A Stock Market Model Based on CAPM and Market Size
RRANK-BASED CAPITAL ASSET PRICING MODEL
KWAME BOAMAH-ADDO, BRANDON FLORES, JAKOB LOVATO, ANDREY SARANTSEV
Abstract.
A rank-based model of competing Brownian particles, introduced in (Banner,Fernholz, Karatzas, 2005), captures the dependence of stock dynamics on size: Small stockshave higher growth rate but higher volatility than large stocks, on average. However, in thismodel, geometric returns of stocks have Gaussian tails. We modify this model using CapitalAsset Pricing Model: The market exposure depends on the rank, and the benchmark movesas a general L´evy process. We show that this model fits real market data. We discussstability and long-term convergence. Introduction
In this article, we modify the model of competing Brownian particles with rank-based driftand diffusion coefficients, introduced in [1] to capture the size effect in the stock market. Thesize of a stock is measured by its market capitalization , or market cap ; that is, total marketvalue, which is computed by multiplying the current market price by the total number ofoutstanding stocks. Quoting the introduction of [1]:Size is one of the most important descriptive characteristics of assets: one canunderstand a great deal about an equity market by observing, and makingsense of, the continual ebb and flow of small-, medium- and large-capitalizationstocks in its midst. Thus it is important to have models which describe (ifnot explain) this flow, and which exhibit stability properties for the resultingdistribution of capital that are in agreement with actual observation.1.1.
Competing Brownian particles.
This model stipulates that the logarithms of stockcapitalizations move as independent Brownian motions with drift and diffusion coefficientsdepending on their current rank relative to other stocks. For example, the top-ranked stockmoves (on the logarithmic scale) as a Brownian motion with drift g and diffusion σ , thesecond-ranked stock moves as a Brownian motion with drift g and diffusion σ , and soon. In general, logarithms of market caps X ( t ) , . . . , X N ( t ) satisfy the following system ofstochastic differential equations:(1) d X i ( t ) = N (cid:88) k =1 X i has rank k at time t )( g k d t + σ k d W i ( t )) , i = 1 , . . . , N, where W , . . . , W N are independent Brownian motions. This is sometimes called a model of competing Brownian particles . This model is interesting both from a theoretical probabilisticviewpoint and its financial application. Let us briefly describe the former and the latter.In the model (1), distances between adjacent particles (the gap process) form an obliquelyreflected Brownian motion in the orthant. Using this, one can prove long-term stabilityas t → ∞ if g < . . . < g N . This means that particles travel more or less close to eachother, instead of splitting into two or more groups, and the gap process has the stationarydistribution. One can estimate convergence rate to this distribution as t → ∞ . As the a r X i v : . [ q -f i n . S T ] A p r KWAME BOAMAH-ADDO, BRANDON FLORES, JAKOB LOVATO, ANDREY SARANTSEV number of particles tends to infinity, we have Law of Large Numbers [23], Central LimitTheorem [24], propagation of chaos [28], and Large Deviations Principle [13]. Remarkableresults can be achieved using time and space scaling, [10, 12]. It was extensively studied in[2, 27]. An important generalization is for L´evy processes instead of W i , [32, 3].1.2. L´evy processes.
A L´evy process L = ( L ( t ) , t ≥
0) is defined by the following prop-erties: L ( t ) − L ( s ) is independent of L ( u ) , ≤ u ≤ s and has distribution dependingonly on t − s , for s < t . One particular example is a Brownian motion W ; in fact, L ( t ) = L (0) + gt + σW ( t ) for real numbers g and σ is the only example of a L´evy pro-cess with continuous trajectories, all other L´evy processes have jumps. If a L´evy process hasonly a.s. finitely many jumps on [0 , t ], we can represent it as a sum of a Brownian motionwith drift and a compound Poisson process:(2) L ( t ) = L (0) + gt + σW ( t ) + Z + . . . + Z N ( t ) , t ≥ , where N = ( N ( t ) , t ≥
0) is a Poisson process with intensity λ , independent of the Brownianmotion W , and Z , Z , . . . i.i.d. independent of W and N . So far in generalizing themodel (1) we used only L´evy processes of the type (2). However, there exist L´evy processeswhich have infinitely many jumps on a finite time horizon, for example stable processes E exp[ iξL ( t )] = exp( − t | ξ | α ) , for α ∈ (0 , , t > , ξ ∈ R . A general L´evy process has a spectral measure ν on R : This measure can be infinite (in caseit is not a process of type (2)), but ν ( R \ [ − ε, ε ]) < ∞ for every ε >
0. If E | L ( t ) | < ∞ , then E exp[ iξL ( t )] = exp (cid:20) − t (cid:90) R ( e iξu − ν (d u ) (cid:21) , t > , ξ ∈ R . L´evy processes are widely used in financial modeling, because they capture a well-known phe-nomenon of heavy tails: Fluctuations of the stock market have tails heavier than predictedby the normal distribution. See, for example, [5].1.3.
Stochastic Portfolio Theory.
For financial applications, relations with StochasticPortfolio Theory (SPT) were studied in [1, 2, 21, 26]. SPT developed the basic insightthat overweighting small stocks and periodically rebalancing portfolio can outperform themarket; [19, 20]. Of particular importance is the capital distribution curve , constructed asfollows. Let S i ( t ) be the market cap of the i th stock at time t , for i = 1 , . . . , N . Then S ( t ) = S ( t ) + . . . + S N ( t ) is the total market cap of our stock market, consisting of N stocks. The market weight of the i th stock is then defined as µ i ( t ) = S i ( t ) S ( t ) , i = 1 , . . . , N. Rank these market weights from top to bottom: µ (1) ( t ) ≥ . . . ≥ µ ( N ) ( t ). The plot(ln n, ln µ ( n ) ( t )) , n = 1 , . . . , N, is called the capital distribution curve . Historically, this curve showed remarkable linearlityand stability over time: CRSP data on December 1929, 1939, . . . , 1999 shows that these 8curves are almost linear, and almost coincide, except at their ends, [19, Chapter 5]. Stabilityof this curve follows from stability of the original system (1) or any modification. Linearitywas proved for competing Brownian particles market model in [11] and shown via simulationsfor a generalized models with L´evy processes in [3]. ANK-BASED CAPITAL ASSET PRICING MODEL 3
Total return and equity premium.
We note that the market cap of a stock is directlyrelated to returns. Indeed, total return is composed of price increase (capital appreciation)and dividends (distributions paid). This distinguishes it from price return , which ignoresdividends and focuses only on price increase. One can adjust the stock price for thesedividends. If one uses this adjusted price for computing the market cap, and ignores changesin the number of shares outstanding (due to buybacks by insiders and new issues), then thechange in market cap C ( t ) leads to total return Q ( s, t ) = ln( C ( t ) /C ( s )) = ln C ( t ) − ln C ( s )from time s to time t > s . We always take geometric rather than arithmetic returns. Forcontinuous time, we can consider the flow of returns d C ( t ). If C is a semimartingale, this isunderstood in the sense of Itˆo calculus.One can adjust this for risk-free rates. Indeed, an investment in stocks returning 10%is not successful if risk-free Treasury bills return 12%. Subtract risk-free returns R (of 1-month or 3-month Treasury bills, money market funds, or another similar instrument) fromtotal stock returns Q . This gives us equity premium P = Q − R , or excess return of riskyinvestments (stocks) over risk-free investments (Treasury bills). This is a reward for risk.Adjust capitalization: Divide it by the cumulative wealth V ( t ) from risk-free investments,if at time t = 0 we invest V (0) = 1. Then R = d V ( t ). This is equivalent to subtractingrisk-free returns from total return and getting equity premium. S ( t ) = C ( t ) V ( t ) ⇒ d ln S ( t ) = d ln C ( t ) − d ln V ( t ) = Q − R = P. However, this model has a significant disadvantage: It implies that returns of individualstocks have sub-Gaussian tails. This contradicts the real observed stock returns, whosedistributions have fat tails, at least in the short run. We modify the model (1) to matchCapital Asset Pricing Model (CAPM).1.5.
Capital Asset Pricing Model.
This classic financial theory states that the only riskwhich matters for pricing is a systematic risk : The β which is the regression coefficient ofits equity premium P upon that of the overall market (the benchmark, usually taken to beStandard & Poor 500 Large-Cap Index or a corresponding index fund) P . Thus(3) P = βP + ε. Non-systematic risk (idiosyncratic, caused by individual events pertaining to this stock)corresponds to ε . Capital Asset Pricing Model states that this risk cannot lead to higherprices, because it can be eliminated by diversifying: Choosing a portfolio of various stocksfrom different industries. Thus β is the only important risk measure. Empirical evidence doesnot support this strong claim, see [16, 18]. Still, CAPM in (3) remains a useful benckmarkmodel. In the article, we give examples of size-based exchange-traded funds which follow (3).1.6. The new model.
We modify (1) to take rank-based β = β k instead of rank-baseddrifts and diffusions g k and σ k from (1). Thus we, in a sense, merge (1) and (3). The ε ( t )term becomes a Brownian motion. The benchmark is taken to be a geometric L´evy process L ( t ): That is, ln S = L is a L´evy process, independent of Brownian motions W , . . . , W N (which can be correlated between each other). Since the overall stock market grows in thelong run, we assume that E [ L ( t ) − L (0)] > KWAME BOAMAH-ADDO, BRANDON FLORES, JAKOB LOVATO, ANDREY SARANTSEV
Organization of the article.
In Section 2, we formally construct this system, andprove its existence and uniqueness. In Section 3, we show that this model fits real mar-ket data for exchange-traded funds. We provide Python code and Excel data on GitHub: github.com/asarantsev/SizeEffect.
In Section 4, we prove stability of the system if β < . . . < β N . Capital distribution curve stability follows from the stability of the system.Section 5 outlines possible future research.2. Notation and Definitions
We introduce a few pieces of notation. The Dirac delta measure at point b is denotedby δ b . For a vector x = ( x , . . . , x N ) ∈ R N , its ranking permutation p x for the vector x isdefined as the unique permutation on { , . . . , N } such that: • x p x ( i ) ≥ x p x ( j ) for i < j (this permutation is ranking the vector x ). • if i < j and x p x ( i ) = x p x ( j ) , then p x ( i ) < p x ( j ) (ties resolved in lexicographic order).Thus, we rank from top to bottom. The dot product of two vectors a = ( a , . . . , a N ) and b = ( b , . . . , b N ) in R N is defined as a · b = a b + . . . + a d b d . The Euclidean norm of a ∈ R N is defined as (cid:107) a (cid:107) := [ a · a ] / . For any vector x = ( x , . . . , x N ) ∈ R N , we define two othervectors: running average ˜ x ∈ R N , and centered vector x ∈ R N :(4) ˜ x k := 1 k ( x + . . . + x k ); x k := x k − ˜ x N , k = 1 , . . . , N. Fix β , . . . , β N ∈ R , and a N × N symmetric positive definite matrix Σ . We operate on afiltered probability space (Ω , F , ( F t ) t ≥ , P ), where the filtration satisfies the usual conditions.All processes in this article are ( F t ) t ≥ -adapted, their trajectories are almost surely right-continuous with left limits.Take a L´evy benchmark process L = ( L ( t ) , t ≥ ≤ s < t , the increment L ( t ) − L ( s ) is independent of F s and its distribution dependsonly on t − s . We assume it has a.s. finitely many jumps on [0 , t ] for every t >
0. Thus ithas representation as a sum of a drift, a Brownian motion, and a compound Poisson processin (2). There exists a modification of this process which is right-continuous with left limits.Take an N -dimensional Brownian motion W = ( W , . . . , W N ) with covariance matrix Σ ,independent of L , such that W ( t ) − W ( s ) is independent of F s for 0 ≤ s < t . Take N processes X i = ( X i ( t ) , t ≥ , i = 1 , . . . , N . Rank them at each time t : X ( k ) ( t ) = X p X ( t ) ( k ) ( t ) , k = 1 , . . . , N ; X (1) ( t ) ≥ . . . ≥ X ( N ) ( t ) . If they satisfy a system of stochastic differential equations:(5) d X i ( t ) = N (cid:88) k =1 p X ( t ) ( k ) = i ) ( β k d L ( t ) + d W k ( t )) , i = 1 , . . . , N, t ≥ . Our market model is defined as follows: market caps of the benchmark (usually taken to beStandard & Poor Composite Index) and of the i th stock at time t ≥ S i ( t ) := e X i ( t ) , i = 1 , . . . , N ; B ( t ) = e L ( t ) . As discussed in the Introduction, d X i ( t ) is total return (or equity premium, depending on thesetup) of the i th stock, and d L ( t ) is the total return (or equity premium) of the benchmark. ANK-BASED CAPITAL ASSET PRICING MODEL 5
The total market cap and market weights are defined as(7) S ( t ) := S ( t ) + . . . + S N ( t ) , µ i ( t ) = S i ( t ) S ( t ) , t ≥ . One can think of d ln S ( t ) as the total return (or equity premium) for the market portfolio,where we buy a certain proportion of the total stock market; or, equivalently, we invest ineach stock in proportion to its market weight. The gap process is defined as(8) Z ( t ) = (cid:0) X (1) − X (2) ( t ) , . . . , X ( N − − X ( N ) ( t ) (cid:1) , t ≥ . The centered system is defined as X ( t ) = ( X ( t ) , . . . , X N ( t )) using (4). The vector of rankedmarket weights is µ () := (cid:0) µ (1) , . . . , µ ( N ) (cid:1) . The state spaces of X, Z, µ, µ () , respectively, are X ( t ) ∈ Π := { x ∈ R N | x + . . . + x N = 0 } , Z ( t ) ∈ [0 , ∞ ) N − ,µ ( t ) ∈ ∆ := { x ∈ [0 , ∞ ) N | x + . . . + x N = 1 } , µ () ( t ) ∈ { x ∈ ∆ | x ≥ . . . ≥ x N } . Real Market Data
Deciles.
We consider annual total (nominal) returns from Ibbotson Yearbook 1926–2015 [14]. Take Treasury short-term bill returns and returns of stock market deciles. At thebeginning of each year, split all stocks into the top 10%, next 10%, etc. up to bottom 10%,by market capitalization. For each decile, construct a capitalization-weighted portfolio ofthese stocks. That is, invest in each stock in proportion to its market cap. Let P k ( t ) be theequity premium for the k th decile in the t th year. For k = 2 , . . . ,
10, we regress P k ( t ) = α k + β k P ( t ) + ε k ( t ) , ε k ( t ) ∼ N (0 , σ k ) i.i.d.For each k = 2 , . . . ,
10, the residuals ε k ( t ) fit the normal distribution, judging by the quantile-quantile plots and standard normality tests (Shapiro-Wilk and Jarque-Bera p -values are allgreater than 5%). Confidence intervals for α k all contain 0; that is, we cannot reject thehypothesis that α k = 0 for p = 5%. But confidence intervals for β k are all contained in(1 , ∞ ). Thus β k > p = 5%. Remark . Note that we do not see exchanges of ranks here: Bottom-decile particle alwaysstays at the bottom, and the rank of every particle is constant. Indeed, there is rebalancingat regular time intervals: Stocks in k th decile at the beginning of a time period might moveto another decile by the end of this period. Thus the k th decile at the beginning of the nextperiod will contain different stocks. Remark . We fit this model for size-based stock portfolios, not for individual stocks, whichwe think it impossible: Individual stocks’ fluctuations have too much noise.Here we provide a table for values of β k , σ k , k = 2 , . . . ,
10, as well as the correlationmatrix. We see that all residuals are significantly correlated (according to Pearson t -test),thus we cannot assume that in our model (5), driving Brownian motions are independent.This is in contrast to the original model (1). k β k σ k KWAME BOAMAH-ADDO, BRANDON FLORES, JAKOB LOVATO, ANDREY SARANTSEV
Slope coefficient (market exposure), standard error, and residuals correlation matrix.3.2.
Size-Based Index Funds.
Take 4 BlackRock iShares index exchange-traded funds:
IVV
Large-Cap Standard & Poor 500,
IJH
Mid-Cap Standard & Poor 400,
IJR
Small-CapStandard & Poor 600, and
IWV
Russell 3000 Total Stock Market Index. Time range: June2000 – February 2020 total monthly returns. Risk-free monthly returns are determined by3-month Treasury bill rate. Regress equity premia P L ( t ), P M ( t ) and P S ( t ) of the large-cap, mid-cap, and small-cap fund upon the equity premia P ( t ) of the total stock marketfund. Here, t is the current month. Residuals for the small cap do not pass normality test.Residuals for the mid cap do pass it, albeit barely. Residuals for the large cap also pass it.Remarks 1, 2 apply here as well. The correlation matrix is . − . − . − .
79 1 .
00 0 . − .
74 0 .
74 1 . Total Stock Market Benchmark.
In this section, we show that real-world equitypremium for a broad index composed of Center for Research in Securities Prices large-capitalization stocks 1926–2015, found in [14], can be modeled by a L´evy process withfinitely many jumps on a finite time horizon, whether we move in steps of a month or a year.First, define the hyperbolic distribution: it has Lebesgue density on the real line f ( x ; α, β, δ, µ ) = γ αδK ( γδ ) exp (cid:104) − α (cid:112) δ + ( x − µ ) + β ( x − µ ) (cid:105) , γ := (cid:112) α − β , where K is the modified Bessel function of the second kind. Monthly time steps.
Take monthly index (total nominal) returns, obtained from [14], 1926–2015, Appendix A1. Subtract risk-free returns, measured by 3 month secondary market rate,obtained from Federal Reserve Economic Data web site, 1935–2015. Thus we have monthlyequity premia for 81 years, total 81 ·
12 = 972 data points. We obtain estimates α = 40 . , β = − . , δ = 0 . , µ = 1 . . Two-sample Kolmogorov-Smirnov test gives us p = 0 . Annual time steps.
Take annual index (total nominal) returns and subtract risk-freereturns, measured by annual total return on short-term government bonds. Both data seriesare obtained from [14], years 1926–2015, 90 data points. We obtain estimates α = 24 . , β = − . , δ = 0 . , µ = 0 . . Two-sample Kolomorogov-Smirnov test gives us p = 0 .
95. This indicates that the fit is good.The empirical vs theoretical density and the quantile-quantile plot are shown in Figure 1.
ANK-BASED CAPITAL ASSET PRICING MODEL 7 (a)
Densities (b)
QQ plot
Figure 1.
Goodness-of-fit of the hyperbolic law for the annual equity pre-mium, 1926–2015Thus both on time horizons of a month and a year this hyperbolic distribution fits theequity premium data for large stocks well. This distribution is infinitely divisible, as shownin [6]. Thus we can create a corresponding L´evy process L = ( L ( t ) , t ≥ , t ]. Indeed, for the spectral measure ν of this process (infinite but with finite moments),and drift g and diffusion σ , we have:(9) E [exp( iξL ( t ))] = exp (cid:20) tgξ + tσ ξ t (cid:90) R ( e iuξ − ν (d u ) (cid:21) , ξ ∈ R , t > . We can take finite measures ν n defined as restrictions of ν on R \ [ − n − , n − ]: ν n ( B ) := ν ( B \ [ − n − , n − ]) , B ⊆ R . This is equivalent to removing all jumps of size less than or equal to 1 /n from the trajectoryof L . Replace in (9) ν with ν n and L with L n :(10) E [exp( iξL n ( t ))] = exp (cid:20) tgξ + tσ ξ t (cid:90) R ( e iuξ − ν n (d u ) (cid:21) , ξ ∈ R , t > . As n → ∞ , ν n → ν weakly. Therefore, the right-hand side of (10) converges to the right-hand side of (9), and thus L n ( t ) → L ( t ) weakly. We have proved that we can find g , σ ,and a finite measure ν n on R (can be decomposed as ν n = λQ , where λ > Q is aprobability measure on R ) such that the theoretical distribution of L (1) for Z k ∼ Q fits themarket data. Thus we can fit the data into the model (5) with L given by (2).4. Main Results and Proofs
Theorem 1.
The system of equations (5) has a unique in law weak solution for any giveninitial condition X (0) = x ∈ R N .Proof. This follows from the results of [3], since our system (5) is a particular case of suchsystem from [3, Definition 4, Remark 1]. (Note that in [3] we rank particles from bottom
KWAME BOAMAH-ADDO, BRANDON FLORES, JAKOB LOVATO, ANDREY SARANTSEV to top and here in the other order.) Namely, we have: Jump measure Λ from [3, (16)] issupported on the line (cid:96) := { ( x , . . . , x n ) | R N | x i = β i y, y ∈ R } , Λ = λ · Q ◦ π − , π : R → (cid:96), π ( β y, . . . , β N y ) = y. (11)Using (2), we can represent the drifts g k from [3, (12)] as g k := β k g , k = 1 , . . . , N , andthe covariance matrix for the Brownian motions W , . . . , W N from [3, Remark 1] can becomputed as follows: W k from [3, Remark 1] is equal to β k σW + W k from (5), with W being the Brownian motion in the decomposition (2), independent of W k . Thus the kl thelement of the covariance matrix is equal to β k β l σ + Σ kl , k, l = 1 , . . . , N. This is a nonsingular matrix. Thus we are under assumptions of [3], and weak existence withuniqueness in law holds. The method of the proof is piecing out , described in [22, 7, 9]. (cid:3)
Now impose stability assumptions: The spectral measure ν = λQ of L satisfies(12) (cid:90) + ∞−∞ x Q (d x ) < ∞ . Since ν is finite, the first moment of ν is also finite. We impose another assumption:(13) g + λm > , m := (cid:90) + ∞−∞ x Q (d x ) > . Assumption (13) is equivalent to E [ L ( t ) − L (0)] > t > β , . . . , β N (14) β > . . . > β N > . Take a continuous-time Markov process X = ( X ( t ) , t ≥
0) on a state space S withtransition kernel P t ( x, · ). A stationary distribution, or invariant measure, is any probabilitymeasure Π on S such that X (0) ∼ Π implies X ( t ) ∼ Π for every t ≥
0. This process is called ergodic if it has a unique stationary distribution Π, and(15) sup E (cid:12)(cid:12) P t ( x, E ) − Π( E ) (cid:12)(cid:12) → t → ∞ . Theorem 2.
Under assumptions (12) , (13) , and (14) , the gap process Z , the centered process X , the process of market weights µ ( · ) and the process of ranked market weights are ergodicon their respective state spaces.Proof. We again use [3]. Let us check assumptions of [3, Theorem 1]. Condition [3, (15)]follows from (11) and (12). Compute the first moment of the jump measure for the k thcomponent: This is λ (cid:90) R β k u Q (d u ) = λβ k m . ANK-BASED CAPITAL ASSET PRICING MODEL 9
Thus the implied drifts m k from [3] are equal to m k := β k g + β k λm , k = 1 , . . . , N , and thecorresponding running averages ˜ m k are equal to˜ m k := 1 k ( m + . . . + m k ) = ˜ β k ( g + λm ) , ˜ β k := 1 k ( β + . . . + β k ) , k = 1 , . . . , N. (16)Condition [3, (13)] takes the form(17) ˜ m k < ˜ m N , k = 1 , . . . , N − . We invert the sign compared to [3] because our ranking is in the reverse order. Condition (17)follows from (13), (14), (16). (cid:3) Possible future research
To find theoretically, as in [11], or simulate, as in [3], the capital distribution curve for alarge number N of stocks, we need some assumptions on the correlations between drivingBrownian motions W k from (5). From the real-world data, it is not clear what such assump-tions would be. For example, as noted in Section 3, sometimes these correlations are positiveand close to 1, and sometimes they are negative, and it is hard to find patterns.Another possible future research topic is adding intercept α to (3), and thus generalizeour modification (5) of (1) further. This α is excess risk-adjusted return: The extra return(or loss, if it is negative) after you adjust for market exposure β .Finally, in cases when the regression residuals in (3) are not normal, one could try to fita heavy-tailed distribution for them. Then we can generalize (5) for the case when not only L , but also W k are L´evy processes. References [1]
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