2. Contents:
• Measurement of risk return.
• Modelling Risk Factors.
• Risk and volatility Measurement.
• Measuring risk using Value-at-Risk.
• Forecasting correlation & Volatility during market crash.
• Extreme Value Theory.
• Risk & Forecasting issues in asset prices (exchange rates & interest rates)
3. Measurement of risk and return
Return
Returns are always calculated as annual rates of return, or the percentage of return
created for each unit (Rupee) of original value.
If an investment earns 5 percent, for example, that means that for every Rs. 100
invested, you would earn Rs. 5 per year (because Rs. 5 = 5% of Rs. 100).
5. For example,
If you buy a share of stock for Rs. 100, and it pays no dividend, and a
year later the market price is Rs. 105,
then your return = [0 + (105 − 100)] ÷ 100 = 5 ÷ 100 = 5%.
If the same stock paid a dividend of Rs. 2,
then your return = [2 + (105 − 100)] ÷ 100 = 7 ÷ 100 = 7%.
6. Cont…
• Risk
Investment risk is the idea that an investment will not perform as expected,
(that its actual return will deviate from the expected return.)
Risk is measured by the amount of volatility, that is, the difference between
actual returns and average (expected) returns.
This difference is referred to as the standard deviation.
--Returns with a large standard deviation (showing the greatest variance
from the average) have higher volatility and are the riskier investments.
7. Modelling Risk Factors
Refers to the risk of loss due to using a financial model with
fundamental errors or incorrect use of a model for decision-making.
The entities can assess and reduce the risk using an effective
validation process, evaluating fundamental correctness and output
analysis.
8.
9. Cont..
• The main types are specification risk, implementation risk, and
model application risk.
• Model Risk Management (MRM) controls risks indicated by the
possible adverse effects of choices made using flawed or inappropriate
models.
• The entities can assess and reduce the risk using an effective
validation process, evaluating fundamental correctness and output
analysis.
10. VOLATILITY MEASUREMENT
• Volatility is a statistical measure of the dispersion of returns
for a given security or market index.
• In most cases, the higher the volatility, the riskier the security.
Volatility is often measured from either the standard
deviation or variance between returns from that same security
or market index.
when the stock market rises and falls more than one percent over
a sustained period of time, it is called a "volatile" market.
11. Calculate Volatility
Variance and standard deviation (the standard deviation is the square
root of the variance).
Since volatility describes changes over a specific period of time you
simply take the standard deviation and multiply that by the square root
of the number of periods.
Vol = σ√T
where:
•v = volatility over some interval of time
•σ =standard deviation of returns
•T = number of periods in the time horizon
12. For simplicity,
let's assume we have monthly stock closing prices of Rs. 1 through Rs. 10.
For example, month one is Rs. 1, month two is Rs. 2, and so on. To calculate
variance, we need to follow the following steps;
1. Find the mean of the data set.
2. Calculate the difference between each data value and the mean.
3. Square the deviations. This will eliminate negative values.
4. Add the squared deviations together.
5. Divide the sum of the squared deviations by the number of data values.
13.
14. Cont..
• If prices are randomly sampled from a normal distribution, then about 68% of all data values will
fall within one standard deviation.
• Ninety-five percent of data values will fall within two standard deviations (2 x 2.87 in our
example), and 99.7% of all values will fall within three standard deviations (3 x 2.87).
• In this case, the values of Rs. 1 to Rs. 10 are not randomly distributed on a bell curve; rather. they
are uniformly distributed.
• Therefore, the expected 68%–95%º–99.7% percentages do not hold. Despite this limitation, traders
frequently use standard deviation, as price returns data sets often resemble more of a normal (bell
curve) distribution than in the given example.
15. Types of Volatility
• Historical volatility
• Implied volatility
• Future realized volatility
Historical volatility is the standard deviation of the annualised
daily returns of the underlying.
• What the market expects in the future.
• This is the price you trade in the market, and you will be a
price taker since the market as a whole is large.
• Some technical or quant indicators can be applied to
implied volatility to predict its direction but what happens in
future could still be completely different.
• what happens in the future.
• This is not known in advance but looking at the past future
realized volatility can give an indication of how good the
market predicted the volatility.
16. Different measures of volatility
1. EWMA,
2. ARCH
3. & GARCH processes
• The Exponentially Weighted Moving Average (EWMA) is a quantitative or
statistical measure used to model or describe a time series.
• The EWMA is widely used in finance, the main applications being
technical analysis and volatility modeling.
1. EWMA
17. Formula
Where:
•Alpha = The weight decided by the user
•r = Value of the series in the current period
.
The EWMA is a recursive function, which means that the current observation is calculated using
the previous observation
The above equation can be rewritten in terms of older weights, as shown below:
It can be further expanded by going back another period:
The process continues until we reach the base term EWMA0
18. The EWMA can be calculated for a given
day range like 20-day EWMA or 200-day
EWMA.
•N = number of days for which the n-day moving
average is calculated
For example, a 15-day moving average’s
alpha is given by 2/(15+1), which means
alpha is 0.125. Naturally shorter the
lookback period – more closely, the
EWMA – tracks the original time series.
19. Volatility can be estimated using the EWMA by following the process:
• Step 1: Sort the closing process in descending order of dates, i.e., from the current to the oldest price.
• Step 2: If today is t, then the return on the day t-1 is calculated as (St / St–1) where St is the price of day t.
• Step 3: Calculate squared returns by squaring the returns computed in the previous step.
• Step 4: Select the EWMA parameter alpha. For volatility modeling, the value of alpha is 0.8 or greater. The weights are
given by a simple procedure. The first weight (1 – a); is the weights that follow are given by a * Previous Weight.
• Step 5: Multiply the squared returns in step 3 to the corresponding weights computed in step 4. Sum the above product
to get the EWMA variance.
• Step 6: Finally, the volatility can be computed as the square root of the variance calculated in step 5.
The volatility number is then used to compute risk measures like the Value at Risk (VaR). It can also be
used for option valuation, where volatility is an input parameter to the Black-Scholes-Merton formula.
20. ARCH/Garch Model:
ARCH and GARCH. These model(s) are also called volatility model(s).
These models are exclusively used in the finance industry as many asset prices are conditional
heteroskedastic.
ARCH — Autoregressive Conditional Heteroskedasticity
GARCH — Generalized Autoregressive Conditional Heteroskedasticity
1.These models relate to economic forecasting and measuring volatility.
2. ARCH model is concerned about modeling volatility of the variance of the series.
3. These model(s) deals with stationary (time-invariant mean) and nonstationary (time-varying
mean) variable(s).
4. Some of the real-time examples where ARCH model(s) applied: Stock prices, oil prices, bond
prices, inflation rates, GDP, unemployment rates, etc.,
21. What is the ARCH model?
• ARCH is an Autoregressive model with Conditional Heteroskedasticity .
• “The ARCH process introduced by Engle (1982) explicitly recognizes the difference between
the unconditional and the conditional variance allowing the latter to change over time as a
function of past errors.”
• Autoregressive: The current value can be expressed as a function of the previous values i.e.
they are correlated.
• Conditional: This informs that the variance is based on past errors.
• Heteroskedasticity: This implies the series displays unusual variance (varying variance).
22. Why an ARCH model?
• Volatility in Finance: Degree of variation price series over time as
measured by the standard deviation of the series.
Suppose that Si is the value of a variable on a day ‘i’. The volatility
per day is the standard deviation of ln(Si /Si-1).
• In time series where the variance is increasing in a systematic way,
such as an increasing trend, this property of the series is
called heteroskedasticity. This means changing or unequal variance
across the series.
23. ARCH Model of Order Unity:
ARCH(p) model is simply an AR(p) model applied to the variance of a time series.
• ARCH(1):
A time-series {ϵ(t)} is given at each instance by ……..ϵ(t) = w(t)*σ(t)
where w(t) is the white noise with zero mean and unit variance.
Var(x(t)) = σ²(t) = ⍺0+⍺1 * σ²(t-1)
where ⍺0, ⍺1 are parameters of the model and ⍺0 > 0, ⍺1 ≥ 0 to ensure that the conditional variance
is positive. σ²(t-1) is lagged square error.
• We say that ϵ(t) is an autoregressive conditional heteroskedastic model of order unity, denoted by
ARCH(1).
ϵ(t) = w(t)* σ(t) = w(t)* ⎷(⍺0 + ⍺1 *ϵ²(t-1))
24. Cont.…
• Similarly ARCH(2):
ϵ(t) = w(t)* σ(t) = w(t)* ⎷(⍺0 + ⍺1 * ϵ²(t-1) + ⍺2 * ϵ²(t-2))
• Similarly ARCH(p):
ϵ(t) = w(t) * ⎷(⍺0 + ⍺(p) * ∑ ϵ²(t-i)
where:
“P” is the number of lag squared residual errors to include in the ARCH model.
i = (1,2,3,-,-,-, -, p) tells us the number of logged periods of the square error.
25. Interpretation:
• If the error is high during the period (t-1), it is more likely that the value of
error at the period (t) is also higher.
• vice versa — If the error is low during the period (t-1) then the value inside
sqrt will be low which results in a decreased error in (t).
• Remember, ⍺1 ≥ 0 for the positive variance.
• For the stability condition to hold, ⍺1 < 1, otherwise ϵ(t) will be explosive
(continue to increase over time).
26. Why does this model volatility?
• From variance formula, we can derive the below equation:
Var(ϵ(t)) = ⍺0 + ⍺1 * Var(ϵ(t-1))
• We can say that the variance of the series is simply a linear
combination of the variance of the prior element of the series.
27. What is a GARCH model?
• Generalized Autoregressive Conditional Heteroskedasticity, or GARCH, is an extension of the ARCH model
that incorporates a moving average component together with the autoregressive component.
• Bollerslev (1986, Journal of Econometrics) generalized Engle’s ARCH model and introduced the GARCH
model.
• Introduction of moving average component allows the model:
1. To model the conditional change in variance over time.
2. Changes in the time-dependent variance.
Examples include conditional increases and decreases in the variance.
Thus GARCH is the “ARMA equivalent” of ARCH, which only has an autoregressive component. GARCH
models permit a wider range of behavior more persistent volatility.
28. GACH Model of Order p, q —GARH(p,q):
GARCH(1,1):
Here we are going to consider a single autoregressive lag and a single “moving average” lag. The
model is given by the following:
ϵ(t) = w(t) * σ(t)
ϵ(t) =w(t) * ⎷(⍺0 + ⍺1 *ϵ²(t-1)) + β1 * σ²(t−1)
Similarly GARH(p,q):
A time-series {ϵ(t)} is given at each instance by ϵ(t) = w(t)*σ(t)
and σ²(t) is given by
where α(i) and β(j) are parameters of the model.
⍺0 > 0, ⍺i ≥ 0, i =1, … q, β≥ 0, j = 1, … p imposed to ensure that the conditional variances are positive.
29. Interpretation:
• The large value of β1 causes σ(t) to be highly correlated with σ²(t−1) and gives the
conditional standard deviation process a relatively long-term persistence, at least
compared to its behavior under an ARCH model.
• For p = 0 the process reduces to the ARCH(q) process.
• For p = q = 0, ϵ(t) is simply white noise.
Here we are adding moving average term, that is the value of
σ² at t, σ²(t), is dependent upon previous σ²(t-j) values.
30. VaR(Value at risk) models
Value at risk is a statistical metric that forecasts the highest possible loss
and the probability of it occurring over a particular period.
It is a significant factor in risk management, financial reporting,
financial control, etc.
It measures the magnitude or potential of losses in a portfolio and is
helpful for banks and large corporations to keep an eye on their portfolio
value at risk.
31. Cont..
It is the measurement of probability of the highest possible value that the portfolio is
vulnerable to losing in a given period.
The three factors of VaR cover all these
concerns efficiently; the three factors are:
•Time frame
•Confidence level
•Loss amount or percentage of loss.
32. VaR Methods and Formulas
• The variance-covariance method,
• The Monte Carlo simulation, and
• The historical method are the three methods of calculating VaR.
33. Variance-Covariance Method:
Also known as the parametric method, this method assumes that the
returns generated from a given portfolio are distributed normally and can be
described by standard deviation and expected returns completely.
The Value at Risk formula:
VaR = Market Price * Volatility
• Here, volatility is used to signify a multiple of standard deviation (SD) on a
particular confidence level. Therefore, a 95% confidence will show
volatility of 1.65 to the standard deviation.
34. Cont..
• Let us understand the calculation of VaR through the Parametric method:
• First, assuming an investor holds only one stock in their portfolio, that of MNO Corporations for Rs. 100,000.
Then, taking the volatility daily at 1% and a confidence level of 90%.
• Using the formula, the calculation for this scenario would be:
• VaR = Market Price * Volatility
• = Rs. 100,000 * .01 * 1.65
• = Rs. 1,650
• The calculation signifies a 1% chance that the maximum amount the investor might lose in one day is Rs.
1,650.
35. Confidence # of Standard Deviations (σ)
95% (high) - 1.65 x σ
99% (really high) - 2.33 x σ
Confidence # of σ Calculation Equals
95% (high) - 1.65 x σ - 1.65 x (2.64%) = -4.36%
99% (really high) - 2.33 x σ - 2.33 x (2.64%) = -6.15%
Actual daily standard deviation, which is 2.64%
The average daily return happened to be fairly close to zero, so it's
safe to assume an average return of zero for illustrative purposes.
36. Montecarlo Method
• This is a dynamic method of calculating VaR.
• In this method, the value at risk is calculated by creating a number of random
scenarios for future rates.
• It uses non-linear price models for estimating any changes in the value for each
such scenario and calculating the VaR based on the worst losses.
• This method is ideal to use in complex situations and is also the most flexible.
VaR = -1 x (percentile loss) x (portfolio value)
37. Cont.
• For example,
if the monthly return for three scenarios ranged between -20% and 25% and two between -15% and -
20%, then the chances of the highest possible loss would be -15%.
• Hence, assuming a 95% confidence, the portfolio shall not lose more than 15% in any given
month.
The VaR statistic has three components: a period, a confidence level, and a loss amount, or
loss percentage, and can address these concerns:
•What can I expect to lose in dollars with a 95% or 99% level of confidence next month?
•What is the maximum percentage I can expect to lose with 95% or 99% confidence over the
next year?
38. Historical Method
• The historical method is the easiest and the simplest method of calculating the
VaR.
• It uses the market data for a long period of time like the past 250 days to calculate
the percentage of change in every risk factor and arrange them in the order of
worst to best.
• This method helps in calculating the probability of the worst outcome that helps in
decision-making as the premise of this method is that history repeats itself.
39. Cont..
• This method computes linear and non-linear possibilities accurately while also
displaying the complete picture of the potential profits and losses in the portfolio.
• The formula is as follows:
VaR Formula = Vm (Vi/ V(i-1))
Here,
M signifies the days in the historical data taken into consideration
Vi indicates the number of variables on the day in question (the day i)
40. Extreme Value Theory
Aims to remedy a deficiency with value at risk (i.e., it gives no
information about losses that exceed the VaR) and glaring weakness of
delta normal value at risk (VaR): the dreaded-fat tails. The key is the
idea that the tail has it's own "child" distribution.
The EVT is also used to model the behavior of tips (Maxima) and
or dips (Minima) in a series of asset returns etc.
Recently, Portfolio managers, Investors, Risk managers, Claim managers etc, have
become more concerned over occurrences under “Extreme market conditions”.
41. Cont..
• Approaches for Decision Making:
After the risk has been identified on its nature and type, assessment has
taken place i.e how big the risk is, the probability to occur and when it is
going to happen, then, the decision makers of the Organization
1. Hedging: Eliminate risk by selling into the market place through cash or spot market transactions
or through Forward, Future and Swap contracts.
2. Diversification: This reduces risk by combining risks which are not perfectly correlated to form a
portfolio.
3. Insurance: Risks are transferred from the buyer (Policyholder) to the seller i.e Insurer
42. Risk & Forecasting issues in asset prices
(exchange rates & interest rates)
• Risk prediction models are mathematical models that aim to predict
the probability of future events.
• A prediction model is a statistical technique commonly used to predict
future behavior. Using predictive modeling, you can predict future
outcomes by analyzing historical and current data.
The mean model and other general models such as regression and
ARIMA.
43. Cont…
• Intrinsic risk
• Parameter risk
• Model risk
A random variation in the data and tools that have available. It’s the “noise” in the
system. The intrinsic risk is usually measured by the “standard error of the model,”
which is the estimated standard deviation of the noise in the variable you are trying to
predict. Although there is always some intrinsic risk (the future is always to some
extent unpredictable).
Errors in estimating the parameters of the forecasting model you are using, under
the assumption that you are fitting the correct model to the data in the first
place. This is usually a much smaller source of forecast error than intrinsic risk.
Choosing the wrong model, i.e., making the wrong
assumptions about whether or how the future will resemble
the past. This is usually the most serious form of forecast error,
and there is no “standard error” for measuring it, because
every model assumes itself to be correct.”
44. Four main methods:
• (1) straight-line,
• (2) moving average,
• (3) simple linear regression and
• (4) multiple linear regression
Technique Use
Math
involved
Data needed
1. Straight line Constant
growth rate
Minimum
level
Historical data
2. Moving
average
Repeated
forecasts
Minimum
level
Historical data
3. Simple
linear
regression
Compare one
independent
with one
dependent
variable
Statistical
knowledge
required
A sample of
relevant
observations
4. Multiple
linear
regression
Compare more
than one
independent
variable with
one dependent
variable
Statistical
knowledge
required
A sample of
relevant
observations
45. Straight-line,
• A financial analyst uses historical figures and trends to predict future
revenue growth.
For 2016, the growth rate was 4.0% based on historical performance. We can use the formula =(C7-
B7)/B7 to get this number. Assuming the growth will remain constant into the future, we will use the
same rate for 2017 – 2021.
To forecast future revenues
46. Moving Average
• Moving averages are a smoothing technique that looks at the underlying pattern of a set of data to
establish an estimate of future values. The most common types are the 3-month and 5-month
moving averages.
48. Simple Linear Regression
• Regression analysis is used extensively in finance-related applications. Many typical applications involve
determining if there is a correlation between various stock market indices such as the S&P 500, the Dow
Jones Industrial Average (DJIA), and the Russell 2000 index.
• As an example, suppose we would like to determine if there is a correlation between the Russell 2000 index
and the DJIA. Does the value of the Russell 2000 index depend on the value of the DJIA? Is it possible to
predict the value of the Russell 2000 index for a certain value of the DJIA? We can explore these questions
using regression analysis.
• Table shows a summary of monthly closing prices of the DJIA and the Russell 2000 for a 12-month time
period. We consider the DJIA to be the independent variable and the Russell 2000 index to be the dependent
variable.
50. Cont..
• Assume the DJIA has reached a value of 32,000. Predict the
corresponding value of the Russell 2000 index. To determine this,
substitute the value of the independent variable, x=32,000 and
calculate the corresponding value for the dependent variable, which is
the predicted value for the Russell 2000 index:
yˆ =−1,496.34+0.11(32,000)
y^=2,122.66 Thus the predicted value for the Russell 2000 index is
approximately 2,123 when the DJIA reached a value of
32,000.
51. Multiple Linear Regression
Multiple Regression is a statistical method to forecast the outcome of a particular variable
dependent on several other independent variables. Its purpose is to establish a linear relationship
between a dependent variable and several other independent variables. It is the extended version
of the Simple Linear Regression which uses only one independent variable.
• Assume that there are two independent variables X1 and X2 affecting the value of a dependent
variable Y. Then, the equation for Multiple Linear Regression will be:
Y = α0 + α1X1 + α2X2+ ϵ
• Here,α stands for the Intercept (constant). α1and α2 represent the change in Y due to the changes
in X1 and X2 respectively. On the other hand, ϵ signifies the residual or error.
52. Cont.…
No. X1 X2 Y
1 32 10 37.9
2 19 9 42.2
3 13 5 47.3
4 13 5 47.5
5 5 5 51.5
6 7 3 48.2
7 34 7 40.3
8 20 6 46.7
9 30 1 18.8
10 17 3 25.8
Here, the dataset contains a sample of 10 houses numbered 1 to 10. X1 indicates the age of the houses
and X2 denotes the number of grocery stores near each of them. Then prices per unit area of the houses
are expressed by Y. Therefore, Y is the dependent variable here. On the other hand, X1 and X2 are the
independent variables.
