Materials for Lecture 20

• Read Chapter 9• Lecture 20 CV Stationarity.xlsx• Lecture 20 Changing Risk Over

Time.xlsx• Lecture 20 VAR Analysis.xlsx• Lecture 20 Simple VAR.xlsx

Value at Risk Analysis

• Value at Risk – VAR• Originally VAR used to quantify

market risk, but considered only 1 source of risk

• By year 2000 businesses were integrating their risk management systems across the whole enterprise– Focus on analyzing multiple sources of

risk including market risk• Now market based VAR analyses

measure integrated market and credit risk

Value at Risk Model

• In an intuitive definition “VAR summarizes the worst loss over a target horizon with a given level of confidence”

• VAR defines the quantile of the projected distribution of gains and losses over the target horizon

Value At Risk Model

• If c is the selected confidence level, VAR corresponds to the 1-c lower tail of the probability distribution (the quantile).

Value At Risk (VAR) Model

To estimate the VAR quantile for a risky business use these steps:

1. Develop a stochastic simulation model of the risky business decision

2. Validate stochastic variables and validate the model

3. Pick a ‘c’ value, say, 5%, so 1-c = 95%4. Simulate the model and analyze the KOV5. Calculate the quantile for the c value6. Calculate VAR = Mean – Quantile at 1-c7. Report the results

Value At Risk (VAR) Model

• On selecting the ‘c’ value – literature uses the 95% level

• This is to say we want to know the value of returns which the business will exceed 95% of the time

• If simulating 1,000 iterations, the quantile will be the 50th value, so we can sort the stochastic results and read the 50th value

• Or simply use the PDF in Simetar

VAR in Simetar

• Simulate the KOV and draw a PDF• Change the Confidence level to 0.90 for “c” =

5%• Edit the title of the chart• VAR value is the Lower Quantile

Valuation Models

• A variation on VAR is the traditional valuation model

• Valuation models focus on the mean and the variation below the mean

VAR as Risk Capital

• VAR is the equity capital that should be set aside to cover most all potential losses with a probability of “c”

• Thus, the VAR is the amount of capital reserves that should be held to meet shortfalls

VAR for Comparing Risky Alternatives

• Simulate multiple scenarios and calculate VAR for each alternative

VAR for Calculation with Simetar

• Simulate multiple scenarios and use the SimData statistics and the QUANTILE function

• =QUANTILE(simulated values, 0.05)• Can graph the PDFs and change

confidence level

VAR Shortcomings• VAR analyses generally used in business

gives a false sense of security • The literature assumes Normality for the

random variables, why?– Normal is easy to simulate – Can easily calculate the Quantile if you know

mean and std deviation Q = Mean – (2.035 * Std Dev)

• The chance of a Black Swan is ignored– This understates the Quantile and the equity

capital reserve needed to cover cash flow deficits

– Contributed to the Recession

Overcoming VAR Shortcomings• Modify the probability distributions for the random

variables that affect the business• Incorporate low probability events that could cause

major harm to the business.• Use and EMP distribution and adjust the Probabilities

and Sorted Deviates as a Fraction for Black Swan events– Change the F(X) values for the low probability– Change the minimum Xs

Covariance Stationary & Heteroskedasticy

• Part of validation is to test if the standard deviation for random variables match the historical std dev. – Referred to as “covariance stationary”

• Simulating outside the historical range causes a problem in that the mean will likely be different from history causing the coefficient of variation, CVSim, to differ from historical CVHist:

CVHist = σH / ῩH Not Equal CVSim = σH / ῩS

Covariance Stationary• CV stationarity likely a problem when

simulating outside the sample period:– If Mean for X increases, CV declines, which

implies less relative risk about the mean as time progresses CVSim = σH / ῩS

– If Mean for X decreases, CV increases, which implies more relative risk about the mean as we get farther out with the forecast CVSim = σH / ῩS

• See Chapter 9

CV Stationarity• The Normal distribution is covariance stationary

BUT it is not CV stationary if the mean differs from historical mean

• For example: – Historical Mean of 2.74 and Historical Std Dev of 1.84

• Assume the deterministic forecast for mean increases over time as: 2.73, 3.00, 3.25, 4.00, 4.50, and 5.00

• CV decreases while the std dev is constantSimulation Results

Mean 2.73 3.00 3.25 4.00 4.50 5.00Std. Dev. 1.84 1.84 1.84 1.84 1.84 1.85CV 67.24 61.48 56.65 46.02 40.88 37.04

Min -3.00 -3.36 -2.83 -1.49 -1.45 -1.03

Max 8.10 8.31 8.59 10.50 9.81 11.85

CV Stationarity for Normal Distribution

• An adjustment to the Std Dev can make the simulation results CV stationary if you are simulating a Normal dist.

• Calculate a Jt+i value for each period (t+i) to simulate as:

Jt+i = Ῡt+i / Ῡhistory

• The Jt+i value is then used to simulate the random variable in period t+i as:

Ỹt+i = Ῡt+i + (Std Devhistory * Jt+i * SND)

Ỹt+i = NORM(Ῡt+i , Std Dev * Jt+i)• The resulting random values for all years t+i have

the same CV but different Std Dev than the historical data– This is the result desired when doing multiple year

simulations

CV Stationarity and Empirical Distribution

• Empirical distribution automatically adjusts so the simulated values are CV stationary if the distribution is expressed as deviations from the mean or trend

Ỹt+i = Ῡt+i * [1 + Empirical(Sj , F(Sj), USD)]

Simulation Results

Mean 2.74 3.00 3.25 4.00 4.50 5.00

Std Dev 1.73 1.90 2.05 2.53 2.84 3.16

CV 63.19 63.19 63.18 63.19 63.19 63.19

Min 0.00 0.00 0.00 0.00 0.00 0.00

Max 5.15 5.65 6.12 7.53 8.47 9.42

Empirical Distribution Validation• Empirical distribution automatically adjusts so the

simulated values are CV stationary– This is done by adjusting the standard deviation – This poses a problem for validation

• The correct method for validating Empirical distribution is:– Set up the theoretical mean and standard deviation– Mean = Historical mean * J– Std Dev = Historical mean * J * CV for simulated values / 100

• Here is an example for J = 2.0Test Values for Stoch 2 Applying the correction for the EMP simulationMean 0.253571 Theoretical mean for the the simulation is J * historical meanStd Dev 0.02327 =F3*D5*2/100 Theoretical std dev for the the simulation is

Historical Mean * J * Simulated CVTest of Hypothesis for Parameters for Stoch 2Confidence Level 95.0000%

Given ValueTest ValueCritical ValueP-Valuet-Test 0.253571 2.03 2.25 0.04 Fail to Reject the Ho that the Mean is Equal to 0.253571428571429Chi-Square Test 0.02327 507.41 LB: 439.00 0.78 Fail to Reject the Ho that the Standard Deviation is Equal to 0.0232702471108615

CV Stationarity and Empirical Distribution

CR 1 CR 2 CR 3 CR 4 CR 5 CR 6 CR 7 CR 8 CR 9 CR 100

5

10

15

20

25

30

35Normal for 10 Years with No Risk Adjustment

Average 5th Percentile 25th Percentile

75th Percentile 95th Percentile

CR 1 CR 2 CR 3 CR 4 CR 5 CR 6 CR 7 CR 8 CR 9 CR 100

5

10

15

20

25

30

35Empirical Fan Graph with No Risk Adjustment

Average 5th Percentile 25th Percentile

75th Percentile 95th Percentile

Add Heteroskedasticy to Simulation

• Sometimes we want the CV to change over time– Change in policy could increase the relative risk – Change in management strategy could change relative

risk– Change in technology can change relative risk– Change in market volatility can change relative risk

• Create an Expansion factor or Et+i value for each year to simulate– Et+i is a fractional adjustment to the relative risk

– 0.0 results in No risk at all for the random variable– 1.0 results in same relative risk (CV) as the historical

period – 1.5 results in 50% larger CV than historical period – 2.0 results in 100% larger CV than historical period

• Chapter 9

Add Heteroskedasticy to Simulation

• Simulate 5 years with no risk for the first year, historical risk in year 2, 15% greater risk in year 3, and 25% greater CV in years 4-5– The Et+i values for years 1-5 are, respectively,

0.0, 1.0, 1.15, 1.25, 1.25

• Apply the Et+i expansion factors as follows:– Normal distribution

Ỹt+i = Ῡt+i + (Std Devhistory * Jt+i * Et+i * SND)Ỹt+i =NORM (Ῡt+i , Std Devhistory * Jt+i * Et+i )

– Empirical Distribution if Si are deviations from mean

Ỹt+i = Ῡt+i * { 1 + [Empirical(Sj , F(Sj), USD) * Et+I ]}

Example of Expansion Factors

• What you need to start a business– Legal entity– A product or service– Website– Web sales capability– Website developed

Setting Up an Internet Business

• Legal entity– Need a lawyer – Get a corporate name– Set up a corporation or LLC– File articles of incorporation with the

state– $2,000 plus– Things to consider for corporation

• Transfer of shares• President• General manager and Treasurer• Secretary

Setting Up an Internet Business

• Product or Service to sell• Protect your product• Copyright or patent

– What is the difference• Copyright protects software for life

of author + 16 years– Cost to file copyright is low about $50– http://www.copyright.gov/– Easy to do

Setting Up an Internet Business

• Website– Know what you want – Have a good design– Check out the website host

• What are their annual fees to host website

• What are other sites they host• What are their provisions for 24/7 service

– redundancy in servers • Do they have security sufficient for credit

cards– Get a firm date for delivery and going

live• Build in a penalty for late delivery

Setting Up an Internet Business

• Website development– Use a local website developer– Examine websites they developed

that offer services you require– Can they program the website to

accept credit cards with high level of security

– Are they going to program it locally? Or sub it out to a programmer in a foreign country

– Will they develop it to your time schedule

– What is the cost and cost of changes

Setting Up an Internet Business

• Website sales support is essential• Link Point is a reliable credit card

service – there are others• Need support from website host to

set this up• Credit card sales are credited to

your bank account in 5+ days• What is the monthly access fee?• What are the hidden costs – 5% for

credit transactions plus access fees

Setting Up an Internet Business