2 Asset returns
Statistical analysis of asset returns is fundamental in financial econometrics, quanti- tative investment and risk management. The aim of this section is twofold. First, we explain briefly the meaning of “price” from which returns are derived; this leads to the notion of limit order book (LOB) which is the basis of most electronic exchanges. Second, we study some mathematical properties of simple and log returns. In Sec- tion 3, we will consider (from the viewpoint of exploratory data analysis) statistical properties of stock returns and illustrate, with empirical data, some well-documented stylized facts of asset returns.
2.1 Prices in an electronic exchange
Many, but not all, financial instruments are traded on electronic exchanges.1 Here are some of the biggest stock exchanges (in terms of the market capitalization of the listed stocks):
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Stock Exchange Nasdaq Stock Exchange
Shanghai Stock Exchange Euronext
Japan Exchange Group
Most exchanges adopt some kind of double auction and a matching algorithm to match buyers and seller, resulting in a limit order book (LOB). The actual operation of the LOB is very complex (and differs from exchange from exchange); here we only describe the basic mechanism. Some material is taken from Chapter 1 of Cartea, Jaimungal and Penalva (2015) which you can read for further details.
As suggested by the name, the state of the LOB is described by a collection of limit orders. A limit order is a type of order to purchase or sell a security at a specified price or better, e.g., “buy 100 share of a stock if the price is $50.1 or lower”. A buy order is also called a bid order; a sell order is called an ask order. The other basic order is market order (buy or sell). By definition, a buy market order (say) leads to an immediate trade (provided that there are sufficient limit orders currently available) to buy the at the best given price. If the order is not filled completely by the limit orders at the given price, it continues with the next available (worse) price and so on; this is called “walking the book”. Thus a large trade can exhibit price impact. Consequently, it is usual to break up a large trade (e.g. sell 10000 shares) into many small orders. As time progresses, new orders are posted (or withdrawn) and trades occur. For a liquid stock, the LOB evolves rapidly throughout the day.
In Figure 2.1 we show a snapshot of the limit order book of the stock Amazon on 2012-06-21. This is sample data from the platform LOBSTER which provides limit order book data (see https://lobsterdata.com/info/DataSamples.php). The y- axis (volume) shows the sum of the sizes of all limit orders (bid/ask) at that price. It is also called the depth. Thus, at a given moment, there is a range of prices corresponding to the bid/ask orders. Naturally, the bid prices are less then the ask prices (otherwise the orders matched immediately). The gap between the largest bid price and the smallest ask price is called the bid-ask spread. In the figure, the bid-ask spread is a bit larger than $0.1. The spread is a measure of the liquidity of the stock.
Within a trading day, there are multiple notions of price. Two useful concepts are: The price of the last transaction.
1Certain instruments, such as exotic derivatives, are traded over-the-counter (OTC). 7
LOB Volume for AMZN at 56732.739350707 ms
220.9 221.0 221.1 221.2 221.3
Figure 2.1: Snapshot of the LOB for Amazon on 2012-06-21.
The midprice which is defined by
Midpricet = 12(Pta + Ptb),
where Pta and Ptb are respectively the (best) ask and bid prices at time t. It may serve as a proxy of the “true underlying price”.
For analysis at lower frequencies (daily or monthly), one typically uses the closing price which, by definition, is the last transaction price of the (last) trading day (of the period). Other choices (such as opening price or price at a specific time) may be used as well. In this course, we use the closing price.
Actual limit order books implemented by exchanges are much more complex. For example, there exist many other order types such as:
Immediate-or-Cancel Fill-or-Kill
Also, an asset (such as a stock) may be listed on multiple exchanges, and orders that are not filled in the given exchange are routed to other exchanges to ensure that the transaction is executed at the best price.
2.2 Returns
2.2.1 Simple and log returns
In financial analysis, it is common to focus on the returns. Returns provide a scale- free (i.e., independent of the size of the asset) summary of the performance of the investment, allowing easy comparison across different assets. Also, returns usually have better statistical properties when compared to the original price processes. Let Pt ≥ 0 denote the price or value of an asset at time t.2 For concreteness, we think of
2Depending on the situation, we also use the notation P(t).
0 100 200 300 400 500 600
S&P500 weekly simple return
2000 2005 2010 2015 2020
Figure 2.2: Time series of weekly simple return of the S&P500 from January 2020 to December 2021.
it as a stock or a portfolio. Over a holding period [t0,t1], the simple return is defined
R(t0,t1)=Pt1 −1=Pt1 −Pt0, Pt0 Pt0
which is the percentage change of the price. The quantity 1 + R(t0, t1) = Pt1 is often Pt0
called the gross return. When time is indexed by an integer (day, week, month, etc.),
Rt= Pt −1 Pt−1
be the simple return over [t − 1, t]. Since the price is non-negative, the minimum value of the simple return is −1. In Figure 2.2 we plot the time series of weekly simple returns of the S&P500 (using the data set in Section 1.2). Statistical modeling of this and similar return series will be studied in later sections.
When the price remains strictly positive, we define the log return by r(t0,t1)=log(1+R(t0,t1))=logPt1 −logPt0.
Rearranging, we have Pt1 = Pt0 er(t0,t1). Similarly, we write rt = log(1 + Rt) if t is an integral time index.
The logarithm log(·) satisfies the inequality log(1 + x) ≤ x for x > −1. Thus we always have
r(t0, t1) ≤ R(t0, t1). Using the second order Taylor approximation
log(1+x)=x−12×2+O(|x|3), |x|→0, when |R(t0, t1)| is small we have
r(t0,t1)≈R(t0,t1)− 21R(t0,t1)2.
In Figure 2.3 we illustrate the inequality (2.1) and the approximation (2.2). When
the time interval is short, the simple and log returns are very close to each other. 9
−0.5 0.0 0.5
Simple return
Figure 2.3: Comparison of the simple and log returns. The blue thick curve shows the log return r as a function of the simple return R; it always holds that r ≤ R. The black dashed line is the identity. The blue thin dotted curve shows the approximation (2.2).
Remark 2.1 (Adjusting for dividends). If the asset pays a dividend, say Dt, in the time interval [t − 1, t], we define the simple and log returns by
Rt = Pt +Dt −1, rt =log(1+Rt). Pt−1
2.2.2 Compounding
Frequently, returns over a longer period are computed by compounding the returns over shorter subperiods (e.g. corresponding to the times at which the portfolio is rebalanced). Here, we see that the simple and log returns behave differently. Consider time points t0 < t1 < · · · < tk . For the simple return, we have the identity
1 + R(t0, tk) = (1 + R(ti−1, ti)).
Taking logarithm, we see that the log return is additive over time:
r(t0, tk) =
r(ti−1, ti).
For this reason, it is more common to use the log return when studying long-term portfolio growth.
From the stochastic modeling perspective, additivity is a convenient property. For
example, suppose we model the log returns over disjoint periods as independent normal
random variables, say r(ti−1, ti) ∼ N(γti−1,ti , σt2 ,t ). Then by the independence
assumption we have where
i−1 i r(t0,tk)∼N(γt0,tk,σt2,t ),
μt0,tk = μti−1,ti, σt2,t = σt2 ,t.
0 k i−1 i i=1
Log return
This is not true for the simple return. Suppose further that γti−1,ti and σt2 ,t i−1 i
proportional to the length of the time interval [ti−1,ti], i.e., γti−1,ti = γ(ti −ti−1) and
σ2 =σ2(t −t )forsomeγ∈Randσ>0. Thenthepriceprocess(P)isa
ti−1,ti i i−1 t geometric Brownian motion and Pt is log-normally distributed at each time t.
Remark 2.2. Using the language of stochastic calculus, the geometric Brownian motion can be expressed in the form
dlogPt =γdt+σdWt ⇔dPt =μPtdt+σPtdWt,
where γ = μ− 21σ2 and (Wt) is a Brownian motion (Wiener process). This is the asset price model used in the Black-Scholes option pricing model. We will soon investigate whether this is an adequate model of asset prices.
2.2.3 Returns of a portfolio
When Pt is the value of a portfolio, it is natural to relate the portfolio return with the returns of the individual assets. Over a holding period [t0,t1], suppose that wi is the weight of asset i, i = 1, . . . , N, whose return is Ri(t0, t1).3 By construction, we have
Ni=1 wi = 1. The simple return of the portfolio is given by
wiRi(t0, ti), (2.5) which is a weighted average of the individual simple returns. In matrix notations, we
Rp(t0, t1) =
where w = (w1,…,wN)⊤ and R = (R1,…,RN)⊤. Linearity is useful for statistical analysis. Suppose we model the vector R of asset returns as a random vector with mean vector μ and covariance matrix Σ. Then, given the portfolio weight vector w, the mean and variance of the portfolio return are given by
E[Rp] = w⊤μ, Var(Rp) = w⊤Σw. (2.6)
These formulas are fundamental in Markowitz’s mean-variance approach to portfolio selection.
On the other hand, the log return of a portfolio is nonlinear in the individual log returns of the assets. We assume that wi ≥ 0 (no short selling) and Ri > −1 for all i, so that the log return is finite. Expressing (2.5) in terms of the log returns, we have
rp(t0, t1) = log(1 + Rp(t0, t1))
wi(1 + Ri(t0, ti))
N wieri(t0,t1)
This nonlinear relation makes it more difficult to analyze portfolio selection in terms
of log returns (which is additive over time).
3Strictly speaking, the weight is wi only at time t0. As the asset prices fluctuate, the portfolio weights change even the portfolio is buy-and-hold. For example, suppose we invest $100 each in two stocks. The initial weights are (w1,w2) = (0.5,0.5). If the first stock grows by 50% and the second stock stays the same, the final value of the portfolio is $250 = $150 + $100. The final weights are (0.6, 0.4).
2.2.4 Excess growth rate
The portfolio log return satisfies a mathematical property which has important finan- cial consequences. Before stating it, we recall a fundamental result in mathematical analysis and probability.
Theorem 2.3 (Jensen’s inequality). Let I ⊂ R be an interval and let f : I → R be a convex function, i.e.,
f((1 − λ)x + λy) ≤ (1 − λ)f(x) + λf(y),
for all x, y ∈ I and λ ∈ [0, 1]. (We say that f is strictly convex if the above inequality is strict whenever x ̸= y and λ ∈ (0, 1).) Let X be a random variable with values in I, such that E[X] is finite. Then we have
E[f(X)] ≥ f(E[X]). (2.8) If f is strictly convex, then equality in (2.8) holds if and only if X is almost surely
Proposition 2.4 (Excess growth rate). Suppose wi ≥ 0 and Ri(t0,t1) > −1 for all
rp(t0, t1) ≥
wiri(t0, t1). (2.9)
If wi >0 for all i, then equality holds if and only if r1 =···=rN. In words, for an all-long portfolio the portfolio log return is always greater than or equal to the weighted average log returns of the assets.
We define the excess growth rate of the portfolio over [t0,t1] be the non-negative quantity
γp (t0, t1) = rp(t0, t1) − wiri(t0, t1). (2.10)
Proof. For notational simplicity, we suppress t0,t1 in the expressions. Fix portfolio
weights w1,…,wN and simple returns r1,…,rN. From (2.7), we have
N rp = log wieri .
To apply Jensen’s inequality, interpret the above expression as rp = log E[Y ], where Y is a discrete and positive random variable satisfying P(Y = eri ) = wi. Since log is a strictly concave function, by Jensen’s inequality we have
This gives the inequality (2.9). Since log is strictly concave, if equality holds then Y is almost surely constant. If wi = P(Y = yi) > 0 for all i, this implies that r1 =···=rN.
Example 2.5. Consider N = 2 assets. Over a holding period, suppose (w1,w2) = (12, 12), R1 = 0.1, R2 = −0.05. Then
γp∗ = log (0.5 × 1.1 + 0.5 × 0.95) − 12(log 1.1 + log 0.95) = 0.0027. 12
rp =logE[Y]≥E[logY]=
Normalized monthly prices
2000 2005 2010
Figure 2.4: Normalized monthly prices of Apple and Nvidia from January 2000 to December 2021.
In the literature, the excess growth rate is also called the diversification return and the rebalancing premium.
The excess growth rate can be used to explain why some portfolios outperform others. Here, we give a simple example inspired by Bouchey et al. (2012). Consider monthly prices of the stocks Apple (AAPL) and Nvidia (NVDA) from Janurary 2000 to December 2021. We renormalize them so that the initial value in Janurary 2000 (denoted t = 0) is 1. The time series of the normalized prices, denoted respectively by PAAPL(t) and PNVDA(t), are shown in Figure 2.4. Both stocks grew tremendously in this period. In this exercise, we assume that the shares are infinitely divisible and there is no transaction cost.
Remark 2.6 (Infinite divisibility). Even though we can only buy integral amount of shares, infinite divisibility is a common assumption in quantitative finance. Suppose there are two stocks with P1 = $20 and P2 = $30. With a capital of $100, it is impossible to split equally between the two. We can do e.g. $100 = 3 × $20 + 2 × $30 with effective weights (w1,w2) = (0.6,0.4). But if the capital is $10000, we can split the capital as follows:
$10000 = $5020 + $4980 = 151 × $20 + 166 × $30.
Now the weights are (0.502, 0.498) which are very close to (0.5, 0.5). So, with a large portfolio, it is possible to be very close to a target portfolio. The assumption of infinite divisibility abstracts this technicality away.
With these two stocks, we consider two portfolios. The first portfolio is a simple buy-and-hold portfolio: at time t = 0, we split the initial capital (assumed to be 1 for normalization purposes) equally between the two stocks. Then no further trade is made, i.e., the portfolio is buy-and-hold. It is easy to see that the value of the portfolio at time t is
V1(t) = 12PAAPL(t) + 12PNVDA(t). (2.11) Note that V1(0) = 1 by construction.
Portfolio value
2010 2015 2020
Buy−and−hold Equal−weighted
Figure 2.5: Time series of the portfolio values V1 (buy-and- hold) and V2 (equal-weighted). In this example, the equal- weighted portfolio significantly outperforms the buy-and-hold portfolio.
The second portfolio is the monthly rebalanced equal-weighted portfolio. At the beginning of each month, we rebalance the portfolio in such a way that equal amount of capital is invested into each stock. Using (2.5) with (w1,w2) = (12, 21) over each month, the gross return of the portfolio V2(t) over [t − 1, t] is
V2(t) = 1 PAAPL(t) + 1 PNVDA(t) = 1 + 1(RAAPL(t) + RNVDA(t)). V2(t − 1) 2 PAAPL(t − 1) 2 PNVDA(t − 1) 2
Compounding, we have
V2 (t) = t 1 + 21 (RAAPL (s) + RNVDA (s)) . (2.12)
In Figure 2.5 we plot the values of the two portfolio. Here, the equal-weighted portfolio outperforms significantly the buy-and-hold portfolio (the final values are about 639 and 304, respectively). Following the approach of Pal and Wong (2013), which builds on Stochastic Portfolio Theory introduced in Fernholz (2002), let us try to explain why this is the case during the concept of excess growth rate. Consider the excess growth rate of the equal-weighted portfolio over [t − 1, t]:
γ∗(t) = log1 PAAPL(t)
2 PAAPL(t − 1)
+ 1 PNVDA(t) 2 PNVDA(t − 1)
− 1 log PAAPL(t)
2 PAAPL(t − 1)
Rearranging, we have, using (2.12),
log V2(t) = 1 log PAAPL(t)
V2(t − 1) 2 PAAPL(t − 1) 14
+ γ∗(t).
+ log PNVDA(t) PNVDA(t − 1)
+ log PNVDA(t) PNVDA(t − 1)
Decomposition of log relative value
2000 2005 2010 2015 2020
excess growth log(GM/AM) log(V2/V1)
Figure 2.6: Time series of the decomposition (2.13). Summing over time and noting that the processes start at 0, we have
log V2(t) = 2 log(PAAPL(t)PNVDA(t)) + γ∗(s).
From this and (2.11), we obtain the following decomposition formula for the log relative
value log V2(t): V1 (t)
PAAPL(t)PNVDA(t) + Γ∗(t), 12 (PAAPL(t) + PNVDA(t))
log V2(t) = log V1(t)
where Γ∗(t) = ts=1 γ∗(s) is the cumulated sum of the excess growth rate. To understand this formula, write
log V2(t) = log GM(t) + Γ∗(t), V1 (t) AM(t)
where GM(t) and AM(t) are respectively the geometric and arithmetic means of
PAAPL(t) and PNVDA(t). Each of the terms of this decomposition is plotted in Figure
2.6. Note that Γ∗(t) is increasing in t due to the (relative) volatility of the stocks. On
the other hand, even though the two stocks grew a lot, at any point in term (the nor-
malized) price of Apple is at most a little more than 2 times that of Nvidia. Thus, the
log ratio log GM(t) remains relatively bounded. For this data set, the cumulated excess AM(t)
growth rate mostly dominated the term log GM(t) which represents the relative sizes AM(t)
of the two stocks. The above analysis can be adapted easily to multiple stocks and any strictly positive and constant portfolio weights. In summary, as long as the rela- tive sizes of the stocks remain bounded, and that the stocks are “sufficiently volatile” (in the sense that Γ∗(t) keeps increasing without bound), the constant rebalanced portfolio will eventually outperform a buy-and-hold portfolio (neglecting transaction costs). This is a form of “volatility pumping” where volatility is captured by systematic rebalancing.
References
Cartea, A ́., Jaimungal, S., & Penalva, J. (2015). Algorithmic and High-Frequency Trading. Cambridge University Press.
Bouchey, P., Nemtchinov, V., Paulsen, A., & Stein, D. M. (2012). Volatility harvest- ing: Why does diversifying and rebalancing create portfolio growth?. The Journal of Wealth Management, 15(2), 26–35.
Fernholz, E. R. (2002). Stochastic Portfolio Theory. Springer.
Pal, S., & Wong, T. K. L. (2013). Energy, entropy, and arbitrage. arXiv preprint arXiv:1308.5376.
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