data). Generally, tick-by-tick data are not regularly spaced in time,

which leads to additional challenges for high-frequency data analysis

[1, 2]. Current research of financial data is overwhelmingly conducted

on the homogeneous grids that are defined with filtering and aver-

aging tick-by-tick data.

Another problem that complicates analysis of long financial time

series is seasonal patterns. Business hours, holidays, and even daylight

saving time shifts affect market activity. Introducing the dummy

variables into time series models is a general method to account for

seasonal effects (see Section 5.2). In another approach, ‘‘operational

time’’ is employed to describe the non-homogeneity of business activ-

ity [2]. Non-trading hours, including weekends and holidays, may be

cut off from operational time grids.

2.2 RETURNS AND DIVIDENDS

2.2.1 S

IMPLE AND COMPOUNDED RETURNS

While price P is the major financial variable, its logarithm,

p ¼ log (P) is often used in quantitative analysis. The primary reason

for using log prices is that simulation of a random price innovation

can move price into the negative region, which does not make sense.

In the mean time, negative logarithm of price is perfectly acceptable.

Another important financial variable is the single-period return (or

simple return) R(t) that defines the return between two subsequent

moments t and t1. If no dividends are paid,

R(t) ¼ P(t)=P(t 1) 1(2:2:1)

Return is used as a measure of investment efficiency.

1

Its advantage is

that some statistical properties, such as stationarity, may be more

applicable to returns rather than to prices [3]. The simple return of a

portfolio, R

p

(t), equals the weighed sum of returns of the portfolio

assets

R

p

(t) ¼

X

N

i¼1

w

ip

R

ip

(t),

X

N

i¼1

w

ip

¼ 1, (2:2:2)

where R

ip

and w

ip

are return and weight of the i-th portfolio asset,

respectively; i ¼ 1, ...,N.

Financial Markets 7

The multi-period returns, or the compounded returns, define the

returns between the moments t and t k þ 1. The compounded

return equals

R(t, k) ¼ [R(t) þ 1] [R(t 1) þ 1] ...[R(t k þ 1) þ 1] þ 1

¼ P(t)=P(t k) þ 1(2:2:3)

The return averaged over k periods equals

ˇ

R(t, k) ¼

Y

k1

i¼0

(R(t i) þ 1)

"#

1=k

1(2:2:4)

If the simple returns are small, the right-hand side of (2.2.4) can be

reduced to the first term of its Taylor expansion:

ˇ

R(t, k)

1

k

X

k1

i¼1

R(t, i) (2:2:5)

The continuously compounded return (or log return) is defined as:

r(t) ¼ log [R(t) þ 1] ¼ p(t) p(t 1) (2:2:6)

Calculation of the compounded log returns is reduced to simple

summation:

r(t, k) ¼ r(t) þ r(t 1) þ ...þ r(t k þ 1) (2:2:7)

However, the weighing rule (2.2.2) is not applicable to the log returns

since log of sum is not equal to sum of logs.

2.2.2 DIVIDEND EFFECTS

If dividends D(t þ 1) are paid within the period [t, t þ 1], the simple

return (see 2.2.1) is modified to

R(t þ 1) ¼ [P(t þ 1) þ D(t þ 1) ]=P(t) 1(2:2:8)

The compounded returns and the log returns are calculated in the

same way as in the case with no dividends.

Dividends play a critical role in the discounted-cash-flow (or pre-

sent-value) pricing model. Before describing this model, let us intro-

duce the notion of present value. Consider the amount of cash K

invested in a risk-free asset with the interest rate r. If interest is paid

8 Financial Markets

every time interval (say every month), the future value of this cash

after n periods is equal to

FV ¼ K(1 þ r)

n

(2:2:9)

Suppose we are interested in finding out what amount of money will

yield given future value after n intervals. This amount (present value)

equals

PV ¼ FV=(1 þ r)

n

(2:2:10)

Calculating the present value via the future value is called discounting.

The notions of the present value and the future value determine the

payoff of so-called zero-coupon bonds. These bonds sold at their

present value promise a single payment of their future value at ma-

turity date.

The discounted-cash-flow model determines the stock price via its

future cash flow. For the simple model with the constant return

E[R(t) ] ¼ R, one can rewrite (2.2.8) as

P(t) ¼ E[{P(t þ 1) þ D(t þ 1)}=(1 þ R)] (2:2:11)

If this recursion is repeated K times, one obtains

P(t) ¼ E

X

K

i¼1

D(t þ i)=(1 þ R)

i

"#

þ E[P(t þ K)=(1 þ R)

K

](2:2:12)

In the limit K !1, the second term in the right-hand side of (2.2.12)

can be neglected if

lim

K!1

E[P(t þ K)=(1 þ R)

K

] ¼ 0(2:2:13)

Then the discounted-cash-flow model yields

P

D

(t) ¼ E

X

1

i¼1

D(t þ i)=(1 þ R)

i

"#

(2:2:14)

Further simplification of the discounted-cash-flow model is based on

the assumption that the dividends grow linearly with rate G

E[D(t þ i) ] ¼ (1 þ G)

i

D(t) (2:2:15)

Then (2.2.14) reduces to

Financial Markets 9

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