Capital Asset Pricing Model with Size Factor and Normalizing by Volatility Index

This celebrated model, abbreviated as CAPM, compares returns of a stock portfolio with returns of the benchmark. This model was proposed by [16, 19, 25] The benchmark is usually taken to be the Standard & Poor 500 for the American stock market. The model states that the only factor which matters for a well-diversified portfolio is market exposure, otherwise known by a standard term beta and denoted by $\beta$. The case $\beta=0$ corresponds to the risk-free portfolio, with guaranteed deterministic return. This is usually measured using a benchmark of a short-term rate $r$, for example 1-month Treasury rate. The case $\beta=1$ corresponds to the market portfolio (the benchmark). When $\beta\in(0,1)$, the stock portfolio can be replciated by investing in a portfolio of risk-free bonds and the stock market benchmark, in proportions $1-\beta$ and $\beta$, respectively. In other words, total returns $Q$ (including price changes and dividends) of this portfolio are related to total returns $Q_{0}$ of the benchmark:

(1)

$$Q=(1-\beta)r+\beta Q_{0}.$$

Equivalently, we can rewrite (1) in terms of equity premia $P=Q-r$ of the portfolio and $P_{0}=Q_{0}-r$ of the benchmark:

(2)

$$P=\beta P_{0}.$$

If $\beta>1$, this (2) still holds, and can be interpreted as shorting bonds and investing everything in the benchmark. We can treat $\beta$ as a risk measure: $\beta>1$ means that the portfolio is riskier than the benchmark, and $\beta\in(0,1)$ means the opposite. The case $\beta<0$ does not usually happen in practice, see [1, Chapter 7].

It is not considered a big achievement if a money manager improved returns by increasing $\beta$. In fact, often these managers are expected to maximize excess return: Total returns of a portfolio adjusted for market exposure. This quantity is denoted by $\alpha$ and, accordingly, is called alpha. These two Greek letters $\alpha$ and $\beta$ are standard notation in Finance. This methodology of market exposure and excess return is well-accepted by both finance academics and practitioners. One can include $\alpha$ into (2) as

(3)

$$P=\alpha+\beta P_{0}+\varepsilon,\quad\mathbb{E}\varepsilon=0.$$

We also include an error term $\varepsilon$, since the model might not hold almost surely. This makes (3) a simple linear regression of $P$ upon $P_{0}$.

Subsequent research disproved the strong claims of CAPM that market exposure $\beta$ is the only risk measure and quantity of interest for a diversified stock portfolio, see [8, 9], and [1, Chapter 7]. For one, there are systematic ways to generate excess return $\alpha$ by using several factors. Also, $\beta$ might be unstable in the long run. Still, $\beta$ is an established risk measure, accepted by financial theorists, analysts, and managers. The CAPM is useful as a benchmark model, a starting point for more complicated and real-life models. Calculating $\beta$ for mutual funds and exchange-traded funds is common practice.

The most well-known and accepted factors are size (average market capitalization of portfolio stocks) and value (fundamentals such as earnings, dividends, or book value, compared to price). These are well-accepted by financial academic community and are considered useful by industry practitioners, to the extent that there are multiple size- and value-based funds traded alongside the S&P 500 funds. See [3, 8, 10].

Including a few factors (for example size and value) would enrich (3). In particular, size factor is related to $\beta$ as follows: Well-diversified portfolios of small stocks have equity premia $P$ closely correlated with that $P_{0}$ of large stock benchmark S&P 500. This simple linear regression of $P$ vs $P_{0}$ has very large $R^{2}$ but $\beta$ which is slightly larger as $1$. This can be done, for example, with exchange-traded funds tracking small, mid, and large (=benchmark) stock indices, see [13, Appendix]. The $\beta$ for mid-cap index is 1.15, and for small-cap index is 1.27. A natural idea then is to model $\beta$ as a function of a portfolio size relative to benchmark. We can try the same for $\alpha$, although in [13, Appendix] we have $\alpha$ is not significantly different from zero for both small-cap and mid-cap indices. See more on the size effect in [24], and the discussion in [1, Chapter 8].

Therefore, we developed the following model in [13]: Let $S$ = market capitalization of target portfolio, and $S_{0}$ = market capitalization of benchmark portfolio. Then the relative size (on the log scale) is defined as

(4)

$$C=\ln(S/S_{0})$$

For $C=0$, the relative size is 0 (on the log scale) or 1 (on the absolute scale). This corresponds to the target portfolio having the same properties as the benchmark portfolio, with $\alpha=0$ and $\beta=1$. The simplest model is linear: For some coefficients $a,b$,

(5)

$$\alpha=aC,\quad\beta=bC+1$$

In fact, in [13] we have more general conditions: $\alpha$ and $\beta$ are general functions of $C$. Here, we focus only on linear functions, see [13, Example 1]. Analysis for small-cap and mid-cap funds above shows that $a\approx 0$ but $b<0$. Rewrite (3) as follows by plugging there (5):

(6)

$$P=AC+\left(1+BC\right)P_{0}+\delta,$$

where $\delta$ are IID regression residuals with mean zero. A good way to generalize (6) is to make it dynamic: a time series. To this end, we make the model complete by writing an equation for $S,S_{0},P_{0}$. With regard to the benchmark equity premia $P_{0}$ and market cap $S_{0}$, we shall write this equation in the next subsection. Unlike total returns, which include dividends, price returns are computed only using price movements. The main idea is to take CAPM linear regression, and replace equity premia with price returns. We get

(7)

$$R=aC+\left(1+bC\right)R_{0}+\varepsilon,$$

where $a,b$ are some coefficients (not necessarily the same as the $A,B$ from (6)), and $\varepsilon$ are IID regression residuals with mean zero (not necessarily the same as $\varepsilon$, but possibly correlated with these). We can interpret change in logarithm as price returns:

(8)

$\displaystyle\begin{split}R(t)&=\ln S(t+1)-\ln S(t)=\frac{S(t+1)}{S(t)},\\ R_{0}(t)&=\ln S_{0}(t+1)-\ln S_{0}(t)=\ln\frac{S_{0}(t+1)}{S_{0}(t)}.\end{split}$

In real-life finance, a stock market is stable in the long run: Small stocks grow, on average, faster than large stocks. Thus formerly small stocks might become large, and formerly large stocks might become small. The relative size of a stock portfolio exhibits mean-reversion. In our article [13], we proved this for continuous time and more general systems than (6), (7), under the assumptions that the benchmark follows a lognormal Samuelson (Black-Scholes) model of geometric random walk with growth rate $g$ and total returns $G$, and

(9)

$$a+bg<0.$$

This is consistent with the observation made above that $a\approx 0$ and $b<0$, since $g>0$. Analysis of real-world finance data in [13] showed that $A\approx a$ and $B\approx b$.

However, there is a drawback in our modeling from [13]. We fit regressions (6) and (7) using real-world monthly data from 1926 taken from Kenneth French’s Dartmouth College Financial Data Library. Residuals $\varepsilon,\delta$ are not IID. Absolute values are autocorrelated. This feature is also true for S&P 500 monthly returns themselves, see our recent article [22]. Also, these returns are not Gaussian. We wish to improve the fit of regressions (6) and (7) to make residuals closer to IID Gaussian. For S&P 500 returns, we did this in [22] as follows: We divided these returns by monthly average VIX: The S&P 500 Volatility Index $V$ computed daily by the Chicago Board of Options Exchange. Our main idea of this article [22] is (in our notation, see [22, Subsection 2.2]):

(10)

$$\frac{R_{0}(t)}{V(t)}\sim\mbox{IID Gaussian}.$$

Similarly, it is reasonable to model normalized equity premia as IID Gaussian:

(11)

$$\frac{P_{0}(t)}{V(t)}\sim\mbox{IID Gaussian}.$$

We did not do this in [22], but we complete this work in this article. For the original equity premia of benchmark $P_{0}$, we plot in Figure 1 the quantile-quantile plot and the autocorrelation function for $P_{0}$ and $|P_{0}|$. For the normalized equity premia $P_{0}/V$, we plot these in Figure 2. We see that division by VIX is needed to model equity premia as in (11). Data for $P_{0}$ is taken from Kenneth French’s data library: Top 10% decile, January 1986–October 2024. For short-term rate $r$, we use the 3-month Treasury rate from Federal Reserve Economic Data: start of month data. Data and Python code is available on Github repository: asarantsev/size-capm-vix

The VIX itself is modeled by an autoregression of order 1 on the log scale, see [22]:

(12)

$$\ln V(t)=\alpha+\beta\ln V(t-1)+W(t).$$

Slightly abusing the notation, but following [22], in the rest of the article we use $\alpha$ and $\beta$ as the intercept and the slope of this autoregression (12), instead of excess return and market exposure from the CAPM. This model (12) has good fit, as shown in [22, Section 3]. Innovations $W$ are IID but not Gaussian, with some finite exponential moments. The point estimate $\hat{\beta}=0.88$, and we reject the unit root hypothesis $\beta=1$.

Methodology of [22] stands in contrast to classic stochastic volatility models [4], where the volatility $V$ is not observed directly and must be inferred from the data $R_{0}$ or $P_{0}$.

A newly developed framework for portfolio management in Stochastic Portfolio Theory, see [11]. One question which concerns it is model stability: In a model of $N$ stocks, do they move together in the long run, or do they split into several subsets, moving away from each other in the long run? A related question is analysis of market weights: A market weight $\mu$ of a stock is its market capitalization (size) divided by the sum of all market capitalizations. Rank market weights at each time from top to bottom:

$$\mu_{(1)}(t)\geq\ldots\geq\mu_{(N)}.$$

The plot of $N$ points

$$(\ln n,\ln\mu_{(n)}(t)),\,n=1,\ldots,N$$

is called the capital distribution curve. For real-world markets, this curve is concave and straight at left upper end, see [11, 12]. See also our own plots in Figure 3. Python code and data are available in GitHub repository asarantsev/size-capm-vix The data is from Kenneth French’s Data Library.

If the model is stable, the capital distribution curve is also stable in the long run: see Theorem 2. Previous research analyzed long-term stability and capital distribution curve for various continuous-time models: [7, 20]. These models were designed to capture the observations that, on average, small stocks have higher volatility and growth rates than large stocks. In [7], competing Brownian particle models were analyzed, where drift and diffusion coefficients depend on the rank of the stock, and linearity of the capital distribution curve was reproduced. In [20], the drift and diffusion depend on the market weight of a stock, but the capital distribution curve is not linear. These models do not use CAPM or VIX.

In the previous article by the second coauthor [13], devoted to the CAPM with size but without VIX (in other words, with constant volatility), we also reproduced model stability for its continuous-time version [13, Theorem 2]. Using simulation, we reproduced the linearity of the capital distribution curve. But we did not state and prove rigorous results on this. A natural question is to reproduce these results for our model here. We accomplish both tasks in this article: (a) we simulate the capital distribution curve in Subsection 5.2, which reproduces its linearity; (b) we prove results on convergence to a Poisson point process. This fills a gap in [13].

In this article, we combine the ideas of our articles [13] on CAPM and [22] on VIX to state a reasonable generalization of [13]. This model can be truncated: Include only price changes for the benchmark $R_{0}$ and the target $R$ (or, equivalently, market size $S_{0}$ and $S$), and volatility $V$. Alternatively, it can be completed: Include also equity premia $P_{0},P$ for the benchmark and the target. A truncated model contains 3 time series, and a completed model contains 5 time series.

In [13], we model (7) and (6) but with $P_{0}$ or $S_{0}$ i.i.d. Gaussian, in continuous time. The article [13] does not include VIX $V$. However, this article is concerned with the idea from [22] that division by VIX normalized returns and premia. We replace $P$ and $P_{0}$ with $P/V$ and $P_{0}/V$ in (6). Also, we replace $R$ with $R/V$ and $R_{0}$ with $R_{0}/V$ in (7). We model $(R_{0}/V,P_{0}/V)$ as IID (but maybe not Gaussian), following (10) and (11). As mentioned in the Introduction, in Section 3 we perform statistical data analysis.

We fill two lacunas left in our research [13]. The first lacuna is stability results for discrete time. In Section 4 of this article, we state and prove this for the case of stochastic VIX. Our results work for constant volatility as well, which is the setting of [13], see (9). Next, we check the stability condition numerically. The second lacuna is rigorous results for the capital distribution curve, which are lacking in [13]. There, we present only the simulation for the constant volatility. In Section 5 of this article, we state and prove a convergence result for the capital distribution curve: Theorem 2. Next, in Theorem 3, we show a remarkable result: the curve is

$$(\ln n,X_{(n)}),\quad n=1,\ldots,N,$$

where $X_{(1)}\geq\ldots\geq X_{(N)}$ are order statistics of a conditional normal sample:

$$X_{1},\ldots,X_{N}\mid\mu,\sigma\sim\mathcal{N}(\mu,\sigma^{2})$$

for random $\mu$ and $\sigma$. We perform the simulation in Section 5 for stochastic VIX. In Appendix, we further analyze this curve for $\mu=$ and $\sigma=1$, and show its upper left and lower right ends replicate real-world behavior.

We take monthly data January 1990 – September 2022. Total $T=405$ data points. As discussed in the Introduction and Background sections, we measure volatility $V$ as Chicago Board of Options Exchange VIX: The Volatility Index for S&P 500, monthly average data. This data is taken from Federal Reserve Economic Data (FRED) web site. For equity premia computation, we need short-term Treasury rates, which are also taken from the FRED web site.

The rest of the data is taken from Kenneth French’s Dartmouth College Financial Data Library. It contains equally-weighted portfolios of stocks split into 10 deciles by size: Decile 1 has smallest 10% stocks by size (market capitalization), Decile 2 has the next 10% smallest stocks, etc up to Decile 10, which contains top 10% largest stocks. In practice, during this time span this Decile 10 portfolio closely corresponds to the S&P 500, the classic benchmark for American stocks. (Although the S&P 500 index is size-weighted, not equally-weighted.) We also use Decile 10 as the benchmark. These portfolios are reconstituted at the end of June of each year. For each decile and each month, the data contains average market size, price returns (excluding dividends), and total returns (including dividends).

Our goal is to fit (7). We rewrite it as

$$\frac{R(t)-R_{0}(t)}{V(t)}=aC(t)+bC(t)\frac{R_{0}(t)}{V(t)}+\varepsilon(t).$$

This linear regression does not have an intercept, however. To make the model complete, we add an intercept $m$ and get:

(13)

$$\frac{R(t)-R_{0}(t)}{V(t)}=aC(t)+bC(t)\frac{R_{0}(t)}{V(t)}+m+\varepsilon(t).$$

For the benchmark with price returns $R_{0}$, we take Decile 10. For the target with price returns $R$, we use Deciles 1, …, 9. We fit these 9 linear regressions (13) separately.

For Deciles 3–9, it is reasonable to model residuals using IID Normal. The Student $t$-test gives $p$-values greater than 5% for $m$ and $a$ for each decile. However, the Student $t$-test gives $p$-values greater than 5% for $b$, for all deciles, except Deciles 2 and 9. Thus we see that for most deciles, we can assume $m=a=0$ but $b\neq 0$. Finally, the 95% confidence intervals for $s^{2}$ for each decile contain $1$. Thus, for Deciles 3–9, we could model

$$\frac{R(t)}{V(t)}=(1+bC(t))\frac{R_{0}(t)}{V(t)}+\varepsilon(t),\quad\varepsilon(t)\sim\mathcal{N}(0,1)\ \mbox{IID},$$

and $b$ is between $-0.1$ and $-0.2$ (except Decile 9, where we fail to reject $b=0$). This model is an improvement over [13] where residuals were not IID or Gaussian.

Our goal is to fit (6). Similarly to (7), we rewrite it as

$$\frac{P(t)-P_{0}(t)}{V(t)}=AC(t)+BC(t)\frac{P_{0}(t)}{V(t)}+\delta(t).$$

This linear regression does not have an intercept, however. To make the model complete, we add an intercept $M$ and get:

(14)

$$\frac{P(t)-P_{0}(t)}{V(t)}=AC(t)+BC(t)\frac{P_{0}(t)}{V(t)}+M+\delta(t).$$

For the benchmark with equity premia $P_{0}$, we use Decile 10. And for the target with equity premia $P$, we use Deciles 1, …, 9. We fit these 9 linear regressions (13) separately.

Similarly to regression (13), for Deciles 3–9, it is reasonable to model residuals using IID, but the evidence for normality is much weaker than in (13). The Student $t$-test gives $p$-values greater than 5% for $M$ and $A$ for each decile. However, the Student $t$-test gives $p$-values greater than 5% for $B$, for all deciles, except Deciles 2 and 9. Thus we see that for most deciles, we can assume $M=A=0$ but $B\neq 0$. Finally, the 95% confidence intervals for $s^{2}$ for each decile contain $1$. Thus, for Deciles 3–9, we could model

$$\frac{P(t)}{V(t)}=(1+BC(t))\frac{P_{0}(t)}{V(t)}+\delta(t),\quad\delta(t)\sim\mbox{IID},\quad\mathbb{E}[\delta(t)]=0,\quad\mathbb{E}[\delta^{2}(t)]=1.$$

and $B$ is between $-0.1$ and $-0.2$ (except Decile 9, where we fail to reject $B=0$). Similarly to (13), this model is an improvement over [13] where residuals were not IID.

Take a sequence of five-dimensional vectors:

(15)

$$\mathbf{Y}(t):=(W(t),Z(t),U(t),\delta(t),\varepsilon(t)),\quad t=1,2,\ldots$$

These five components might be correlated between themselves. First, we model $V$ using (12) with innovations $W$. Next, we model $R_{0}$ following (10) for some constant $g\in\mathbb{R}$:

(16)

$$\frac{R_{0}(t)}{V(t)}=g+U(t),$$

but we do not necessarily assume $U$ is Gaussian. We subtract $g$ because Assumption 1 states that $\mathbb{E}[U(t)]=0$, but the left-hand side of (16) has nonzero mean. Similarly, we model $P_{0}$ following (11) for another constant $G\in\mathbb{R}$:

(17)

$$\frac{P_{0}(t)}{V(t)}=G+Z(t),$$

and $Z$ might not be Gaussian. These two equations (16) and (17) model the benchmark. Next, we combine (6) and (7) with these new ideas, following the outline in Section 2. We can replace $P_{0}$ and $P$ with $P_{0}/V$ and $P/V$ in (6):

(18)

$$\frac{P(t)}{V(t)}=M+AC(t)+(1+BC(t))\frac{P_{0}(t)}{V(t)}+\delta(t).$$

Similarly, we can replace $R_{0}$ and $R$ with $R_{0}/V$ and $R/V$ in (7):

(19)

$$\frac{R(t)}{V(t)}=m+aC(t)+(1+bC(t))\frac{R_{0}(t)}{V(t)}+\varepsilon(t).$$

Recall the definition of the relative size process

(20)

$$C(t)=\ln\frac{S(t)}{S_{0}(t)}.$$

Consider a discrete-time process $\mathbf{X}=(\mathbf{X}(0),\mathbf{X}(1),\ldots)$ in $\mathbb{R}^{d}$.

Below, we show that, if $\beta\in(0,1)$, these stationary versions $(V(t),R_{0}(t))$ exist, and Assumption 23 is well-defined. The following is the main result of this article.

To check Assumption 23 is hard. But for real-world values of coefficients found in Section 3,

(31)

$$x:=aV(t)+bR_{0}(t)\quad\mbox{is small.}$$

Indeed, from the data analysis we can assume $a=0$, because the point estimates $\hat{a}$ are not significantly different from zero. Next, $b\in(-0.2,-0.1)$, and $R_{0}(t)$ is the growth rate of the benchmark. A well-known fact for the growth rate (for example, found in [1]) is that per annum, it is of order 10%. Per month, it is on average even less. Thus $bR_{0}(t)$ is small. This completes the explanation of (31).

For $x\approx 0$, we use the first-order approximation: $\ln|1+x|=\ln(1+x)\approx x$. We simplify Assumption 23:

(32)

$$\mathbb{E}[aV(t)+bR_{0}(t)]=a\mathbb{E}[V(t)]+b\mathbb{E}[R_{0}(t)]<0.$$

This is analogous to the result (9) from [13]. Let us quickly show that condition (32) holds for real-world market data. As mentioned above, we can assume $a=0$. Next, we have $\hat{b}<0$ for each of the Deciles 3–9. Finally, $\mathbb{E}[R_{0}(t)]>0$, because $R_{0}$ is the growth rate of the benchmark; together with the US economy, stock market indices in the USA grow in the long run. To show when the left-hand side of (32) is well-defined, we state the following result. The assumptions are reasonable in the light of discussion from Subsection 2.4.

We stated and proved in [13] results on stationarity and ergodicity of a continuous-time analogue of our model, but with constant volatility. We did not state and prove stability result for discrete time. Here, we fill this gap. In the notation of this article, we can assume $V(t)=1$ is constant. This corresponds to $W(t)\equiv 0$.

Assume $Z(t)\sim\mathcal{N}(0,\sigma^{2})$ IID, and $V(t)=1$. This violates Assumption 1 (positive density). But this does not affect the stationarity proof. Let us check Assumption (9). We have: $R_{0}(t)=g+Z(t)$. Then we can rewrite the left-hand side of Assumption 23:

(33)

$\displaystyle\begin{split}\mathbb{E}&\left[\ln|\xi|\right]<0,\quad\xi:=1+a+bR_{0}(t)\sim\mathcal{N}(\mu,\rho^{2}),\\ \mu&:=1+a+bg,\quad\rho:=|b|\sigma.\end{split}$

It is impossible to compute this logarithmic moment directly. But one can do this numerically using Monte Carlo. The set

$$\left\{(\mu,\rho)\in[0,\infty)^{2}:\quad\mathbb{E}\left[\ln|\xi|\right]<0,\quad\xi\sim\mathcal{N}(\mu,\rho^{2})\right\}$$

is shown in Figure 4. Python code is on GitHub repository asarantsev/size-capm-vix in file gaussian-simulation.py

Consider the benchmark and $N$ portfolios $1,\ldots,N$. We model each pair (benchmark, portfolio $k$) with this model. The model must be the same for all $k$. We let $S_{k}(t)$ be the market capitalization (size) for the $k$th portfolio, and $S_{0}(t)$ the market size of the benchmark. We define market weights as

(34)

$$\mu_{k}(t)=\frac{S_{k}(t)}{S_{0}(t)+S_{1}(t)+\ldots+S_{N}(t)},\quad k=0,1,\ldots,N.$$

We can also rank them from top to bottom:

(35)

$$\mu_{(0)}(t)\geq\ldots\geq\mu_{(N)}(t).$$

Market weights and portfolios based on them are the main topic of Stochastic Portfolio Theory in both discrete time, see [6, 21, 28] and continuous time, see [11, 12].

Define by $\delta_{k}$ and $\varepsilon_{k}$ the sequences of innovations for equity premium, as in (18), and price returns, as in (19), for the $k$th portfolio, $k=1,\ldots,N$:

(36)

$\displaystyle\begin{split}\frac{P_{k}(t)}{V(t)}&=M+AC_{k}(t)+(1+BC_{k}(t))\frac{P_{0}(t)}{V(t)}+\delta_{k}(t);\\ \frac{R_{k}(t)}{V(t)}&=m+aC_{k}(t)+(1+bC_{k}(t))\frac{R_{0}(t)}{V(t)}+\varepsilon_{k}(t);\\ R_{k}(t)&=\ln\frac{S_{k}(t)}{S_{k}(t-1)},\quad C_{k}(t)=\ln\frac{S_{k}(t)}{S_{0}(t)}.\end{split}$

Together with (12), (16), (17), this is a time series Markov model for $2N+3$ series.

This is the main stability result. It has the meaning that if we have several portfolios, they stay in the long run as one cloud, and do not split into several clouds.

Thus the ranked market weights process also has a stationary distribution and converges to this distribution in the long run, regardless of the initial conditions. In this stationary distribution, we can plot these ranked market weights versus their ranks on the log scale:

$$\left((\ln(n+1),\ln\mu_{(n)}(t)),\quad n=0,\ldots,N\right)$$

This plot is called the capital distribution curve. With real-world markets, this curve is linear on most span, and concave overall. Moreover, it shows remarkable long-term stability. See the famous picture in [11, Chapter 4] for eight capital dsitribution curves at end of years 1929, 1939, …, 1999; see the same picture as [12, Figure 13.4].

In our previous article [13], we captured the observation that well-diversified portfolios of small stocks have higher risk but higher return than that of large stocks. We reproduced this shape of capital distribution curve in [13] using simulation.

From (37) and (34), we get a simple expression of $\mu_{k}(t)$ from $C_{k}(t)$:

$$\ln\mu_{k}(t)=C_{k}(t)+\ln S_{0}(t)-\ln(S_{0}(t)+\ldots+S_{N}(t)),\,k=1,\ldots,N.$$

Therefore, ranking market weights $\mu_{k}(t),k=1,\ldots,N$ from top to bottom at any fixed time $t$ is equivalent to ranking relative size terms $C_{k}(t),k=1,\ldots,N$ from top to bottom: $C_{(1)}(t)\geq\ldots\geq C_{(N)}(t)$. Thus instead of plotting the (slightly modified) capital distribution curve $(\ln k,\ln\mu_{(k)}(t)),k=1,\ldots,N$, we can plot $(\ln k,C_{(k)}(t)),k=1,\ldots,N$.

The linearity of the capital distribution curve was rigorously proved in [7] for competing Brownian particles and disproved in [20] for volatility-stabilized models. These two types of continuous-time models both capture the property that small stocks have higher risk but higher return than large stocks. But these models do not use CAPM.

Instead of ranking all market weights, we can rank only weights of all portfolios but the benchmark. This technique will miss one portfolio from the capital distribution curve, but will not influence the overall behavior of the said curve. Consider the combined model:

• •

  Volatility $V$ governed by (12), with innovations $W$ having the variance-gamma distribution as in [22, subsection 3.3]

• •

  Benchmark price returns $R_{0}$ governed as (16) with Gaussian $Z$ having mean and standard deviation as in [22, Table 2, Large Price]

• •

  $N$ portfolios with relative size $C_{k}$ as in (37), $k=1,\ldots,N$, each governed by (22) for $a=m=0$ and negative $b$ to be chosen, with innovations $\varepsilon_{k}(t)\sim\mathcal{N}(0,1)$ independent for each $k$

For simplicity, we write all equations with number coefficients, replacing $-b$ by $c$:

	$\displaystyle\ln V(t)$	$\displaystyle=0.346+0.882\ln V(t-1)+W(t),$
	$\displaystyle W(t)$	$\displaystyle=0.0621+0.0621\Gamma(t)+0.1392\sqrt{\Gamma(t)}Y(t),$
	$\displaystyle Y(t)$	$\displaystyle\sim\mathcal{N}(0,1),\quad\Gamma(t)\ \mbox{is Gamma with shape}\ 1/0.6573,\ \mathbb{E}[\Gamma(t)]=1,$
	$\displaystyle R_{0}(t)$	$\displaystyle=V(t)Z(t),\quad Z(t)\sim\mathcal{N}(0.062,0.202^{2}),\quad\mathrm{Corr}(Y(t),Z(t))=\rho,$
	$\displaystyle C_{k}(t+1)$	$\displaystyle=C_{k}(t)(1-cR_{0}(t))+V(t)\varepsilon_{k}(t),\quad\varepsilon_{k}(t)\sim\mathcal{N}(0,1),\quad k=1,\ldots,N,$
		$\displaystyle(Y(t),Z(t)),\Gamma(t),\varepsilon_{1}(t),\ldots,\varepsilon_{N}(t)\ \mbox{independent for fixed}\ t$
		$\displaystyle\bigl{(}Y(t),Z(t),\Gamma(t),\varepsilon_{1}(t),\ldots,\varepsilon_{N}(t)\bigr{)},\ t=1,2,\ldots\quad\mbox{IID}.$

We pick $N=100$ and simulate these equations for $T=400$ time steps, starting from

$$\ln V(0)=3,\quad C_{k}(0)=0,\,k=1,\ldots,N.$$

We do this for $t=T$, to give enough time to converge to the stationary distribution. We see various slopes for different cases: $\rho=0$ or $\rho=-50\%$ (this latter correlation is found in [22, Table 2, Large Price]); and $c=0.1$ or $c=0.2$ (lower and upper bound for $-b$ found in Subsection 3.2). We perform two simulations for each case. Each simulation has the same values of $V$ and $R_{0}$. We show all four curves for each simulation in Figure 5. See the Python code in the file capital-distribution-simulation.png from the GitHub repository asarantsev/size-capm-vix

Fix a constant $k=1,2,\ldots$. Assume the market weights $\mu_{n}$, or, equivalently, relative size $C_{n}$ are in the stationary distribution, which (by Theorem 2) is limiting distribution. To stress this, we write $t=\infty$ for time argument. Thus we have the ranked (sorted, ordered from top to bottom) values of the relative size process in the stationary distribution:

$$C_{(1)}(\infty)>\ldots>C_{(N)}(\infty).$$

Here, $N$ is the overall number of portfolios (excluding the benchmark). We are interested in the joint distribution of these sorted relative size values.

Assume $\mathcal{Z}_{1},\ldots,\mathcal{Z}_{N}\sim\mathcal{N}(0,1)$ IID is the standard normal sample.