Asset Returns

 

One-Period Simple Return

Holding the asset for one period from date t − 1 to date t would result in a simple gross return:

\[ 1+R_t=\frac{P_t}{P_{t-1}} \quad \mbox{or} \quad P_t=P_{t-1}(1+R_t)\] The corresponding one-period simple net return or simple return is \[ R_t=\frac{P_t}{P_{t-1}}-1= \frac{P_t-P_{t-1}}{P_{t-1}} \]

 

Multiperiod Simple Return

Holding the asset for k periods between dates t − k and t gives a k-period simple gross return \[ 1+R_t[k]=\frac{P_t}{P_{t-k}}= \frac{P_t}{P_{t-1}} \times \frac{P_{t-1}}{P_{t-2}}\times \cdots\times \frac{P_{t-k+1}}{P_{t-k}} = \prod\limits_{j=0}^{k-1}(1+R_{t-j}) \] Then \[\begin{equation} R_t[k]=(P_t-P_{t-k})/P_{t-k}. \end{equation}\]

In practice, the actual time interval is important in discussing and comparing returns (e.g., monthly return or annual return). If the time interval is not given, then it is implicitly assumed to be one year. If the asset was held for k years, then the annualized (average) return is defined as \[ \mbox{Annulaized }{R_t[k]}=\left[\prod\limits_{j=0}^{k-1} (1+R_{t-j}) \right]^{1/k}-1\] This is a geometric mean of the k one-period simple gross returns involved and can be computed by \[ \mbox{Annulaized }{R_t[k]}=\exp\left[\frac{1}{k}\sum\limits_{j=0}^{k-1} \ln(1+R_{t-j}) \right]-1 \approx \frac{1}{k}\sum\limits_{j=0}^{k-1} R_{t-j}\]

 

Continuous Compounding

Assume that the interest rate of a bank deposit is 10% per annum and the initial deposit is $1.00. If the bank pays interest once a year, then the net value of the deposit becomes $1(1 + 0.1) = $1.1 one year later. If the bank pays inter- est semiannually, the 6-month interest rate is 10%/2 = 5% and the net value is \(\$1(1 + 0.1/2)^2 = \$1.1025\) after the first year. In general, if the bank pays interest m times a year, then the interest rate for each payment is 10%/m and the net value of the deposit becomes \(\$1(1 + 0.1/m)^m\) one year later.

In general, the net asset value A of continuous compounding is \[A=C\exp(r\times n) \quad \mbox{or} \quad C=A\exp(-r\times n) \] where \(r\) is the inerest rate per annum, \(C\) is the initial capital, and \(n\) is the number of years.

 

Continuously Compound Return

The natural logarithm of the simple gross return of an asset is called the continu- ously compounded return or log return: \[ r_t=\ln(1+R_t)=\ln\frac{P_t}{P_{t-1}}=p_t-p_{t-1} \] where \(p_t=\ln(P_t)\).

Continuously compounded returns \(r_t\) enjoy some advantages over the simple net returns \(R_t\): \[\begin{eqnarray*} r_t[k] &=& \ln (1+R_t[k])=\ln [(1+R_t)(1+R_{t-1})\cdots (1+R_{t-k+1})] \\ &=& \ln(1+R_t)+\ln(1+R_{t-1})+\cdots+ \ln(1+R_{t-k+1}) \\ &=& r_t+r_{-1}+\cdots+r_{t-k+1} \end{eqnarray*}\]

 

Portfolio Return

The simple net return of a portfolio consisting of \(N\) assets is a weighted average of the simple net returns of the assets involved, where the weight on each asset is the percentage of the portfolio’s value invested in that asset. Let \(p\) be a portfolio that places weight \(w_i\) on asset \(i\). Then the simple return of \(p\) at time t is \(R_{p,t} =\sum_{i=1}^N w_i R_{it}\) , where \(R_{it}\) is the simple return of asset \(i\)

The continuously compounded returns of a portfolio, however, do not have the above convenient property. If the simple returns \(R_{it}\) are all small in magnitude, then we have \(r_{p,t}\approx \sum_{i=1}^N w_i r_{it}\) , where \(r_{p,t}\) is the continuously compounded return of the portfolio at time \(t\). This approximation is often used to study portfolio returns.

 

Divident Payment

If an asset pays dividends periodically, we must modify the definitions of asset returns. Let Dt be the dividend payment of an asset between dates \(t − 1\) and \(t\) and \(P_t\) be the price of the asset at the end of period \(t\) . Thus, dividend is not included in \(P_t\) . Then the simple net return and continuously compounded return at time t become \[R_t = \frac{P_t +D_t}{P_{t-1}} −1, \quad r_t =\ln(P_t +D_t)−\ln(P_{t−1})\]

 

Excess Return

Excess return of an asset at time t is the difference between the asset’s return and the return on some reference asset. The reference asset is often taken to be riskless such as a short-term U.S. Treasury bill return. The simple excess return and log excess return of an asset are then defined as \[ Z_t=R_t-R_{0t}, z_t=r_t-r_{0t},\] where \(R_{0t}\) and \(r_{0t}\) are the simple and log returns of the reference asset, respectively. In the finance literature, the excess return is thought of as the payoff on an arbitrage portfolio that goes long in an asset and short in the reference asset with no net initial investment.

 

Remark.

A long financial position means owning the asset. A short position involves selling an asset one does not own. This is accomplished by borrowing the asset from an investor who has purchased it. At some subsequent date, the short seller is obligated to buy exactly the same number of shares borrowed to pay back the lender. Because the repayment requires equal shares rather than equal dollars, the short seller benefits from a decline in the price of the asset. If cash dividends are paid on the asset while a short position is maintained, these are paid to the buyer of the short sale. The short seller must also compensate the lender by matching the cash dividends from his own resources. In other words, the short seller is also obligated to pay cash dividends on the borrowed asset to the lender.

 

Summary fo Ralationship

\[ r_t=\ln(1+R_t)\] If the return \(R_t\) and \(r_t\) are in percentages, then \[ r_t=100\ln \left(1+\frac{R_t}{100} \right), \quad R_t=100 (e^{r_t/100}-1)\] \[ 1+R_t[k]=(1+R_t)(1+R_{t-1})\cdots(1+R_{t-k+1})\] \[ r_t[k]=r_t+r_{t-1}+\cdots+ r_{t-k+1}\] If the continuously compounded interest rate is \(r\) per annum, then the relationship between present and future values of an asset is \[A =C\exp(r\times n), \quad C=A\exp(-r\times n) \]

 

Example

If the monthly log return of an asset is 4.46%, then the corre- sponding monthly simple return is \(100[\exp(4.46/100) − 1] = 4.56\%\). Also, if the monthly log returns of the asset within a quarter are 4.46%, −7.34%, and 10.77%, respectively, then the quarterly log return of the asset is \((4.46 − 7.34 + 10.77)\% = 7.89\%\).

 

Distribution Properties of Returns

To study asset returns, it is best to begin with their distributional properties. The objective here is to understand the behavior of the returns across assets and over time. Consider a collection of N assets held for T time periods, say, \(t = 1,...,T\). For each asset \(i\), let rit be its log return at time \(t\). The log returns under study are \(\{r_{it};i = 1,...,N;t = 1,...,T\}\). One can also consider the simple returns \(\{R_{it};i = 1,...,N;t = 1,...,T\}\) and the log excess returns \(\{z_{it};i = 1,...,N;t = 1,...,T\}\).

 

Review of Statistical Distribution and Their Moments

  1. Joint Distribution \[ F_{X,Y}(x,y,\theta)=P(X\le x, Y\le y, \theta)\] \[F_{X,Y}(x,y,\theta)=\int_{-\infty}^x \int_{-\infty}^yf_{x,y}(w,z)dz dw \]

  2. Marginal Distribution \[F_X(x,\theta) =F_{X,Y}(x,\infty,\cdots,\infty, \theta)\] For a given probability \(p\), the smallest real number \(x_p\) such that \(p \le F_X(x_p)\) is called quantile of the random variable \(X\). More specifically, \[ x_p=\inf\limits_x\{x| p \le F_X(x)\}. \]

  3. Conditional Distribution \[F_{X|Y\le y}(x,\theta)=\frac{P(X\le x, Y\le y, \theta)}{P(Y \le y, \theta )}\] and \[f_{x|y}(x,\theta)=\frac{f( x, y, \theta)}{f_y(y, \theta )}\] where \[f_y(y,\theta)=\int_{-\infty}^\infty f_{x,y}(x,y,\theta)d x \]

  4. Moments of a Random Variables

The \(\ell\)th moment of a continuous random variable \(X\) is defined as \[ m'_\ell =\mbox{E}(X^\ell)=\int_{-\infty}^\infty x^\ell f(x)dx \] The \(\ell\) central moment of \(X\) is defined as
\[ m_\ell =\mbox{E}[(X-\mu_x)^\ell)=\int_{-\infty}^\infty (x-\mu_x)^\ell f(x)dx \]

  • Second coentral moment denoted by \(\sigma_x^2\), measures the variability of \(X\) and is called the variance of \(X\). The positive square root, \(\sigma_x\), of variance is the standard deviation of \(X\).

  • The third central moment measures the symmetry of \(X\) with respect to its mean, whereas the fourth central moment measures the tail behavior of \(X\).

  • In statistics, skewness and kurtosis, which are normalized third and fourth central moments of \(X\), are often used to summarize the extent of asymmetry and tail thickness. Specifically, the skewness and kurtosis of \(X\) are defined as \[ S(x)=\mbox{E}\left[\frac{(X-\mu_x)^3}{\sigma_x^3} \right],\quad K(x)=\mbox{E}\left[\frac{(X-\mu_x)^4}{\sigma_x^4} \right] \]

  • The quantity \(K(x) − 3\) is called the excess kurtosis because \(K(x) = 3\) for a normal distribution. Thus, the excess kurtosis of a normal random variable is zero. A distribution with positive excess kurtosis is said to have heavy tails, implying that the distribution puts more mass on the tails of its support than a normal distribution does.

  • In application, skewness and kurtosis can be estimated by their sample counterparts. Let \(\{x_1 , . . . , x_T \}\) be a random sample of \(X\) with \(T\) observations. The sample mean and sample variance are \[\hat{\mu}=\frac{1}{T}\sum\limits_{t=1}^T x_t, \quad \hat{\sigma}_x^2=\frac{1}{T-1}\sum\limits_{t=1}^T (x_t-\hat{\mu}_x)^2 \] the sample skewness is \[ \hat{S}(x)=\frac{1}{(T-1)\hat{\sigma}_x^3}\sum\limits_{t=1}^T (x_t-\hat{\mu}_x)^3 \] and the sample kurtosis is \[ \hat{K}(x)=\frac{1}{(T-1)\hat{\sigma}_x^4}\sum\limits_{t=1}^T (x_t-\hat{\mu}_x)^4 \]

  • Under the normality assumption, \(\hat{S}(x)\) and \(\hat{K} (x) − 3\) are distributed asymptotically as normal with zero mean and variances 6/T and 24/T, respectively.

  • Given an asset return series \(\{r_1,...,r_T\}\), to test the skewness of the returns, we consider the null hypothesis \(H_0 : S(r) = 0\) versus the alternative hypothesis \(H_a : S(r) \ne 0\). The t-ratio statistic of the sample \[ t=\frac{\hat{S}(r)}{\sqrt{6/T}}\] The decision rule is as follows. Reject the null hypothesis at the α significance level, if \(|t|>Z_\alpha/2\), where \(Z_\alpha/2\) is the upper \(100(\alpha/2)\)th quantile of the standard normal distribution. Alternatively, one can compute the p value of the test statistic t and reject \(H_0\) if and only if the \(p\) value is less than \(\alpha\).

  • Similarly, one can test the excess kurtosis of the return series using the hypotheses \(H_0 :K(r)−3=0\) versus \(H_a :K(r)−3 \ne 0\).The test statistic is \[t= \frac{\hat{K}(r)−3}{\sqrt{24/T}},\] which is asymptotically a standard normal random variable. The decision rule is to reject \(H_0\) if and only if the p value of the test statistic is less than the significance level \(\alpha\).

  • Combine the two prior tests and use the test statistic \[ JB= \frac{\hat{S}^2(r)}{6T}+\frac{[\hat{K}(r)−3]^2}{24/T}, \] which is asymptotically distributed as a chi-squared random variable with 2 degrees of freedom, to test for the normality of \(r_t\). One rejects \(H_0\) of normality if the p value of the JB statistic is less than the significance level.

 

Example

Consider the daily simple returns of the International Business Machines (IBM) stock. The sample skewness and kurtosis of the returns are parts of the descriptive (or summary) statistics that can be obtained easily using various statistical software packages. R is used in the demonstration, where d-ibm3dx7008.txt is the data file name.

library(fBasics)
## Warning: package 'fBasics' was built under R version 4.0.3
## Loading required package: timeDate
## Warning: package 'timeDate' was built under R version 4.0.3
## Loading required package: timeSeries
## Warning: package 'timeSeries' was built under R version 4.0.3
da=read.table("d-ibm3dx7008.txt",header=T)  #Load the data.
# header=T means 1st row of the data file contains
# variable names. The default is header=F, i.e., no names.

dim(da) # Find size of the data: 9845 rows and 5 columns.
## [1] 9845    5
da[1,] # See the first row of the data
ibm=da[,2] # Obtain IBM simple returns

sibm=ibm*100 # Percentage simple returns

basicStats(sibm) # Compute the summary statistics
# Alternatively, one can use individual commands as follows:

mean(sibm)
## [1] 0.04016126
var(sibm)
## [1] 2.864705
sd(sibm)
## [1] 1.692544
skewness(sibm)
## [1] 0.06139878
## attr(,"method")
## [1] "moment"
kurtosis(sibm, method="excess") # the value of the kurtosis is computed by the "moment" method and a value of 3 will be subtracted
## [1] 9.916359
## attr(,"method")
## [1] "excess"
kurtosis(sibm, method="moment")
## [1] 12.91636
## attr(,"method")
## [1] "moment"
s1=skewness(sibm)

t1=s1/sqrt(6/9845) # Compute test statistic > t1

t1
## [1] 2.487093
## attr(,"method")
## [1] "moment"
pv=2*(1-pnorm(t1)) # Compute p-value.

pv
## [1] 0.01287919
## attr(,"method")
## [1] "moment"
#Turn to log returns in percentages

libm=log(ibm+1)*100

t.test(libm) # Test mean being zero.
## 
##  One Sample t-test
## 
## data:  libm
## t = 1.5126, df = 9844, p-value = 0.1304
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
##  -0.007641473  0.059290531
## sample estimates:
##  mean of x 
## 0.02582453
#The result shows that the hypothesis of zero expected return cannot be rejected at the 5% or 10% level.

normalTest(libm,method='jb') # Normality test
## 
## Title:
##  Jarque - Bera Normalality Test
## 
## Test Results:
##   STATISTIC:
##     X-squared: 60921.9343
##   P VALUE:
##     Asymptotic p Value: < 2.2e-16 
## 
## Description:
##  Tue Jan 19 12:01:29 2021 by user: HPENG

 

Distribution of Returns

The most general model for the log returns \(\{r_{it};i = 1,\ldots,N;t = 1,\ldots,T\}\) is its joint distribution function: \[ F_r(r_{11}, \ldots, r_{N1};r_{12}, \ldots, r_{N2}; \ldots; r_{1T}, \ldots, r_{NT}, \mathbf{Y}; \theta), \] where \(\mathbf{Y}\) is a state vector consisting of variables that summarize the environment in which asset returns are determined and \(\theta\) is a vector of parameters that uniquely determines the distribution function \(F_r(·)\).

 

Normal Distribution

A traditional assumption made in financial study is that the simple returns \(\{R_{it} |t = 1,\ldots , T \}\) are independently and identically distributed as normal with fixed mean and variance. This assumption makes statistical properties of asset returns tractable. But it encounters several difficulties.

  • First, the lower bound of a simple return is −1. Yet the normal distribution may assume any value in the real line and, hence, has no lower bound.

  • Second, if \(R_{it}\) is normally distributed, then the multiperiod simple return \(R_{it} [k]\) is not normally distributed because it is a product of one-period returns.

  • Third, the normality assumption is not supported by many empirical asset returns, which tend to have a positive excess kurtosis.

 

Lognormal Distribtuion

Another commonly used assumption is that the log returns \(r_t\) of an asset are independent and identically distributed (iid) as normal with mean \(\mu\) and variance \(\sigma^2\).

  • The simple returns are then iid lognormal random variables with mean and variance given by \[ \mathrm{E}(R_t)=\exp\left(\mu+\frac{\sigma^2}{2}\right)-1, \quad \mbox{Var}(R_t)=\exp(2\mu+\sigma^2)[\exp(\sigma^2)-1]\]

  • Alternatively, let \(m_1\) and \(m_2\) be the mean and variance of the simple return Rt, which is lognormally distributed. Then the mean and variance of the corresponding log return \(r_t\) are \[\mbox{E}(r_t)=\ln \left[\frac{m_1+1}{\sqrt{1+m_2/(1+m_1)^2}} \right], \quad \mbox{Var}(r_t)=\ln \left[1+\frac{m_2}{(1+m_2)^2} \right] \]

  • Because the sum of a finite number of iid normal random variables is normal, \(r_t [k]\) is also normally distributed under the normal assumption for \(\{r_t \}\). In addition, there is no lower bound for \(r_t\), and the lower bound for \(R_t\) is satisfied using \(1 + R_t = \exp(r_t )\).

  • However, the lognormal assumption is not consistent with all the properties of historical stock returns. In particular, many stock returns exhibit a positive excess kurtosis.

 

Stable Distribution

  • The stable distributions are a natural generalization of normal in that they are stable under addition, which meets the need of continuously compounded returns \(r_t\) . Furthermore, stable distributions are capable of capturing excess kurtosis shown by historical stock returns.

  • However, nonnormal stable distributions do not have a finite variance, which is in conflict with most finance theories. In addition, statistical modeling using nonnormal stable distributions is difficult. An example of nonnormal stable distributions is the Cauchy distribution, which is symmetric with respect to its median but has infinite variance.

 

Scale Mixture of Normal Distribution

The log return \(r_t\) is normally distributed with mean \(\mu\) and variance \(\sigma^2\) [i.e., \(r_t \sim N (\mu, \sigma^2)\) ]. However, \(\sigma^2\) is a random variable that follows a positive distribution (e.g., \(\sigma^2\) follows a gamma distribution). An example of finite mixture of normal distributions is \[ r_t \sim (1-X)N(\mu, \sigma_1^2)+XN(\mu, \sigma_2^2),\] where \(X\) is a Bernolli random variable such that \(P(X=1)=\alpha=1-P(X=0)\) with \(0<\alpha<1\), \(\sigma_1^2\) is small and \(\sigma_2^2\) is relatively large.

The following R program and Figure show the probability density functions of a finite mixture of normal, Cauchy, and standard normal random variable. The finite mixture of normal is (1 − X)N(0, 1) + X × N(0, 16) with X being Bernoulli such that P (X = 1) =0.05, and the density of Cauchy is \[ f(x)=\frac{1}{\pi(1+x^2)}, -\infty <x<\infty\]

x <- seq(-4,4,by=0.05)

f_n <- dnorm(x,0,1)

f_m <- 0.95*dnorm(x,0,1)+0.05*dnorm(x,0,16)

f_c <- 1/pi/(1+x^2)

plot(x=x,y=f_n, type="l", col=2, lty=1, ylab='f(x)')

lines(x=x,y=f_m, type="l", col=4, lty=2)

lines(x=x,y=f_c, type="l", col=7, lty=3)

legend(2,0.35, c('Normal', 'Mixture','Cauchy'), cex=0.8, col=c(2,4,7), lty=c(1:3))

 

Multivariate Returns

Let \(r_t = (r_{1t} , \ldots , r_{Nt} )′\) be the log returns of \(N\) assets at time \(t\). The analysis is then focused on the specification of the conditional distribution function \(F(r_t |r_{t−1}, \ldots, r_1, \theta)\).

The mean vector and covariance matrix of a random vector \(X = (X_1, \ldots , X_p)\) are defined as \[\mbox{E}(\mathbf{X}) = \mu_x = [\mbox{E}(X_1), . . . , \mbox{E}(X_p)]′,\] \[\mbox{Cov}(\mathbf{X}) =\Sigma_x = \mbox{E}[(\mathbf{X} − \mu_x)(\mathbf{X} − \mu_x)′].\]

When the data \(\{x_1, \ldots , x_T \}\) of \(\mathbf{X}\) are available, the sample mean and covariance matrix are defined as \[\hat{\mu}_x=\frac{1}{T}\sum\limits_{t=1}^T \mathbf{x}_t, \quad \hat{\Sigma}_x=\frac{1}{T-1}\sum\limits_{t=1}^T (\mathbf{x}_t-\hat{\mu}_x)(\mathbf{x}_t-\hat{\mu}_x)'. \]  

Likelihood Function of Returns

If the conditional distribution \(f(r_t |r_{t−1},\ldots , r_1, \theta)\) is normal with mean \(\mu_t\) and variance \(\sigma_t^2\), then \(\theta\) consists of the parameters in \(\mu_t\) and \(\sigma_t^2\), and the likelihood function of the data is \[ f(r_1,\ldots, r_T; \theta)=f(r_1; \theta)\prod\limits_{t=2}^T \frac{1}{\sqrt{2\pi}\sigma_t}\exp \left[-\frac{(r_t-\mu_t)^2}{2\sigma_t^2} \right], \] where \(f(r_1,\theta)\) is the marginal density function of the first observation \(r_1\).

  • The value of \(\theta\) that maximizes this likelihood function is the maximum-likelihood estimate (MLE) of \(\theta\).

  • Since the log function is monotone, the MLE can be obtained by maximizing the log-likelihood function, \[ \ln f(r_1,\ldots, r_T,\theta) =\ln f(r_1,\theta)-\frac12 \sum\limits_{t=2}^T \left[ \ln(2\pi) +\ln(\sigma_t^2)+\frac{(r_t-\mu_t)^2}{\sigma_t^2} \right], \] which is easier to handle in practice.

  • The log-likelihood function of the data can be obtained in a similar manner if the conditional distribution \(f(r_t |r_{t−1},\ldots, r_1; \theta)\) is not normal.

 

Empirical Properties of Returns

The following figureshows the time plots of monthly simple returns and log returns of IBM stock from January 1926 to December 2008.

ibm_return <- read.table('m-ibm3dx2608.txt', head=TRUE)
attributes(ibm_return)
## $names
## [1] "date"   "ibmrtn" "vwrtn"  "ewrtn"  "sprtn" 
## 
## $class
## [1] "data.frame"
## 
## $row.names
##   [1]   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18
##  [19]  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36
##  [37]  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54
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## [775] 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792
## [793] 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810
## [811] 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828
## [829] 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846
## [847] 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864
## [865] 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882
## [883] 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900
## [901] 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918
## [919] 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936
## [937] 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954
## [955] 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972
## [973] 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990
## [991] 991 992 993 994 995 996
par(mfrow=c(2,1), mar=c(4,4,1,1), oma=c(0,0,0,0))
plot(as.Date.character(ibm_return[,1], "%Y%m%d"), ibm_return[,2], type="l", ylab="Simple Return", xlab="Year")

ibm_logreturn <-log(1+ibm_return[,2])

plot(as.Date.character(ibm_return[,1], "%Y%m%d"), ibm_logreturn, type="l", ylab="log Return", xlab="Year")

Figure: Time plots of monthly returns of IBM stock from January 1926 to December 2008. Upper panel is for simple returns, and lower panel is for log returns.

 

Descriptive Statistics for Daily and Monthly Simple and Log Returns of Selected Indexes and Stocks

Table above provides some descriptive statistics of simple and log returns for selected U.S. market indexes and individual stocks. The returns are for daily and monthly sample intervals and are in percentages. The data spans and sample sizes are also given in the Table.

 

From the table, we make the following observations.

  1. Daily returns of the market indexes and individual stocks tend to have high excess kurtoses. For monthly series, the returns of market indexes have higher excess kurtoses than individual stocks.

  2. The mean of a daily return series is close to zero, whereas that of a monthly return series is slightly larger.

  3. Monthly returns have higher standard deviations than daily returns.

  4. Among the daily returns, market indexes have smaller standard deviations than individual stocks. This is in agreement with common sense.

  5. The skewness is not a serious problem for both daily and monthly returns.

  6. The descriptive statistics show that the difference between simple and log returns is not substantial.

 

#The following figure shows the empirical density functions of monthly simple and logreturns of IBM stock from 1926 to 2008.

 par(mfrow=c(1,2),  oma=c(0,0,0,0))

mu1 <- mean(ibm_return[,2])
sd1 <- sd(ibm_return[,2])
x <- seq(-0.4,0.6, by=0.001)
f1 <- dnorm(x,mean=mu1,sd=sd1)
plot(x,f1,type="l", col=2, lty=2, xlab="Simple Return", ylab="Density", ylim=c(0,7),  xlim=c(-0.4,0.6))

f1_den<-density(ibm_return[,2], bw = "sj")
lines(f1_den,type="l", lty=1, col=4)

x <- seq(-0.4,0.4, by=0.001)
mu2 <- mean(ibm_logreturn)
sd2 <- sd(ibm_logreturn)
f2 <- dnorm(x,mean=mu2,sd=sd2)
plot(x,f2,type="l", col=2, lty=2, xlab="Log Return", ylab="Density", ylim=c(0,7),  xlim=c(-0.4,0.4))

f2_den<-density(ibm_logreturn, bw = "sj")
lines(f2_den,type="l", lty=1, col=4)

The plots indicate that the normality assumption is questionable for monthly IBM stock returns.

  1. The empirical density function has a higher peak around its mean.
  2. But fatter tails than that of the corresponding normal distribution.

In other words, the empirical density function is taller and skinnier, but with a wider support than the corresponding normal density.

 

Processes Considered

Besides the return series, we also consider the volatility process and the behavior of extreme returns of an asset. The volatility process is concerned with the evolution of conditional variance of the return over time. This is a topic of interest because

 

par(mfrow=c(2,1), mar=c(4,4,0,0), oma=c(0,0,0,0))

US_rate10 <- read.table("m-gs10.txt", head=TRUE)
date10<-paste(US_rate10[,1],US_rate10[,2],US_rate10[,3], sep="/")
plot(as.Date.character(date10,"%Y/%m/%d"), US_rate10[,4],type="l", xlab="Year (a)", ylab="Rate")


US_rate1 <- read.table("m-gs1.txt", head=TRUE)
date1<-paste(US_rate1[,1],US_rate1[,2],US_rate1[,3], sep="/")
plot(as.Date.character(date1,"%Y/%m/%d"), US_rate1[,4],type="l", xlab="Year (b)",  ylab="Rate")

Time plots of monthly U.S. interest rates from April 1953 to February 2009: (a) 10-year Treasury constant maturity rate and (b) 1-year maturity rate.

 

par(mfrow=c(2,1), mar=c(4,4,0,0), oma=c(0,0,0,0))
jpus <- read.table("d-jpus.txt", head=TRUE)
date_jpus<-paste(jpus[,1],jpus[,2],jpus[,3], sep="/")
plot(as.Date.character(date_jpus,"%Y/%m/%d"), jpus[,4],type="l", xlab="Year", sub="(a)",  ylab="Yeus")

n <-dim(jpus)[1]
change <- jpus[2:n,4]-jpus[1:(n-1),4]
plot(as.Date.character(date_jpus[2:n],"%Y/%m/%d"), change,type="l", xlab="Year", sub="(b)",  ylab="Change")

Time plot of daily exchange rate between U.S. dollar and Japanese yen from January 4, 2000, to March 27, 2009: (a) exchange rate and (b) changes in exchange rate.

Descriptive Statistics of Selected U.S. Financial Time Series