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Unformatted text preview: Value': at Risk In Chapter 15 we examined measures such as delta, gamma, and vega for describing
different aspects of the risk in a portfolio of derivatives. A financial institution usually
calculates each of these measures each day for every market variable to which it is
exposed, Often there are hundreds, or even thousands, of these market variables.
A deltagammavega analysis, therefore, leads to a huge number of different risk
measures being produced each day. These risk measures provide valuable information
for a trader who is responsible for managing the part of the financial institution's
portfolio that is dependent on the particular market variable. However, they do not
provide a way of measuring the total risk to which the financial institution is exposed.
Value at Risk (VaR) is an attempt to provide a single number summarizing the total
risk in a portfolio of financial assets. It has become widely used by corporate treasurers
and fund managers as well as by financial institutions. Central bank regulators also use
VaR in determining the capital a bank is required to keep to reflect the market risks it is
bearing.
In this chapter we; explain the VaR measure and describe the two main approaches
for calculating it. These are known as the historical simulation approach and the modelbuilding approach, 18.1 THE VaR MEASURE
When using the valueatrisk measure, we are interested in making a statement of the
following form:
We are X percent certain that we will not lose more than V dollars in the next N days.
The variable V is the VaR of the portfolio. It is a function of two parameters: the time
horizon (N days) and the confidence level (X%), It is the loss level over N days that we
are X% certain will not be exceeded. Bank regulators require banks to calculate VaR
with N = 10 and X = 99 (see the discussion in Business Snapshot 18.1).
In general, when N days is the time horizon and X% is the confidence level, VaR is
the loss corresponding to the (100  X)th percentile of the distribution of the change in
the value of the portfolio over the next N days. For example, when N = 5 and X = 97,
VaR is the third percentile of the distribution of changes in the value of the portfolio
435 436 CHAPTER 18 Business Snapshot 18.1
The Basel Committee on Bank Supervision is a committee of the world's bank
regulators that meets regularly in Basel, Switzerland. In 1988 it published what has
become known as The 1988 BIS Accord, or simply The Accord. This isan agreement
between the regulators on how the capital a bank is required to hold for credit risk
should be calculated. Several years later the Basel Committee published The 1996
Amendment, which was implemented in 1998 and required banks to hold capital for
market risk: as well as credit risk. The Amendment distinguishes between a bank's
oans
trading book and its banking book. The banking book consists primarily
and is not usually revalued on a regular basis for managerial and a
ting
purposes. The trading book consists of the myriad of different instrument that are
traded by the bank (stocks, bonds, swaps, forward contracts, options, etc.) and is
normally revalued daily..
 The 1996 BIS Amendment calculates capital for the trading book using the VaR
measure with N = 10 and X = 99. This means that it focuses on the revaluation loss
over a 10day period that is expected to be exceeded only 1% of the time. The capital
it requires the bank to hold is k times this VaR measure (with an adjustment for what
are termed specific risks). The multiplier k is chosen on a bankbybank basis by the
regulators and must be at least 3.0. For a bank with excellent welltested VaR
estimation procedures, it is likely that k will be set equal to the minimum value
of ~.O. For other banks it may be higher.
over the next 5 days. VaR is illustrated for the situation where the change in the value of
the portfolio is approximately normally distributed in Figure 18.1.
VaR is an attractive measure because it is easy to understand. In essence, it asks
the simple question "How bad can things get?" This is the question all senior
managers want answered. They are very comfortable with the idea of compressing
all the Greek letters for all the market variables underlying a portfolio into a single
number.
If we accept that it is useful to have a single number to describe the risk of a portfolio,
an interesting question is whether VaR is the best alternative. Some researchers have
Calculation ofVaR from the probability distribution of the change in the
portfolio value; confidence level is X%. Figure 18.1 437 Value at" Risk Alternative situation to Figure 18.1. \TaR is the same, but the potential
'
loss is larger.
..,.._ Figure 18.2 VaR argued that VaR may tempt traders to choose a portfolio with a return distribution
similar to that in Figure 18.2. The portfolios in Figures 18.1 and 18.2 have the same
VaR, but the portfolio in Figure 18.2 is much riskier because potential losses are much
larger.
A measure that deals with the problem we have just mentioned is Conditional VaR
(CVaR):1 Whereas VaR asks the question "How bad can things get?", CVaR asks
"If things do get bad, how much can we expect to lose?" CVaR'is the expected loss
during an Nday period conditional that we are in the (l00  X)% left tail of the
distribution. For example, with X = 99 and N = 10, CVaR is the average amount we
lose over a lOday period assuming that a 1% worstcase event occurs.
In spite of its weaknesses, VaR (not CVaR) is the most popular measure of risk
among both regulators and risk managers. We will therefore devote most of the rest of
this chapter to how it can be measured. The Time Horizon
In theory, VaR has two parameters. These are N, the time horizon measured in days,
and X, the confidence interval. In practice analysts almost invariably set N = 1 in the
first instance. This is because there is not enough data to estimate directly the behavior
of market variables over periods of time longer than 1 day. The usual assumption is
Nday VaR = Iday VaR x .IN This formula is exactly true when the changes in the value of the portfolio on successive
days have independent identical normal distributions with mean zero. In other cases it
is an approximation.
As explained in Business Snapshot 18.1, regulators require a bank's capital to be at
least three times the lOday 99% VaR. Given the way a lOday VaR is calculated, this
minimum capital level is, to all intents and purposes 3 x .vTO = 9.49 times the Iday
99% VaR.
1 This measure, which is also known as expected shortfall or tail loss, was suggested by P. Artzner,
F. Delbaen, J.M. Eber, and D. Heath, "Coherent Measures of Risk," Mathematical Finance, 9 (1999):
20328. These authors define certain properties that a good risk measure should have and show that the
standard VaR measure does not have all of them. 438 18.2 CHAPTER 18 HISTORICAL SIMULA nON
Historical simulation is one popular way of estimating VaR. It involves using past data
in a very direct way as a guide to what might happen in the future. Suppose that we
wish to calculate VaR for a portfolio using a Iday time horizon, a 99% confidence
level, and 500 days of data. The first step is to identify the market variables affecting
the portfolio. These will typically be exchange rates, equity prices, interest rates, and so
on. We then collect data on the movements in these market variables over the most
recent 509 days. This provides us with 500 alternative scenarios for what can happen
between today and tomorrow. Scenario 1 is where the percentage changes in the values
of all variables are the same as they were on the first day for which we have collected
data; scenario 2 is where they are the same as on the second day for which we have
data; and so on. For each scenario we calculate the dollar change in the value of the
portfolio between today and tomorrow. This defines a probability distribution for daily
changes in the value of our portfolio. The fifthworst daily change is the first percentile
of the distribution. The estimate of VaR is the loss when we are at this first percentile
point. Assuming that the last 500 days are a good guide to what could happen during
the next day, we are 99% certain that we will not take a loss greater than our VaR
estimate.
The historical simulation methodology is illustrated in Tables 18.1 and 18.2.
Table 18.1 shows observations on market variables over· the last 500 days. The
observations are taken at some particular: point in time during the day (usually the
close of trading). We denote the first day for which we have data as Day 0; the second as
Day 1; and so on. Today is Day 500; tomorrow is Day 501.
Table 18.2 shows the values of the market variables tomorrow if their percentage
changes between today and tomorrow are the same as they were between Day i  I and
Day i for 1 ::( i ::( 500. The first row in Table 18.2 shows the values of market variables
tomorrow assuming their percentage changes between today and tomorrow are the
same as they were between Day 0 and Day I; the second row shows the values of market
variables tomorrow assuming their percentage changes between Day 1 and Day 2 occur;
and so on. The 500 rows in Table 18.2 are the 500 scenarios considered. Table 18.1 Data for VaR historical simulation calculation. Day Market
variable 1 Market
variable 2 Market
variable n 0
2
3 20.33
20.78
21.44
20.97 0.1132
0.1159
0.1162
0.1184 65.37
64.91
65.02
64.90 498
499
50.0 25.72
25.75
25.85 0.1312
0.1323
0.1343 62.22
61.99
62.10 1 439 Value at Risk Table 18.2 Scenarios generated f9r tomorrow (Day SOl) using data in Table 18.1. Scenario
number Market
variable 1 Market
variable 2 Market
variable n Portfolio IlaUle
($ millions) Change in value
($ millions) 1
2
3 26.42
26.67
25.28 0.1375
0.1346
0.1368 61.66
62.21
61.99 23.71
23.12_
22.94 0.21
0.38
0.56 499
500 25.88
25.95 0.1354
0.1363 61.87
62.21 23.63
22.87 0.13
0.63 Define Vi as the value of a market variable on Day i and suppose that today is Day In.
The ith scenario assumes that the value of the market variable tomorrow will be In our example, Tn = 500. For the first variable, the value today; Vsoo, is 25.85. Also
Vo = 20.33 and VI = 20.78. It follows that the value of the first market variable in the
first scenario is
7585
_. 20.78 x 20.33 = _.76 47 The penultimate column of Table 18.2 shows the value of the portfolio tomorrow for
each of the 500 scenarios. We suppose the value of the portfolio today is $23.50 million.
This leads to the numbers in the final column for the change in the value between
today and tomorrow for all the different scenarios. For Scenario 1 the change in value
is +$210,000, for Scenario 2 it is .,$380,000, and so on.
We are interested in the Ipercentile point of the distribution of changes in the
portfolio value. Because there are a total of 500 scenarios in Table 18.2 we can estimate
this as the fifth worst number in the final column of the table. Alternatively, we can use
the techniques of what is known as extreme value theory to smooth the numbers in the left
tail of the distribution in an attempt to obtain a more accurate estimate of the 1% point
of the distribution. 2 As mentioned in the previous section, the Nday VaR for a 99%
confidence level is calculated as .IN times the Iday VaR.
Each day the VaR estimate in our example would be updated using the most recent
500 days of data. Consider, for example, what happens on Day 501. We find out new
values for all the market variables and are able to calculate a new value for our
portfolio. 3 We then go through the procedure we have outlined to calculate a new
VaR. We use data on the market variables from Day 1 to Day 501. (This gives us the
required 500 observations on the percentage changes in market variables; the DayO
See P. Embrechts, C. Kluppelberg, and T. Mikosch. Modeling Extremal Events for Insurance and Finance.
New York: Springer, 1997; A.I. McNeil, "Extreme Value Theory for Risk Managers," in Internal Modellilg
and CAD II. London, Risk Books, 1999, and available from www.math.ethz.ch/~mcneil. 2 3 Note that the portfolio's composition may have changed between Day 500 and Day 501. 440 CHAPTER 18
values of the market variables are no longer used.) Similarly, on Day 502, we use data
from Day 2 to Day 502 to determine VaR, and so on. 18.3 MODELBUILDING APPROACH
The main alternative to historical simulation is the modelbuilding approach. Before
getting into the details of the approach, it is appropriate to mention one issue
concerned with the units for measuring volatility. Daily Volatilities
In option pricing we usually measure time in years, and the volatility of an asset is
usually quoted as a "volatility per year". When using the modelbuilding approach to
 calculate VaR, we usually measure time in days and the volatility oran asset is usually
quoted as a "volatility per day".
What is the relationship between the volatility per year used in option pricing and the
volatility per day used in VaR calculations? Let us define O"year as the volatility per year
of a certain asset and O"day as the equivalent volatility per day of the asset. Assuming 252
trading days in a year, we can use equation (13.2) to write the standard deviation of the
continuously compounded return on the asset in 1 year as either O"year or O"day .J252. It
follows that
O"year = O"day J?5?
or
O"year
O"day = .J252 so that daily volatility is about 6% of annual volatility.
As pointed out in Section 13.4, O"day is approximately equal to the standard
deviation of the percentage change in the asset price in one day. For the purposes
of calculating VaR we assume exact equality. We define the daily volatility of an asset
price (or any other variable) as equal to the standard deviation of the percentage
change in one day.
Our discussion in the next few sections assumes that we have estimates of daily
volatilities and correlations. In Chapter 19, we discuss how the estimates can be
produced. SingleAsset Case
We now consider how VaR is calculated using the modelbuilding approach in a very
simple situation where the portfolio consists of a position in a single stock. The portfolio
we consider is one consisting of $10 million in shares of Microsoft. We suppose that
N = 10 and X = 99, so that we are interested in the loss level over 10 days that we are
99% confident will not be exceeded. Initially, we consider a Iday time horizon.
We assume that the volatility of Microsoft is 2% per day (corresponding to about
32% per year). Because the size of the position is $10 million, the standard deviation of
daily changes in the value of the position is 2% of $10 million, or $200,000.
~t is customary in the modelbuilding approach to assume that the expected change in
a market variable over the time period considered is zero. This is not strictly true, but it Value at Risk 441 is a reasonable assumption. The expected change in the price of a market variable over
a short time period is generally small when compared with the stqndard deviation of the
change. Suppose, for example, that Microsoft has an expected return of 20% per
annum. Over a Iday period, the expected return is 0.20/252, or about 0.08%, whereas
the standard deviation of the return is 2%. Over a 10day period, the expected return is
0.08 x 10, or about 0.8%, whereas the standard deviation of the return is 2.JTO, or
about 6.3%.
So far, we have established that the change in the value of the portfolio of Microsoft
shares over a Iday period has a standard deviation of $200,000 and (at least approximately) a mean of zero. We assume that the change is normally distributed. 4 From the
tables at the end of this book, N( 2.33) = 0.01. This means that there is a 1%
probability that a normally distributed variable will decrease in value by more than
2.33 standard deviations. Equivalently, it means that we are 99% certain that a normally
distributed variable will not decrease in value by more than 2.33 standard deviations.
The Iday 99% VaR for our portfolio consisting of a $10 million position in Microsoft
is therefore
2.33 x 200,000 = $466,000
As discu'ssed earlier, the Nday VaR is calculated as ,IN times the Iday VaR. The
.
10day 99% VaR for Microsoft is therefore
466,000 x .JTO = $1,473,621 Consider next a portfolio consisting of a $5 million position in AT&T, and suppose the
daily volatility of AT&T is 1% (approximately 16% per year). A similar calculation to
that for Microsoft shows that the standard deviation of the change in the value of the
portfolio in 1 day is
5,000,000 x 0.01 = 50,000
Assuming the change is normally distributed, the Iday 99% VaR is
50,000 x 2.33 = $116,500
and the 10day 99% VaR is
116,500 x .JTO = $368,405 TwoAsset Case
Now consider a portfolio consIstmg of both $10 million of Microsoft shares and
$5 million of AT&T shares. We suppose that the returns on the two shares have a
bivariate normal distribution with a correlation of 0.3. A standard result in statistics
tells us that, if two variables X and Y have standard deviations equal to ax and ay with
the coefficient of correlation between them equal to p, the standard deviation of X + Y
is given by To be consistent with the option pricing assumption in Chapter 13, we could assume that the price of
Microsoft is lognormal tomorrow. Because I day is such a short period of time, this is almost
indistinguishable from the assumption we do makethat the change in the stock price between today and
tomorrow is normal. 4 442 CHAPTER 18
To apply this result, we set X equal to the change in the value of the position in
Microsoft over a Iday period and Y equal to the change in the value of the position in
AT&T over a Iday period, so that
erx = 200,000 and ery = 50,000 The standard deviation of the change in the value of the portfolio consisting of both
stocks over a Iday period is therefore
)200,0002 + 50, 000 2 + 2 x 0.3 x 200,000 x 50,000 = 220,227
The mean change is assumed to be zero. The change is normally distributed. So the
Iday 99% VaR is therefore
220,227 x 2.33 = $513,129
_The 10day 99% VaR is .JTO times this, or $1,622,657. The Benefits of Diversification
In the example we have just considered:
1. The lOday 99% VaR for the portfolio of Microsoft shares is $1,473,621.
2. The lOday 99% VaR for the portfolio of AT&T shares is $368,405.
3. The 10day 99% VaR for the portfolio of both Microsoft and AT&T shares is $1,622,657.
The amount
(1,473,621
represents the benefits
correlated, the VaR for
VaR for the Microsoft
perfect correlation leads + 368,405)  1,622,657 = $219,369 of diversification. If Microsoft and AT&T were perfectly
the portfolio of both Microsoft and AT&T would equal the
portfolio plus the VaR for the AT&T portfolio. Less than
to some of the risk being "diversified away". 5 18.4 LINEAR MODEL
The examples we have just considered are simple illustrations of the use of the linear
model for calculating VaR. Suppose that we have a portfolio worth P consisting of n
assets with an amount Cti being invested in asset i (1 ::;;; i ::;;; n). We define /).xi as the
return on asset i in 1 day. It follows that the dollar change in the value of our
investment in asset i in 1 day is Cti /).Xi and
/I /)'P = I:Cti /).Xi (18.1) i=l where /)'P is the dollar change in the value of the whole portfolio in 1 day. Harry Markowitz was one of the first researchers to study the benefits of diversification to a portfolio
mallager. He was awarded a Nobel prize for this research in 1990. See H. Markowitz, "Portfolio Selection,"
Journal oj Finance, 7, I (March 1952): 7791. 5 Value at Risk 443 In the example considered in the previous section, $10 million was invested in the first
asset (Microsoft) and $5 million was invested in the second f!.sset (AT&T), so that
(in millions of dollars) ell = 10, el2 = 5, and
/::,.P = lO/::"XI + 5/::"X2 If we assume that the /::"Xi in equation (18.1) are multivariate:normal, /::,.P is normally
distributed. To calculate VaR, we therefore need to calculate only the mean and
standard deviation of /::"P. We assume, as discussed in the previous section, that the
expected value of each /::"Xi is zero. This implies that the mean of /::,.P is zero.
To calculate the standard deviation of /::,.P, we defirJe (Ji as the daily volatility of the
ith asset and Pij as the coefficient of correlation between returns on asset i and asset j.
This means that (Ji is the standard deviation of /::"Xi, and Pij is the coefficient of
correlation between /::"Xi and /::"Xj. The variance of /::,.P, which we will denote by (J~,
is given by
11 (J~ = 11 LLPiPielPi(Jj
i=1 j==1 This equation can also be written
Il 1J (J~ = Lel~(Jl +2 LLpiPielP;(Jj
;=1 (18.2) ;=1 j<; The standard deviation of the change over N days is (Jp..Jjij, and the 99% VaR for an
Nday time horizon is 2.33(Jp..Jjij.
In the example considered in the previous section, (JI = 0.02, (J2 = 0.01, and
Pl2 = 0.3. As already noted, ell = 10 and el2 = 5, so that
(J~ = 102 X 0.022 + 52 X 0.01 2 + 2 x 10 x 5 x 0.3 x 0.02 x 0.01 = 0.0485 and (Jp = 0.220. This is the standard deviation of the change in the portfolio value per
day (in millions of dollars). The lOday 99% VaR is 2.33 x 0.220 x .JIO =
$1.623 million. This agrees with the calculation in the previous section. Handling Interest Rates
It is out of the question to define a separate market variable for every single bond price or interest rate to which a company is exposed. Some simplifications are necessary when
the modelbuilding approach is used. One possibility is to assume that only parallel
shifts in the yield curve occur. It is then necessary to defirJe only one market variable:
the size of the parallel shift. The changes in the value of bond portfolio can then be
calculated using the duration relationship
/::,.P=DP/::,.y where P is the value of the portfolio, /::,.P is the its change in P in one day, D is the
modified duration of the portfolio, and /::"y is the parallel shift in 1 day.
This approach does not usually give enough accuracy. The procedure usually
followed is to choose as market variables the prices of zerocoupon bonds with standard
maturities: 1 month, 3 months, 6 months, 1 year, 2 years, 5 years, 7 years, 10 years, and
30 years. For the purposes of calculating VaR, the cash flows from instruments in the 444 CHAPTER 18
portfolio are mapped into cash flows occurring on the standard maturity dates.
Consider a $1 million position in a Treasury bond lasting 1.2 years that pays a coupon
of 6% semiannually. Coupons are paid in 0.2, 0.7, and 1.2 years, and the principal is
paid in 1.2 years. This bond is, therefore, in the first instance regarded as a $30,000
position in 0.2year zerocoupon bond plus a $30,000 position in a 0.7year zerocoupon bond plus a $1.03 million position in a 1.2year zerocoupon bond. The
position in the 0.2year bond is then replaced by an equivalent position in Imonth
and 3month zerocoupon bonds; the position in the 0.7year bond is replaced by an
equivalent position in 6month and Iyear zerocoupon bonds; and the position in the
1.2yeai bond is replaced by an equivalent position in Iyear and 2year zerocoupon
bonds. The result is that the position in the 1.2year couponbearing bond is for VaR
purposes regarded as a position in zerocoupon bonds having maturities of 1 month,
3 months, 6 months, 1 year, and 2 years.
This procedure is known as caslzjlow mapping. One way of doing it is explained in the
 appendix at the end of this chapter. Applications of the Linear Model
The simplest application of the linear model is to a portfolio with no derivatives
consisting of positions in stocks, bonds, foreign exchange, and commodities. In this
case, the change in the value of the portfolio is linearly dependent on the percentage
changes in the prices of the assets comprising the portfolio. Note that, for the purposes
ofVaR calculations, all asset prices are measured in the domestic currency. The market
variables considered by a large bank in the United States are therefore likely to include
the value of the Nikkei 225 index measured in dollars, the price of a lOyear sterling
<,
zerocoupon bond measured in dollars, and so on.
An example of a derivative that can be handled by the linear model is a forward
contract to buy a foreign currency. Suppose the contract matures at time T. It can be
regarded as the exchange of a foreign zerocoupon bond maturing at time T for a
domestic zerocoupon bond maturing at time T. For the purposes of calculating VaR,
the forward contract is therefore treated as a long position in the foreign bond
combined with a short position in the domestic bond. Each bond can be handled using
a cashflow mapping procedure.
Consider next an interest rate swap. As explained in Chapter 7, this can be regarded
as the exchange of a floatingrate bond for a fixedrate bond. The fixedrate bond is a
regular couponbearing bond. The floatingrate bond is worth par just after the next
payment date. It can be regarded as a zerocoupon bond with a maturity date equal to
the next payment date. The interest rate swap therefore reduces to a portfolio of long
and short positions in bonds and can be handled using a cashflow mapping procedure. The Linear Model and Options
We now consider how the linear model can be used when there are options. Consider
first a portfolio consisting of options on a single stock whose current price is S. Suppose
that the delta of the position (calculated in the way described in Chapter 15) is 8. 6 Since 8
6 Normally we denote the delta and gamma of a portfolio by
lower case Greek letters 8 and y to avoid overworking J::,.. J::,. and f. In this section and the next, we use the Value at Risk is the rate of change of the value of the portfolio with S, it is approximately true or
(18.3) where /::,.S is the dollar change in the stock price in 1 day and /::,.P is, as usual, the dollar
change in the portfolio in 1 day. We define /::,.X as the percentage change in the stock
price in 1 day, so that
/::,.S /::,.X =  S It follows that an approximate relationship between /::,.P and /::,.x is /::,.P = So /::,.x When we have a position in several underlying market variables that includes options,
we can derive an approximate linear relationship between /::,.P and the /::"xi similarly.
This relationship is
II /::,. P = L (18.4) SiDi /::"Xi i=1 where Si is the value of the ith market variable and Di is the delta of the portfolio with
respect to the ith market variable. This corresponds to equation (18.1):
II (18.5) /::,.P = Lai /::"xi
i=1 with ai = SiDi' Equation (18.2) can therefore be used to calculate the standard deviation
of /::,.P.
Example 18.1 A portfolio consists of options on Microsoft and AT&T. The options on Microsoft have a delta of 1,000, and the options on AT&T have a delta of 20,000. The
Microsoft share price is $120, and the AT&T share price is $30. From equation (18.4), it is approximately true that
/::,.P = 120 x 1,000 x /::"XI or
/::,.P = l20,000/::"xl + 30 x 20,000 x /::"x2 + 600,000/::"X2 where /::"Xj and /::"X2 are the returns from Microsoft and AT&T in 1 day and /::,.P is
the resultant change in the value of the portfolio. (The portfolio is assumed to be
equivalent to an investment of $120,000 in Microsoft and $600,000 in AT&T.)
Assuming that the daily volatility of Microsoft is 2% and the daily volatility of
AT&T is 1% and the correlation between the daily changes is 0.3, the standard
deviation of /::,.P (in thousands of dollars) is
/(120 x 0.02)2 + (600 X 0.01)2 + 2 x 120 x 0.02 x 600 x 0.01 x 0.3 = 7.099 Since N( 1.65) = 0.05, the 5day 95% VaR is 1.65 x .J5 x 7,099 = $26,193. 446 18.5 CHAPTER 18 QUADRATIC MODEL
When a portfolio includes options, the linear model is an approximation. It does not
take account of the gamma of the portfolio. As discussed in Chapter 15, delta is defined
as the rate of change of the portfolio value with respect to an underlying market
variable and gamma is defined as the rate of change of the delta with respect to the
market variable. Gamma measures the curvature of the relationship between the
portfolio value and an underlying market variable.
Figure 18.3 shows the impact of a nonzero gamma on the probability distribution of
the value of the portfolio. When gamma is positive, the probability distribution tends to
be positively skewed; when gamma is negative, it tends to be negatively skewed.
Figures 18.4 and 18.5 illustrate the reason foi this result. Figure 18.4 shows the relationship between the value of a long call option and the price of the underlying asset. A long
call is an example of an option position with positive gamma. The figure shows that,
 when the probability distribution for the price of the underlying asset at the end of 1 day
is normal, the probability distribution for the option price is positively skewed. 7
Figure 18.5 shows the relationship between the value of a short call position and the
price of the underlying asset. A short call position has a negative gamma. In this case, we
see that a normal distribution for the price of the underlying asset at the end of 1 day gets
mapped into a negatively skewed distribution for the value of the option position.
The VaR for a portfolio is critically dependent on the left tail of the probability
distribution of the portfolio value. For exa.mple, when the confidence level used is 99%,
the VaR is the value in the left tail below which there is only 1% of the distribution. As
indicated in Figures l8.3(a) and 18.4, a positive gamma portfolio tends to have a less
heavy left tail than the normal distribution. If we assume the distribution is normal, we
will tend to calculate a VaR that is too high. Similarly, as indicated in Figures 18.3(b)
and 18.5, a negative gamma portfolio tends to have a heavier left tail than the normal
distribution. If we assume the distribution is normal, we will tend to calculate a VaR
that is too low.
For a more accurate estimate of VaR than that given by the linear model, we can use
both delta and gamma measures to relate!:lP to the !:lxi' Consider a portfolio
Probability distribution for value of portfolio: (a) positive gamma;
(b) negative gamma. Figure 18.3 (a) (b) 7 A$ mentioned in footnote 4, we can use the nonnal distribution as an approximation to the lognonnal
distribution in VaR calculations. Value at Risk 447 Translation of normal probability distribution for asset into probability
distribution for value of a long call on a s s e t .  ' Figure 18.4 Value of
long call Underlying asset Translation of normal probability distribution for asset into probability
distribution for va,lue of a short call on asset. Figure 18.5 Value of
short call Underlying asset 448 CHAPTER 18 ° dependent on a single asset whose price is S. Suppose and yare the delta and gamma
of the portfolio. From the appendix to Chapter 15, the equation
b.P = ° b.S + ~y(b.S)2 is an improvement over the approximation in equation (18.3).8 Setting
b.S b.x =  S reduces this to
(18.6) More generally for a portfolio with n underlying market variables, with each instrument
in the portfolio being dependent on only one of the market variables; equation (18.6)
becomes
1l b.P = L 11 SiOi b.xi i=1 + L ~S?Yi (b.xii
i=l where Si is the value of the ith market variable, and Oi and Yi are the delta and gamma
of the portfolio with respect to the ith market variable. When individual instruments in
the portfolio may be dependent on more than one market variable, this equation takes
the more general form
(18.7) where Yij is a "cross gamma" defined as Equation (18.7) is not as easy to work with as equation (18.5), but it can be used to
calculate moments for b.P. A result in statistics known as the CornishFisher expansion
can be used to estimate percentiles of the probability distribution from the moments. 9 18.6 MONTE CARLO SIMULATION
As an alternative to the approaches described so far, we can implement the modelbuilding approach using Monte Carlo simulation to generate the probability distribution
8 The Taylor series expansion in the appendix to Chapter 15 suggests the approximation e Ilt + 8 IlS + ~y(IlS)2
ignored. In practice, the e Ilt IlP = when terms of higher order than At are
ignored. term is ·so small that it is usually 9 See Technical Note 10 on the author's website for details of the calculation of moments and the use of
CornishFisher expansions. When there is a single underlying variable, E(IlP) = 0.5S 2ri', E(llp 2) =
S2 1l2i' + 0.75S 4 r 2a4 , and E(llp 3 ) = 4.5S 4 1l2ra4 + 1.875S 6 r 3a6 , where S is the value of the variable and a
is its daily volatility. Sample Application E in the DerivaGem Application Builder implements the CornishFisher expansion method for this case. Value at Risk 449 for I::!.P. Suppose we wish to calculate a Iday VaR for a portfolio. The procedure is
as follows:
.1. Value the portfolio today in the usual way using the current values of market
variables.
2. Sample once from the multivariate normal probability distribution of the I::!.Xi. 1O
3. Use the values of the I::!.xi that are sampled to determine the value of each market
variable at the end of one day.
4. Revalue the portfolio at the end of the day in the usual way.
5. Subtract the value calculated in Step 1 from the value in Step 4 to determine a
sample I::!.P.
6. Repeat Steps 2 to 5 many times to build up a probability distribution for I::!.P.
The VaR is calculated as the appropriate percentile of the probability distribution of
I::!.P. Suppose, for example, that we calculate 5,000 different sample values of I::!.P in the
way just described. The Iday 99% VaR is the value of I::!.P for fhe 50th worst outcome;
the Iday VaR 95% is the value of I::!.P for the 250th worst outcome; and so on. lI The
Nday Va:R is usually assumed to be the Iday VaR multiplied by JNY
The drawback of Monte Carlo simulation is that it tends to be slow because a
company's complete portfolio (which might consist of hundreds of thousands of
different instruments) has to be revalued many times. 13 One way of speeding things
up is to assume that equation (18.7) describes the relationship between I::!.P and the I::!.xi'
We can then jump straight from Step 2 to Step 5 in the Monte Carlo simulation and
avoid the need for a complete revaluation of the portfolio. This is sometimes referred to
as the partial simulation approach. 18.7 COMPARISON OF APPROACHES
We have discussed two methods for estimating VaR: the historical simulation approach
and the modelbuilding approach. The advantages of the modelbuilding approach are
that results can be produced very quickly and it can be used in conjunction with volatility
updating schemes such as those we will describe in the next chapter. The main disadvantage of the modelbuilding approach is that it assumes that the market variables
have a multivariate normal distribution. In practice, daily changes in market variables
often have distributions that are quite different from normal (see, e.g., Table 16.1).
The historical simulation approach has the advantage that historical data determine
the joint probability distribution of the market variables. It also avoids the need for
cashflow mapping (see Problem 18.2). The main disadvantages of historical simulation
10 One way of doing so is given in Chapter 17. As in the case of historical simulation, extreme value theory can be used to "smooth the tails" so that
better estimates of extreme percentiles are obtained.
II 12 This is only approximately true when the portfolio includes options, but it is the assumption that is made
in practice for most VaR calculation methods. 13 An approach for limiting the number of portfolio revaluations is proposed in F. Jamshidian and Y. Zhu
"Scenario simulation model: theory and methodology," Finance and Stochastics, 1 (1997),4367. 450 CHAPTER 18
are that it is computationally slow and does not easily allow volatility updating schemes
to be used. 14
One disadvantage of the modelbuilding approach is that it tends to give poor results
for lowdelta portfolios (see Problem 18.21). 18.8 STRESS TESTING AND BACK TESTING
ad~ition to calculating VaR, many companies carry out what is known as a stress
test of their portfolio. Stress testing involves estimating how the portfolio would have In performed under some of the most extreme market moves seen in the last 10 to 20 years.
For example, to test the impact of an extreme movement in US equity prices, a
company might set the percentage changes in all market variables equal to those on
October 19, 1987 (when the S&P 500 moved by 22.3 standard deviations). If this is
considered to be too extreme, the company might choose January 8, 1988 (when the
S&P 500 moved by 6.8 standard deviations). To test the effect of extreme movements in
UK interest rates, the company might set the percentage changes in all market variables
equal to those on April 10, 1992 (when lOyear bond yields moved by 7.7 standard
deviations).
The scenarios used in stress testing are also sometimes generated by senior management. One technique sometimes used is to ask senior management to meet periodically
,and "brainstorm" to develop extreme· scenarios that might occur given the current
economic environment and global uncertainties.
Stress testing can be considered as a way of taking into account extreme events that
do occur from time to time but that are virtually impossible according to the probability distributions assumed for market variables. A 5standarddeviation daily move
in a market variable is one such extreme event. Under the assumption of a normal
distribution, it happens about once every 7,000 years, but, in practice, it is not
uncommon to see a 5standarddeviation daily move once or twice every 10 years.
Whatever the method used for calculating VaR, an important reality check is back
testing. It involves testing how well the VaR estimates would have performed in the past.
Suppose that we are calculating a Iday 99% VaR. Back testing would involve looking
at how often the loss in a day exceeded the Iday 99% VaR that would have been
calculated for that day. If this happened on about 1% of the days, we can feel
reasonably comfortable with the methodology for calculating VaR. If it happened
on, say, 7% of days, the methodology is suspect. 18.9 PRINCIPAL COMPONENTS ANALYSIS
One approach to handling the risk arising from groups of highly correlated market
variables is principal components analysis. This takes historical data on movements in
the market variables and attempts to define a set of components or factors that explain
the movements.
14 For a way of adapting the historical simulation approach to incorporate volatility updating, see J. Hull
and A. White. "Incorporating volatility updating into the historical simulation method for valueatrisk,"
Journal oj Risk I, No.1 (1998): 519. Value at Risk 451
Factor loadings for US Treasury data. Table 18.3 PCl 6m
12m
2y
3y
4y
5y
7y
lOy
30y PC2 PC3 PC4 PC5 PC6 PCl '':PC8 PCg PClO 0.21
0.26
0.32
0.35
0.36
0.36
0.36
0.34
0.31
0.25 0.57
0.49
0.32
0.10
0.02
0.14
0.17
0.27
0.30
0.33 0.50
0.23
0.37
0.38
0.30
0.12
0.04
0.15
0.28
0.46 0.47
0.37
0.58
0.17
0.27
0.25
0.14
0.01
0.10
0.34 0.39
0.70
0.52
0.04
0.07
0.16
0.08
0.00
0.06
0.18 0.02
0.01
0.23
0.59
0.24
0.63
0.10
0.12
0.01
0.33 0.01
0.04
0.04·.
0.56
0.79
0.15
0.09
0.13
0.03
0.09 0.00
0.02
0.05
0.12
0.00
0.55
0.26
0.54
0.23
0.52 0.01
0.01
0.00
0.12
0.09
0.14
0.71
0.00
0.63
0.26 0.00
0.00
0.01
0.05
0.00
0.08
0.48
0.68
0.52
0.13 The approach is best illustrated with an example. The market variables we will
consider are 10 US Treasury rates with maturities between 3 months and 30 years.
Tables 18.3 and 18.4 shows results produced by Frye for these market variables using
1,543 daily observations between 1989 and 1995. 15 The first column,in Table 18.3 shows
the maturities of the rates that were considered. The remaining 10 columns in the table
show the 10 factors (or principal components) describing the rate moves. The first
factor, shown in the column labeled PC1, corresponds to a roughly parallel shift in the
yield curve. When we have one unit of that factor, the 3month rate increases by
0.21 basis points, the 6month rate increases by 0.26 basis points, and so on. The
second factor is shown in the column labeled PC2. It corresponds to a "twist" .or
"steepening" of the yield curve. Rates between 3 months and 2 years move in one
direction; rates between 3 years and 30 years move in the other direction. The third
factor corresponds to a "bowing" of the yield curve. Rates at the short end and long end
of the yield curve move in one direction; rates in the middle move in the other direction.
The interest rate move for a particular factor is known asfactor loading. In our example,
the first factor's loading for the tbieemonth rate is 0.21. 16
Because there are 10 rates and 10 factors, the interest rate changes observed on any
given day can always be expressed as a linear sum of the factors by solving a set of
10 simultaneous equations. The quantity of a particular factor in the interest rate
changes on a particular day is known as the factor score for that day.
The importance of a factor is measured by the standard deviation of its factor score.
The standard deviations of the factor scores in our example are shown in Table 18.4 and
Table 18.4 Standard deviation of factor scores (basis points). PCl PC2 PC3 PC4 PC5 PC6 PCl PC8 PC9 PClO 17.49 6.05 3.10 2.17 1.97 1.69 1.27 1.24 0.80 0.79 See J. Frye, "Principals of Risk: Finding VAR through FactorBased Interest Rate Scenarios," in VAR:
Understanding and Applying Value at Risk, pp. 27588. London: Risk Publications, 1997. 15 16 The factor loadings have the property that the sum of their squares for each factor is 1.0. 452 CHAPTER 18 the factors are listed in order of their importance. The numbers in Table 18.4 are
measured in basis points. A quantity of the first factor equal to one standard deviation,
therefore, corresponds to the 3month rate moving by 0.21 x 17.49 = 3.67 basis points,
the 6month rate moving by 0.26 x 17.49 = 4.55 basis points, and so on.
The technical details of how the factors are determined are not covered here. It is
sufficient for us to note that the factors are chosen so that the factor scores are
uncorrelated. For instance, in our example, the first factor score (amount of parallel
shift) is uncorrelated with the second factor score (amount of twist) across the
1,543 days. The variances of the factor scores (i.e., the squares of the standard
deviations) have the property that they add up to the total variance of the data. From
Table 18.4, the total variance of the original data (i.e., sum of the variance of the
observations on the 3month rate, the variance of the observations on the 6month rate,
and so on) is
17.492 + 6.052 + 3.10 2 + ... + 0.79 2 = 367.9
From this it can be seen that the first factor accounts for 17.492 /367.9 = 83.1 % of the
variance in the original data; the first two factors account for (17.49 2 + 6.052 )/367.9 =
93.1 % of the variance in the data; the third factor accounts for a further 2.8% of the
variance. This shows most of the risk in interest rate moves is accounted for by the first
two or three factors. It suggests that we can relate the risks in a portfolio of interest
rate dependent instruments to movements in these factors· instead of considering all
10 interest rates. The three most important factors from Table 18.3 are plotted in
Figure 18.6. 17
•
The three most important factors driving yield curve movements. Figure 18.6 Factor loading
0.6 0.4 I
I
I __ I .~ e== ...:: ::.::: _ •••;;:.:::=:::._.......=.:~:..::..:.::..: I 0.2 _ / I
I .. ,'.,', , I
I •: ' "
, / Maturity (years) I o ++:;.~.!f'1tI1t~
I / 1,t 0.2 I ~,: 15 20 25 30 I I ,.
~
~ 0.6 10 / I. 1: 0.4 ,5 : ,"" ·
·
·
·
· ,
I . 17 Similar results to those described here, in respect of the nature of the factors and the amount of the total
risk they account for, are obtained when a principal components analysis is used to explain the movements in
almost any yield curve in any country. 453 Value at Risk Using Principal Components Analysis to Calculate VaR
To illustrate how a principal components analysis can be used to calculate VaR, suppose
We have a portfolio with the exposures to interest rate moves shown in Table 18.5.
A Ibasispoint change in the Iyear rate causes the portfolio value to increase by
$10 million, a Ibasispoint change in the 2year rate causes it to increase by $4 million,
and so on. We use the first two factors to model rate moves. (As mentioned above, this
captures over 90% ofthe uncertainty in rate moves.) Using the data in Table 18.3, our
exposure to the first factor (measured in millions of dollars per factor score basis point) is
10 x 0.32 + 4 x 0.35  8 x 0.36  7 x 0.36 + 2 x 0.36 = 0.08
and our exposure to the second factor is
10 x (0.32) + 4 x (0.10)  8 x 0.02 7 x 0.14+ 2 x 0.17 = 4.40 Suppose that fl and 12 are the factor scores (measured in basis points). The change in
the portfolio value is, to a good approximation, given by
/::".P = 0.08fl  4.4012
The factor scores are uncorrelated and have the standard deviations given in Table 18.4.
The standard deviation of /::".P is therefore J 0.08 2 x 17.492 + 4.40 2 X 6.05 2 = 26.66 Hence, the Iday 99% VaR is 26.66 x 2.33 = 62.12. Note that the data in Table 18.5
are such that we have very little exposure to the first factor and significant exposure to
the second factor. Using only one factor would significantly understate VaR (see
Problem 18.13). The durationbased method for handling interest rates, mentioned in
Section 18.4, would also significantly understate VaR as it considers only parallel shifts
in the yield curve.
A principal components analysis can in theory be used for market variables other
than interest rates. Suppose that a financial institution has exposures to a number of
different stock indices. A principal components analysis can be used to identify factors
describing movements in the indices and the most important of these can be used to
replace the market indices in a VaR analysis. How effective a principal components
analysis is for a group of market variables depends on how closely correlated they are.
As explained earlier in the chapter, VaR is usually calculated by relating the actual
changes in a portfolio to percentage changes in market variables (the /::"'Xi)' For a VaR
calculation, it may therefore be most appropriate to carry out a principal components
analysis on percentage changes in market variables rather than actual changes.
Change in portfolio value for a lbasispoint
rate move ($ millions). Table 18.5 iyear
rate 2year
rate 3year
rate 4year
rate 5year
rate +10 +4 8 7 +2 454 CHAPTER 18 SUMMARY
A value at risk (VaR) calculation is aimed at making a statement of the form: "We are
X percent certain that we will not lose more than V dollars in the next N days." The
variable V is the VaR, X% is the confidence level, and N days is the time horizon.
One approach to calculating VaR is historical simulation. This involves creating a
database consisting of the daily movements in aU market variables over a period of
time. The first simulation trial assumes that the percentage changes in each market
variabl~ are the same as those on the first day covered by the. database; the second
simulation trial assumes that the percentage changes are the same as those on the
second day; and so on. The change in the. portfolio value, !:::'P, is calculated for each
simulation trial, and the VaR is calculated as the appropriate percentile of the
probability distribution of !:::'P.
An alternative is the model~building approach. This is relatively straightforward if
two assumptions can be made:
1. The change in the value of the portfolio (!:::'P) is linearly dependent on percentage
changes in market variables.
2. The percentage changes in market variables are multivariate normally distributed. The probability distribution of !:::'P is then normal, and there are analytic formulas for
relating the standard deviation of !:::'P to the volatilities and correlations of the underlying market variables. The VaR can be calculated from wellknown properties of the
normal distribution.
When a portfolio includes options, !:::'P is not linearly related
the percentage
changes in market variables. From knowledge of the gamma of the portfolio, we can
derive an approximate quadratic relationship between !:::'P and percentage changes in
market variables. Monte Carlo simulation can then be used to estimate VaR.
In the next chapter we discuss how volatilities and correlations can be estimated and
monitored. FURTHER READING
Artzner P., F. Delbaen, J.M. Eber, and D. Heath. "Coherent Measures of Risk," Mathematical
Finance, 9 (1999): 20328.
Basak, S., and A. Shapiro. "ValueatRiskBased Risk Management: Optimal Policies and Asset
Prices," Review of Financial Studies, 14, 2 (2001): 371405.
Beder, T. "VaR: Seductive but Dangerous," Financial Analysts Journal, 51, 5 (1995): 1224.
Boudoukh, J., M. Richardson, and R. Whitelaw. "The Best of Both Worlds," Risk, May 1998:
6467.
Dowd, K. Beyond Value at Risk: The New Science of Risk Management. New York: Wiley, 1998.
Duffie, D., and J. Pan. "An Overview of Value at Risk," Journal of Derivatives, 4, 3 (Spring
1997): 749.
Embrechts, P., C. Kluppelberg, and T. Mikosch. Modeling Extremal Events for Insurance and
Finance. New York: Springer, 1997.
Frye, J. "Principals of Risk: Finding VAR through FactorBased Interest Rate Scenarios" in
VAR: Understanding and Applying Value at Risk, pp. 27588. London: Risk Publications, 1997. Value at Risk 455 Hendricks, D. "Evaluation of ValueatRisk Models Using Historical Data,"Economic Policy
Review, Federal Reserve Bank of New York, 2 (April 1996): 3969 . •
r
Hopper, G. "Value at Risk: A New Methodology for Measuring Portfolio Risk," Business
Review, Federal Reserve Bank of Philadelphia, July/August 1996: 1929.
Hua P., and P. Wilmott, "Crash Courses," Risk, June 1997: 6467.
Hull, J. c., and A. White. "Value at Risk When Daily Changes in Market Variables Are Not
Normally Distributed,': Journal oj Derivatives, 5 (Spring 1998): 919.
Hull, J. c., and A. White. "Incorporating Volatility Updating into the Historical Simulation
Method for Value at Risk," Journal oj Risk, 1, 1 (1998): 519.
Jackson, P., D. J. Maude, and W. Perraudin. "Bank Capital and Value at Risk." Journal oj
Derivatives, 4, 3 (Spring 1997): 7390.
Jamshidian, F., and Y. Zhu. "Scenario Simulation Model: Theory and Methodology," Finance
and Stochastics, 1 (1997): 4367.
Jorion, P. Value at Risk, 2nd edn. McGrawHill, 200!.
Longin, F. M. "Beyond the VaR," Journal oj Derh'ath'es, 8, 4 (Summer 2001): 3648.
Marshall, c., and M. Siegel. "Value at Risk: Implementing a Risk Measurement Standard,"
Journal oj Derivatives 4, 3 (Spring 1997): 91111.
McNeil, A: J. "Extreme Value Theory for Risk Managers," in Internal Modeling and CAD II,
London: Risk Books, 1999. See also: www.math.etbz.ch/~mcneil.
Neftci, S. N. "Value at Risk Calculations, Extreme Events and Tail Estimation," Journal oj
Derivatives, 7, 3 (Spring 2000): 2338.
Rich, D. "Second Generation VaR and RiskAdjusted Return on Capital," Journal oj
Derivatives, 10,4 (Summer 2003): 5161. Questions and Problems (Answers in Solutions Manual)
18.1. Consider a position consisting of a $100,000 investment in asset A and a $100,000
investment in asset B. Assume that the daily volatilities of both assets are 1% and that
the coefficient of correlation between .their returns is 0.3. What is the 5day 99% VaR for
the portfolio?
18.2. Describe three ways of handling instruments that are dependent on interest rates when
the modelbuilding approach is used to calculate VaR. How would you handle these
instruments when historical simulation is used to calculate VaR?
18.3. A financial institution owns a portfolio of options on the US dollarsterling exchange
rate. The delta of the portfolio is 56.0. The current exchange rate is 1.5000. Derive an
approximate linear relationship between the change in the portfolio value and the
percentage change in the exchange rate. If the daily volatility of the exchange rate is
0.7%, estimate the 10day 99% VaR.
18.4. Suppose you know that the gamma of the portfolio in the previous question is 16.2. How
does this change your estimate of the relationship between the change in the portfolio
value and the percentage change in the exchange rate?
18.5. Suppose that the daily change in the value of a portfolio is, to a good approximation,
linearly dependent on two factors, calculated from a principal components analysis. The
delta of a portfolio with respect to the first factor is 6 and the delta with respect to the
second factor is 4. The standard deviations of the factor are 20 and 8, respectively.
What is the 5day 90% VaR? 456 CHAPTER 18 18.6. Suppose that a company has a portfolio consisting of positions in stocks,~bonds, foreign
exchange, and commodities. Assume that there are no derivatives. Explain the
assumptions underlying (a) the linear model and (b) the historical simulation model for
calculating VaR.
18.7. Explain how an interest rate swap is mapped into a portfolio of zerocoupon bonds with
standard maturities for the purposes of a VaR calculation.
18.8. Explain the difference between value at risk and conditional value at risk.
18.9. Explain why the linear model can provide only approximate estimates of VaR for a
portfolio containing options.
18.10. Verify that the 0.3year zerocoupon bond in the cashflow mapping example in the
appendix to this chapter is mapped into a $37,397 position in a 3month bond and a
$11,793 position in a 6month bond.
18.1 L ~uppose that the 5year rate is 6%, the 7year rate is 7% (both expressed with annual
compounding), the daily volatility of a 5year zerocoupon bond is 0.5%, and the daily
volatility of a 7year zerocoupon bond is 0.58%. The correlation between daily returns
on the two bonds is 0.6. Map a cash flow of $1,000 received at time 6.5 years into a
position in a 5year bond and a position in a 7year bond using the approach in the
appendix. What cash flows in 5 and 7 years are equivalent to the 6.5year cash flow?
18.12. Some time ago a company entered into a forward contract to buy £1 million for
$1.5 million. The contract now has 6 months to maturity. The daily volatility of a
6month zerocoupon sterling bond (when its price is translated to dollars) is 0.06% and
the daily volatility of a 6month zerocoupon dollar bond is 0.05%. The correlation
between returns from the two bonds is 0.8. The current exchange rate is 1':53. Calculate
the standard deviation of the change in the dollar value of the forward contract in 1 day.
What is the 10day 99% VaR? Assume that the 6month interest rate in both sterling and
dollars is 5% per annum with continuous compounding.
18.13. The text calculates a VaR estimate for the example in Table 18.5 assuming two factors.
How does the estimate change if you assume (a) one factor and (b) three factors.
18.14. A bank has a portfolio of options on an asset. The delta of the options is 30 and the
gamma is 5. Explain how these numbers can be interpreted. The asset price is 20 and
its volatility is 1% per day. Adapt Sample Application E in the DerivaGem Application
Builder software to calculate VaR.
18.15. Suppose that in Problem 18.14 the vega of the portfolio is 2 per 1% change in the
annual volatility. Derive a model relating the change in the portfolio value in 1 day to
delta, gamma, and vega. Explain without doing detailed calculations how you would use
the model to calculate a VaR estimate. Assignment Questions
18.16. A company has a position in bonds worth $6 million. The modified duration of the
portfolio is 5.2 years. Assume that only parallel shifts in the yield curve can take place
and that the standard deviation of the daily yield change (when yield is measured in
percent) is 0.09. Use the duration model to estimate the 20day 90% VaR for the
portfolio. Explain carefully the weaknesses of this approach to calculating VaR. Explain
two alternatives that give more accuracy. Value at "Risk 457 18.17. Consider a posItIOn consIstmg of a $300,000 investment in gold and a $500,000
investment in silver. Suppose that the daily volatilities of these two assets are 1.8%
and 1.2%, respectively, and that the coefficient of correlation between their returns is 0.6.
What is the 10day 97.5% VaR for the portfolio? By how much does diversification
reduce the VaR?
18.18. Consider a portfolio of options on a single asset. Suppose that the delta of the portfolio
is 12, the value of the asset is $10, and the daily volatility of the a'sset is 2%. Estimate the
Iday 95% VaR for the' portfolio from the delta. Suppose next that the gamma of the
portfolio is 2.6. Derive a quadratic relationship between the change in the portfolio
value and the percentage change in the underlying asset price in one day. How would
you use this in a Monte Carlo simulation?
18.19. A company has a long position in a 2year bond and a 3year bond, as well as a short
position in a 5year bond. Each bond has a principal of $100 and pays a 5% coupon
annually. Calculate the company's exposure to the Iyear, 2year, 3year, 4year, and
5year rates. Use the data in Tables 18.3 and 18.4 to calculate a 20day 95% VaR on the
assumption that rate changes are explained by (a) one factor, (b) two factors, and (c) three
factors. Assume that the zerocoupon yield curve is flat at 5%.
18.20. A bank has written a call option on one stock and a put option on another stock. For
the first option the stock price is 50, the strike price is 51, the volatility is 28% per
annum, and the time to maturity is 9 months. For the second option the stock price is
20, the strike price is 19, the volatility is 25% per annum, and the time to maturity is
1 year. Neither stock pays a dividend, the riskfree rate is 6% per annum, and the
correlation between stock price returns is 0.4. Calculate a lOday 99% VaR:
(a) Using only deltas
(b) Using the partial simulation approach
(c) Using the full simulation approach
18.21. A common complaint of risk managers is that the modelbuilding approach (either linear
or quadratic) does not work well when delta is close to zero. Test what happens when delta
is close to zero by usiJ;lg Sample Application E in the DerivaGem Application Builder
software. (You can do this by experim.enting with different option positions and adjusting
the position in the underlying to give a delta of zero.) Explain the results you get. CHAPTER 18 458 APPENDIX
CASHFLOW MAPPING
In tllis appendix we explain one procedure for mapping cash flows to standard maturity
dates. We will illustrate the procedure by considering a simple example of a portfolio
consisting of a long position in a single Treasury bond with a principal of $1 million
maturing in 0.8 years. We suppose that the bond provides a coupon of 10% per annum
payable semiannually. This means that the bond provides coupon payments of $50,000
in 0.3 years and 0.8 years. It also provides a principal payment of $1 nlillion in 0.8
years. The Treasury bond can therefore be regarded as a position in a 0.3year zerocoupon bond with a principal of $50,000 and a position in a 0.8year zerocoupon bond
with a principal of $1,050,000.
The position in the O.3year zerocoupon bond is mapped into an equivalent position
 in 3month and 6month zerocoupon bonds. The position in the 0.8year zerocoupon
bond is mapped into an equivalent position in 6month and Iyear zerocoupon bonds.
The result is that the position in the 0.8year couponbearing bond is, for VaR
purposes, regarded as a position in zerocoupon bonds having maturities of 3 months,
6 months, and 1 year. The Mapping Procedure
Consider the $1,050,000 that will be received in 0.8 years. We suppose that zero rates,
daily bond price volatilities, and correlations between bond returIJ.s are as shown in
Table 18.6.
.
The first stage is to interpolate between the 6month rate of 6.0% and the Iyear rate
of 7.0% to obtain a 0.8year rate of 6.6%. (Annual compounding is assumed for all
rates.) The present value of the $1,050,000 cash flow to be received in 0.8 years is
1,050,000 = 997662
1.066°·8
'
We also interpolate between the 0.1 % volatility for the 6month bond and the 0.2%
volatility for the Iyear bond to get a 0.16% volatility for the 0.8year bond.
Table 18.6 Data to illustrate mapping procedure. Maturity: Zero rate (% with annual compounding):
Bond price volatility (% per day):
Correlation between daily returns 3month bond
6month bond
Iyear bond 3month 6momh Iyear 5.50
0.06 6.00
0.10 7.00
0.20 3month
bond 6mont!z
bond Iyear
bond 1.0
0.9
0.6 0.9
1.0
0.7 0.6
0.7
1.0 V alue at Risk
Table 18.7 459 The cashflow mapping result.
$50,000 received
ill 0.3 years Position in 3month bond ($):
Position in 6month bond ($):
Position in Iyear bond ($): $1,050,000 received
ill 0.8 years 37,397
11,793 319,589
678,074 Total 37,397
331,382
678,074 Suppose we allocate Ci of the present value to the 6month bond and I  Ci of the
present value to the Iyear bond. Using equation (18.2) and matching variances, we
obtain
0.0016 2 = 0.00l2 Ci 2 + 0.002\1  Cii + 2 x 0.7 x 0.001 x 0.002Ci(1  Ci)
This is a quadratic equation that can be solved in the usual way to give Ci = 0.320337.
This means that 32.0337% of the value should be allocated to a 6month zerocoupon
bond and 67.9663% of the value should be allocated to a Iyear zero coupon bond. The
0.8year bond worth $997,662 is therefore replaced by a 6month pond worth
997,662 x 0.320337 = $319,589
and a Iyear bond worth
997,662 x 0.679663 = $678,074
This cashflow mapping scheme has the advantage that it preserves both the value and
the variance of the cash flow. Also, it can be shown that the weights assigned to the two
adjacent zerocoupon bonds are always positive.
For the $50,000 cash flow received at time 0.3 years, we can carry out similar
calculations (see Problem 18.10). It turns out that the present value of the cash flow
is $49,189. It can be mapped into a position worth $37,397 in a 3month bond and a
position worth $11,793 in a 6month bond.
The results of the calculations are summarized in Table 18.7. The 0.8year couponbearing bond is mapped into a position worth $37,397 in a 3month bond, a position
worth $331,382 in a 6month bond, and a position worth $678,074 in a Iyear bond.
Using the volatilities and correlations in Table 18.6, equation (18.2) gives the variance
of the change in the price of the 0.8year bond with 11 = 3, Cil = 37,397, Ci2 = 331,382,
Ci3 = 678,074,0"1 = 0.0006, 0"2 = 0.001,0"3 = 0.002, and P12 = 0.9, P13 = 0.6, P23 = 0.7.
This variance is 2,628,518. The standard deviation of the change in the price of the
bond is therefore )2,628,518 = 1,621.3. Because we are assuming that the bond is the
only instrument in the portfolio, the lOday 99% VaR is
1621.3 x
or about $11,950. .JlO x 2.33 = 11,946 ...
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 Spring '11
 DonBlasius
 Math, Derivative, The Land

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