"Frontmatter". In: Analysis of Financial Time Series

6 FINANCIAL TIME SERIES AND THEIR CHARACTERISTICSBecause the repayment requires equal shares rather than equal dollars, the short sellerbenefits from a decline in the price of the asset. If cash dividends are paid on the assetwhile a short position is maintained, these are paid to the buyer of the short sale. Theshort seller must also compensate the lender by matching the cash dividends from hisown resources. In other words, the short seller is also obligated to pay cash dividendson the borrowed asset to the lender; see Cox and Rubinstein (1985).Summary of RelationshipThe relationships between simple return R t and continuously compounded (or log)return r t arer t = ln(1 + R t ), R t = e r t− 1.Temporal aggregation of the returns produces1 + R t [k] =(1 + R t )(1 + R t−1 ) ···(1 + R t−k+1 ),r t [k] =r t + r t−1 +···+r t−k+1 .If the continuously compounded interest rate is r per annum, then the relationshipbetween present and future values of an asset isA = C exp(r × n),C = A exp(−r × n).1.2 DISTRIBUTIONAL PROPERTIES OF RETURNSTo study asset returns, it is best to begin with their distributional properties. Theobjective here is to understand the behavior of the returns across assets and overtime. Consider a collection of N assets held for T time periods, say t = 1,...,T .For each asset i, letr it be its log return at time t. The log returns under studyare {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 }.1.2.1 Review of Statistical Distributions and Their MomentsWe briefly review some basic properties of statistical distributions and the momentequations of a random variable. Let R k be the k-dimensional Euclidean space. Apoint in R k is denoted by x ∈ R k . Consider two random vectors X = (X 1 ,...,X k ) ′and Y = (Y 1 ,...,Y q ) ′ .LetP(X ∈ A, Y ∈ B) be the probability that X is in thesubspace A ⊂ R k and Y is in the subspace B ⊂ R q . For most of the cases consideredin this book, both random vectors are assumed to be continuous.