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Likelihood of Loss and the Value at Risk
Module Author
Saurav Roychoudhury Assistant Professor of Finance
School of Management Capital University
sroychou@capital.edu 614-236-7230
Year: 2010
Funding Source: NSF (DUE 0618252)
Any opinions, findings, and conclusions or recommendations expressed in this material
are those of the author and do not necessarily reflect the views of the National Science
Foundation.
2
Module Prerequisites The module covers topics in finance, statistics and economics. Students should have basic
knowledge of probability and statistics and know the basic properties of Normal and Log-
normal distributions and have done preliminary hypothesis testing. A background in
economics and finance is helpful but not necessary. A course in computational science
(Computational Science I) is beneficial in giving students the background in modeling
techniques prior to beginning this module. It is expected that the students are familiar
with working with Excel 2007.
Required math level and Science Level: Intermediate.
Overview
Financial loss impairs our ability to invest, save, consume, borrow, retire and sustain our
well-being. It is possible to model, at least within a reasonable bound, the likelihood of
loss and the maximum loss that can befall an investment. The mathematical models can
be solved by applying statistical techniques, using historical data, and Monte-Carlo
simulations. This module uses mathematical models to estimate the probability or
likelihood of a given loss and the maximum potential loss to an investment over a defined
period of time. The module uses common statistical techniques to industry standard
Value-at-Risk models.
This module is designed primarily for undergraduate students in finance, economics, and
statistics. However the module can also cater to undergraduate students in mathematics,
physics, computer science, engineering and geology who wish to take a course in
computational finance. The module can be used as an integrated part of a computational
finance course, a stand-alone component or an add-on to a typical investments or a risk
management course in finance. The module can be taught in both the classroom and the
computer lab. For advanced students, the module has a section which uses matrix algebra.
For more details on using matrix algebra to construct portfolios in excel, refer to the
3
computational finance module “Allocation of investment portfolios along the efficient
frontier”.
Keywords
Value-at-Risk, VaR, Likelihood of Loss, Monte Carlo Simulation, Lognormal returns,
Portfolio hedging.
Introduction
There is a saying in life that there is no gain without pain. It is very true in financial
markets, In order to get a higher return than a return on a government bond or a bank
certificate of deposit, an investor will have to put money in risky securities which can
range from a highly rated municipal bond to stocks and the riskiest of options. Financial
loss matters as it affects everyone and one of the most important and profound question
that confronts any investor is what is the worst that can happen? An investor can be a
person saving for retirement or on the down payment on a house or both; investor can be
a commercial bank, an insurance company, the government, a large corporation, a small
business or simply a speculator. An investor could face bankruptcy; financial traders who
trade with huge amounts of borrowed money can face financial ruin if they fail to “exit”
their position beyond a limit, a bank can have its equity wiped out if its assets drop below
a certain amount.
Problem Statement
Given the risks, the questions which an investor may typically ask can be:-
What is the most I can lose on this investment?
What is the likelihood that I may lose half of my investment in a given year?
How much money do I have to set aside so that my equity is not wiped out?
Am I saving enough for my retirement?
This module will attempt to provide answer to these questions within the framework of
mathematical models.
4
But first things first, we need to know some basic background material before we can
start answering these questions.
Background Information
Returns
Simple Periodic Returns
Rates of return are the building blocks of quantitative finance. A simple periodic rate of
return, on any investment can be calculated by using the general formula
Example 1: If an investment of $100 grows to $115 in year 1 and $125 in year 2, what
are the 1 period simple periodic returns for years 1 & 2.
Solution: Using the above equation for simple periodic return,
Year
Investment
Value
Simple
Periodic
Return
0 100.00$
1 115.00$ 15.00% <--=(115/100) -1
2 120.00$ 4.35% <--=(120/115) - 1
Stock returns are one of the most widely used financial returns. The data on stocks prices
are also easily available. When investors invest in a company stock they expect return in
the form of dividends1 and capital gains. The stock return in any time period t, is simply
the ratio of the sum of the dividends, tD , plus the capital gains2 to the stock price at time
t-1. Algebraically, the simple periodic return is given as
1 Some companies like CSCO have never paid a dividend. Investors buy non-dividend paying stocks
because they tend to compensate the investors with higher stock prices. 2 Capital gains or loss from a stock at time t is the difference between the stock price at time t and price at
time t-1 i.e., )( 1 tt PP .
5
There are several good websites from where students can download price information.
Some of them are MSN money, NASDAQ.com, Bloomberg.com. Yahoo! Finance has a
wealth of financial information in the website. It is well structured and it automatically
adjusts the price data to account for dividends and stock splits3. So, the periodic return
formula using Yahoo! Finance data reduces to:
(1)
For details on how to download stock price data from Yahoo and compute the monthly
stock returns, refer to the sub-section “Downloading Stock data from Yahoo! Finance and
calculating the daily stock returns using Excel” under the section “Empirical Data.”
Continuous Returns
If the periodic return is and our initial investment is $1 then the end of period value or
simple gross return is $1 ( ). It is convenient to convert the periodic returns to
continuous returns as the statistical properties of continuous returns are more tractable.
We can convert any periodic return into its compounded return by taking the logarithm of
1 plus the periodic return. The continuously compounded return is given as
( ) (
) (
) (2)
For example, if the continuous return ( ) = 9.53%. It is easy to
convert back from to . If we compound continuously for one year by 9.53%, the
periodic return for the year equals – .
3 Companies may decide to split their stock in to two or more parts. For example, if Microsoft price is $40,
and the company announces a 2:1 stock split, then each stock of $40 would be now counted as 2 stocks of
$20 each. A company may also wish to do a reverse split usually when they think their stock value is too
low. A 1:2 reverse split would bundle two unit stocks into a single unit. If Microsoft goes for a reverse split
in our example, then 2 stocks of $40 would be bundled into a single stock of $80. Yahoo! Finance takes
into account the effect of any such stock splits using a split adjustment factor.
6
When we wish to emphasize the distinction between and we shall refer to as
the simple return. Continuously compounded return, , enjoy some advantages over
simple periodic return . If we would like to find multi-period continuously
compounded returns, it is simply the sum of the continuously compounded one-period
returns. Compounding, a multiplicative operation, is converted to an additive operation
by taking logarithms.
( ) ( ) ( ) ( ) ( )
Example 2: If an investment of $100 grows to $115 in year 1 and $125 in year 2. What
are the continuous returns for each of the years 1 & 2. Also, show that the continuously
compounded 2 year return is the sum of continuous returns from year 1 and year 2.
Solution: Compute the continuously compounded returns for years 1 and 2 using equation
(2). Adding both the 1 period returns will give us 18.23%. This is similar to the 2-year
continuous return using equation (2).
Year
Investment
Value
Continuous
Return
0 100.00$
1 115.00$ 13.98% <--=LN(115/100)
2 120.00$ 4.26% <--=LN(120/115)
Sum of 1 Period Returns 18.23% <--=13.98% + 4.26%
2-year Continuous Returns 18.23% <--=LN(120/100)
Note: The sum of two 1 period simple periodic returns will not be equal to the 2 year
simple return (Try this as an exercise).
Analysis of Returns Data
In order to quantify the risk, it is important to analyze the returns data. Let us start with
the data on the US stock market returns from 1927 to 2008. The data is in the Excel file
NSFII-01.xlsx and has been compiled from the data available at Kenneth French‟s
7
website.4 The returns data was computed from the data on all NYSE, AMEX, and
NASDAQ5 firms in the database maintained by the Center for Research in Equity Prices
(CRSP).
In order to visually analyze raw data, a histogram is often used. In excel, the Data
analysis ToolPak‟s histogram tool calculates the frequency distribution of a range of data
to produce a bar chart that shows the distribution graphically.
1. Choose Data/Data Analysis/Histogram.
2. The Histogram dialog asks for an “Input Range” and a “Bin Range”. In our
example, the input range is the data under „Stock returns‟ and the bin range will
be the data under „Return Intervals (%). For each number in the bin range, the
histogram counts the number of observations that are greater than or equal to the
bin value, and less than the next highest bin value.
3. Figure below shows the dialog box already filled in. Use the output options group
to select the location of the output. In our example we choose the output to be
displayed in a “New Worksheet”.
4 The data is available at http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/Data_Library/f-
f_factors.html
5 NASDAQ data is only available from 1971 onwards.
8
4. The next step is to select what kind of options you want to use for the frequency
distribution. If you choose “Chart Output”, Excel automatically generates a chart
for the frequency distribution as shown in figure.
The frequency distribution of historical stock market annual returns shows that there was
one year where the return was less than -40%, two returns were between -40% and -30%,
three returns between -30% and -20% and seven returns between -20% and -10%. Out of
0
2
4
6
8
10
12
14
16
18
Fre
qu
en
cy
Return Intervals in %
Frequency distribution of the US Stock Returns: 1927-2008
Frequency
9
82 years of data, the US stock market returned less than -10% in 13 of the years. A very
simple way to estimate the likelihood of a return of less than -10% would be to divide 13
by 82, which is 15.9%. This approach becomes more precise as we increase the number
of observations and data can be described using a distribution. There are two major
advantages of presenting the data in a distribution. Firstly, we can summarize even the
largest data sets into one distribution and get a measure of what values occur most
frequently and the range of high and low values. Secondly, if we have large number of
returns that are independent of each other and from the same population, then the return
distribution can be approximated by a normal distribution6. A normal distribution has
convenient properties which makes it amenable to mathematical modeling. The entire
distribution can be described (and generated) by the first two moments; that is by mean
and standard deviation. The observations are distributed symmetrically around the mean.
The area under the normal distribution defined by one standard deviation on either side of
the mean comprises 68% of all observations. A two standard deviation distance from the
mean covers approximately 95% and three standard deviations covers about 98% of the
data.
In our dataset, the population mean and population standard deviation of the US stocks
returns from 1927-2008 is 11.39% and 20.62% respectively. If we assume the returns are
6 This is the basis for the Central Limit theorem.
10
normally distributed as shown in the figure below, then from the properties of normal
distribution, 68% of the returns will lie approximately between -9.23% (i.e., 11.39% -
20.62%) and 32.01 % (i.e., 11.39% + 20.62%). 95% of the stock market annual returns
will lie between -29.85 percent and 52.63 percent (i.e., 11.39% ± 2×20.62%).
Model
Likelihood of Loss
How does the Normal distribution relate to likelihood of Loss? Suppose we wish to
estimate the likelihood that an investment in the US stock market will produce a loss of
10% or more in a single year based on the historical mean and standard deviation. We
begin converting the periodic annual returns into their continuous counterparts by adding
one to the returns and taking the natural logarithms of these values. We estimate the mean
of these continuous returns as 11.39% and the standard deviation as 20.62%. Next step is
to convert our target return of -10% to its continuous equivalent, which equals -10.54%
11
[i.e., ln(1-0.10)]. We then create what is called a standard normal variable by
standardizing the difference between the target return ( ) and the mean ( ) by dividing it
by the standard deviation ( ).
( )
The standard normal distribution is a special case of the normal distribution where it
rescales the actual mean and standard deviation to a normal distribution with a mean zero
and a standard deviation of one. It shares all the properties of a normal distribution (see
figure below).
Standard Normal Distribution
In our example, the standardized difference z equals -1.06 [i.e., (-10.54-11.39)/20.62].
The standardized value, -1.06, is also called the normal deviate or the standardized
variable. It measures how many standard deviations away from the mean is the target
return. By properties of a normal distribution, a normal deviate of -1 standard deviation is
34% below the mean. Thus, there is a 16 percent (50%-34%) chance of experiencing a
return that is one standard deviation or further below the mean. Therefore, a probability
of return that is -1.064 standard deviation units below the mean will be little less than
16%. If we look in the Standard Normal Distribution table under -1.06, it will show a
12
value of 14.38%.7 We can also use the excel function NORMSDIST(z) where z =-1.06 to
come up with the same result8.
Problem 1: The large company stocks have a historical mean of 13 percent and a standard
deviation of 20 percent. If the returns on large company stocks are normally distributed,
what will be the likelihood that large company stocks will have a return of -25 percent or
lower next year?
Problem 2: An investor has a $100 million portfolio. The million dollar portfolio has a mean
of 15% and a standard deviation of 45%. The investor wants that his end of year investment
should not fall below $900,000. What is the likelihood that the investment will be less than
the cutoff value of $900,00?
So far we have based our analysis on the assumption that the return distribution is
normal. The normal distribution is the most frequently used model to teach Value-at-
Risk. For any distribution, the mean and the standard deviation are the first two moments,
the third moment measures the skewness (which measures both the direction and the
magnitude of any asymmetry) and the fourth moment is described by the kurtosis which
measures the fatness of the tails of the distribution. For a normal distribution, the kurtosis
has a value of 3 and the skewness has a value of zero implying perfect symmetry. Sample
estimates of skewness for US stock returns tend to be negative for stock indices but close
to zero or positive for individual stocks. As evident from the figure below, our dataset is
clearly not symmetric, the first four moments of our dataset is shown as below along with
a histogram line plot9. The distribution is negatively skewed and platykurtotic (negative
excess kurtosis).
7 In the standard normal distribution table, we look down the left column to -1.0 and across the top row to
0.06. The intersection shows a z value of -1.06 corresponds to a probability of 14.46%.
8 You can create your own standard normal distribution table by using the NORMSDIST function in Excel.
NORMSDIST(Z) return the area under the standard normal distribution curve from a negative infinity to
the Z value for a normal distribution with a mean of 0 and standard deviation of 1.0. See NSFII_05.xlsx. 9 You can find the moments by using Excel‟s Data/Data Analysis/Descriptive Statistics. See
NSFII_05.xlsx.
13
In real world, the investment returns in general are not precisely normally distributed,
even theoretically. First, most financial assets like stocks have limited liability, so that the
largest loss an investor can realize is his total investment and no more. This implies that
the return will never be less than -100%. But since the normal distribution constitute the
entire real line, the lower bound of -100% is clearly violated by normality. Second, if
one-period returns are assumed to be normal then multiperiod returns cannot also be
normal as they are products of the single period returns. For example, consider a 15%
return compounded over two consecutive years. The cumulative two-year return equals
32.25 % [(1+0.15) × (1+0.15) -1 = 0.3225)]. However, if we consider a -15% return
compounded over two consecutive years, the cumulative two-year return equals -27.75
percent [(1-0.15) × (1-0.15) -1 = -0.2775)]. This asymmetry that results from
compounding is captured precisely by the lognormal distribution which is often
considered to be an alternative for normal distribution. In the lognormal model, the
natural logarithm of gross asset return, is assumed to be normally distributed, which
implies that single period gross simple return, is distributed as lognormal variate (i.e.,
( )). The lognormal distribution is skewed to the right. Also, if gross
returns are lognormally distributed, the gross returns cannot fall below negative 100
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percent. These properties of the lognormal distribution make it a more realistic
characterization of the behavior of market returns than the normal distribution.
Value at Risk
Value at Risk (VaR) describes the quintile of the projected distribution of losses (or
gains) over a target time horizon. In other words, it estimates the worst loss over a time
horizon with a given level of confidence. VaR can be measured both in percentage terms
or currency terms though currency terms are more heavily used as it is more intuitive to
understand.
The generic VaR statistic is defined as a one-sided confidence interval on portfolio losses
[ ( ̃) ]
where ( ̃) is the change in the market value of the portfolio, expressed as a
function of the forecast horizon and the vector changes in the random variable(s) ̃.
The interpretation is that, over a specified number of trading days, the value of the
portfolio will decline by no more than VaR at % of time. For example, if the VaR on a
portfolio is $ 10 million at a one-week, 99% confidence level, there is a only a 1% ( %)
chance that the value of the portfolio will drop more than $ 10 million over any given
week. The VaR can be reported as a dollar value or in percentage terms. The choice of
% depends on the risk tolerance of the investor, securities trader, fund manager, hedge
fund or the bank. Typical values of range from 1% to 15%.
There are three broad methods of calculating VaR, and there are numerous variations
within each method. VaR can be estimated by running hypothetical portfolios through
historical data or can be computed analytically by making assumptions about return
distribution and risks, or from Monte Carlo simulations. We describe the three methods
in this section using 3 months of daily returns of Microsoft stock10
.
10
The daily returns of Microsoft stock from April 1, 2010 to July 1, 2010 is in the Excel file
NSFII_02_msft.xslx
15
1. Historical Simulation
Historical simulation represents the simplest way of estimating the VaR. It is a non-
parametric approach where the returns are not assumed to be based on any statistical
distribution. The VaR for a portfolio is estimated by using actual historical returns data to
create a hypothetical time series of returns on that portfolio. The approach is based on the
naïve assumption that history repeats itself, with the period used providing a full and
complete picture of the risks to the portfolio. This method puts an equal weight to the
time series data when it comes to measuring the VaR, that is, it treats the change in
portfolio from today to tomorrow as similar to it was in the past. In its simplest form, the
VaR is calculated by looking at the percentiles of the data. It is defined as the
percentile of the potential losses that can occur within a given portfolio during a specified
time period. For example, if the dataset has 100 daily time series observations, then the
daily VaR at 5% will be the 5th
lowest observation in the dataset.
The historical approach suffers from some obvious problems because it assumes that past
prices are a good predictor of future prices and it uses single sample paths of prices to
compute VaR which may not correctly reflect the future. Also, if the sample period does
not include periods of high volatility it is likely to underestimate the VaR.
Historical Simulation method: 1 Risky Asset
Example 3: Calculate VaR at 1%, 5%, 10% and 15% for a 1 day horizon for Microsoft
stock.
Solution:
Step 1: We use three months of Microsoft stock returns to calculate the daily VaR.
Step 2: Find the value that corresponds to the appropriate VaR. Excel‟s PERCENTILE
function returns the percentile of values in an array. For example, if 5% is α
percentile, then 95% of the values in the array would be greater than the value returned
by the PERCENTILE function. The EXCEL syntax is =PERCENTILE( ). To
16
create an array for Microsoft returns, see the appendix A2 on “How to create an Array
Name”.
To calculate the daily VaR at 10% for Microsoft stock, the formula is
=PERCENTILE( ). This returns -3.96% which implies that based on the data
there is a 95% confidence that the worst loss won‟t exceed -3.96%. In other words, only
one out of twenty times you would expect Microsoft stock to fall by more than or equal
to -3.96% in any given trading day. The detailed solution is provided in the “Historical”
tab in the excel workbook NSFII_02_Msft.xlsx.
The historical simulation method is heavily dependent on the time series of returns. If
Microsoft returns were from a period when there was very high volatility (as in later half
of 2008), the VaR numbers would have a higher negative value.
Problem 3: Compute the daily VaR at 1%,5% and 10% using the Historical Simulation
Method for Microsoft stock using there months data from September 2008 to November
2008. Are the numbers similar to the example? Why or why not?
2 Risky Assets
One of the primary objectives of creating a multi- asset portfolio is reducing or
containing risk through diversification. The risk depends on the number of risky assets
and how diverse the risky assets are. More the number of stocks in a stock portfolio, the
lower would be the portfolio risk. For example, an S&P index portfolio will typically
have a lower risk than a portfolio of 10 random stocks. Similarly, more diverse are the
stocks in the portfolio (indicated by low or negative covariance between the assets) the
higher is the benefit from reducing risk. A portfolio of auto or technology stocks will
have a higher risk than a portfolio which has a more diverse composition- multi-sector,
international, young and old firms etc.
Example 4: The excel file NSFII_2 VaR.xlsx contains time series monthly stock price
data on risky assets “A” and “B” from January 2002 to December 2006. A minimum
17
variance portfolio suggests that the portfolio weights on A and B to be 43% and 57%
respectively. Given the information, find the monthly VaR for the portfolio.
Solution:
Step 1: Find the historical portfolio returns from January 2002 to December 2006
assuming the portfolio weights to remain constant.
Step 2: Use Excel‟s PERCENTILE function to find the cut-off points from the historical
portfolio returns corresponding to the VaR of 1%, 5%, 10% & 15%.
2. Delta Normal VaR
One of the popular analytical methods of estimating the Value at Risk is by using a
parametric method. In its purest form the Delta Normal method assumes that the returns
of the risky asset or portfolio are assumed to follow a normal distribution. The lognormal
distribution is also used as a possible alternative. The Delta Normal method follows
directly from the discussion on the “Likelihood of Loss” section. The VaR statistic is
defined as a one-sided confidence interval on portfolio losses when returns are distributed
as normal or lognormal.
Delta Normal method: 1 Risky Asset
(3)
Example 5: Calculate VaR at 1%, 5%, 10% and 15% for a 1 day horizon for Microsoft
stock using the Delta Normal method.
Solution:
Step 1: We use three months of Microsoft stock returns to calculate the mean and
standard deviation of daily returns.
Step 2 : Calculate the value corresponding to the . The NORMSINV returns the
inverse of the standard normal cumulative distribution for probability ( ) corresponding
18
to the normal distribution with standard deviation . The value tells us how many
standard deviations away from the mean is the .
For = 5%, the value is -1.645 standard deviations to the left of the mean. The VaR
at 5% will be [ ( ) ] which implies that there is a
95% confidence that the worst daily return will not exceed -3.22%. The detailed solution
is provided in “Delta Normal” tab in the excel workbook NSFII_02_Msft.xlsx.
Problem 4: A portfolio has a mean of 15% and a standard deviation of 45%. What is the
annual VaR at 1%? With 1 percent probability what is the maximum loss at the end of the
year. Assume normal distribution.
Problem 5*: A hedge fund has a $100 million portfolio. The hedge portfolio has a mean of
15% and a standard deviation of 45%. What will be the VaR if the returns follow a lognormal
distribution? Also calculate the daily VaR at 5%.
2 Risky Assets
Finding the risk for a 2 Asset portfolio
The portfolio risk is measured by portfolio standard deviation. A single asset standard
deviation can be easily obtained by using the excel formula STDEVP, however for a
portfolio it is a bit more complicated.
Let us assume that there are only 2 risky assets, A and B, available for consideration in an
investment portfolio. The expected portfolio return is given by
(4)
where, and are the average returns and and are the respective portfolio
weights of the two assets A and B.
We also assume that the portfolio weights should add up to 1 (i.e., ) and
that they are non negative ( )
The portfolio variance is given by
( )
19
and the standard deviation
√
( ) (5)
Note: The Covariance between A and B can also be expressed as ( )
where is the correlation coefficient between assets A and B.
Example 6: Find the Delta Normal VaR for the 2 stock portfolio using the data in
NSFII_2 VaR.xlsx.
Solution:
Step 1: Find the portfolio return and standard deviation using equations (4) & (5).
Step 2: Find the Z values corresponding to 1%, 5%, 10% and 15% using NORMSINV
function.
Step 3: Find the respective VaRs corresponding to 1%, 5%, 10% and 15% using equation
(3)
Monte-Carlo method
One of the most important characteristic of asset returns is their randomness. The return
of Microsoft stock over the next day is unknown today, the best guess of tomorrows price
will be today‟s price. A single point like an average may not be the best estimate for a
group of data points. However, if we have thousands and thousands of points to consider,
we greatly expand our horizon of possible outcomes. One way to do is by generating
random numbers to reveal possible outcomes in an uncertain environment and
simultaneously attach probabilities to each outcome. This technique was named after the
town of Monte Carlo in Monaco, where the primary casino attractions are games of
chance such as roulette wheels, dice, and slot machines. When we roll a die, we know
that a 1, 2, 3, 4, 5, or 6 will come up (given a fair die!), but we don't know which number
will appear for any particular roll. That‟s randomness, and the random behavior in dice is
similar to how Monte Carlo simulation works. One of the earliest applications of Monte
Carlo simulation occurred in the late 1940s, when scientists at the Manhattan Project at
Los Alamos National Laboratory used the method to predict the range of possible nuclear
20
fission results. While most spreadsheet analysts are not rocket scientists, the principles of
how to forecast results with uncertainty are almost exactly the same.
Steps for Monte Carlo Simulation
We can use Excel‟s () in step 1 to generate random numbers between 0 and 1.
However, there is no way to record the results from the simulated trials as Excel will
recalculate every time the Excel sheet is refreshed or used. There are commercially
available products like @Risk and Crystal ball which allows us to perform Monte Carlo
simulations within the Excel framework but they are expensive. Dr. Roger Myerson who
is a professor of Economics at the University of Chicago has developed an excel add-in
called SIMTOOLS which can be used to conduct Monte-Carlo simulations in excel. The
add-in is available free of cost at his website. For instructions on how to install and use
SIMTOOLS, refer to the Appendix A3 on “How to use Simtools”.
In the following example, we will show how to use Monte Carlo method using the
SIMTOOLS add-in.
Step 1: Generate a Random Number (between 0 and 1)
and also select the appropraie distributions
Step 2: Convert Random Number to Sampled Value of Input using the selected
distribution
Step 3: Sampled Value is used in this trial of
spreadsheet model for calculation
Step 4: Record the simulation output(s) from
the trial
21
Monte-Carlo method: 1 Risky Asset
Example 7: Calculate VaR at 1%, 5%, 10% and 15% for a 1 day horizon for Microsoft
stock using the Monte Carlo method.
Solution:
Step 1: Select a specific distribution for Microsoft asset returns and estimate the
parameters of that distribution. As a simple case we use a normal distribution with
parameters similar to the delta normal method (mean = -0.36% and standard deviation=
1.74%). We use Excel‟s () which generates an evenly distributed random real
number between 0 and 1. Once the distribution has been selected we use a pseudo-
random number generator that will generate N hypothetical outcomes.
Step 2: To convert random numbers to sample values of the input we use Excel‟s
NORMINV(Probability, Mean, Standard Deviation) function which returns the inverse
of the normal cumulative distribution for the specified mean and standard deviation with
the random probability value generated by RAND(). However, the mean values
calculated will give a different output every time the worksheet is refreshed or by
pressing F9.
Step 4: In order to store the results for each number of trials we use SIMTOOLS. For
example, we want to create a table which will store results from 3000 trials. We start by
selecting a “counter” column which will be the leftmost column for our simulation. In
the figure below the Counter column starts from Simtools in cell B7. The cell on the right
(C7) would be the formula reference cell for the value we want to simulate (D5). Once
both the cells (B7 and C7) are highlighted we write B7:C3007 on the name tab and press
Shift and Enter simultaneously to highlight the entire range.
22
Once the column is highlighted, go to the ribbon item “Add-ins” and select
Simtools/Simulation table.11
This will populate the columns B8:C3007 with 3000
simulated results of msft daily stock returns. To find the 5% VaR, simply select the 5%
percentile value using the PERCENTILE function (as in the Historical simulation
method). If there are 3000 observations, it will select the 150th
smallest observation. This
gives a VaR value of -3.15%.The detailed solution is provided in the “Monte Carlo” tab
in the excel workbook NSFII_02_Msft.xlsx.
11
For some Excel 2007 features refer to Appendix A1- “Ribbon feature in Excel 2007”
23
Here is a summary of the results of the three methods to find the daily VaR for Microsoft
stock returns.
A Comparison of 3 methods to Estimate VaR
α
Delta
Normal
Historical
Simulation
Monte-Carlo
method
1%
-4.40% -4.17% -4.31%
5% -3.22% -3.96% -3.15%
10% -2.58% -2.35% -2.54%
15% -2.16% -1.89% -2.17%
Here is a summary of the results of the three methods to find the daily VaR for Microsoft
stock returns.
Problem 6: Compute the daily VaR at 1%, 5% and 10% using the Delta Normal and Monte
Carlo method (assume Normal distribution) for Microsoft stock using three months data
from September 2008 to November 2008. Are the numbers similar to the example? Why or
why not?
2 Risky Assets
Example 8: Find the VaR by Monte Carlo approach for the 2 stock portfolio using the
data in NSFII_2 VaR.xlsx. Assume returns are normally distributed.
Solution:
We use the same basic framework from the Delta Normal method. We assume the
portfolio returns to follow a normal distribution with an expected portfolio return of
and standard deviation of .
Step 1: Generate random portfolio return by using Excel‟s ( () )
Step 2: Do 5000 iterations of step 1 using the SIMTOOLS to create a simtable.
Step 3: Find the respective cutoff points corresponding to the VaR of 1%,5%,10% & 15%
using Excel‟s PERENTILE function to come up with estimates of VaR.
Lognormal distribution is often used as an alternative to Normal distribution as discussed
earlier in this module. In the lognormal model, the natural logarithm of gross asset return
is assumed to be normally distributed, which implies that single period gross simple
return is distributed as a lognormal variate.
Example 9: Find the VaR by Monte Carlo approach for the 2 stock portfolio using the
data in NSFII_2 VaR.xlsx. Assume lognormal distribution.
25
Solution:
Step 1: Convert the mean and standard deviation to the lognormal equivalent we apply
the following formula
√
Step 2: Generate random portfolio return by using the Simtools formula
( () ) . The one is added at the end to convert the gross
portfolio return to net portfolio return.
Repeat Steps 2 & 3 from the normal distribution example to arrive at the required VaR
numbers.
3 Risky Assets
It is easy to modify the mean and standard deviation of a 2 asset portfolio to a three asset
portfolio (A, B and C). The average portfolio return can be written as
(6)
With the condition
and
And the portfolio standard deviation can be written as
√
( ) ( ) ( )
(7)
But as the number of assets in a portfolio increases, the calculation of portfolio standard
deviation becomes quite tedious. We can resort to matrix algebra to solve this problem.
The example below is on 3 risky assets but the formula can be generalized to include any
number of risky assets.
Matrix method
The expected return, and portfolio weights are both column vectors
= [
] and [
]
26
The expected return on the portfolio is given by the matrix multiplication
(8)
Where is the transpose of the vector .
Note, the above equation is equivalent to equation (6) in the 3 asset case.
For the portfolio variance (and standard deviation), we have to first define the variance-
covariance matrix , where | | ( is the row element and is the column
element)
(example,
)
And ( ) (example, ( ) )
For a 3 – asset case, the variance covariance matrix is the shaded area is the matrix.
Asset A Asset B Asset C
Asset A
Asset B ( )
Asset C ( ) ( )
The portfolio variance is given as
(9)
Example 10: a) Jack has a portfolio consisting of three assets A,B, and C with expected
average returns of 17%, 6% and 12% and standard deviations of 30%, 25% and 45% ,
respectively. The variance-covariance matrix for the three assets is shown below
Asset A Asset B Asset C
Asset A 0.37 Asset B -0.07 0.1
Asset C 0.05 -0.01 0.25
Find the VaR (at 1%, 5%, 10% and 15%) in percentage terms for the 3 assets using the
Delta Normal as well as the Monte Carlo Method.
27
Solution: See NSFII_04.xlsx for details.
Delta Normal method.
Monte Carlo Method
b)What is the maximum amount that Jack can lose in a year with VaR at 1%, 5%, 10%
and 15% if he had a $100,000 of initial investment.
See NSFII_04.xlsx for details.
28
Problem 7: The file NSFII_03.xlsx has stock returns data on 13 stocks from June 2001 to
June 2007. Create a portfolio on JPM, ORCL and HAL stocks with the portfolio weights of
33%, 33% and 34%, respectively. Calculate the VaR at 1%, 5%, 10% and 15% using the Delta
Normal and Monte Carlo method. Assume normal distribution.
Assessment of the model As past stock market data is readily available, it would be easy to assess how efficient are
the models in forecasting the potential loss. An interesting exercise would be to create a
weekly or monthly VaR of a portfolio and see how many times the VaR would fall within
the prescribed limit during a semester.
Empirical Data
Historical data on stocks and bonds are widely available on a host of financial websites.
My recommendation is Yahoo finance! at http://finance.yahoo.com/. The historical data
on stock indices which go back till 1926 is available free at Kenneth French‟s website at
http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/Data_Library/f-f_factors.html
All the data that has been used in the module is available in the provided excel
workbooks.
Downloading Stock data from Yahoo! Finance and calculating the daily stock
returns using Excel.
Step 1: Type the url http://finance.yahoo.com on the address bar of your web browser
and click on „Finance‟.
29
Step 2: It would open a page similar to as shown above. In the “Get Quotes” box, type
the ticker symbol for the stock you want to look up. If you do not know or remember the
ticker symbol of a particular stock you can always click on the „Symbol Lookup‟ button
which would assist you finding the correct symbol. When you have entered your stock
symbol (in our example it MSFT for Microsoft), press enter or click on “Get Quotes”.
30
Step 3: It would take you to a page similar to as shown above. You can choose
„Historical Prices‟ to get Microsoft‟s price history or you can click on the picture of the
price chart and then scroll down to arrive at „Historical Prices‟.
Step 4: Indicate the time period and frequency for the data we want. We have chosen a
three month time frame from April 1 to July 1, 2010. We chose the „Daily‟ option to
download daily price data.
31
Step 5: If we scroll down the webpage it gives us an option to download the data in Excel
spreadsheet format. Yahoo allows you to save the spreadsheet file as „Table.csv‟ which is
a comma separated value file and can be opened directly by Excel. We changed the name
to „msft.csv‟ to save it in our computer.
32
Step 6: To compute monthly returns we keep only the columns „date‟ and „Adj. Close‟.
The adjusted closing stock price in yahoo accounts for any dividends and stock splits
which may have occurred during our sample period. The date is tabulated in descending
order and we need to sort it in ascending order in order to calculate returns. We use the
Excel „sort‟ function to sort the date in ascending order. At this point it is preferable to
save it as a „Microsoft Excel workbook‟ file instead of a „Comma separated value‟ file as
the Excel workbook file preserves any formulas we work on while the .csv file does not.
Step 7:The stock returns can be calculated using the formula for continuous periodic
returns
(
⁄ ). In our example Microsoft daily stock return corresponding to
4/5/2010 is given by typing the Excel formula “=LN )2/3( BB ” in cell C3. Here tP
Price on 4/5/2010 and 1tP Price on 4/5/2010. Copying the formula from cell C3 to the
end of your date range will give you the time series data on Microsoft stock returns.
There is a simple way to copy the returns without having to drag the cell C3. Put the
cursor on the bottom right of cell C3 till the plus sign appears (as shown in the figure).
33
Then double-click on the plus sign and the column would populate itself. This method
works only if there is a non-empty column on the left.
Problem Solutions
1. Assuming Normality, a return of -25 percent is [-25 – (-13)]/20 = -1.9 standard
deviations below the mean. If we look up the cumulative probability distribution table of
a normal distribution or use the excel function NORMSDIST(z), this will give us a value
of 2.87%. In other words, the likelihood that a portfolio of large company stocks will fall
by 25 percent or more next year is 2.87 percent.
2. The likelihood that the portfolio will fall below $900,000 is 28.9 percent. See
NSFII_01.xlsx [tab „LOL‟] for solution.
34
3. The VaR for Microsoft returns using the data from 2008 will be higher because of high
volatility during that period.
4. We have (1%) =-2.33. Using Equation 3, we get VaR (@1% = -89.68%.
Maximum loss = $100 89.68% = $89.68 million
Here is an alternate way to calculate the VaR using the Delta Normal method using
Solver. We first find the cut-off value by setting the target cell equal to α. The target cell
gives the probability that the normal variate is worth less than the cut-off. The solution is
available at NSFII_01.xlsx [tab „VaR‟]
35
Once we have the cut-off value, we find VaR by subtracting it from the investment, as
shown below.
5. Use () function to find the inverse of a lognormal distribution. See file
NSFII_01.xlsx [tab “Lognormal Distribution”] for complete solution.
6. The VaR for 2008 will be higher because of high volatility during that period.
7. See file NSFII_03.xlsx for a detailed solution.
Suggestions to Instructors It is important that the students learn about the significance of quantifying risk. Give
examples of different areas where the concept of VaR and the likelihood of loss can be
applied in real life. The students should be able to distinguish between the three methods
and use their own value judgment to choose the appropriate method. The module can be
done over three weeks if the students have the necessary background in probability and
statistics or they have already done the module on portfolio analysis (similar to my
Computational finance module Allocation of investment portfolios along the efficient
frontier).
There are 10 solved examples in this module. The problems are shown as “Problem” and
typed in red with an Arial Narrow font. There are seven problems in the module and the
36
detailed solutions are provided. The challenging problems are denoted with an asterix.
The examples and the problems are all available in the multiple Excel files supplied with
this teaching module.
References
Das, Sanjiv and Lynagh, Stephen, Value at Risk, HBS Number: 9-297-069 Type: Case
(Gen Exp) Publication Date: 1/31/1997, Harvard Business Publishing.
Damodaran, Aswath, Strategic Risk Taking, Wharton School Publishing, October 2009,
ISBN-10: 0-13-704377-5.
Duffie, D. and J. Pan, An Overview of Value at Risk, Working Paper, Stanford
University, 1997
J.P. Morgan RiskMetrics Monitor, Second Quarter 1996. RiskMetrics – Technical
Document, J.P. Morgan, December 17, 1996;
37
APPENDIX
A1. Ribbon feature in Excel 2007.
Excel 2007 features a new “Ribbon” feature which groups small icons for common tasks
rather than the old drop down menu style. For instance, if we want to change the
formatting of a cell or increase or decrease the number of decimal spaces, it can be done
with just a click.
To learn more about the new features in Excel 2007, refer to the Computer Worldi article
on Excel.
A2. How to create an Array Name
It is very convenient to name a series of data in a column or a row as an array in excel.
An array can be a single row (called a one-dimensional horizontal array), a column (a
one-dimensional vertical array), or multiple rows and columns (a two-dimensional array).
Instead of selecting the whole row or column we can use the array name in the formula.
You can create the array name by selecting the data and right click to choose “Name a
Range” or you can highlight the data and then type the preferred name in the “Name
Box” as shown below for Microsoft daily returns data (array name= msft).
38
How to use Simtools
Simtools is an excel add-in developed to conduct Monte-Carlo simulations in excel. The
add-in was developed by Dr. Roger Myerson who is a professor of Economics at the
University of Chicago. The Simtools add-in is available as free software and can be
downloaded from http://home.uchicago.edu/~rmyerson/addins.htm#simt
The add-in needs to be saved in one of system directories. It can be either copied to
“..…\Program Files\Microsoft Office\Office12\Library” directory or the “…..\Documents
and Settings\….\Application Data\Microsoft\AddIns” directory or to the Excel
MacroLibrary folder (in a Macintosh). You can then can add Simtools in excel by the
Tools>Add-Ins command, checking the SimulationTools box. For more details on how to
select add-ins, here is a step-by-step tutorial at
http://peltiertech.com/WordPress/installing-an-add-in-in-excel-2007/
Normal Distribution
There are two Excel functions that are heavily used with Simtools. They are RAND and
NORMINV. Every RAND() takes a random value drawn from a uniform distribution on
the interval between 0 and 1 such each variable is independent. New random values of
39
the RANDs are drawn every time the spreadsheet is recalculated. The
NORMINV(probability, mean, standard deviation) function can be used to make random
variables that have a normal distribution. By putting a RAND() value in the place of the
probability parameter generates a normal random variable which has an approximately
two-thirds of being between one standard deviation from the mean. For example, if we
enter the formula =NORMINV(RAND(),10,5) into any cell, then its value becomes a
normal random variable with mean 10 and standard deviation 5, and the variables lie
between 5 and 15 about two-thirds of the time.
Lognormal Distribution
LNORMINV(probability, mean, stdevn) returns the inverse cumulative distribution for a
lognormal random variable, parameterized by its mean and standard deviation. So the
formula =LNORMINV(RAND(),10,5) generates a lognormal random variable that has
mean 10 and standard deviation 5. The value of a lognormal random variable can be any
nonnegative number.
Triangular Distribution
TRIANINV(probability, lowerbound, mostlikely, upperbound) returns the inverse
cumulative for a triangular probability density on the interval from the lowerbound to the
upperbound, with mode at the mostlikely value. So =TRIANINV(RAND(),5,10,20)
yields a random variable that takes values between 5 and 20, with a probability density
that is highest at 10.
Some useful functions
CORAND(correlarray, randsource) returns a vector of uniform 0-to-1 random variables
(individually like RANDs) that are correlated appropriately to serve as seeds for
constructing random variables with the given correlations. CORAND is an array
function, designed to return values simultaneously to a selected range of cells in a row,
when entered with CTRL-SHIFT-ENTER.
40
The correlation array should be symmetric and must have 1s on the diagonal from
top-left to bottom-right. For a non-symmetric array, CORAND only looks at the portion
on and below the diagonal.) The correlarray parameter can also be a single number, in
which case CORAND functions as if the correlarray parameter were an array,
returning two random values with the given correlation.
CORRELPR(values1, values2, probabilities). For a probability distribution with
corresponding values of two random variables, CORRELPR returns the correlation
coefficient of the random variables. The first two parameters (values1, values2) are
ranges which contain one cell for each possible combination of values of the two random
variables, listing possible values of the first variable in the values1 range and the second
variable in the values2 range. The third parameter (probabilities) is a range containing
the corresponding probabilities of these value pairs. The cells in the probabilities range
must contain nonnegative numbers that sum to 1. The values1 and values2 ranges must
each have the same number of rows and the same number of columns as the probabilities
range.
COVARPR(values1, values2, probabilities). For a probability distribution with
corresponding values of two random variables, COVARPR returns the covariance of the
random variables. The first two parameters (values1, values2) are ranges which contain
one cell for each possible combination of values of the two random variables, listing
possible values of the first variable in the values1 range and the second variable in the
values2 range. The third parameter (probabilities) is a range containing the
corresponding probabilities of these value pairs. The cells in the probabilities range must
contain nonnegative numbers that sum to 1. The values1 and values2 ranges must each
have the same number of rows and the same number of columns as the probabilities
range.
PRODS(values) multiplies each pair of values in the given range and returns the
products as a square array. The values must be given in one row or one column. To
illustrate the use of this function, suppose that a range named "correls" lists the
41
correlations of the random returns per share of various stocks, a range named "stdevns"
lists the standard deviations of these stock returns, and a range named "shares" lists the
numbers of shares of these stocks in some investment portfolio; then the standard
deviation of the total returns of the portfolio is
SUMPRODUCT(PRODS(shares),PRODS(stdevns),correls)^0.5
Simulation Table
The Simulation Table tabulates output from repeated MonteCarlo simulations of a
spreadsheet model with random variables. To use SimulationTable, Excel's Calculation
property must be set to Automatic (see Tools>Options>Calculation). Before using the
SimulationTable macro, a range must be selected in which the output to be tabulated is in
the top row, but not in the top-left cell. The output from repeated recalculations of the
model then fills the lower rows of the selected range below these output cells. The
leftmost column of the selected range is filled with fractile numbers, indicating (in each
row of the simulation table) what fraction of the simulation data is above this row.
SimulationTable stores the output data as values that are not recalculated whenever the
spreadsheet changes.
i http://www.computerworld.com/s/article/9028228/Excel_2007_Cheat_Sheet
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