Description
model building approach or the variance-covariance approach to calculate value at risk with the help of examples
Market Risk VaR: ModelBuilding Approach
The Model-Building Approach
The main alternative to historical simulation is to make assumptions about the probability distributions of the returns on the market variables and calculate the probability distribution of the change in the value of the portfolio analytically This is known as the model building approach or the variance-covariance approach
Microsoft Example
We have a position worth $10 million in Microsoft shares The volatility of Microsoft is 2% per day (about 32% per year) We use N=10 and X=99
Microsoft Example continued
The standard deviation of the change in the portfolio in 1 day is $200,000 The standard deviation of the change in 10 days is
200 ,000 10 = $632,456
Microsoft Example continued
We assume that the expected change in the value of the portfolio is zero (This is OK for short time periods) We assume that the change in the value of the portfolio is normally distributed Since N(–2.33)=0.01, the VaR is
2.33 × 632,456 = $1,473,621
AT&T Example
Consider a position of $5 million in AT&T The daily volatility of AT&T is 1% (approx 16% per year) The SD per 10 days is
50,000 10 = $158,144
The VaR is
158114 × 2.33 = $368,405 ,
Portfolio
Now consider a portfolio consisting of both Microsoft and AT&T Suppose that the correlation between the returns is 0.3
S.D. of Portfolio
A standard result in statistics states that
2 ? X +Y = ? 2 + ? Y + 2?? X ? Y X
In this case ?X = 200,000 and ?Y = 50,000 and ? = 0.3. The standard deviation of the change in the portfolio value in one day is therefore 220,227
VaR for Portfolio
The 10-day 99% VaR for the portfolio is
220,227 × 10 × 2 . 33 = $ 1, 622 , 657
The benefits of diversification are (1,473,621+368,405)–1,622,657=$219,369
The Linear Model
We assume The daily change in the value of a portfolio is linearly related to the daily returns from market variables The returns from the market variables are normally distributed
The Linear Model
Suppose that we have a portfolio P consisting of n assets with an amount ?i being invested in asset. So
n
?P = ? ? i ?xi
i =1
For our example it would be
?P = 10?x1 + 5?x2
Markowitz Result for Variance of Return on Portfolio
n n
Variance of Portfolio Return = ?? ? ij wi w j ? i ? j
i =1 j =1
wi is weight of ith instrument in portfolio ? i2 is variance of return on ith instrument in portfolio ? ij is correlation between returns of ith and jth instrument s
When Linear Model Can be Used
Portfolio of stocks Portfolio of bonds Portfolio of currencies Forward contracts Interest-rate swap
The Linear Model and Options
Consider a portfolio of options dependent on a single stock price, S. Define
?= ?P ?S
and
?x = ?S S
Linear Model and Options continued
As an approximation
?P = ? ?S = S? ?x
Similarly when there are many underlying market variables
?P = ? Si ?i ?xi
i
where ?i is the delta of the portfolio with respect to the ith asset
Example
Consider an investment in options on Microsoft and AT&T. Suppose the stock prices are 120 and 30 respectively and the deltas of the portfolio with respect to the two stock prices are 1,000 and 20,000 respectively As an approximation
? P = 120 × 1,000 ? x1 + 30 × 20 ,000 ? x2
where ?x1 and ?x2 are the percentage changes in the two stock prices
Example
Assuming the daily volatilities of Microsoft and AT&T are 2% and 1% respectively. The correlation between the daily changes is 0.3. The portfolio standard deviation = 7.099 The five day 95% VaR = 1.65*sqrt(5)*7.099 = $26,193
But the distribution of the daily return on an option is not normal
The linear model fails to capture skewness in the probability distribution of the portfolio value.
But the distribution of the daily return on an option is not normal
Positive Gamma
Negative Gamma
Translation of Asset Price Change to Price Change for Long Call
Long Call
Asset Price
Translation of Asset Price Change to Price Change for Short Call
Asset Price
Short Call
Model Building vs Historical Simulation
Thank you!!!
doc_496948813.pdf
model building approach or the variance-covariance approach to calculate value at risk with the help of examples
Market Risk VaR: ModelBuilding Approach
The Model-Building Approach
The main alternative to historical simulation is to make assumptions about the probability distributions of the returns on the market variables and calculate the probability distribution of the change in the value of the portfolio analytically This is known as the model building approach or the variance-covariance approach
Microsoft Example
We have a position worth $10 million in Microsoft shares The volatility of Microsoft is 2% per day (about 32% per year) We use N=10 and X=99
Microsoft Example continued
The standard deviation of the change in the portfolio in 1 day is $200,000 The standard deviation of the change in 10 days is
200 ,000 10 = $632,456
Microsoft Example continued
We assume that the expected change in the value of the portfolio is zero (This is OK for short time periods) We assume that the change in the value of the portfolio is normally distributed Since N(–2.33)=0.01, the VaR is
2.33 × 632,456 = $1,473,621
AT&T Example
Consider a position of $5 million in AT&T The daily volatility of AT&T is 1% (approx 16% per year) The SD per 10 days is
50,000 10 = $158,144
The VaR is
158114 × 2.33 = $368,405 ,
Portfolio
Now consider a portfolio consisting of both Microsoft and AT&T Suppose that the correlation between the returns is 0.3
S.D. of Portfolio
A standard result in statistics states that
2 ? X +Y = ? 2 + ? Y + 2?? X ? Y X
In this case ?X = 200,000 and ?Y = 50,000 and ? = 0.3. The standard deviation of the change in the portfolio value in one day is therefore 220,227
VaR for Portfolio
The 10-day 99% VaR for the portfolio is
220,227 × 10 × 2 . 33 = $ 1, 622 , 657
The benefits of diversification are (1,473,621+368,405)–1,622,657=$219,369
The Linear Model
We assume The daily change in the value of a portfolio is linearly related to the daily returns from market variables The returns from the market variables are normally distributed
The Linear Model
Suppose that we have a portfolio P consisting of n assets with an amount ?i being invested in asset. So
n
?P = ? ? i ?xi
i =1
For our example it would be
?P = 10?x1 + 5?x2
Markowitz Result for Variance of Return on Portfolio
n n
Variance of Portfolio Return = ?? ? ij wi w j ? i ? j
i =1 j =1
wi is weight of ith instrument in portfolio ? i2 is variance of return on ith instrument in portfolio ? ij is correlation between returns of ith and jth instrument s
When Linear Model Can be Used
Portfolio of stocks Portfolio of bonds Portfolio of currencies Forward contracts Interest-rate swap
The Linear Model and Options
Consider a portfolio of options dependent on a single stock price, S. Define
?= ?P ?S
and
?x = ?S S
Linear Model and Options continued
As an approximation
?P = ? ?S = S? ?x
Similarly when there are many underlying market variables
?P = ? Si ?i ?xi
i
where ?i is the delta of the portfolio with respect to the ith asset
Example
Consider an investment in options on Microsoft and AT&T. Suppose the stock prices are 120 and 30 respectively and the deltas of the portfolio with respect to the two stock prices are 1,000 and 20,000 respectively As an approximation
? P = 120 × 1,000 ? x1 + 30 × 20 ,000 ? x2
where ?x1 and ?x2 are the percentage changes in the two stock prices
Example
Assuming the daily volatilities of Microsoft and AT&T are 2% and 1% respectively. The correlation between the daily changes is 0.3. The portfolio standard deviation = 7.099 The five day 95% VaR = 1.65*sqrt(5)*7.099 = $26,193
But the distribution of the daily return on an option is not normal
The linear model fails to capture skewness in the probability distribution of the portfolio value.
But the distribution of the daily return on an option is not normal
Positive Gamma
Negative Gamma
Translation of Asset Price Change to Price Change for Long Call
Long Call
Asset Price
Translation of Asset Price Change to Price Change for Short Call
Asset Price
Short Call
Model Building vs Historical Simulation
Thank you!!!
doc_496948813.pdf