INFOSYS BPO

Summer Internship Project

Developing a framework for Value at Risk(VaR) modeling

Soumyadeep Mukherjee

7/31/2008

The purpose of this project was to create a general framework with examples that will help to build value at risk(VaR) models for portfolios consisting of various financial instruments

1.Value at Risk(VaR) : Value at risk(VAR) is a tool to measure market risk. It is

defined as the maximum loss over a target horizon such that there is a low, pre

specified probability that the actual loss will be larger.

To measure VAR, we first need to mention two quantitative parameters,

The confidence level which is the probability that loss will not exceed what

specified in VaR.

The time horizon

Longer the horizon, greater the VAR measure. Similarly, higher the confidence level,

the greater the VAR measure.

2.Risk factors

Risk factors refer to variables, a change in value of which affects the value of the

portfolio. For example, since a change in interest rate causes a change in value of

bond, interest rate may be considered as a risk factor for bond.

One of the fundamental principles behind risk management is - divide to conquer. It

would be infeasible to model all financial instruments the portfolio to their individual

source of risk. The art of risk management consists of choosing a set of limited risk

factors that hopefully will span or cover the whole spectrum of risks. Instruments are

then decomposed into these elemental risk factors by a process called mapping.

Figure shows the process where three underlying risk factors were identified for 5

instruments and then there aggregate risk is calculated at the portfolio level.

3.VaR methods

The various methods used for calculating VaR differ by

Distributional assumptions –

o When we make assumptions about on the distribution of risk factors

and parameters associated with the distribution it is called parametric

method.

o When we make no assumptions about the distributions of risk factors

but only depend on the data available, it is called non parametric

method.

Linearity assumption –

o When we assume that the relationship between portfolio value and

risk factor is linear it is called local valuation method. This linearity

assumption makes computations simpler.

The distribution of the change price is the same as that of the change in

risk factors. This is particularly convenient for portfolios with

numerous sources of risks, because linear combinations of normal

distributions are normally distributed. Hence if we assume normal

distributions for risk factors, the price also has a normal distribution.

o Full valuation methods don’t assume any linearity of relationships

between risk factors and price and hence can accommodate non linear

relationships. Here for a change in value of the risk factors we fully

reprice the portfolio value.

The three most widely used methods for VaR computation are –

1. Historical simulation

2. Monte Carlo simulation

3. Variance Covariance method

3. a Historical Simulation - Historical simulation method for VaR computation is

based on the assumption that future returns can be predicted from the past returns of

risk factors. In this method the accuracy of the VaR computation is highly dependent

on the number of data points that are used. This method has been explained as

follows.

Let us assume that we have N risk factors and data for M periods. We will use this

historical data to calculate VaR.

Data may look like as follows.

Risk factor 1 Risk factor 2 …….. Risk factor N

Period 1 A1 B1 … X1

Period 2 A2 B2 …. X2

….. …. … …. ….

Period M Am Bm ….. Xm

M different scenarios are generated using the historical data based on the percentage

change in value of risk factors between period t and period t+1. For example the

percentage difference between A1 and A2 is used to generate scenario 1 for risk

factor1.Scenario 1 for all the risk factors are generated and is used for calculating the

simulated value of the risk factors in future period. Then the value of the portfolio is

calculated using the simulated value of risk factors.

Then the difference in the portfolio value is calculated. Similarly, M scenarios are

generated. Depending upon the confidence level specified, change in the portfolio

value is considered. These steps may be condensed in a flowchart as follows.

3.b Monte Carlo Simulation

This method is based on the assumption that we have some information about the

distribution of risk factors. Then using this distribution we can draw randomly a large

number of scenarios and price the portfolio in each scenario.

For example for equity portfolios we assume the underlying risk factor i.e. stock price

follows lognormal distribution. To simulate the stock price values a large no of

possible values of the lognormal variable are generated. For each of these simulated

values the portfolio value is calculated. Also we calculate the profit & loss of the

portfolio for each of the simulated scenarios.

A rich set of scenarios will give a very good approximation for the distribution of

P&L value of the portfolio. The lowest q – quantile of this distribution can be used as

an approximation for VaR.

Data for M periods for N factors

Use % change between period day t and day t+1 to generate the scenario using N risk factors

Calculate the portfolio value using the generated risk factor values in scenario

Calculate the change in the portfolio value (portfolio value in scenario – portfolio value in period M)

If #scenario < M

Arrange the change in portfolio value in ascending order

Depending upon confidence, consider the [(100% - conf)* M] th value

Increase t

3.c Variance Covariance method

This method is based on the assumption that the short term changes in the market

parameters and in the value of the portfolio are normal. This method also reflects the

fact that the market parameters are not independent; however it is restricted to the first

degree of dependence – correlation.

Price of a portfolio is a function of the market data, say P(x), where x is the vector of

the market data. The current parameters of the market are known x0, however

tomorrow the market will move to a new vector x1. The important simplifying

assumption of the variance covariance approach is that the changes of the parameter

vector are assumed to be normally distributed. Then we can write by Taylor’s series –

If the portfolio is linear in market parameters (as in case of equity portfolios) then the

first derivative will be constant and the second derivative will be zero. If the portfolio

is not linear in market parameters then we need to calculate the sensitivity of the

portfolio value for a change in each of the market parameters.

Once we have got the sensitivity measures, the RHS becomes a linear combination of

normal variables since we assumed change in market parameters follow normal

distribution. Hence change in portfolio value also is normally distributed. We find out

its mean and standard deviation from the mean and standard deviation of market

parameters. Then for VaR we need only to select the appropriate quantile of the

normally distributed variable.

VaR (dP) = NormsInv (confidence level) *stdev (dP) - E (dP)

There are various variants of variance covariance method –

Delta normal method: Here we consider only the first derivative of the Taylor

expansion. Hence it is useful when portfolio value is a linear combination of risk

factors or second derivatives are very small.

Delta Gamma method: Here we consider the second derivative also. Hence it is useful

when there is a non linear relationship like in case of options. It gives more accurate

VaR values.

Q&A on VaR

What's the difference between EaR, VaR, and EVE?

Earnings at Risk typically looks only at potential changes in cash flows/earnings over

the forecast horizon. Value at risk looks at the change in the entire value over the

forecast horizon. Economic Value of Equity also looks at value change, but typically

over a longer forecast horizon than VAR (up to 1 year). In a trading environment,

where profit and loss are equivalent to changes in value, EaR and VaR should be the

same.

What is market risk?

Market risk is usually defined as the risk to loss in a financial instrument from an

adverse movement in market prices or rates. What's adverse? Well, it depends. If you

own a bond, then a rise in interest rates is adverse, but if you have lent/sold a bond, it

is a fall in rates that is adverse. Generally people classify sources of market risk into

four categories, interest rates, equities, foreign exchange and commodities.

What is Stress Testing?

I think of stress testing as measure of risk exposure that's complementary to VaR.

Stress testing is a measure of potential loss as a result of a plausible event in an

abnormal market environment. Two types of stress testing are popular. The first is

based on economic scenarios. Pretend your portfolio experiences the 1987 or 1997

stock market crash again. The second is "matrix" based. Change a bunch of

assumptions about correlations and variances and see what happens. Neither is

statistical in nature, in contrast to VaR. That is, you don't know the probability of any

particular scenario.

What is Backtesting?

Backtesting is a statistical process for validating the accuracy of a VaR model.

Banking regulators require backtesting for banks that use VaR for regulatory capital.

It involves a comparison between the number of times the VaR model under-predicts

the subsequent day's loss, versus the number of time such an under-prediction is

expected. If losses exceeding VaR have a 1 in 100 chance of ocurring, then we expect

to see 2 or 3 of those in a year. There is a lot of debate about whether backtesting is

meaningful, because it is difficult to validate a model based on a few extreme events -

not enough data.

What do regulators think of VaR?

Love-hate, I think. Love first. Banking regulators internationally have agreed to allow

banks to use VaR models to calculate regulatory capital. Don't ask why banks have

minimum capital set by regulators, as that is a different FAQ. In the USA, the

securities regulator allows corporates to use VaR to express their exposure to market

risk in their annual and quarterly regulatory public financial filings. Now hate.

Regulators aren't sure that VaR is the "right" measure of risk? Nor are they sure how

much weight should be given to it in risk management. They really aren't sure

whether VaR should be extended to the measurement of other kinds of risk, such as

credit risk.

What is CVaR, Conditional Value at Risk?

Unfortunately, the term is not used consistently by all authors. Conditional value at

risk (cvar) is most often used to refer to a measure of the risk of loss beyond the VaR.

I.e., if the VaR of a portfolio is DM 5,000, then what is the expected loss beyond DM

5,000 (or "mean excess loss"), given that an observed loss is greater than the VaR.

However, some use the term to mean the estimation of VaR from "conditional" asset

return distributions (a conditional distribution is one that takes into account changes in

the shape of the distributions through time).

What is the proper relation between the VaR in a portfolio and the amount of

capital that should be held against it?

There are many considerations, if capital is to be based on VaR. VaR doesn't tell you

how big your losses could be on a bad day, it only defines what distingishes a bad day

from other days. If you have two portfolios with exposures to risks of different

markets, but the portfolios nevertheless have the same VaR, then it may be wrong to

keep the same capital against each portfolio, because one may have much worse

performance given a VaR exceedance day. Also, since VaR looks at only a particular

forecast horizon, and a bad economic environment may extend beyond that horizon,

the relationship between VaR and a business-continuity-threatening type of market

event is murky at best. Finally, the relationship between the amount of risk taken and

the amount of capital to be held may come down to the nature of the trading and the

risk appetite of the "owners" of the capital. If the portfolio has significant nonlinear

risks, then the relation between the capital and VaR is even more difficult to judge, as

it is sometimes the case that the nonlinearities are greatest beyond the VaR (e.g., in a

trading book, with a portfolio of barrier options, where the barriers are not hit within

the set of market moves resulting in the VaR). I could go on, for example, the

relationship between VaR and the cost of capital under the investment rule of

shareholder wealth maximization is not clear - whereas if it were clear, then we could

deduce the amount of capital just sufficient to support a given level of risk. And, the

impact of VaR-based capital requirements on the incentives of those taking the risks

is not all all clear. Having said all that, which should be pretty discouraging, I will

hazard that a one day VaR equal to about 3% of the trading capital is a pretty good

sized risk in a normal environment.

4.Example models

4.a Methodology for Calculating VaR for Equity portfolio

Objective : To calculate the VaR (Value at risk) for a portfolio consisting of three

stocks (TCS, TATA motors, Infosys ) using the three different methods of VaR

calculation i.e. delta normal method, Historical simulation method and Monte Carlo

simulation method.

Underlying risk factor: For equity the underling risk factor is the market price of the

shares of the stock.

Data: We collected data for past M periods for share prices of the three stocks. The

data will look like -

Date TCS TATA Motors Infosys

0 A0 B0 C0

1 A1 B1 C1

2 A2 B2 C2

3 A3 B3 C3

4 A4 B4 C4

…… …… …… …..

M Am Bm Cm

Historical Simulation: Let the share prices on the day of calculation of VaR be At, Bt

and Ct. We calculate the simulated share prices for each of the M scenarios as follows

–

Ai,sim = (Ai/Ai-1)*At

Bi,sim = (Bi/Bi-1)*Bt

Ci,sim = (Ci/Ci-1)*Ct

Now for each of these M scenarios we calculate the value of the portfolio as

Pi,sim = Ai,sim * Wa + Bi,sim * Wb + Ci,sim * Wc.

Where Wa, Wb and Wc are the weights of TCS, TATA motors and Infosys stock

respectively.

This will give a portfolio value for each of the M scenarios.

Let at time t the portfolio value is Pt. We calculate the difference between today’s

portfolio value and portfolio value we calculated for each of the M scenarios.

This is denoted by ΔPi = Pi,sim – Pt

Next we arrange these ΔPi values in increasing order. The first percentile will be our

VaR value.

Variance Covariance method

In variance covariance method we calculate the variance of each of the stocks based

on the shares prices collected. Let these variance be – V(A) , V(B) and V(C).

Also we calculate the pairwise covariance between stocks. Let those be Cov(A,B),

Cov(B,C) and Cov(C,A)

Then the variance of the portfolio is calculated as =

V(P) = Wa * Wa*V(A) + Wb * Wb * V(B) + Wc * Wc * V(C) + 2*Wa * Wb *

Cov(A,B) + 2*Wb * Wc * Cov(B,C) + 2*Wc*Wa*Cov(C,A)

Standard deviation of portfolio = SD(P) = Sqrt(V(P))

VaR for the portfolio for 99% confidence = 2.33 * SD(P)

Monte Carlo simulation: We assume the stock prices follow lognormal distribution.

Hence log of the returns on stock prices follows normal distribution.

In monte carlo method we generate a large no of possible prices for the share of the

stocks and calculate the portfolio value over those large number of combinations and

then select the bottom 1 percentile as the VaR value.

Log of the returns on stock prices follow normal distribution. Hence we simulate the

log returns by generating random values of standard normal variable. But the random

numbers should be correlated i.e. they should have the same correlation as for each

pair of stocks.

For example, the correlation between random values generated for TCS and infosys

stocks should be the same as the correlation between the actual historical log returns

between the two stocks.

This is accomplished as follows –

1. The variance covariance matrix is decomposed into a choleskey matrix by

following the choleskey decomposition method. The choleskey matrix is a

Upper triangular matrix U such that transpose(U) * U = variance covariance

matrix.

2. Then if we multiply the nomal random returns generated by this choleskey

matrix then we will get correlated random returns. Using this correlated

random returns we calculate the next day’s share price as follows – Next day’s

price = today’s price * exp (correlated random return)

3. Using the next day’s price for the three stocks we calculate the next day’s

simulated price for the portfolio Pi,sim.

4. Let today the portfolio value be Pt. We calculate the difference between

today’s portfolio value and portfolio value we calculated for each of the M

scenarios. This is denoted by ΔPi = Pi,sim – Pt

We do the above process( 1 to 4) a large number of time to generate a large number

of simulated ΔPi values.

Next we arrange these ΔPi values in increasing order. The first quantile will be our

VaR value.

4.b Methodology for Calculating VaR for Equity Options portfolio

Objective : To calculate the VaR (Value at risk) for a portfolio consisting of three

equity options (GE, Microsoft and Pfizer) using the three different methods of VaR

calculation i.e. Variance Covariance method, Historical simulation method and Monte

Carlo simulation method.

Underlying risk factor: There are six factors affecting the price of a stock option –

1. The current stock price, S0

2. The Strike price, K

3. The time to expiration, T

4. The volatility of stock price, σ

5. The risk free rate, r

6. The dividend expected during the life of option

Model - Given the data on the above risk factors we need to calculate the option prices

to know what maximum change in option price is possible. The maximum change in

prices of the individual options multiplied by the no options in the portfolio gives the

maximum change in value of the portfolio of stock option.

Value at Risk ( VaR for portfolio of stock options with p% confidence)

= ∑ (change in the price of stock option with p % confidence) * no of options in the

portfolio

We calculated the value of the individual options in the portfolio using Black Scholes

model.

The assumptions behind Black Scholes model are as follows –

1. Stock prices have log normal distribution with constant mean and volatility

2. The short selling of securities with the used proceeds is permitted

3. There are no transaction costs or taxes. All securities are perfectly divisible.

4. There are no dividends during the life of derivative

5. There is no risk less arbitrage opportunity.

6. Security trading is continuous.

7. The risk free rate of interest r is constant and same for all maturities.

Apart from the above we assume that all the options are European options.

The Black Scholes formula for the prices at time zero of a European call option on a

non dividend paying stock and European put option on a non dividend paying stock

are -

The function N(x) is the cumulative probability distribution function of a standardized

normal distribution. In other words it is the probability that a variable with with

standard normal distribution φ (0, 1) will be less than x.

Option Greeks -

Options greeks are measures that tells how the option value changes w.r.t the change

of the risk factors that we mentioned earlier. There are a no of such measures –

Delta

The delta of a portfolio of options or other derivatives dependent on a single asset

whose price is S is given by where Π is the value of the portfolio.

It is the rate of change of option value with respect to change in the underlying stock

price.

Theta

The theta of a portfolio of options is the rate of change of the value of the portfolio

with respect to the passage of time with all else remaining the same. Theta is

sometimes referred to as the time delay of the portfolio.

Gamma

The gamma of portfolio of options on an underlying asset, Γ, is the rate of change of

the portfolio’s delta with respect to the price of the underlying asset. It is the second

partial derivative of the portfolio with respect to the asset price.

Γ =

If gamma is small then delta changes slowly, and adjustments to keep a portfolio delta

neutral need to be made relatively infrequently. However if gamma is large in

absolute terms, delta is highly sensitive to the price of the underlying asset. It is then

quite risky to leave a delta neutral portfolio unchanged for a long time.

Vega

The vega of a portfolio of derivatives, ν, is the rate of change of the value of the

portfolio with respect to the volatility of the underlying asset.

Ν =

If vega is high in absolute terms, the portfolio’s value is very sensitive to small

changes in volatility. If vega is low in absolute terms, volatility changes have

relatively little impact on the value of the portfolio.

Rho

The rho of s portfolio of options is the rate of change of the value of the portfolio with

respect to the interest rate.

ρ =

It measures the sensitivity of the value of the portfolio to interest rates.

Data: We collected data for past M periods for share prices of the three stocks. The

data will look like -

Date GE Microsoft Pfizer

0 A0 B0 C0

1 A1 B1 C1

2 A2 B2 C2

3 A3 B3 C3

4 A4 B4 C4

…… …… …… …..

M Am Bm Cm

Also we collected the Greek values for the options that we had chosen for the above

three stocks.

Historical Simulation: Let the share prices on the day of calculation of VaR be At, Bt

and Ct. We calculate the simulated share prices for each of the M scenarios as follows

–

Ai,sim = (Ai/Ai-1)*At

Bi,sim = (Bi/Bi-1)*Bt

Ci,sim = (Ci/Ci-1)*Ct

Now for each of these M scenarios we calculate the value of the corresponding

options using Black scholes formula. Let the no options of the three type in the

portfolio be – An, Bn and Cn respectively.

For each of the options we calculate the simulated VaR value with 99% confidence.

This is calculated by taking the difference of the calculated option prices from today’s

price in each of the M scenarios and taking the 1 percentile among the M values

generated.

Let those simulated VaR value be – ΔA, ΔB and ΔC respectively. Then VaR value

with 99% confidence for the portfolio is calculated as -

Pi,sim = An * ΔA + Bn* ΔB + Cn* ΔC.

Variance Covariance method

In parametric method for a portfolio of options, in order to account for the non

linearity of the security value change, some risk factors (i.e. stock price) second order

terms are considered. Nevertheless, not all risk factors second order terms are

included in the VaR analysis. For a short time horizon, those second order terms are

insignificant and this makes VaR computation practical.

1. First we get the grrek values for all the options in the portfolio as well as the

volatility of the underlying stock from the Bloomberg database.

2. Next we calculate the security price change dV and portfolio P&L dP using

those greek values. Price change dV for each equity derivative can be

calculated as

3. For portfolio P&L, E[(dP)] = ∑ E(dV) and E[(dP2)] = ∑ E(dV2)

4. stddev(dV) = [E(dV2) – (E(dV))2]1/2

Stddev (dP) = [E (dP2) – (E(dP))2]1/2

5. VaR for both security and portfolio level is calculated as follows –

VaR (dV) = NormsInv (confidence level) *stdev (dV) - E (dV)

VaR (dP) = NormsInv (confidence level) *stdev (dP) - E (dP)

VaR for the portfolio for 99% confidence = 2.33 * stdev (dP) - E (dP)

Monte Carlo simulation: We assume the stock prices follow lognormal distribution.

Hence log of the returns on stock prices follows normal distribution.

In Monte Carlo method we generate a large no of possible prices for the share of the

stocks and calculate the portfolio value over those large numbers of combinations and

then select the bottom 1 percentile as the VaR value.

Log of the returns on stock prices follow normal distribution. Hence we simulate the

log returns by generating random values of standard normal variable. But the random

numbers should be correlated i.e. they should have the same correlation as for each

pair of stocks.

For example, the correlation between random values generated for GE and Microsoft

stocks should be the same as the correlation between the actual historical log returns

between the two stocks.

This is accomplished as follows –

1. The variance covariance matrix is decomposed into a choleskey matrix by

following the choleskey decomposition method. The choleskey matrix is a

Upper triangular matrix U such that transpose (U) * U = variance covariance

matrix.

2. Then if we multiply the normal random returns generated by this choleskey

matrix then we will get correlated random returns. Using this correlated

random returns we calculate the next day’s share price as follows – Next day’s

price = today’s price * exp (correlated random return)

3. Using the next day’s price for the three stocks we calculate the next day’s

simulated price for the option.

Given stock prices, we calculate the value of the corresponding options using

Black Scholes formula. Also we take the difference of the simulated option

values from that of today. Let those differences be ΔAi, ΔBi and ΔCi

respectively.

We do the above process (1 to 4) a large number of times to generate a large number

of simulated ΔAi, ΔBi and ΔCi values.

Next we arrange these ΔAi, ΔBi, ΔCi values in increasing order. The first quantile will

be our VaR value for each of the options. Let those simulated VaR value be – ΔA, ΔB

and ΔC respectively. Then VaR value with 99% confidence for the portfolio is

calculated as -

Pi,sim = An * ΔA + Bn* ΔB + Cn* ΔC

4.c Methodology for Calculating VaR for bond portfolio

Objective : To calculate the VaR (Value at risk) for a portfolio consisting of a single

treasury bond using the three different methods of VaR calculation i.e. Variance

Covariance method, Historical simulation method and Monte Carlo simulation

method.

Underlying risk factor: For bond the only factor affecting its price is interest rate.

Since a normal coupon bond has different cash flows at different points in time, we

need to take the spot rates at those different points in time to discount the cash flows

to price the bond.

These spot rates are calculated by interpolating the key rates available on standard

financial database. These key rates are spot rates after some standard time differences

like – 3 month spot rate, 6 month spot rate, 1 year spot rate etc.

For example if we have the next coupon payment 4 months from now, we will find

spot rate after 4 month by interpolating 3 month and 6 month spot rates.

Hence each of these key rates is a risk factor for VaR calculation of bond. For our

model we had some eight key rates and hence eight risk factors.

Data: We collected data for past M periods for eight key rates. The data will look like

-

Date 1 month 3 month 6 month 1 yr 2 yr 3 yr 5 yr 7 yr

0 A0 B0 C0 D0 E0 F0 G0 H0

1 A1 B1 C1 D1 E1 F1 G1 H1

2 A2 B2 C2 D2 E2 F2 G2 H2

3 A3 B3 C3 D3 E3 F3 G3 H3

4 A4 B4 C4 D4 E4 F4 G4 H4

…… …… …… ….. ….. ….. ….. ….. …..

M Am Bm Cm Dm Em Fm Gm Hm

Also we collected the Greek values for the options that we had chosen for the above

three stocks.

Historical Simulation: Let the share prices on the day of calculation of VaR are At, Bt

Ct, Dt, Et, Ft, Gt and Ht. We calculate the simulated share prices for each of the M

scenarios as follows –

Ai,sim = (Ai/Ai-1)*At

Bi,sim = (Bi/Bi-1)*Bt

Ci,sim = (Ci/Ci-1)*Ct

Di,sim = (Di/Di-1)*Dt

Ei,sim = (Ei/Ei-1)*Et

Fi,sim = (Fi/Fi-1)*Ft

Gi,sim = (Gi/Gi-1)*Gt

Hi,sim = (Hi/Hi-1)*Ht

Now for each of these M scenarios we calculate the value of the rates at cash flow

points using interpolation. Then we use the rates at cash points to discount the cash

flows at those points to get the price of the bond.

So this will generate M sets of possible price of the bond the next day.

Monte Carlo simulation: We assume the log of the returns on interest rates follows

normal distribution.

In Monte Carlo method we generate a large no of possible values of the interest rates

and then calculate the value of the bond value over those large numbers of

combinations.

Log of the returns on interest rates follow normal distribution. Hence we simulate the

log returns by generating random values of standard normal variable. But the random

numbers should be correlated i.e. they should have the same correlation as for each

pair of key rates.

For example, the correlation between random values generated for 1 month and 3

month should be the same as the correlation between the actual historical log returns

between the two rates.

This is accomplished as follows –

1. The variance covariance matrix is decomposed into a choleskey matrix by

following the choleskey decomposition method. The choleskey matrix is a

Upper triangular matrix U such that transpose (U) * U = variance covariance

matrix.

2. Then if we multiply the normal random returns generated by this choleskey

matrix then we will get correlated random returns.

3. Using this correlated random returns we calculate the next day’s share price as

follows – Next day’s rate= today’s rate* exp (-.5*daily volatility of rate* time

horizon+ sqrt(time horizon) *correlated return)

4. Using the next day’s simulated rates for the key points we calculate the

simulated rates at cash flow points. We calculate the change in the rates at the

cash flow points and then multiply them by the key rate duration at those cash

flow points. Holding all other maturities constant, key rate duration measures

the sensitivity of a security or the value of a portfolio to a 1% change in yield

for a given maturity.

It is calculated as KRD = where P+

and P- are the value of the bond when yield goes down and up by ∆y.

Hence Percentage change in price of the bond =

We do the above process (1 to 4) a large number of times to generate a large number

of simulated interest rate values and hence percentage change in price of bond.

Next we arrange these percentage change values in increasing order. The first quantile

will be our VaR value for each of the bond portfolio.