i, can be written as:

(3)   \begin{equation*} <!-- /wp:paragraph --> <!-- wp:paragraph --> w_i = \dfrac{h_iS_i}{V_t}. <!-- /wp:paragraph --> <!-- wp:paragraph --> \end{equation*}

Then the portfolio dynamics can be rewritten as:

(4)   \begin{equation*} <!-- /wp:paragraph --> <!-- wp:paragraph --> dV_t = V_t * \sum_{i=1}^N w_i \dfrac{dS_i(t)}{S_i(t)}. <!-- /wp:paragraph --> <!-- wp:paragraph --> \end{equation*}

Now, suppose we want to construct a portfolio conisting of 2 stocks following the same GBM. We assume that the two stocks are not correlated and we want to invest 50% in each share, meaning that the portfolio weights will not change (i.e. h_i = 1/2). Then, after some simple simplifications, the portfolio dynamics can be written as:

(5)   \begin{equation*} <!-- /wp:paragraph --> <!-- wp:paragraph --> dV_t = V_t [\mu dt + \dfrac{1}{2} \sigma (dW_1(t) + dW_2(t))]. <!-- /wp:paragraph --> <!-- wp:paragraph --> \end{equation*}

There is a small problem in this equation, we have two Wiener processes where we should have one. However, this can be solved by noticing that dW_1(t) + dW_2(t) ~ N(0, 2dt). Then using the fact that we can define dW_t = \dfrac{dW_1(t) + dW_2(t)}{\sqrt(2)} ~ N(0, dt) we can write

(6)   \begin{equation*} <!-- /wp:paragraph --> <!-- wp:paragraph --> dV_t = V_t [ \mu dt + \dfrac{1}{2}\sqrt(2) \sigma dW_t. <!-- /wp:paragraph --> <!-- wp:paragraph --> \end{equation*}

So we see that there is some diversification. The expected return is the same as the two stocks but the variance is lower. You can play around with different strategies for h_i to find the portfolio dynamics and the associated diversification benefits that you could receive. However, if stocks are correlated with each other, their Wiener processes will also be correlated. This will result in a lower reduction of volatility when you combine the stocks in a portfolio.

Last but not least

This was a short overview of how portfolio dynamics is used to model the value of a portfolio. It is a very powerful tool that is used by many quantitative analysts who are responsible for modelling the value of existing portfolios. Moreover, you can also find the best strategy that fits your risk appetite. For example, you could try to find the optimal strategy that minimizes the volatility of a portfolio or maximizes return. However, it all depends on your level of risk aversion and how much risk you are willing to take.


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