Risk and financial portfolio analytics - A technical Introduction

Risk and Financial Portfolio
Analytics: A Technical
Introduction
Oleksandr Romanko, Ph.D.
Senior Research Analyst, Risk Analytics – Business Analytics, IBM
Adjunct Professor, University of Toronto
Toronto SMAC Meetup
January 15, 2015

© 2015 IBM Corporation2
Please note:
IBM Risk Analytics statements regarding its plans, directions, and intent are subject to
change or withdrawal without notice at IBM’s sole discretion.
Information regarding potential future products is intended to outline our general product
direction and it should not be relied on in making a purchasing decision.
The information mentioned regarding potential future products is not a commitment,
promise, or legal obligation to deliver any material, code or functionality. Information about
potential future products may not be incorporated into any contract. The development,
release, and timing of any future features or functionality described for our products
remains at our sole discretion.
Performance is based on measurements and projections using standard IBM benchmarks
in a controlled environment. The actual throughput or performance that any user will
experience will vary depending upon many factors, including considerations such as the
amount of multiprogramming in the user's job stream, the I/O configuration, the storage
configuration, and the workload processed. Therefore, no assurance can be given that an
individual user will achieve results similar to those stated here.

About me
Dr. Oleksandr Romanko
 Senior Research Analyst, Quantitative Research at Risk Analytics,
Business Analytics, IBM, with the company since 2010
 Ph.D. in Computer Science from McMaster University
 Author of over 10 papers and reports
 Adjunct professor at University of Toronto
and lecturer at McMaster University
 Research areas:
 business analytics, operational research, optimization, finance
 portfolio optimization, multi-objective optimization
 market and credit risk modeling and optimization
 numerical methods for risk management
 design of numerical algorithms and their software implementation

Making the world work better – pioneering the science
2008
19731969
1981

IBM Centennial: 100 Years of Innovation

© 2015 IBM Corporation
 Analytics Jobs

Data science
9

 Business Analytics

Predictive Analytics
What will happen?
Descriptive Analytics
What has happened?
Prescriptive Analytics
What should we do?
What is analytics?
Data Insight Action
DecideAnalyze
Business Value
11
Analytics is the scientific process of deriving insights from
data in order to make decisions

History of analytics

Movies

Applications of big data analytics
Homeland Security
FinanceSmarter Healthcare Multi-channel
sales
Telecom
Manufacturing
Traffic Control
Trading Analytics Fraud and Risk
Log Analysis
Search Quality
Retail: Churn, NBO

 Cloud

Bluemix
www.bluemix.net

Bluemix

 Applied Statistics

What kind of data are we dealing with?
 Types of data
• Quantitative
• Categorical (ordered, unordered)
 Data collection
• Independent observations (one observation per subject)
• Dependent observations (repeated observation of the same subject, relationships
within groups, relationships over time or space)
 Type of data drives the direction of your analysis
• How to plot
• How to summarize
• How to draw inferences and conclusions
• How to issue predictions
19

Quantitative data
 Examples: financial return, temperature, age, income
 Quick check: “Does it makes sense to calculate an average?”
 Appropriate summary statistics:
– Mean and Median
– Standard Deviation
– Percentiles
 More advanced predictive methods: Regression, Time Series Analysis, …
 Plot your data!
20

Summarizing quantitative data
 One-number summaries
– Mean
Average, obtained by summing all observations and dividing by the number of obs.
– Median
The center value, below and above which you will find 50% of the observations.
 Summarizing your data with one number may not tell the whole story:
21
Median = 19.8 Median = 19.8 Median = 10.5

Flaw of averages
“Plans based on
average assumptions
are wrong on average”
Average depth 3 ft

“Most observations fall within ±2 standard
deviations of the mean.”
Standard deviation

23
If the data is normally distributed
95 % of observations
Standard Deviation = 4.2
~95% of observations
between 11.4 and 28.2

Descriptive statistics - example
 Random sample of 5000 customers of a credit card company
24
Amount spent on
primary card last
month
Debt to income
ratio (x100)
N
Valid 5000 5000
Missing 0 0
Mean 1683.7340 9.9578
Median 1690.0670 8.8000
Std. Deviation 210.26680 6.42317
Minimum .00 .00
Maximum 2482.72 43.10

Percentiles
 Generalizations of the median (50th percentile).
 The pth is the data point below which p percent of the observations fall.
 Often used to compare a single observation to a general population.
 Examples:
– Standardized test scores
If you scored in the 93th percentile, your score was higher than that of 93% of test
takers.
– Finance and risk management
If your portfolio value-at-risk 95% is $10M, your portfolio loss will not exceed $10M
with probability 95%.
25

Percentiles - example
 Percentiles can be another way of describing how spread out data values are.
Example: 5-Number Summary
Minimum – 25th percentile – Median – 50th percentile - Maximum
26
Amount spent on
primary card last
month
Debt to income
ratio (x100)
Minimum .00 .00
Percentiles
25 1567.4658 5.1250
50 1690.0670 8.8000
75 1814.5430 13.5000
Maximum 2482.72 43.10

Distributions: Normal distribution
27

Distributions
28

Distributions
29

Distributions
Estimate of the probability distribution of global mean temperature resulting
from a doubling of CO2 relative to its pre-industrial value, made from
100000 simulations

 Simulation Modeling

Sums of random variables
 For any random variable and a constant
 Expectation of the sum of two random variables is equal to the sum of
expectations
and, therefore
 Example: expected return of a portfolio
 For the variance

 How to compute the
probability distribution of the
sum of random variables?
 We cannot add PDFs or
PMFs
 The formula involves non-
trivial integration and is
known as convolution:
 Use simulation to evaluate
such complex integrals

Simulation modeling – example 1
 We want to invest $1000 in the US stock market for 1 year:
 Invest into the S&P 500 market index (index fund)
 Value of investment at the end of year 1:
 Market return over the time period [0,1) is
 Generate scenarios for the market return over the year and compute
 decide on the number of scenarios and the set of scenarios for
 generate scenarios
 use historic scenarios
 draw randomly from historic scenarios (bootstrapping)
 draw random numbers from the assumed distribution (Monte Carlo)
 visualize and analyze the approximate probability distribution of
 In our example we assume that the return of the market over the next year
follow Normal distribution

 Between 1977 and 2007, S&P 500 returned 8.79% per year on average with a
standard deviation of 14.65%
 Generate 100 scenarios for the market return over the next year (draw
100 random numbers from a Normal distribution with mean 8.79% and standard
deviation of 14.65%):
 Compute and plot
Number of values 100
Mean $ 1,087.90
Std Deviation $ 146.15
Skewness 0.0034442
Kurtosis 2.871695
Mode $ 1,118.96
5% Perc $ 837.40
95% Perc $ 1,324.00
Minimum $ 708.81
Maximum $ 1,458.52

Simulation modeling – example 1 in Matlab
600 700 800 900 1000 1100 1200 1300 1400 1500
0
5
10
15
20
25
Value at time 1
Frequency
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
600
700
800
900
1000
1100
1200
1300
1400
1500
Time
Value
Simulated Value Paths
v0 = 1000; % initial capital
Ns = 100; % number of scenarios
% Generate Normal random variables
r01 = normrnd(0.0879, 0.1465, Ns, 1);
% Distribution of value at the end of year 1
v1 = (1 + r01) * v0;
% Plot a histogram of the distribution of outcomes for v1
[frequencyCounts, binLocations] = hist(v1, 10);
bar(binLocations, frequencyCounts);
% Plot simulated paths over time
time = 0:1:1;
plot(time,[v0*ones(100,1) v1],'Linewidth',2);

Why use simulation?
 Example 1 illustrates very basic Monte Carlo simulation system
 Simulation allows us to evaluate (approximately) a function of a random variable
 in example 1 the function is simple
 given distribution of , in some cases we can compute distribution of in closed
form, e.g., if followed a Normal distribution, then also follows a Normal
distribution with mean and standard deviation
 if was not Normally distributed, or if the output variable were a more complex
function of the input variable , it would be difficult and practically impossible to
derive the probability distribution of from the probability distribution of
 Other advantages of simulation:
 simulation enables visualizing probability distribution resulting from compounding
probability distributions of multiple input variables (example 2)
 simulation allows incorporating correlations between input variables (example 3)
 simulation is a low-cost tool for checking the effect of changing a strategy on an output
variable of interest (example 4)
 Next, we extend example 1 to illustrate such situations

 You are planning for retirement and decide to invest in the market for the next
30 years (instead of only the next year as in example 1). Your initial capital is
still
 Assume that every year your investment returns from investing into the
S&P 500 will follow a Normal distribution with the mean and standard deviation
as in example 1.
 Value of investment after 30 years:
 The return over 30 years will depend on the realization of 30 random variables
 Observations:
 sum of Normal random variables is Normal
 here we have multiplication of Normal random variables, is it Normal?

 Between 1977 and 2007, S&P 500 returned 8.79% per year on average with a
standard deviation of 14.65%
 Simulate 30 columns of 100 observations each of single period returns:
Mean $ 12,587.62
Std Deviation $ 10,948.39
Skewness 3.349066
Kurtosis 28.24214
Mode $ 4,458.97
5% Perc $ 2,655.55
95% Perc $ 32,481.38
Minimum $ 609.75
Maximum $194,355.00

0 1 2 3 4 5 6
x 10
4
0
5
10
15
20
25
30
35
40
Value after 30 years
Frequency
0 5 10 15 20 25 30
0
1
2
3
4
5
6
x 10
4
Time
Value
r_speriod30 = normrnd(0.0879, 0.1465, Ns, 30);
v30 = v0 * prod(1 + r_speriod30, 2);
[frequencyCounts, binLocations] = hist(v30, 10); bar(binLocations, frequencyCounts);
time = 0:1:30; v_t = v0*ones(Ns,1);
for(t=1:30) v_t = [v_t v0 * prod(1 + r_speriod30(:,1:t), 2)]; end
plot(time,v_t,'Linewidth',2);

0 2 4 6 8 10 12 14
x 10
4
0
50
100
150
200
250
300
350
400
450
500
Frequency
0 5 10 15 20 25 30
0
2
4
6
8
10
12
14
x 10
4
Time
Value
r_speriod30 = normrnd(0.0879, 0.1465, Ns, 30);
v30 = v0 * prod(1 + r_speriod30, 2);
[frequencyCounts, binLocations] = hist(v30, 100); bar(binLocations, frequencyCounts);
time = 0:1:30; v_t = v0*ones(Ns,1);
for(t=1:30) v_t = [v_t v0 * prod(1 + r_speriod30(:,1:t), 2)]; end
plot(time,v_t,'Linewidth',2);

 You are planning for retirement and decide to invest in the market for the next
30 years. Your initial capital is
 You have an opportunity to invest in stocks and Treasury bonds:
 allocate 50% of your capital to the stock market (S&P 500 index fund) today
 allocate 50% of your capital to bonds today
 Assume that every year your investment returns from investing into the
S&P 500 and Treasury bonds will follow a Normal distribution with the mean
and standard deviation as in example 2 (for S&P 500), mean 4% and standard
deviation 7% for bonds. Assume correlation -0.2 between the stock market and
the Treasury bond market.
 Covariance matrix:
 Value of investment after 30 years:

 Simulate 30 years of 100 observations each of single period correlated returns:
Mean $ 7,892.80
Skewness 2.921482
Kurtosis 20.48869
Mode $ 5,050.96
5% Perc $ 2,951.82
95% Perc $17,457.43
Minimum $ 1,408.63
Maximum $79,729.34

0 1 2 3 4 5 6 7 8
x 10
4
0
200
400
600
800
1000
1200
Frequency
0 5 10 15 20 25 30
0
1
2
3
4
5
6
7
8
x 10
4
Time
Value
mu = [0.0879; 0.04]; % expected return
sigma = [0.1465^2, -0.0021; -0.0021, 0.07^2]; % covariance matrix
% Generate correlated Normal random variables
stockRet = ones(Ns,1);
bondsRet = ones(Ns,1);
for iYear = 1:30
scenarios = mvnrnd(mu, sigma, Ns);
stockRet = stockRet .* (1 + scenarios(:,1));
bondsRet = bondsRet .* (1 + scenarios(:,2));
end
v30 = 0.5*v0*stockRet + 0.5*v0*bondsRet;

 Using scenario generation procedure from example 3 for decision-making
 Compare portfolios:
 50-50 portfolio allocation in stocks and bonds (Strategy A)
 30-70 portfolio allocation in stocks and bonds (Strategy B)
Mean $ 1,865.13
Skewness 3.506451
Kurtosis 40.18968
Mode $ 687.75
5% Perc $ -254.41
95% Perc $ 6,027.23
Minimum $-1,829.78
Maximum $45,972.08

 Mean-Variance
Portfolio Selection

Measuring risk and portfolio selection
 Consider n assets with random returns:
 proportion invested in asset i
 exp. return and standard dev. of
the return of asset i
 variance-covariance matrix
 Portfolio expected return and variance:
 Set of admissible portfolios:
Portfolio Return ( )
Probabilitydensity
0
Variance
(standard deviation)
Mean
return
Portfolio return distribution ( ) is assumed to be Gaussian (Normal)

 Consider n assets with random returns:
 proportion of total funds invested in asset i
 expected return and standard deviation of
the return of asset i
 correlation coefficient of i’s and j’s returns
 vector of expected returns
 variance-covariance matrix (PSD)
 Expected return and variance of the resulting portfolio:
 Set of admissible portfolios:
Portfolio selection
49

Portfolio selection
A feasible portfolio x is efficient if it has:
 maximal expected return among all portfolios with the same variance,
 minimum variance among all portfolios with the same expected return.
Mean-variance optimization (Markowitz, 1952):
Alternative formulations
Solving for all the values of V, R, or  gives efficient portfolios:

Portfolio selection
Portfolio optimization problem – efficient frontier:

Portfolio selection
Extensions of mean-variance model: introduce transaction costs
 Mean-variance portfolio optimization problem – two
objectives:

Portfolio selection
 Mean-variance portfolio optimization problem – efficient
frontier and portfolio composition:

Questions

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configuration, and the workload processed. Therefore, no assurance can be given that an individual user will achieve results similar to those stated here.
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Risk and financial portfolio analytics - A technical Introduction

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