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ECMT2130
Financial Econometrics:
Introduction to Financial Econometrics
Sinan Deng
The University of Sydney
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Overview
Introduction to
Financial
Econometrics
Major Asset Markets
Market Efficiency
Efficient Market Hypothesis
Prices and Returns
Financial Market Predictability
The Random Walk Model
Covariance and Correlation
Expectation and Variance Operator
Portfolio Return and Risk
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Major Asset Markets
1. Securities Markets
Ø Bond markets
Ø Equity markets
2. Foreign exchange markets
3. Derivatives markets
Ø Forwards
Ø Swaps
Ø Futures
Ø Options
4. Money markets
5. Commodity markets
6. Physical asset markets
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Market Efficiency
1. Allocative efficiency refers to the basic concept in economics known as
Pareto efficiency. Briefly, a Pareto efficient allocation is such that any
reallocation of resources that makes one or more individuals better off
results in at least one individual being made worse off.
2. Operational efficiency mainly concerns the industrial organization of
capital markets. That is, the study of operational efficiency examines
whether the services supplied by financial organizations (e.g. brokers,
dealers, banks and other financial intermediaries) are provided
according to the usual criteria of industrial efficiency (for example, such
that price equals marginal cost for the services rendered)
3. Informational efficiency refers to the extent that asset prices reflect the
information available to investors.
4. Portfolio efficiency is a narrower concept than the others. An efficient
portfolio is one such that the variance of the return on the portfolio is as
small as possible for any given level of expected return.
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Informational Efficiency
• Eugene Fama’s (1970) identified three levels of market efficiency:
1. Weak-form efficiency: Prices of the securities instantly and fully reflect
all information of the past prices. This means future price movements
cannot be predicted by using past prices, i.e past data on stock prices is
of no use in predicting future stock price changes.
2. Semi-strong-form efficiency: Asset prices fully reflect all of the publicly
available information. Therefore, only investors with additional inside
information could have an advantage in the market. Any price anomalies
are quickly found out and the stock market adjusts.
3. Strong-form efficiency: Asset prices fully reflect all of the public and
inside information available. Therefore, no one can have an advantage
in the market in predicting prices since there is no data that would
provide any additional value to the investors.
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Efficient Market Hypothesis
• Eugene Fama’s (1970) summary “A Market in which prices always ’fully
reflect’ available information is called ’efficient’.”
• The efficient market hypothesis (EMH) states that it is impossible for
investors to purchase undervalued stocks or sell stocks for inflated prices.
• No one could outperform the market by using the same information that is
already available to all investors, except through luck.
• Another theory related to the efficient market hypothesis created by Louis
Bachelier is the "random walk" theory, which states that prices in the
financial markets evolve randomly.
• Therefore, identifying trends or patterns of price changes in a market
can't be used to predict the future value of financial instruments.
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Gross and Net Returns
• The gross return of stocks at time t+1 is defined as
!"# = !"# + !"#!
where !"# is the price of the stock at time t+1 and !"# is the dividend at
time t+1.
• The arithmetic net return is
!"# = !"# − 1 = !"# + !"# − !!
• If we assume the stock does not pay dividend, then
!"# = $!"#$! and !"# = !"# − 1 = $!"#%$!$!
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Log Returns
• The log return is defined as
!"#& = ln !"# = ln !"#! = !"# − !
• Note that !"#& = ln !"# = ln 1 + !"# ≈ !"#
when !"# is small.
• Therefore, the log return is approximately net return, when the net return
is “small”.
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Multi-Period Returns
• Suppose we hold the asset for k periods from time t to t + k, the k-period
gross return is !→!"( = !"(!
• Then it follows that !→!"( = !"(! = !"(!"(%#×!"(%#!"(%)×⋯×!"#!= !"(!"(%#…!"# =1*+#( !"*
• The k-period gross return !→!"( is the product of the k one-period gross
return: The multi-period return is often called a compound return.
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Compound Interest
• Suppose Sinan deposits $100 into the bank, and the interest rate of the
deposit is 5% per annum.
• If the bank pays interest annually, the end-of-year value of the deposit is $100×(1 + 5%) = $105.
• If the bank pays interest semi-annually, then the end-of-year value of the
deposit is $100× 1 + ,%) ) = $105.0625.
• In general, if the bank pays interest n times a year, then the end-of-year
value of the deposit is $100× 1 + ,%. .
• What is the end-of-year value of the deposit if the 5% annual interest
rate is paid continuously, when → ∞? lim.→/$100× 1 + 5% . = $100×,%
• The continuous compounding will yield the end-of-year value $105.1271
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Compound Interest
• Let !"# be the gross return for an investment from t to t + 1. What is the
equivalent continuously compounded return? That is, what is the rate of
return !"# earned continuously over [, + 1] which yields the same
gross return !"#? We should have!"# = lim.→/ 1 + !"# . = 0!"#
or, !"# = ln !"#
• Log return is the continuously compounded return.
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Compound Interest
• “Compound interest is the eighth wonder of the
world. He who understands it, earns it; he who
doesn’t, pays it.” – Albert Einstein
• "Money makes money. And the money that
money makes makes money" - Benjamin Franklin
• Compound Interest - Compound interest is an
interest arrangement in which the amount of
interest that is accrued each period is
determined by the current balance, rather than
just the original principal.
• A key feature of compound interest is that you
make interest on interest. While the interest
"rate" stays the same, your principal each period
changes.
• https://www.youtube.com/watch?v=qDxDCtZ9U
kE
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Financial Prices
• Financial prices are not predictable.
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Financial Returns
• Financial returns are predictable.
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Expectation and Variance Operators
• The expectation operator takes a random variable and gives you its
average value; the variance operator takes a random variable and gives
you its variance.
Expectation
Operator
• =
• + = +
• =
• + = +
• + = + ()
• + = + ()
Variance
Operator
• = 0
• + =
• = )
• + = )
• + = + + 2(, )
• + = ) + ) +2(, )
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The Random Walk Model
• Given a conditioning set F which contains known information, we can
define the conditional distribution of a random variable X conditional on
F. Conditional moments are based on the conditional distribution.
• Let ℱ! = !, !%#, … = !, !%#, … to be the information set at time t.
• We form the forecasts of future prices or returns conditional on the
information set ℱ!. For example, one step ahead price forecast is (!"#|ℱ!). Two step ahead return forecast is (!")|ℱ!).
• A famous model for asset returns is the random walk model. Let ! be log
return and ! be log price.!"# = ! + !"#
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The Random Walk Model
• Depending on the dependence structure in ! , we have different
versions of random walk models. The most restrictive one is to assume ! is an i.i.d. sequence. The least restrictive one is to assume ! is a
white noise. While the most famous one is to assume ! is a “martingale
difference sequence” (m.d.s.),(!"#|ℱ!) = 0, ∀
• In this case, the log price sequence ! is called a “martingale”. This
model is consistent with the Efficient Market Hypothesis, as(!"#|ℱ!) = (!"# − !|ℱ!) = !"# ℱ!) − ! = 0 !"# ℱ!) = !
• The best forecast of tomorrow’s price is today’s price.
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Covariance and Correlation
• Suppose we have two different variables. How do we measure whether
these variables are related or associated with one another? Two
measures for doing this are the covariance and correlation.
• For two jointly distributed real-valued random variables X and Y with
finite second moments, the covariance is defined as the expected
value (or mean) of the product of their deviations from their individual
expected values.12 = , = − − = − []
• If the (real) random variable pair , can take on the
values 3, 3 for = 1,… , with equal probabilities 3 = #., then the
covariance can be equivalently written in terms of the
means , = #.∑3+#. (3 − ())(3 − ())
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Covariance and Correlation
• The population correlation coefficient 12 between two random
variables X and Y with expected values 1 and 2 and standard
deviations 1 and 2 is defined as:
12 = , = (, )12
• The closer the coefficient is to either −1 or 1, the stronger the correlation
between the variables.
• If the variables are independent, correlation coefficient is 0, but the
converse is not true because the correlation coefficient detects only linear
dependencies between two variables.