CS代写 FM321: Risk Management and Zhu

Leture 1: Introduction
FM321: Risk Management and Zhu
27 September 2022
LSE Finance

Lecturer: Dr Linyan Professor of Finance; PhD Economics, UCSD Oﬀice hour: Thu 14:00 – 15:00 in MAR 7.22

Lecturer: Dr Linyan Professor of Finance; PhD Economics, UCSD Oﬀice hour: Thu 14:00 – 15:00 in MAR 7.22
Class teachers:
Dr Mesut Tastan Oﬀice hour: Wed 12:30 – 13:30 in MAR 6.38
Oﬀice hour: Mon 15:30 – 16:30 in MAR 6.38

Main topics
Introduction to financial returns and risks Volatility as a risk measure
Univariate volatility models Multivariate volatility models
Other risk measures beyond volatility Implementing risk forecasts Backtesting and stress-testing

There is no textbook that covers the topics in the course exactly the way I plan to.
Financial Risk Forecasting by Jon Danielsson is the main textbook (required), and closely approximates what we will do for most topics.
A list of readings is available in the course syllabus for those wishing to study the subject further.

Classes and formative assignments
A formative assignment will be given every week on Thursday.
The assignments will ask you to implement the techniques discussed
in the lectures.
They will not be graded, but they will be discussed in the classes the following week (no classes in Week 1).
Please make every effort to attempt them before class time. They will be based on R.

Assessment
5% First summative assignment,
available on Tue 18 Oct 2022 (Week 4) due on Sun 30 Oct 2022, 11:59pm
5% Second summative assignment
available on Tue 15 Nov 2022 (Week 8) due on Sun 27 Nov 2022, 11:59pm
Summative assignments follow the same format as formative ones.
40% Course project
available in Week 11
due in Jan 2023
50% In-class assessment (ICA) Tue 6 Dec 2022
closed-book
All course material is examinable unless stated otherwise.

Course Introduction
Key concepts: price and returns Stylized facts about financial returns

Price and returns
Suppose you have an equity. How to measure its performance?

Price and returns
Suppose you have an equity. How to measure its performance?
Measure of performance needs to take into account the percentage change of the value of financial assets from all sources.
For an equity, total return includes capital gain and dividend yield.

Price and returns
Time is discrete: t = 1,…,T.
Pt: price of an equity at the end of period t
Dt: dividend paid out from the equity during period t Two types of returns:
Simple (or arithmetic) returns:
Rt+1 = Pt+1 +Dt+1 −1 Pt
Compound (or geometric / logrithmic) returns:
rt+1 = log Pt+1+Dt+1 Pt
log: natural log (base e ≈ 2.7182818 . . . )

Simple vs. log return
On a day-to-day basis, simple returns are used for a number of purposes (e.g., accounting, or communicating with clients and investors)
In analytical work, logarithmic returns are preferred for data at high frequencies:
Log returns aggregate over time in a simple manner (e.g., the logarithmic return over one-month is the sum of the logarithmic return over the days in that month)
Log returns are symmetric
Log returns are not bounded below, so one can fit symmetric distributions to the data
In many derivatives pricing models, log returns are preferred for theoretical reasons

Simple vs log return
For small returns (typical with short periods), log returns approximate simple returns well.
Recall Taylor expansion:
log(1+x) ≈ x when x is small
rt+1 = log = log
Pt Pt+1+Dt+1 − 1 + 1
≈ Pt+1 +Dt+1 −1 Pt

For equities, prices are typically given ex-dividends, so those need to
be adjusted;
In general, need to adjust for a wide range of possible activities that can affect the computation of returns, such as:
Stock splits
Reverse splits
Stock Buybacks
Price data adjusted for these effects is usually readily available
For simplicity, from here on, we’ll assume that our price data is adjusted for the effects of dividends and corporate actions, so the terms referring to dividends will drop out of the formulas

Portfolio returns
N assets: n = 1,2,…,N
Pn,t: price of asset n at the end of period t wn: weight of asset n in the portfolio
Price of the portfolio:
Pt = Simple return of the portfolio:
Rt = However, not true for log returns:
rt = log Pt ̸= Pt−1
wnlog Pn,t Pn,t−1
XN

Equity indices
Show how the price of a representative portfolio of securities in a particular market or asset class has evolved over time
Multiple weighting schemes are used in practice:
Price-weighted (e.g., Dow Jones Industrial Average)
Market capitalization or market value weighted (e.g., S&P500)

S&P 500 Price Index

S&P 500 daily returns
Figure: Sample: 3 Jan 1950 to 8 Aug 2016

S&P 500 daily returns
Summary statistics:
Note how small the average is compared to the standard deviation!

Unit conversion
It is typically a good idea to express descriptive statistics (not the original data) on returns in a standard unit (percent per year)
With returns that are independent and additive, mean and variance of returns are proportional to the number of periods (T), so standard deviation grows at the rate √T.
We will use this convention in general. The usual factors are:
Fromdailytoannual: T=252(orT=250,orT=260) From monthly to annual: T = 12
From quarterly to annual: T = 4

Stylized facts
Volatility clusters
Time-varying correlations

Stylized fact 1

Volatility
Volatility, defined as the standard deviation of returns, is the most commonly used measure of risk
Distinction 1: Volatility can be
Unconditional volatility
Conditional volatility
q 2 σ= E rt−E(rt)
Distinction 2:
q 2 σt = Et−1 rt − Et−1(rt)
True population volatility: hypothetical quantity that is unknown. Sample unconditional volatility
vut1XT 2 1XT σˆ= T−1 rt− ̄r ̄r=T rt
t=1 t=1 Sample conditional volatility: GARCH etc.

Volatility
Daily volatility
σdaily =tT−1 Annualized volatility
σann = √252σdaily

Volatility clusters
Suppose we use the annualized volatility equation and calculate volatility over a decade
Then we see that volatility comes in many cycles

Volatility clusters
Calculate volatility year by year:
Both long-run and short run We call these volatility clusters

Volatility clusters
Shocks to volatility tends to be persistent, but eventually die out:

Correlation
Covariance measures the covariation of two series of returns. Sample covariance:
d t=1 Cov(rx, ry) =
PT (rx,t − ̄rx)(ry,t − ̄ry)
T − 1 Correlation is a standardized measure of covariation:
ρbx,y ≡Corr(rx,ry)=
It is always the case that
We presented the unconditional sample quantities. Can also study the population or the conditional quantities.
− 1 ≤ ρˆ x , y ≤ 1
Cov(rx, ry) σˆ x σˆ y

Autocorrelation
Measures how a time series is correlated with its own lagged values:
1lag: ρˆ1 =Corr(rt,rt−1) d
klag: ρˆk =Corr(rt,rt−k)
If autocorrelation is statistically significant, it shows evidence of
predictability in the time series
The autocorrelation function (ACF) of a time series shows the measured autocorrelation in the series as a function of the number of lags k (k = 0,1,2,…)

ACF of S&P 500 daily returns
Sample: 1928 – mid 2022 Source: Jon Danielsson

Test for autocorrelation
One can test the significance of individual coeﬀicients (ρˆk) Alternatively, one can test the joint significance of the first K
autocorrelation coeﬀicients with the Ljung-Box statistic:
We will test for both returns (rt), predictability in mean (price
forecasting or alpha)
And returns squared r2t , predictability in volatility.
J k = T ( T + 2 ) ρˆ k ∼ χ 2K

Test for autocorrelation
Sample: 1928 – mid 2022 Source: Jon Danielsson

Test for autocorrelation
Testing for the presence of serial correlation yields the following results:
number of lags Returns 100 Squared returns 100
test stat 18.7 46
p-value 0.606 0.00129
There is strong evidence of serial correlation for squared returns

Stylized fact 2

Definition: a series exhibits fat tails if the likelihood of extreme outcomes in that series is higher than that of a normally-distributed random variable with the same mean and standard deviation.

Probabilty distribution of N(0,1)

Student-t distribution
The Student-t is convenient when we need a fat tailed distribution
The degrees of freedom, (ν), of the Student–t distribution indicate how fat the tails are ν = ∞ implies the normal
ν < 2 superfat tails For a typical stock 3 < ν < 5 A fat-tailed distribution implies that the probability of non-extreme outcomes is lower than that of the normal. Test for non-normality One can use statistical methods to test for normality Jarque-Bera test: may point to fat tails if null of normality is rejected: JB = T6 × Skewness2 + T4 (Kurtosis − 3)2 ∼ χ2 Tests only for an implication of normality For the S&P 500, Kurtosis = 30.07 (fat tails) Skewness = -1.01 (negatively skewed, or left-skewed) so JB = 514530, whereas the critical value at significance level 5% is approximately 5.98. Visualize non-normality One can also use graphical methods to inspect the data QQ-Plots plot quantiles of the target distribution (e.g., the empirical distribution of S&P500 returs) against quantiles of a reference distribution Plot would be linear if the target distribution is a linear function of the reference one Deviations from linearity reveal the existence of heavy tails. S&P 500 vs. normal distribution S&P 500 vs. student-t(4) distribution Implications of fat tails Extreme outcomes tend to have a disproportional influence on economic and financial matters, so the assumption of normality tends to lead to underestimation of risk . . . . . . but use of non-normal distributions is usually complicated, and tends to require more data. Stylized fact 3 Time-varying correlations Correlations tend to rise sharply at times of crisis. Great Financial Crisis Many funds using quant strategies saw correlations in their holdings increase a lot, so riskiness of portfolios also increased Quote by ( ’s CFO) on the firms’ Alpha Fund: ”We were seeing things that were 25-standard deviation moves several days in a row.” Under a normal distribution, the probability of an event of this magnitude is less than 3 × 10−138; the age of the universe is about 5 × 1012 days, so these events should happen once in about 10125 universes. Maybe distribution of returns isn’t normal after all. Stylized facts The conditional volatility of financial returns exhibits cyclical patterns (market shocks tend to lead to periods of high volatility; the effect of shocks is persistent but eventually dies out) Financial returns are not normally distributed, and exhibit fat tails The correlations of returns series varies over time, and rises in times of crises We will develop models that help explain and analyze these patterns during the course 程序代写 CS代考加微信: powcoder QQ: 1823890830 Email: powcoder@163.com

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