HWESER
C ritical Concepts for the 2018 FRM® Exam
MARKET RISK MEASUREMENT
AND MANAGEMENT
Value at Risk (VaR)
VaR for a given confidence level occurs at the
cutoff point that separates the tail losses from the
remaining distribution.
Historical simulation approach: order return
observations and find the observation that
corresponds to the VaR loss level.
Parametric estimation approach: assumes a
distribution for the underlying observations.
• Normal distribution assumption:
VaR = (—pr + or x za )
• Lognormal distribution assumption:
VaR = (1 — e^r ~ a* XZ
ol )
Expected Shortfall
Provides an estimate o f tail loss by averaging the
VaRs for increasing confidence levels in the tail.
Weighted Historical Simulation
Approaches
• Age-weighted: adjusts the most recent (distant)
observations to be more (less) heavily weighted.
• Volatility-weighted: replaces historic returns with
volatility-adjusted returns; actual procedure of
estimating VaR is unchanged.
• Correlation-weighted: updates the variance-
covariance matrix between assets in the portfolio.
• Filtered historical simulation: relies on
bootstrapping of standardized returns based on
volatility forecasts; able to capture conditional
volatility, volatility clustering, and/or data
asymmetry.
Backtesting VaR
• Compares the number of instances when losses exceed
the VaR level (exceptions) with the number predicted
by the model at the chosen level of confidence.
• Failure rate: number of exceptions/number of
observations.
• The Basel Committee requires backtesting at the
99% confidence level over one year; establishes zones
for the number of exceptions with corresponding
penalties (increases in the capital multiplier).
Mapping
Mapping involves finding common risk factors
among positions in a given portfolio. It may be
difficult and time consuming to manage the risk
of each individual position. One can evaluate the
value of portfolio positions by mapping them
onto common risk factors.
Pearson Correlation Coefficient
Commonly used to measure the linear
relationship between two variables:
„ _ covXY
PXY
--
---------
ctx cty
Spearman’s Rank Correlation
Step 1: Order the set pairs of variables X and Y
with respect to set X.
Step 2: Determine the ranks o f X. and Y for each
time period i.
Step 3 : Calculate the difference of the variable
rankings and square the difference.
«£<*?
PS = 1 ~
Where n is the number of observations for each
variable and d. is the difference between the
l
ranking for period i.
Kendall’s t
T _ nc — nd
n(n — 1) / 2
Where the number of concordant pairs is
represented as nc (pair rankings in agreement),
and the number of discordant pairs is represented
as nd (pair rankings not in agreement).
Mean Reversion
• Implies that over time variables or returns regress
back to the mean or average return.
• Mean reversion rate, a, is expressed as:
St — St_! = a (|x -S t_1)
• The (3 coefficient of a regression is equal to the
negative of the mean reversion rate.
Autocorrelation
• Measures the degree that a variables current value
is correlated to past values.
• Has the exact opposite properties of mean
reversion.
• The sum of the mean reversion rate and the one-
period autocorrelation rate will always equal one.
Correlation Swap
• Used to trade a fixed correlation between two or
more assets with a realized correlation.
• Realized correlation for a portfolio of n assets:
P real ized
• Payoff for correlation swap buyer:
notional amount x ( p ^ - p6 J
Gaussian Copula
• Indirectly defines a correlation relationship
between two variables.
• Maps the marginal distribution of each variable
to a standard normal distribution (done on
percentile-to-percentile basis).
• The new joint distribution is a multivariate
standard normal distribution.
• A Gaussian default time copula can be used for
measuring the joint probability of default between
two assets.
Regression-Based Hedge
PV01N >
DV01r
/
where:
FR = face amount of hedging instrument
FN = face amount of initial position
Bond Valuation Using Binomial Tree
Using backward induction, the value of a bond at
a given node in a binomial tree is the average of the
present values of the two possible values from the
next period. The appropriate discount rate is the
forward rate associated with the node under analysis.
There are three basic steps to valuing an option on
a fixed-income instrument using a binomial tree:
Step 1: Price the bond value at each node using
the projected interest rates.
Step 2 : Calculate the intrinsic value of the
derivative at each node at maturity.
Step 3: Calculate the expected discounted value of
the derivative at each node using the risk-
neutral probabilities and work backward
through the tree.
Interest Rate Expectations
Expectations play an important role in
determining the shape of the yield curve and
can be illustrated by examining yield curves that
are flat, upward-sloping, and downward-sloping.
If expected 1 -year spot rates for the next three
years are rv r2, and r , then the 2-year and 3-year
spot rates are computed as:
r(2) = V(l + r1)(l + r2)-l
r(3) =
yj ( 1 + q)(l + r2)(l + r3) — 1
Convexity Effect
All else held equal, the value of convexity
increases with maturity and volatility.
Term Structure Models
Model 1: assumes no drift and that interest rates
are normally distributed:
dr = odw
Model 2: adds a positive drift term to Model 1
that can be interpreted as a positive risk premium
associated with longer time horizons:
dr = Xdt + odw
where:
X = interest rate drift
Ho-Lee Model: generalizes drift to incorporate
time-dependency:
dr = X(t)dt + odw
Vasicek Model: assumes a mean-reverting process
for short-term interest rates:
dr = k(0 - r)dt + odw
where:
k = a parameter that measures the speed of
reversion adjustment
0 = long-run value o f the short-term rate
assuming risk neutrality
r = current interest rate level
Model 3: assigns a specific parameterization of
time-dependent volatility:
dr = X(t)dt + oe~“'dw
where:
o = volatility at t = 0, which decreases
exponentially to 0 for a > 0
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