Value at Risk : a specific real estate model Direct real estate value at Risk
Value at Risk - Final Report
-
Upload
mukherjeesoumyadeep61164 -
Category
Documents
-
view
226 -
download
0
Transcript of Value at Risk - Final Report
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.