financial models i
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Finalcial Models 1TRANSCRIPT
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Risk Management and Financial Returns
Elements of Financial Risk Management
Chapter 1
Elements of Financial Risk Management
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Why should firms manage risk?Classic portfolio theory: Investors can
eliminate firm-specific risk by diversifying holdings to include many different assets
Investors should hold a combination of the risk-free asset and the market portfolio.
Firms should not waste resources on risk management, as investors do not care about firm-specific risk.
Modigliani-Miller: The value of a firm is independent of its risk structure.
Firms should simply maximize expected profits regardless of the risk entailed.
Elements of Financial Risk Management
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Evidence on RM practices
Large firms tend to manage risk more actively than small firms, which is perhaps surprising as small firms are generally viewed to be more risky.
However smaller firms may have limited access to derivatives markets and furthermore lack staff with risk management skills.
Elements of Financial Risk Management
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Does RM improve Firm Performance?The overall answer to this question appears
to be YES.Analysis of the risk management practices
in the gold mining industry found that share prices were less sensitive to gold price movements after risk management.
Similarly, in the natural gas industry, better risk management has been found to result in less variable stock prices.
A study also found that RM in a wide group of firms led to a reduced exposure to interest rate and exchange rate movements.
Elements of Financial Risk Management
5Does RM improve Firm Performance?
Researchers have found that less volatile cash flow result in lower costs of capital and more investment.
A portfolio of firms using RM outperformed a portfolio of firms that did not, when other aspects of the portfolio were controlled for.
Similarly, a study found that firms using foreign exchange derivatives had higher market value than those who did not.
The evidence so far paints a fairly rosy picture of the benefits of current RM practices in the corporate sector.
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Can RM Techniques be improved?Evidence on the RM systems in some of the
largest U.S. commercial banks is less cheerful.On average risk forecasts tend to be overly
conservative-perhaps a virtue-but at certain times the realized losses far exceeded the risk forecasts. Importantly, the excessive losses tended to occur on consecutive days.
One is able to forecast an excessive loss tomorrow based on the observation of an excessive loss today.
This serial dependence unveils a potential flaw in current financial sector RM practices.
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A brief Taxonomy of RisksMarket Risk: the risk to a financial portfolio
from movements in market prices such as equity prices, foreign exchange rates, interest rates and commodity prices.
In financial sector firms market risk should be managed. E.g. option trading desk.
In nonfinancial firms market risk should perhaps be eliminated.
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Defining Asset ReturnsThe daily geometric or “log” return on an
asset is defined as
The arithmetic return is instead defined as
Elements of Financial Risk Management
ttt SSR lnln 11
1111 tttttt SSSSSr
9Defining Returns
The two returns are typically fairly similar, as can be seen from
the approximation holds because
whenis close to 1. One advantage of the log return is that we can
easily calculate the compounded return for K-days
Elements of Financial Risk Management
11111 1lnlnlnln ttttttt rrSSSSR
1)ln( xxx
K
k
K
kktktkttKtKtt RSSSSR
1 11:1 )ln()ln()ln()ln(
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Exercise Asset ReturnsOpen file Data Chapter2Data.xlsCompute the returns for Close (S&P 500
price) and VIX (S&P 500 volatility)
Answer
Elements of Financial Risk Management
ttt SSR lnln 11
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Stylized Facts of Asset Returns
We can consider the following list of so-called stylized facts which apply to most stochastic returns.
Each of these facts will be discussed in detail in the first part of the book.
We will use daily returns on the S&P500 from 1/1/97 to 12/31/01 to illustrate each of the features.
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Stylized Fact 1Daily returns have very little autocorrelation.
We can write
Returns are almost impossible to predict from their own past.
Fig 1.1 shows the correlation of daily S&P500 returns with returns lagged from one to 100 days.
We will take this as evidence that the conditional mean is roughly constant.
Elements of Financial Risk Management
1,2,3,...for 0, 11 tt RRCorr
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Exercise: Stylized Fact 1Using same data, compute autocorrelation
for S&P returns. Do it for lag 1 to 100.
Use function correl(). Use $D$4 type of writing to ensure upper cells remain still when you drag the formula!
Answer
Elements of Financial Risk Management
1,2,3,...for 0, 11 tt RRCorr
14Autocorrelations of Daily S&P Returns for Lags 1 through 100 1/1/97-12/31/01Figure 1.1
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Stylized Fact 2The unconditional distribution of daily returns
have fatter tail than the normal distribution. Fig.1.2 shows a histogram of the daily
S&P500 return data with the normal distribution imposed.
Notice how the histogram has longer and fatter tails, in particular in the left side, and how it is more peaked around zero than the normal distribution.
Fatter tails mean a higher probability of large losses than the normal distribution would suggest.
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Exercise: Stylized Fact 2Using same data and function frequency,
construct 11 equally spaced bins from min(return) to max(return), compute probabilities and graph the histogram
Remember to use CTRL+ALT+ENTER to “drag” formula
Answer
Elements of Financial Risk Management
17Figure 1.2Histogram of Daily S&P 500 Returns and the Normal DistributionJan 1, 2001 - Dec 31, 2010
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Stylized Fact 3The stock market exhibits occasional, very
large drops but not equally large up-moves. Consequently the return distribution is
asymmetric or negatively skewed. This is clear from Figure 1.2 as well.
Other markets such as that for foreign exchange tend to show less evidence of skewness.
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Stylized Fact 4The standard deviation of returns
completely dominates the mean of returns at short horizons such as daily.
It is typically not possible to statistically reject a zero mean return.
Our S&P500 data has a daily mean of 0.0353% and a daily standard deviation of 1.2689%.
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Exercise: Stylized Fact 4Compute mean and standard deviation of
the returns. Use function average() and stdev()
Answer:◦Daily mean of 0.0353% ◦Daily standard deviation of 1.2689%.
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Stylized Fact 5Variance measured for example by squared
returns, displays positive correlation with its own past.
This is most evident at short horizons such as daily or weekly.
Fig 1.3 shows the autocorrelation in squared returns for the S&P500 data, that is
• Models which can capture this variance dependence will be presented in Chapters 4 & 5
Elements of Financial Risk Management
smallfor ,0),( 21
21 tt RRCorr
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Exercise: Stylized Fact 5Compute returns squared for S&P500Compute autocorrelation (Use same
formulas than before… copy-paste)
Answer:
Elements of Financial Risk Management
smallfor ,0),( 21
21 tt RRCorr
23Figure 1.3Autocorrelation of Squared Daily S&P 500 ReturnsJan 1, 2010 - Dec 31, 2010
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Stylized Fact 6Equity and equity indices display negative
correlation between variance and returns.This often termed the leveraged effect,
arising from the fact that a drop in stock price will increase the leverage of the firm as long as debt stays constant. (Lvg=D/E)
This increase in leverage might explain the increase variance associated with the price drop. We will model the leverage effect in Chapters 4 and 5.
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Exercise: Stylized Fact 6Using S&P 500 returns and VIX returns,
compute the correlation using 250 days of data. Then compute the moving correlation with 250 days and graph it.
Answer:
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Stylized Fact 7
Correlation between assets appears to be time varying.
Importantly, the correlation between assets appear to increase in highly volatile down-markets and extremely so during market crashes.
We will model this important phenomenon in Chapter 7.
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Stylized Fact 8
Even after standardizing returns by a time-varying volatility measure, they still have fatter than normal tails.
We will refer to this as evidence of conditional non-normality.
It will be modeled in Chapters 6 and 9.
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Stylized Fact 9
As the return-horizon increases, the unconditional return distribution changes and looks increasingly like the normal distribution.
Issues related to risk management across horizons will be discussed in Chapter 8.
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Exercise: Stylized Fact 9Compute returns with lag 5, 10 & 15 for
the data.Hint: Use the copy-paste function “smartly”Answer:
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A generic model of asset returnsBased on the above of stylized facts our model of
individual asset returns will take the generic form
• The conditional mean return is thus μt+1 and the conditional variance
• The random variable zt+1 is an innovation term, which we assume is identically and independently distributed (i.i.d.) as D(0,1).
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A generic model of asset returns JP Morgan’s RiskMetrics model for dynamic volatility
• The volatility for tomorrow, time t+1, is computed at the end of today, time t, using the following simple updating rule:
• On the first day of the sample, t = 0, the volatility can be set to the sample variance of the historical data available.
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From asset returns to portfolio returns• The value of a portfolio with n assets at time t is the
weighted average of the asset prices using the current holdings of each asset as weights:
• The return on the portfolio between day t+1 and day t is then defined as when using arithmetic returns
• When using log returns return on the portfolio is:
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Introducting the VaR risk measureValue-at-Risk - What loss is such that it will only be
exceeded p·100% of the time in the next K trading days?
VaR is allways defined in dollars, denoted by $VaR $VaR loss is implicitly defined from the probability of
getting an even larger loss as in
Note by definition that (1−p)100% of the time, the $Loss will be smaller than the VaR.
Also note that for this course we will use VaR based on log returns defined as
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Introducing the VaR risk measureNow we are (1−p)100% confident that we will get a
return better than −VaR. Knowing that the $VaR of a portfolio is $500,000
does not mean much unless we know the value of the portfolio
The two VaRs are related as follows:
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Introducing the VaR risk measureSuppose our portfolio consists of just one securityFor example an S&P 500 index fundNow we can use the Risk-Metrics model to provide the VaR
for the portfolio. Let VaRP
t+1 denote the p .100% VaR for the 1-day ahead return, and assume that returns are normally distributed with zero mean and standard deviation sPF,t+1. Then:
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Introducing the VaR risk measureTaking -1() on both sides of the preceding equation
yields the VaR as
If we let p = 0.01 then we get -1P= -1
0.01= -2.33
If we assume the standard deviation forecast, sPF,t+1 for tomorrow’s return is 2.5% then:
If the value of the portfolio today is $2 million, the $VaR would simply be
.51
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37Figure 1.4Value at Risk (VaR) from the Normal DistributionReturn Probability Distribution
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Portfolio Return
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1-day 1% VaR = 0.05825
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Introducing the VaR risk measureConsider a portfolio whose value consists of 40
shares in Microsoft (MS) and 50 shares in GE. To calculate VaR for the portfolio, collect historical
share price data for MS and GE and construct the historical portfolio pseudo returns
Constructing a time series of past portfolio pseudo returns enables us to generate a portfolio volatility series using for example the RiskMetrics approach where
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Introducing the VaR risk measureWe can now directly model the volatility of the portfolio
return, RPF,t+1, call it sPF,t+1, and then calculate the VaR for the portfolio as
• We assume that the portfolio returns are normally distributed
40Figure 1.51-day, RiskMetrics 1% VaR in S&P500 PortfolioJan 1, 2001 - Dec 31, 2010
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Drawbacks of VaRExtreme losses are ignored - The VaR number only tells
us that 1% of the time we will get a return below the reported VaR number, but it says nothing about what will happen in those 1% worst cases.
VaR assumes that the portfolio is constant across the next K days, which is unrealistic in many cases when K is larger than a day or a week.
Finally, it may not be clear how K and p should be chosen.