Preface to Mathematics Monograph Series
Preface
Chapter1 Asset Returns1
1.1 Returns1
1.1.1 One-period simple returns and gross returns1
1.1.2 Multiperiod returns2
1.1.3 Log returns and continuously compounding2
1.1.4 Adjustment for dividends4
1.1.5 Bond yields and prices5
1.1.6 Excess returns6
1.2 Behavior of?nancial return data7
1.2.1 Stylized features of?nancial returns12
1.3 E±cient markets hypothesis and statistical models for returns16
1.4 Tests related to e±cient markets hypothesis20
1.4.1 Tests for white noise20
1.4.2 Remarks on the Ljung-Box test22
1.4.3 Tests for random walks23
1.4.4 Ljung-Box test and Dickey-Fuller test26
1.5 Appendix: Q-Q plot and Jarque-Bera test26
1.5.1 Q-Q plot26
1.5.2 Jarque-Bera test27
1.6 Further reading and software implementation28
1.7 Exercises29
Chapter2 Linear Time Series Models31
2.1 Stationarity31
2.2 Stationary ARMA models33
2.2.1 Moving average processes34
2.2.2 Autoregressive processes38
2.2.3 Autoregressive and moving average processes45
2.3 Nonstationary and long memory ARMA processes50
2.3.1 Random walks50
2.3.2 ARIMA model and exponential smoothing52
2.3.3 FARIMA model and long memory processes53
2.3.4 Summary of time series models54
2.4 Model selection using ACF， PACF and EACF55
2.5 Fitting ARMA models: MLE and LSE59
2.5.1 Least squares estimation59
2.5.2 Gaussian maximum likelihood estimation61
2.5.3 Illustration with gold prices63
2.5.4 A snapshot of maximum likelihood methods67
2.6 Model diagnostics: residual analysis69
2.6.1 Residual plots69
2.6.2 Goodness-of-?t tests for residuals72
2.7 Model identi?cation based on information criteria73
2.8 Stochastic and deterministic trends75
2.8.1 Trend removal76
2.8.2 Augmented Dickey-Fuller test77
2.8.3 An illustration79
2.8.4 Seasonality82
2.9 Forecasting84
2.9.1 Forecasting ARMA processes84
2.9.2 Forecasting trends and momentum of?nancial markets89
2.10 Appendix: Time series analysis in R97
2.10.1 Start up with R97
2.10.2 R-functions for time series analysis98
2.10.3 TSA{ an add-on package99
2.11 Exercises100
Chapter3 Heteroscedastic Volatility Models104
3.1 ARCH and GARCH models105
3.1.1 ARCH models105
3.1.2 GARCH models110
3.1.3 Stationarity of GARCH models113
3.1.4 Fourth moments115
3.1.5 Forecasting volatility118
3.2 Estimation for GARCH models120
3.2.1 Conditional maximum likelihood estimation120
3.2.2 Model diagnostics122
3.2.3 Applications of GARCH modeling124
3.2.4 Asymptotic properties131
3.2.5 Least absolute deviations estimation132
3.3 ARMA-GARCH models136
3.4 Extended GARCH models137
3.4.1 EGARCH models138
3.4.2 Asymmetric power GARCH143
3.4.3 Excess returns and GARCH-in-Mean146
3.4.4 Integrated GARCH model147
3.5 Stochastic volatility models148
3.5.1 Probabilistic properties149
3.5.2 Parameter estimation149
3.5.3 Leverage e?ects152
3.6 Appendix: State space models153
3.6.1 Linear models153
3.6.2 Kalman recursions for Gaussian models153
3.6.3 Nonlinear models156
3.6.4 Particle?lters158
3.7 Exercises160
Chapter4 Multivariate Time Series Analysis163
4.1 Stationarity and auto-correlation matrices163
4.1.1 Stationary vector processes163
4.1.2 Sample cross-covariancecorrelation matrices165
4.2 Vector autoregressive models168
4.2.1 Stationarity169
4.2.2 Parameter estimation170
4.2.3 Model selection and diagnostics173
4.2.4 Illustration with real data175
4.2.5 Granger causality179
4.2.6 Impulse response functions182
4.3 Cointegration185
4.3.1 Unit roots and cointegration186
4.3.2 Engle-Granger method and error correction models187
4.3.3 Johansen''s likelihood method191
4.3.4 Illustration with real data195
4.4 Exercises199
Chapter5 E±cient Portfolios and Capital Asset Pricing Model201
5.1 E±cient portfolios201
5.1.1 Returns and risks of portfolios201
5.1.2 Portfolio optimization202
5.1.3 E±cient portfolios and Sharpe ratios205
5.1.4 E±cient frontiers206
5.1.5 Challenges of implementation207
5.2 Optimizing expected utility function208
5.3 Capital asset pricing model210
5.3.1 Market portolio210
5.3.2 Capital asset pricing model212
5.3.3 Market? and its applications214
5.4 Validating CAPM215
5.4.1 Econometric formulation215
5.4.2 Maximum likelihood estimation216
5.4.3 Testing statistics218
5.5 Empirical studies223
5.5.1 An overview223
5.5.2 Fama-French portfolios224
5.5.3 Further remarks227
5.6 Cross-sectional regression227
5.7 Portfolio optimization without a risk-free asset228
5.8 CAPM with unknowing risk free rate236
5.8.1 Validating the Black version of CAPM237
5.8.2 Testing statistics237
5.9 Complements240
5.9.1 Proof of5.43240
5.9.2 Proof of5.48240
5.10 Exercises241
Chapter6 Factor Pricing Models245
6.1 Multifactor pricing models245
6.1.1 Multifactor models245
6.1.2 Factor pricing models249
6.2 Applications of multifactor models250
6.3 Model validation with tradable factors251
6.3.1 Existence of a risk-free asset252
6.3.2 Estimation of risk premia252
6.3.3 Testing statistics253
6.3.4 An empirical study using Fama-French portfolios256
6.3.5 Absence of a risk-free asset258
6.4 Macroeconomic variables as
內容試閱：
Chapter 1
Asset Returns The primary goal of investing in a -nancial market is to make pro-ts without taking excessive risks. Most common investments involve purchasing -nancial assets such as stocks， bonds or bank deposits， and holding them for certain periods. Posi- tive revenue is generated if the price of a holding asset at the end of holding period is higher than that at the time of purchase for the time being we ignore transaction charges. Obviously the size of the revenue depends on three factors: i the initial capital i.e. the number of assets purchased， ii the length of holding period， and iii the changes of the asset price over the holding period. A successful investment pursues the maximum revenue with a given initial capital， which may be measured explicitly in terms of the so-called return . A return is a percentage de-ned as the change of price expressed as a fraction of the initial price. It turns out that asset returns exhibit more attractive statistical properties than asset prices themselves.
Therefore it also makes more statistical sense to analyze return data rather than price series.
1.1 Returns
Let Pt denote the price of an asset at time t. First we introduce various de-nitions for the returns for the asset.
1.1.1 One-period simple returns and gross returns
Holding an asset from time t ? 1 to t， the value of the asset changes from Pt?1 to Pt. Assuming that no dividends paid are over the period. Then the one-period simple return is de-ned as
It is the pro-t rate of holding the asset from time t ? 1 to t. Often we write Rt = 100Rt%， as 100Rt is the percentage of the gain with respect to the initial capital Pt?1. This is particularly useful when the time unit is small such as a day or an hour; in such cases Rt typically takes very small values. The returns for lessrisky assets such as bonds can be even smaller in a short period and are often quoted in basis points ， which is 10; 000Rt. The one period gross return is de-ned as Pt=Pt?1 = Rt +1. It is the ratio of the new market value at the end of the holding period over the initial market value. 1.1.2 Multiperiod returns
The holding period for an investment may be more than one time unit. For any integer k 1， the returns for over k periods may be de-ned in a similar manner.
For example， the k-period simple return from time t ? k to t is and the k-period gross return is Pt=Pt?k = Rtk + 1. It is easy to see that the multiperiod returns may be expressed in terms of one-period returns as follows:
If all one-period returns Rt; ;Rt?k+1 are small， 1.3 implies an approximation
This is a useful approximation when the time unit is small such as a day， an hour or a minute.
1.1.3 Log returns and continuously compounding
In addition to the simple return Rt， the commonly used one period log return is
de-ned as
Note that a log return is the logarithm with the natural base of a gross return and log Pt is called the log price. One immediate convenience in using log returns is that the additivity in multiperiod log returns， i.e. the k period log return rtk ?
logPt=Pt?k is the sum of the k one-period log returns:
An investment at time t ? k with initial capital A yields at time t the capitalwhere ?r = rt +rt?1 +? ? ?+rt?k+1=k is the average one-period log returns. In this book returns refer to log returns unless speci-ed otherwise.
Note that the identity 1.6 is in contrast with the approximation 1.4 which is only valid when the time unit is small. Indeed when the values are small， the two returns are approximately the same:
However， rt Rt. Figure 1.1 plots the log returns against the simple returns for the Apple Inc share prices in the period of January 1985 { February 2011. The returns are calculated based on the daily close prices for the three holding periods: a day， a week and a month. The -gure shows that the two de-nitions result almost the same daily returns， especially for those with the values between ?0.2 and 0.2. However when the holding period increases to a week or a month， the discrepancy between the two de-nitions is more apparent with a simple return always greater than the corresponding log return.
Figure 1.1 Plots of log returns against simple returns of the Apple Inc share prices in January 1985 { February 2011. The blue straight lines mark the positions where the two returns are identical. The log return rt is also called continuously compounded return due to its close link with the concept of compound rates or interest rates . For a bank deposit
account， the quoted interest rate often refers to as `simple interest ''. For example， an interest rate of 5% payable every six months will be quoted as a simple interest of 10% per annum in the market. However if an account with the initial capital $1 is held for 12 months and interest rate remains unchanged， it follows from 1.2 that the gross return for the two periods is i.e. the annual simple return is 1:1025 ? 1 = 10:25%， which is called the compoundreturn
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