Financial Econometrics Slides-01: Return Properties -Introduction
Preliminaries Financial Jargon: Definitions Asset Return What is your investment plan? Recap!
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Preliminaries Financial Jargon: Definitions Asset Return What is your investment plan? Recap!
Financial Econometrics
Slides-01: Return Properties -Introduction
School of Economics
Preliminaries Financial Jargon: Definitions Asset Return What is your investment plan? Recap!
Introduction
Financial time series (FTS) analysis is concerned with theory and practice of
asset valuation over time.
Comparison with other Time Series analysis: similarity and difference? Highly
related, but with some added uncertainty, because FTS must deal with the
ever-changing business & economic environment and the fact that volatility is
not directly observed. Objective of the course
• to learn ways to get financial information from web directly and to process
the information.
• to provide some basic knowledge of financial time series data such as
skewness, heavy tails, and measure of dependence between asset returns
• to introduce some statistical tools & econometric models useful for
analyzing these series.
Preliminaries Financial Jargon: Definitions Asset Return What is your investment plan? Recap!
Examples of kind of problems you can solve
• Testing whether financial markets are weak-form informationally efficient.
• Testing whether the CAPM or APT represent superior models for the
determination of returns on risky assets.
• Measuring and forecasting the volatility of bond returns.
• Explaining the determinants of bond credit ratings used by the ratings
• Modelling long-term relationships between prices and exchange rates
• Testing technical trading rules to determine which makes the most money.
• Testing the hypothesis that earnings or dividend announcements have no
effect on stock prices.
• Testing whether spot or futures markets react more rapidly to news.
• Forecasting the correlation between the returns to the stock indices of two
countries.
Preliminaries Financial Jargon: Definitions Asset Return What is your investment plan? Recap!
What are special characteristics of Financial Data?
1 Frequency & quantity of data
• Stock market prices are measured every time there is a trade or somebody
posts a new quote.
• Recorded asset prices are usually those at which the transaction took place.
No possibility for measurement error but financial data are noisy.
Preliminaries Financial Jargon: Definitions Asset Return What is your investment plan? Recap!
Examples of FTS
• to analyze high-dimensional asset returns, including co-movement
Examples of financial time series
1. Daily log returns of Apple stock: 2004 to 2013 (10 years)
2. The VIX index
3. CDS spreads: Daily 3-year CDS spreads of JP Morgan from July
20, 2004 to September 19, 2014.
4. Quarterly earnings of Coca- : 1983-2009
Seasonal time series useful in
• earning forecasts
• pricing weather related derivatives (e.g. energy)
• modeling intraday behavior of asset returns
5. US monthly interest rates (3m & 6m Treasury bills)
Relations between the two series? Term structure of interest
6. Exchange rate between US Dollar vs Euro
Fixed income, hedging, carry trade
7. Size of insurance claims
Values of fire insurance claims (×1000 Krone) that exceeded 500
from 1972 to 1992.
8. High-frequency financial data:
Tick-by-tick data of Caterpillars stock: January 04, 2010.
Preliminaries Financial Jargon: Definitions Asset Return What is your investment plan? Recap!
2004 2006 2008 2010 2012 2014
Daily log returns of Apple stock
Figure 1: Daily log returns of Apple stock from 2004 to 2013
2006 2008 2010 2012 2014
CDS of JPM: 3−yr spread
Figure 3: Time plot of daily 3-year CDS spreads of JPM: from July 20, 2004 to September
Preliminaries Financial Jargon: Definitions Asset Return What is your investment plan? Recap!
VIXCLS [2004−01−02/2014−03−07]
Last 14.11
Jan 02 2004 Jan 03 2007 Jan 04 2010 Jan 02 2013
Figure 4: CBOE Vix index: January 2, 2004 to March 7, 2014.
1985 1990 1995 2000 2005 2010
EPS of Coca Cola: 1983−2009
Figure 5: Quarterly earnings per share of Coca-
Preliminaries Financial Jargon: Definitions Asset Return What is your investment plan? Recap!
2000 2002 2004 2006 2008 2010
Dollars per Euro
Figure 6: Daily Exchange Rate: Dollars per Euro
2000 2002 2004 2006 2008 2010
ln−rtn: US−EU
Figure 7: Daily log returns of FX (Dollar vs Euro)
Preliminaries Financial Jargon: Definitions Asset Return What is your investment plan? Recap!
What Is a Stock?
What is a stock?
A stock (also known as equity) is a security that represents the ownership of a
fraction of a corporation. This entitles the owner of the stock to a proportion
of the corporation’s assets and profits equal to how much stock they own.
Units of stock are called shares.
• Stocks are bought and sold predominantly on stock exchanges, though
there can be private sales as well, and they are the foundation of nearly
every portfolio.
• Historically, they have outperformed most other investments over the long
run. Historical returns
https://pages.stern.nyu.edu/~adamodar/New_Home_Page/datafile/histretSP.html
Preliminaries Financial Jargon: Definitions Asset Return What is your investment plan? Recap!
Assets and equities
What is a financial asset?
Financial assets represent investments in the assets and securities of other
institutions. Financial assets include stocks, sovereign and corporate bonds,
preferred equity, and other hybrid securities. Financial assets are valued
depending on how the investment is categorized and the motive behind it.
What Is Equity?
Equity, typically referred to as shareholders’ equity (or owners’ equity for
privately held companies), represents the amount of money that would be
returned to a company’s shareholders if all of the assets were liquidated and all
of the company’s debt was paid off in the case of liquidation. In the case of
acquisition, it is the value of company sales minus any liabilities owed by the
company not transferred with the sale.
Preliminaries Financial Jargon: Definitions Asset Return What is your investment plan? Recap!
More Definitions…
What Is Earnings Per Share (EPS)?
Earnings per share (EPS) is calculated as a company’s profit divided by the
outstanding shares of its common stock. The resulting number serves as an
indicator of a company’s profitability. It is common for a company to report
EPS that is adjusted for extraordinary items and potential share dilution.
The higher a company’s EPS, the more profitable it is considered to be.
Example: Exxon Mobil Investopia Example: Exxon Mobil Yahoo Finance
https://www.investopedia.com/terms/e/equity.asp
https://www.investopedia.com/markets/quote?tvwidgetsymbol=xom
Preliminaries Financial Jargon: Definitions Asset Return What is your investment plan? Recap!
Asset Returns: Definition
Let Pt be the price of an asset at time t, and assume no dividend.
• One-period simple return:
Gross return
Pt = Pt−1(1 +Rt)
Simple return:
• Multiperiod simple return
1 +Rt(k) =
= (1 +Rt)(1 +Rt−1) · · · (1 +Rt−k+1)
j=0 (1 +Rt−j)
Preliminaries Financial Jargon: Definitions Asset Return What is your investment plan? Recap!
Asset Returns: Example
Table below gives five daily closing prices of Apple stock in December 2011.
12/8 to 12/9 1 + Rt = 393.62/390.66 ≈ 1.0076 so that the daily
simple return is 0.76%, which is (393.62− 390.66)/390.66.
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
Time interval is important! Default is one year.
Annualized (average) return:
Annualized[Rt(k)] =
(1 + Rt−j)
An approximation:
Annualized[Rt(k)] ≈
Continuously compounding: Illustration of the power of compound-
ing (int. rate 10% per annum)
Type #(payment) Int. Net
Annual 1 0.1 $1.10000
Semi-Annual 2 0.05 $1.10250
Quarterly 4 0.025 $1.10381
Monthly 12 0.0083 $1.10471
Weekly 52 0.1
Daily 365 0.1
Continuously ∞ $1.10517
A = C exp[r × n]
where r is the interest rate per annum, C is the initial capital, n is
the number of years, and exp is the exponential function.
• The 1-day simple return of holding the stock from 12/8 to 12/9:
• The 3-day simple return for holding the stock from 12/02 to 12/07:
• The 5-day simple return for holding the stock from 12/02 to 12/09:
Preliminaries Financial Jargon: Definitions Asset Return What is your investment plan? Recap!
Annulalized Asset Returns
12/8 to 12/9 1 + Rt = 393.62/390.66 ≈ 1.0076 so that the daily
simple return is 0.76%, which is (393.62− 390.66)/390.66.
Date 12/02 12/05 12/06 12/07 12/08 12/09
Price($) 389.70 393.01 390.95 389.09 390.66 393.62
Time interval is important! Default is one year.
Annualized (average) return:
Annualized[Rt(k)] =
(1 + Rt−j)
An approximation:
Annualized[Rt(k)] ≈
Continuously compounding: Illustration of the power of compound-
ing (int. rate 10% per annum)
Type #(payment) Int. Net
Annual 1 0.1 $1.10000
Semi-Annual 2 0.05 $1.10250
Quarterly 4 0.025 $1.10381
Monthly 12 0.0083 $1.10471
Weekly 52 0.1
Daily 365 0.1
Continuously ∞ $1.10517
A = C exp[r × n]
where r is the interest rate per annum, C is the initial capital, n is
the number of years, and exp is the exponential function.
Preliminaries Financial Jargon: Definitions Asset Return What is your investment plan? Recap!
Annulalized Asset Returns
Present value:
C = A exp[−r × n]
Continuously compounded (or log) return
rt = ln(1 + Rt) = ln
= pt − pt−1,
where pt = ln(Pt).
Multiperiod log return:
rt(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)
= rt + rt−1 + · · · + rt−k+1.
Example Consider again the Apple stock price.
1. What is the log return from 12/8 to 12/9:
A: rt = ln(393.62)− ln(390.66) = 7.5%.
2. What is the log return from day 12/2 to 12/9?
A: rt(4) = ln(393.62)− ln(389.7) = 1%.
Portfolio return: N assets
Example: An investor holds stocks of IBM, Microsoft and Citi-
Group. Assume that her capital allocation is 30%, 30% and 40%.
Use the monthly simple returns in Table 1.2 of the text. What is the
mean simple return of her stock portfolio?
Preliminaries Financial Jargon: Definitions Asset Return What is your investment plan? Recap!
Annulalized Asset Returns
Example Consider again the Apple stock price.
• What is the log return from 12/8 to 12/9?
• What is the log return from day 12/2 to 12/9?
• What is the log return from day 12/6 to 12/8?
Preliminaries Financial Jargon: Definitions Asset Return What is your investment plan? Recap!
Market Index and Return
• Market index: Pm,t =
witPi,t, t = 1, 2, · · ·
weight wit depends on outstanding shares of stock i, etc
• Log return:
rm,t = 100%× ln
Topic 1. Features of Some Financial Time Series
• Financial time series
– Market index and return
• Market index: � ,� = ∑ “#,��#,�
#%� , � = 1,2…
weight “#,� depends on outstanding shares of stock &, etc
• Log return: � ,� = 100% × ln
eg. S&P/ASX200 Index and Return
02 03 04 05 06 07 08 09 10 11
RETURN PRICE
UNSW Business School,
Slides-01, Financial Econometrics 10
Preliminaries Financial Jargon: Definitions Asset Return What is your investment plan? Recap!
Portfolio Return
An investor holds stocks of IBM, Microsoft and Citi- Group. Assume that her
capital allocation is 30%, 30% and 40%. What is the mean simple return of
her stock portfolio?
Assume monthly simple returns for IBM, microsoft and Citi-Group, 1.35%,
2.62% and 1.17% respectively.
• Portfolio Return: Rp,t =
witRi,t, t = 1, 2, · · · , where N is the
number of assets held by investor and wit is wealth allocation.
Preliminaries Financial Jargon: Definitions Asset Return What is your investment plan? Recap!
Adjusted Returns
1 Adjusting for dividends (Total Returns)
rt = ln (1 +Rt) = ln
= ln(Pt +Dt)− ln(Pt−1)
2 Adjusting for inflation (Real Returns)
3 Adjusting for Risk (Excess Returns)
Zt = Rt −Rft
zt = ln(Zt) = ln(Rt −Rft) ̸= rt − rft
BUT if Zt is small: zt ≈ rt − rft
Preliminaries Financial Jargon: Definitions Asset Return What is your investment plan? Recap!
Dividends, Excess returns
Answer: E(Rt) = 0.3× 1.35 + 0.3× 2.62 + 0.4× 1.17 = 1.66.
Dividend payment:
− 1, rt = ln(Pt + Dt)− ln(Pt−1).
Excess return: (adjusting for risk)
Zt = Rt −R0t, zt = rt − r0t
where r0t denotes the log return of a reference asset (e.g. risk-free
interest rate).
Relationship:
rt = ln(1 + Rt), Rt = e
If the returns are in percentage, then
rt = 100× ln(1 +
), Rt = [exp(rt/100)− 1]× 100.
Temporal aggregation of the returns produces
1 + Rt(k) = (1 + Rt)(1 + Rt−1) · · · (1 + Rt−k+1),
rt(k) = rt + rt−1 + · · · + rt−k+1.
These two relations are important in practice, e.g. obtain annual
returns from monthly returns.
Example: If the monthly log returns of an asset are 4.46%, −7.34%
and 10.77%, then what is the corresponding quarterly log return?
Answer: 4.46− 7.34 + 10.77 = 7.89%.
Example: If the monthly simple returns of an asset are 4.46%,
−7.34% and 10.77%, then what is the corresponding quarterly simple
Answer: R = (1+0.0446)(1−0.0734)(1+0.1077)−1 = 1.0721−1
= 0.0721 = 7.21%
Preliminaries Financial Jargon: Definitions Asset Return What is your investment plan? Recap!
If the monthly log returns of an asset are 4.46%, −7.34% and 10.77%, then
1 what is the corresponding quarterly log return?
2 what is the corresponding quarterly simple return?
(1) Quarterly log return:
4.46− 7.34 + 10.77 = 7.89%.
(2) Quarterly simple return:
R = (1 + 0.0446)(1− 0.0734)(1 + 0.1077)− 1
= 1.0721− 1
= 0.0721 = 7.21%
Preliminaries Financial Jargon: Definitions Asset Return What is your investment plan? Recap!
How to inverst $100?
How much will you inverst?
https://www.investopedia.com/articles/investing/100615/investing-100-month-stocks-30-years.asp
Preliminaries Financial Jargon: Definitions Asset Return What is your investment plan? Recap!
https://app.sli.do/event/2646HUC4YsFip27iyVgzk2
Preliminaries
Financial Jargon: Definitions
Asset Return
Simple/Gross Return
Annulaized/Compounded Retun
Portfolio Return
Adjusted return
What is your investment plan?
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