FINANCIAL DATA AND THEIR PROPERTIES

2 FINANCIAL DATA AND THEIR PROPERTIES
exponential smoothing for forecasting and methods for model comparison. Chapter 3
considers some applications of the models introduced in Chapter 2 in the form of case
studies. The goal is to help readers understand better data analysis, empirical modeling,
and making inference. It also points out the limitations of linear time series models
in long-term prediction. Chapter 4 focuses on modeling conditional heteroscedasticity
(i.e., the conditional variance of an asset return). It introduces various econometric
models for describing the evolution of asset volatility over time. The chapter also discusses alternative methods to volatility modeling, including use of daily high and low
prices of an asset. In Chapter 5, we demonstrate some applications of volatility models
using, again, some case studies. All steps for building volatility models are given, and
the merits and weaknesses of various volatility models are discussed, including the
connection to diffusion limit of continuous time models. Chapter 6 is concerned with
analysis of high frequency financial data. It starts with special characteristics of high
frequency data and gives models and methods that can be used to analyze such data.
It shows that nonsynchronous trading and bid-ask bounce can introduce serial correlations in a stock return. It also studies the dynamic of time duration between trades and
some econometric models for analyzing transaction data. In particular, we discuss the
use of logistic linear regression and probit models to study the stock price movements
in consecutive trades. Finally, the chapter studies the realized volatility using intraday
log returns. Chapter 7 discusses risk measures of a financial position and their use
in risk management. It introduces value at risk and conditional value at risk to quantify the risk of a financial position within a holding period. It also provides various
methods for calculating risk measures for a financial position, including RiskMetrics, econometric modeling, extreme value theory, quantile regression, and peaks over
thresholds.
The book places great emphasis on application and empirical data analysis. Every chapter contains real examples, and, in many occasions, empirical
characteristics of financial time series are used to motivate the development of
econometric models. In some cases, simple R scripts are given on the web page
for specific analysis. Many real data sets are also used in the exercises of each
chapter.
1.1 ASSET RETURNS
Most financial studies involve returns, instead of prices, of assets. Campbell et al.
(1997) give two main reasons for using returns. First, for average investors, return of
an asset is a complete and scale-free summary of the investment opportunity. Second,
return series are easier to handle than price series because the former have more
attractive statistical properties. There are, however, several definitions of an asset
return.
Let P
t be the price of an asset at time index t. We discuss some definitions of
returns that are used throughout the book. Assume for the moment that the asset pays
no dividends.
Tsay, R. S. (2012). An introduction to analysis of financial data with r. Retrieved from http://ebookcentral.proquest.com
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ASSET RETURNS 3
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 + Rt =
P
t
P
t-1
or P
t = Pt-1(1 + Rt). (1.1)
The corresponding one-period simple net return or simple return is
R
t =
P
t
P
t-1
– 1 =
P
t – Pt-1
P
t-1
. (1.2)
For demonstration, Table 1.1 gives five daily closing prices of Apple stock in
December 2011. From the table, the 1-day gross return of holding the stock from
December 8 to December 9 is 1 + Rt = 393.62/390.66 ≈ 1.0076 so that the corresponding daily simple return is 0.76%, which is (393.62-390.66)/390.66.
Multiperiod Simple Return. Holding the asset for k periods between dates
t – k and t gives a k-period simple gross return
1 + Rt[k] =
P
t
P
t-k
=
P
t
P
t-1
×
P
t-1
P
t-2
× · · · ×
P
t-k+1
P
t-k
= (1 + Rt)(1 + Rt-1) · · · (1 + Rt-k+1)
=
k-1
j=0
(1 + Rt-j ).
Thus, the k-period simple gross return is just the product of the k one-period simple
gross returns involved. This is called a compound return. The k-period simple net
return is R
t[k] = (Pt – Pt-k)/Pt-k.
To illustrate, consider again the daily closing prices of Apple stock of Table 1.1.
Since December 2 and 9 are Fridays, the weekly simple gross return of the stock is
1 + Rt[5] = 393.62/389.70 ≈ 1.0101 so that the weekly simple return is 1.01%.
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
AnnualizedRt[k] =
⎡⎣
k-1
j=0
(1 + Rt-j )
⎤⎦
1/k
– 1.
TABLE 1.1. Daily Closing Prices of Apple Stock from December 2 to 9, 2011
Date 12/02 12/05 12/06 12/07 12/08 12/09
Price($) 389.70 393.01 390.95 389.09 390.66 393.62
Tsay, R. S. (2012). An introduction to analysis of financial data with r. Retrieved from http://ebookcentral.proquest.com
Created from baruch on 2020-04-24 18:40:17.
Copyright © 2012. John Wiley & Sons, Incorporated. All rights reserved.
4 FINANCIAL DATA AND THEIR PROPERTIES
This is a geometric mean of the k one-period simple gross returns involved and can
be computed by

1 k
where exp(x) denotes the exponential function and ln(x) is the natural logarithm of the
positive number x. Because it is easier to compute arithmetic average than geometric
mean and the one-period returns tend to be small, one can use a first-order Taylor
expansion to approximate the annualized return and obtain
AnnualizedRt[k] ≈
1 k
k-1
j=0
R
t-j . (1.3)
Accuracy of the approximation in Equation (1.3) may not be sufficient in some applications, however.
Continuous Compounding. Before introducing continuously compounded
return, we discuss the effect of 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, 1 year
later. If the bank pays interest 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, 1 year later.
Table 1.2 gives the results for some commonly used time intervals on a deposit of
$1.00 with interest rate of 10% per annum. In particular, the net value approaches
TABLE 1.2. Illustration of the Effects of Compounding: the Time Interval is 1 Year and the
Interest Rate is 10% Per Annum
Number of Interest Rate Net
Type Payments per Period Value
Annual 1 0.1 $1.10000
Semiannual 2 0.05 $1.10250
Quarterly 4 0.025 $1.10381
Monthly 12 0.0083 $1.10471
Weekly 52
0.1
52
$1.10506
Daily 365
0.1
365
$1.10516
Continuously ∞ $1.10517
Tsay, R. S. (2012). An introduction to analysis of financial data with r. Retrieved from http://ebookcentral.proquest.com
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Copyright © 2012. John Wiley & Sons, Incorporated. All rights reserved.
ASSET RETURNS 5
$1.1052, which is obtained by exp(0.1) and referred to as the result of continuous
compounding. The effect of compounding is clearly seen.
In general, the net asset value A of continuous compounding is

which is referred to as the present value of an asset that is worth A dollars n years
from now, assuming that the continuously compounded interest rate is r per annum.
Continuously Compounded Return. The natural logarithm of the simple
gross return of an asset is called the continuously compounded return or log return:
r
t = ln(1 + Rt) = ln
P
t
P
t-1
= pt – pt-1, (1.6)
where pt = ln(Pt). Continuously compounded returns rt enjoy some advantages over
the simple net returns Rt. First, consider multiperiod returns. We have
r
t[k] = ln(1 + Rt[k]) = ln[(1 + Rt)(1 + Rt-1) · · · (1 + Rt-k+1)]
= ln(1 + Rt) + ln(1 + Rt-1) + · · · + ln(1 + Rt-k+1)
= r
t + rt-1 + · · · + rt-k+1.
Thus, the continuously compounded multiperiod return is simply the sum of continuously compounded one-period returns involved. Second, statistical properties of log
returns are more tractable.
To demonstrate, we again consider the daily closing prices of Apple stock of
Table 1.1. The daily log return from December 8 to December 9 is rt = log(393.62) –
log(390.66) ≈ 0.75% and the weekly log return from December 2 to December 9 is
r
t[5] = log(393.62) – log(389.70) ≈ 1.00%. One can easily verify that the weekly
log return is the sum of the five daily log returns involved.
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 wi on asset i. Then, the simple return of p at time t
is R
p,t = N i=1 wi Rit, where Rit 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 Rit are all small in magnitude, then
we have r
p,t ≈ N i=1 wi rit, where rp,t is the continuously compounded return of the
portfolio at time t. This approximation is often used to study portfolio returns.
Tsay, R. S. (2012). An introduction to analysis of financial data with r. Retrieved from http://ebookcentral.proquest.com
Created from baruch on 2020-04-24 18:40:17.
Copyright © 2012. John Wiley & Sons, Incorporated. All rights reserved.
6 FINANCIAL DATA AND THEIR PROPERTIES
Dividend Payment. If an asset pays dividends periodically, we must modify
the definitions of asset returns. Let D
t be the dividend payment of an asset between
dates t – 1 and t, and Pt 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 =
P
t + Dt
P
t-1
– 1, rt = ln(Pt + Dt) – ln(Pt-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
Zt
= R
t – R0t, zt = rt – r0t, (1.7)
where R0t and r0t 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 of Relationship. The relationships between simple return Rt and
continuously compounded (or log) return rt are
r
t = ln(1 + Rt), Rt = ert – 1.
If the returns R
t and rt are in percentages, then
r
t = 100 ln 1 +
R
t
100
, Rt = 100 ert/100 – 1 .
Temporal aggregation of the returns produces
1 + Rt[k] = (1 + Rt)(1 + Rt-1) · · · (1 + Rt-k+1),
Tsay, R. S. (2012). An introduction to analysis of financial data with r. Retrieved from http://ebookcentral.proquest.com
Created from baruch on 2020-04-24 18:40:17.
Copyright © 2012. John Wiley & Sons, Incorporated. All rights reserved.

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