Page 1

2009 Financial Risk Manager (FRM) Level 1 (L1) Formula Sheets Prepared by David Harper, CFA, FRM, CIPM

This document is for only for paid members of Bionic Turtle Copyright @ 2009 by Bionic Turtle, LLC

2 / 55

Value‐at‐risk (VaR)

VaR summarizes the worst loss over a target horizon that will not be exceeded with a

given level of confidence. VaR is given by:

( ) 1P L VaR c

Portfolio Variance and Covariance

The variance of the two-asset portfolio is given by:

2 2 2 2 2

2 2 2 2 2

2 cov( , )

2

a b a a b b a b

a b a a b b a b a b

w w w w a b

w w w w

cov( , )cov( , ) a b

a b

a ba b

Portfolio variance with several assets:

2 2

1 1

( ) ( , )N N N

Portfolio i i i j i ji i j i

w Var R w w Cov R R

Capital asset pricing model (CAPM)

The capital asset pricing model (CAPM) is given by:

( ) [ ( ) )]i F i M FE R R E R R

CAPM tells us that the expected excess return of a risky security is equal to the systematic risk of

that security measured by its beta times the market's risk premium. The key insight of the CAPM

is that a security’s risk premium is proportional to its systematic risk.

Security Market Line (SML)

The security market line (SML) plots expected return against beta:

F m F( ) [ ( ) ]i iE R R E R R

Beta

Beta is a measure of an asset’s sensitivity to movements in the market. A security’s beta is the

covariance of the return of the security with the return of the market portfolio divided by the

variance of the return of the market portfolio:

2

Cov( , )i Mi

M

R R

3 / 55

Price of Risk (beta)

The price of risk is excess expected return of the market portfolio above the risk-free rate:

Mprice of risk = E(R )FR

Quantity of Risk (beta)

The quantity of risk is beta (). Beta is the measure of the asset’s sensitivity to the market.

Specifically, it is the covariance between the asset’s return and the market’s return divided by the

market’s variance (the division is a way to standardize the beta into a unit-less metric).

m

m

cov( , )Beta

var( )i

i

R R

R

Diversification and Risk Management

The value of firm’s equity is future expected cash flows discounted at CAPM rate:

EquityF m F

Expected [future cash flows]Value

1+R [ ( ) ]E R R

Treynor Measure, Sharpe Measure, and Jensen’s Alpha

( )P FP

P

E R RT

The Treynor measure: excess return divided by

portfolio beta ():

( )

( )P F

PP

E R RS

R

The Sharpe measure: excess return divided by

portfolio volatility (standard deviation):

( ) ( ( ) )P F P P M FE R R E R R Jensen’s alpha is the excess return equated to

alpha plus expected systematic return:

Tracking Error (TE)

Tracking error (TE) is the standard deviation of the difference between the portfolio return

and the benchmark return:

( )P BTE R R

Ex ante tracking error can be given by:

2 2 2 22TE P P B BTEV

4 / 55

Information ratio (IR)

The information ratio (IR, aka, the appraisal ratio) is given by:

( ) ( )

( )P B

P B

E R E RIR

R R

Sortino Ratio

The Sortino ratio is given by:

P

2

0

E(R )Sortino ratio =

1 T

PttR MARPt

MAR

R MART

Arbitrage Pricing Theory (APT)

APT postulates a multiple-factor model of excess returns. APT assumes that there are K factors

such that the excess returns can be expressed as:

,1

K

n n k k nk

r X b u

X(n,k) = exposure of stock (n) to factor (k). These exposures are factor loadings. In practice, we will assume that the exposures are known before the returns are observed.

b(k) = the factor return for factor k. These factor returns are either attributed to the factors at the end of the period or observed during the period.

u(n) = stock (n)'s specific return; i.e., the return that cannot be explained by the factors. This is also called the idiosyncratic return to the stock. The excess return contains an unexplained specific (idiosyncratic) return

In regard to expected excess return

,1

{ }K

n n n k kk

f E r X m

m(k) = the factor forecast for factor k.

The factor forecast is simply the sum of [Exposure(k) * Factor(k)].

5 / 55

Discrete random variables (probability function)

A discrete random variable (X) assumes a value among a finite set including x1, x2, x3 and so

on. The probability function is expressed by:

( ) ( )k kP X x f x

The function must meet two conditions:

x

1st condition: ( ) 0

2nd condition: ( ) 1

f x

f x

Continuous random variable

A continuous random variable (X) has an infinite number of values within an interval:

( ) ( )b

aP a X b f x dx

The function must meet two conditions:

1st condition: ( ) 0

2nd condition: ( ) 1

f x

f x dx

Note that instead of an “in between” interval, a continuous variable can be expressed in

cumulative terms; i.e., what is the probability that X “is less than” some value?

( ) ( ) ( ) ( )x

F x P X x f u du x

Bayes’ Theorem

( | ) ( )( | )

( )

P U G P GP G U

P U

We can expand the denominator into the elaborated Bayes’ formula:

( | ) ( )( | )

( | ) ( ) ( | ) ( )

P U G P GP G U

P U G P G P U G P G

6 / 55

Conditional probability

What is the probability of B occurring, given that A has already occurred?

( )( | ) ( ) ( | ) ( )

( )

P A BP B A P A P B A P A B

P A

Mathematical expectation

In the case of a discrete random variable, expected value is given by:

1 1 2 2( ) ( ) ( ) ( ) ( )n nE X x f x x f x x f x xf x

In the case of a continuous random variable, expected value is given by:

( ) ( )E X xf X dx

Variance and standard deviation

2 2( ) ( ) ( )Xvar X x f x dx

But if X is a discrete random variable, the variance is given by:

2 2Variance( ) [( ) ]XX E X

And the standard deviation, which is simply the square root of the variance, is given by:

2Standard Deviation = ( ) [( ) ]X var X E X

Important variance formula

Variance is also conveniently expressed as the difference between the expected value of X^2 and

the square of the expected value of X:

2 2 2Variance( ) [( ) ] ( ) [ ( )]X E X E X E X

7 / 55

Properties of variance if X and Y are independent

2constant2 2 2

2 2 2

2 2

2 2 2

2 2

2 2 2 2 2

2 2 2

only if independentonly if independent

only if

1. 02 .2 .3.4.5.6.7. ( ) (

indepen t)

den

X Y X Y

X Y X Y

X b X

aX X

aX b X

aX bY X Y

X

ab

aaa b

E X E X

Chebyshev’s Inequality

Chebyshev’s inequality provides a shorthand method for specifying a cumulative probability

without our need to know the underlying distribution (conditional on a finite variance):

2 2

1 1( ) , or ( ) 1P X k P X k

k k

Coefficient of variation

Because the standard deviation depends on the units of measurement, the coefficient of variation

is used to measure relative variation. In other words, the coefficient of variation (like the

correlation coefficient) is a unit-less number.

coeff. of variation (V)= (100)X

Xu

Covariance

The covariance (a.k.a., covariance of joint distributions) is given by:

cov( , ) [( )( )]XY X YX Y E X Y

Properties of covariance

2

2 2 2

2 2 2

1. If X &Y are independent, cov( , ) 0

2. cov( , ) cov( , )

3. cov( , ) var( ). In notation,

4. If X &Y are not independent,

2

2

XY

XX X

X Y X Y XY

X Y X Y XY

X Y

a bX c dY bd X Y

X X X

8 / 55

Correlation Coefficient

The correlation coefficient is the covariance (X,Y) divided by the product of the each variable’s

standard deviation. The correlation coefficient translates covariance into a unitless metric

that runs from -1.0 to +1.0:

cov( , )

StandardDev( ) StandardDev( )XY

X Y XYX Y

X Y

X Y

Properties of correlation:

Correlation has the same sign (+/-) as covariance

Corrleation measures the linear relationship between two variables

Between -1.0 and +1.0, inclusive

The correlation is a unit-less metric

Zero covariance → zero correlation (But the converse not necessarily true. For example, Y=X^2 is nonlinear )

Define, calculate and interpret the mean and variance of a set of random variables.

The variance of the sum of correlated variables is given by the following:

2 2 2

2 2 2

2 , and given that

2

X Y X Y XY XY X Y

X Y X Y X Y

The variance of the difference of correlated variables is given by:

2 2 2

2 2 2

2 and given that

2

X Y X Y XY XY X Y

X Y X Y X Y

Describe the difference between conditional and unconditional expectation.

An unconditional expectation is the expected value of the variable without any restrictions (or

lacking any prior information).

A conditional expectation is an expected value for the variable conditional on prior information

or some restriction (e.g., the value of a correlated variable). The conditional expectation of Y,

conditional on X = x, is given by:

( | )E Y X x

The conditional variance of Y, conditional on X=x, is given by:

var( | )Y X x

9 / 55

Moments of a distribution

The k-th moment about the mean () is given by:

1( )

k-th momentn k

iix

n

In this way, the difference of each data point from the mean is raised to a power (k=1, k=2, k=3,

and k=4). There are the four moments of the distribution:

If k=1, refers to the first moment about zero: the mean.

If k=2, refers to the second moment about the mean: the variance.

If k=3, refers to the third moment about the mean: skewness

If k=4, refers to the fourth moment about the mean: peakedness.

Skewness (asymmetry)

Skewness refers to whether a distribution is symmetrical. An asymmetrical distribution is

skewed and will be either positively (to the right) or negatively (to the left) skewed. The measure of

“relative skewness” is given by the equation below, where zero indicates symmetry (no skewness):

3

3 3

[( ) ]Skewness =

E X

Kurtosis

Kurtosis measures the degree of “peakedness” of the distribution, and consequently of “heaviness

of the tails.” A value of three (3) indicates normal peakedness. The normal distribution has

kurtosis of 3, such that “excess kurtosis” equals (kurtosis – 3).

4

4 4

[( ) ]Kurtosis =

E X

Note that technically skew and kurtosis are not, respectively, equal to the third and fourth

moments; rather they are functions of the third and fourth moments.

10 / 55

Summary Population Sample

Mean X

n

1

n

ii

X

Xn

Variance 2 2

1

1( )

n

x ii

X Xn

2 2

1

1( )

1

n

x ii

s X Xn

Covariance 1( )( )XY i iX X Y Y

n

1sample ( )( )

1XY i iX X Y Y

n

Correlation XY

X Y

sample sample XY

X YS S

Skew 3

3 3

3 3

Skewness =

[( ) ]E X

3

3

3

Sample Skewness =

( )( 1)

X XN

S

Kurtosis 4

4 4

4 4

Kurtosis =

[( ) ]E X

4

4

4

Sample Kurtosis =

( )( 1)

=

X XN

S

11 / 55

Normal distribution

Here is the probability density function (PDF) for a normally distributed random variable:

2 2( ) 21( )

2

xf x e

Key properties of the normal include:

Symmetrical around its mean value (symmetry = 0)

Peaks at mean but descends rapidly to tails

~68% at 1, ~95% at 2, ~99.7% at 3

Only requires (fully described by) two parameters, mean and variance

A linear combination (function) of two normally distributed random variables is itself normally distributed

Skew = 0, Kurtosis = 3 (excess kurtosis = 0)

Key locations on the normal distribution are noted below. In the FRM curriculum, the choice

of one-tailed 5% significance and 1% significance (i.e., 95% and 99% confidence) is

common, so please pay particular attention to the yellow highlights:

% of all observations (two-tailed)

% of observations “to the left” (one-tailed)

Intervals – multiple of the standard deviation

Interval – mathematically expressed (two-tailed)

VAR -worst expected loss at the given confidence

~ 50% ~ 25% 2/3 ˆ ˆ0.67u

~ 68% ~ 34% 1 ˆ ˆu

~ 90% ~ 5.0 % 1.645 (~1.65) ˆ ˆ1.65u ˆ ˆ1.65

~ 95% ~ 2.5% 1.96 ˆ ˆ1.96u

~ 98% ~ 1.0 % 2.327 (~2.33) ˆ ˆ2.33u ˆ ˆ2.33

~ 99% ~ 0.5% 2.58 ˆ ˆ2.58u

Standard normal distribution

A normal distribution is fully specified by two parameters, mean and variance (or standard

deviation). We can transform a normal into a unit or standardized variable:

X

X

XZ

This unit or standardized variable is normally distributed with zero mean and variance of

one (1.0).

12 / 55

Sampling distribution of means

If either: (i) the population is infinite and random sampling, or (ii) we have a finite population

and sampling with replacement, then the variance of the sampling distribution of means is

given by:

22 2[( ) ]

XE X

n

If the population is size (N), if the sample size n N, and if sampling is conducted “without

replacement,” then the variance of the sampling distribution of means is given by:

22

1X

N n

n N

Standard error of a sample mean.

The standard error is simply the standard deviation of the sampling distribution of the

estimator.

In the case of a sample mean, according to the central limit theorem, the variance of the

estimator is the population variance divided by the sample size. The standard error is the square

root of this quantity:

2X Xse

n n

If the population is distributed with mean and variance 2 but the distribution is not a

normal distribution, then the standardized variable given by Z below is “asymptotically normal;

i.e., as (n) approaches infinity () the distribution becomes normal.

~ (0,1)

XXZ N

n

13 / 55

Student’s t‐distribution…

As the degrees of freedom (d.f.) increases, the t-distribution converges with the normal

distribution. It is similar to the normal, except it exhibits heaver tails (the lower the d.f.., the

heavier the tails). The student’s t variable is given by:

X

x

Xt

S n

Properties of the t-distribution:

Like the normal, it is symmetrical

Like the standard normal, it has mean of zero (mean = 0)

Its variance = k/(k-2) where k = degrees of freedom. Note, as k increases, the variance approaches 1.0. Therefore, as k increases, the t-distribution approximates the standard normal distribution.

Both the normal (Z) and student’s t (t) distribution characterize the sampling distribution of

the sample mean. The difference is that the normal is used when we know the population

variance; the student’s t is used when we must rely on the sample variance. In practice, we don’t

know the population variance, so the student’s t is typically appropriate.

X X

X XX XZ t

n nS

Chi‐square distribution

For the chi-square distribution, we observe a sample variance and compare it to a

hypothetical population variance. This variable has a chi-square distribution with (n-1) degrees

of freedom:

22( 1)2

( 1) ~ n

sn

Properties of the chi-square distribution:

Nonnegative (>0)

Skewed right, but as d.f. increases it approaches normal

Expected value (mean) = k, where k = degrees of freedom

Variance = 2k, where k = degrees of freedom

The sum of two independent chi-square variables is also a chi-squared variable

14 / 55

F distribution

The F ratio is the ratio of sample variances, with the greater sample variance in the numerator:

2

2x

y

sF

s

Properties of F distribution:

Nonnegative (>0)

Skewed right

Like the chi-square distribution, as d.f. increases, approaches normal

The square of t-distributed r.v. with k d.f. has an F distribution with 1,k d.f.

m * F(m,n)=χ2

Critical t-values

The critical t-values show what percentage of the area under the student’s t distribution curve lies

between the values. The random variable is given by:

X

x

Xt

S n

It follows the student’s t distribution with (n-1) degrees of freedom (d.f.). The confidence interval

is given by:

x xX

S SX t X t

n n

Population regression function (PRF)

The population regression function (PRF) describes the population regression line (PRL). We

don’t observe the PRF; instead, we try to infer it with the sample regression function (SRF). The

population regression function is given by:

1 2( | )i iE Y X B B X

Its stochastic equivalent adds the stochastic (or random) error term:

1 2i i iY B B X u

B1 = intercept = parameter or regression coefficient

B2 = slope = parameter or regression coefficient

15 / 55

Sample regression function (SRF)

Stochastic PRF 1 2i i iY B B X u

Sample regression function (SRF) 1 2

ˆi iY b b X

Stochastic sample regression function (SRF) 1 2i i iY b b X e

b1 = intercept = parameter or regression coefficient

b2 = slope = parameter or regression coefficient

ei = the residual term

Residual sum of squares (RSS)

Each residual is the difference between the observed and predicted Y; the residual sum of squares

(RSS) is the sum of the square of these residuals:

2 2

21 2

ˆRSS = ( )

( )

i i

i i

e Y Y

Y b b X

Standard errors in OLS

Assume the sample regression function (SRF):

1 2i i iY b b X e

It contains two estimators (estimates of the population parameters). Each estimate has a

standard error, a measure of its variability.

The intercept is given by b1. Its standard error is the square root of its variance:

22

1 1 12var( ) ( ) var( )i

i

xb se b b

n x

The slope coefficient is given by b2. Its standard error is the square root of its variance:

2

2 2 22var( ) ( ) var( )

i

b se b bx

The standard error of the regression, SER (a.k.a., standard error of estimate), is given by the

square root of: RSS/(n-2):

2 22 2 in a two-variable mod lˆ eˆi ie e

nk

k n k

16 / 55

Sum of squares

We can break the regression equation into three parts:

Explained sum of squares (ESS),

Residual sum of squares (RSS), and

Total sum of squares (TSS).

The explained sum of squares (ESS) is the squared distance between the predicted Y and the

mean of Y:

2

1

ˆ( )n

ii

ESS Y Y

The residual sum of squares (RSS) is the summation of each squared deviation between the

observed (actual) Y and the predicted Y:

2

1

ˆ( )n

i ii

RSS Y Y

The ordinary least square (OLS) approach minimizes the RSS. The RSS and the standard error of

regression (SER) are directly related; the SER is the standard deviation of the Y values around

the regression line. The residual sum of squares (RSS) is the square of the error term. It is

directly related to the standard error of the regression (SER):

22 2

1

ˆ( ) ( )n

ii i

i

RSS eRSS Y Y RSS SER n k SER

n k n k

Or, equivalently:

2 22 2 in ˆ a two-variable m dˆ o eli ie

kk

e

n n k

The standard error of the regression (SER) is a function of the residual sum of squares (RSS):

2

1

ˆ

Standard Error of the Regression (SER) =

n

ii

n k

17 / 55

Coefficient of Determination & Correlation Coefficient

The coefficient of determination (R^2 or r^2) measures the proportion or percentage of the total variation in Y explained by the regression model:

2 22

2 2

( ) ( ) explained variation1

total variation( ) ( )est esty y y y

ry y y y

The coefficient of determination can be directly inferred from the ANOVA and its sum of squares:

TSS ESS RSS

2 1ESS RSS

RTSS TSS

The correlation coefficient is the square root of the coefficient of determination:

2 1ESS RSS

r RTSS TSS

Test of hypothesis for the slope (b2)

To test the hypothesis that the regression coefficient (b2) is equal to some specified value (), we

use the fact that the statistic

2

2

test statistic ( )

bt

se b

This has a student's distribution with n - k degrees of freedom; k=2 in a two-variable model.

Jarque-Bera

The Jarque-Bera is a popular test of normality that incorporates both skew and kurtosis. It is given

by the following:

2

2 3

6 4

KnJB S

n = the sample size, S = skewness (not sample variance!), K = kurtosis

The JB value is a random variable that follows the chi-square distribution with 2 degrees of

freedom (d.f. = 2).

18 / 55

Prediction Error

The predication error is the difference between the predicted Y and the true mean value. Like the

regression coefficients, the predicted Y has a sampling distribution. The variance of the

predicted Y is given by:

2

022

1ˆvar YXi

X X

n x

Distinguish between simple and multivariate regression

A simple regression is a two-variable regression, one dependent regressed against one

independent variable:

1 2 2( )t tE Y B B X

1 2 2 3 3( )t t tE Y B B X B X

A multivariate regression has more than one independent variable:

1 2 2t t tY B B X

1 2 2 3 3t t t tY B B X B X

Partial slope coefficient

The three-variable (two independents + one dependent) linear regression is given by:

1 2 2 3 3t t t tY B B X B X

In this case, B2 and B3 are partial slope coefficients (or partial regression coefficients). This

means,

In the case of B2, B2 measures the change in the mean value Y, E(Y), per unit of change in X2, holding constant the value of X3.

In the case of B3, B3 measures the change in the mean value Y, E(Y), per unit of change in X3, holding constant the value of X2.

The partical slope coefficients are, therefore, measures of direct sensitivity; i.e., what change in

the dependent variable can be directly attributable to a particular independent variable.

19 / 55

Explain the assumptions of the multiple linear regression model.

A8.1. Linear in parameters

A8.2. X2 and X3 uncorrelated with disturbance term

A8.3. Expected value of error term is zero

A8.4 Constant variance (homoskedasticity)

A8.5. No autocorrelation (a.k.a., serial correlation) between error terms

A8.6. No collinearity between X2 and X3. If the explanantory variables are correlated, such a violation is called multicollinearity.

A8.7. Error term is normally distributed

Standard errors in multilinear regression

The variances and standard errors are given by the following:

2 2 2 22 3 3 2 2 3 2 3 2

1 2 2 22 3 2 3

21var( )

( )

t t t t

t t t t

X x X x X X x xb

n x x x x

1 1( ) var( )se b b

23 2

2 2 2 22 3 2 3

var( )( )

t

t t t t

xb

x x x x

2 2( ) var( )se b b

222

3 2 2 22 3 2 3

var( )( )

t

t t t t

xb

x x x x

3 3( ) var( )se b b

Relationship between the number of Monte Carlo replications and the standard error

of the estimated values.

The relationship between the number of replications and precision (i.e., the standard error of

estimated values) is not linear: to increase the precision by 10X requires approximately 100X more

replications. The standard error of the sample standard deviation:

ˆ1 ( ) 1ˆ( )

2 2

SESE

T T

Therefore to increase VaR precision by (1/T) requires a multiple of about T2 the number of

replications; e.g., x 10 precision needs x 100.

20 / 55

Correlated random variables

The following transforms two independent random variables into correlated random variables:

1 1

22 1 2

1 2

1 2

(1 )

, : independent random variables

: correlation coefficient

, : correlated random variables

Periodic returns (e.g., 1 period = 1 day)

Assume that one period equals one day. You can either compute the “continuously compounded

daily return” or the “simple percentage change.” If Si-1 is yesterday’s price and Si is today’s price,

the continuously compounded return (ui) is given by:

1

ln ii

i

Su

S

The simple percentage change is given by:

1

1

i ii

i

S Su

S

Volatility weighting schemes

Un-Weighted Scheme

2 2

1

1 m

n n ii

um

Weighted Scheme

2 2

1

m

n i n ii

u

Alphas are weights so

they must sum to one.

21 / 55

Estimate volatility using the EWMA model.

In exponentially weighted moving average (EWMA), the weights decline (in constant proportion,

given by lambda).

Exponentially weighted moving average (EWMA):

2 0 21

1 22

2 23

(1 )

(1 )

(1 )

n n

n

n

u

u

u

Recursive version of EWMA:

2 2 21 1(1 )n n nu

RiskMetricsTM is a branded EWMA:

2 2 21 1(0.94) (0.06)n n nu

RiskMetricsTM Approach

RiskMetrics is a branded form of the exponentially weighted moving average (EWMA) approach:

21 1(1 )t t th h r

The optimal (theoretical) lambda varies by asset class, but the overall optimal parameter used by

RiskMetrics has been 0.94. In practice, RiskMetrics only uses one decay factor for all series:

0.94 for daily data

0.97 for monthly data (month defined as 25 trading days)

RiskMetricsTM is EWMA

with a lambda (smoothing

constant) of 0.94

In EWMA weights also sum

to one. However, they

decline in constant ratio

(lambda).

Lambda is the

“persistence

parameter” or

“smoothing constant”

22 / 55

GARCH(p,q) model

EWMA is a special case of GARCH(1,1) where gamma = 0 and (alpha + beta = 1)

2 2 21 1(1 )n n nu

GARCH (1,1) is the weighted sum of a long run-variance (weight = gamma), the most recent

squared-return (weight = alpha), and the most recent variance (weight = beta)

2 2 21 1n L n nV u

2 2 21 1n L n nV u

Mean reversion in the GARCH(1,1) model.

Long-run average variance as a function of omega and the weights (alpha, beta):

1LV

Explain how GARCH models perform in volatility forecasting.

The forecasted volatility forward (k) days is given by:

2 2[ ] ( ) ( )kn k L n LE V V

Discuss how correlations and covariances are calculated, and explain the consistency

condition for covariances.

Correlations play a key role in the calculation of value at risk (VaR). We can use similar methods

to EWMA for volatility. In this case, an updated covariance estimate is a weighted sum of

The recent covariance; weighted by lambda

The recent cross-product; weighted by (1-lambda)

1 1 1cov cov (1 )n n n nx y

(Weighted) Long-

run variance

Lagged, squared

return (1)

Lagged variance (1)

23 / 55

Bernoulli

A Bernoulli variable is discrete and has two possible outcomes:

1 if C defaults in I

0 elseX

Binomial

A binomial distributed random variable is the sum of (n) independent and identically

distributed (i.i.d.) Bernoulli-distributed random variables. The probability of observing (k)

successes is given by:

!( ) (1 ) ,

( )! !k n kn n n

P Y k p pk k n k k

Poisson

The Poisson distribution depends upon only one parameter, lambda λ, and can be interpreted as

an approximation to the binomial distribution. A Poisson-distributed random variable is usually

used to describe the random number of events occurring over a certain time interval. The

lambda parameter (λ) indicates the rate of occurrence of the random events; i.e., it tells us

how many events occur on average per unit of time.

( )!

k

P N k ek

Exponential

The exponential distribution is popular in queuing theory. It is used to model the time we have

to wait until a certain event takes place. According to the text, examples include “the time

until the next client enters the store, the time until a certain company defaults or the time until

some machine has a defect.”

1( ) , , 0xf x e x

24 / 55

Weibull

Weibull is a generalized exponential distribution; i.e., the exponential is a special case of the

Weibull where the alpha parameter equals 1.0.

( ) 1 , 0

x

F x e x

Gamma distribution

11

( ) , 0( )

x xf x e x

Generalized extreme value (GEV) fits block maxima

The Generalized extreme value (GEV) distribution is given by:

1

exp (1 ) 0( )

exp( ) 0y

yH y

e

The (xi) parameter is the “tail index;” it represents the fatness of the tails. In this expression, a

lower tail index corresponds to fatter tails.

Peaks over threshold (POT)

Peaks over threshold (POT) collects the dataset of losses above (or in excess of) some threshold.

( ) ( | )UF y P X u y X u

Peaks over threshold (POTS):

1

,

1 (1 ) 0

( )

1 exp( ) 0

x

G xx

25 / 55

Describe the hazard rate of an exponentially distributed random variable.

The parameter 1/ has a natural interpretation as hazard rate or default intensity.

1( ) , 0

( ) 1 , 0

x

x

f x e x

F x e x

1

( )

( ) 1

x

x

f x e

F x e

Basis

Basis = Spot Price Hedged Asset – Futures Price Futures Contract = S0 – F0

Minimum variance hedge ratio

If the spot and future positions are perfectly correlated, then a 1:1 hedge ratio results in a perfect

hedge. However, this is not typically the case. The optimal hedge ratio (a.k.a., minimum variance

hedge ratio) is the ratio of futures position relative to the spot position that minimizes the

variance of the position. Where is the correlation and is the standard deviation, the optimal

hedge ratio is given by:

* S

F

h

And the number of futures contracts is given by N* when NA is the size of the position being

hedged and QF is the size of one futures contract:

** A

F

h NN

Q

Optimal number of futures contracts needed to hedge an exposure

When futures are used, a small adjustment, known as “tailing the hedge” can be made to allow for

the impact of daily settlement. The only difference here is to replace the units with values.

Instead of using quantities, as in:

** A

F

h QN

Q

26 / 55

We use the dollar value of the position being hedged and the dollar value of one futures

contract, as in:

** A

F

h VN

V

Stock index futures contracts to change a stock portfolio’s beta

Given a portfolio beta (), the current value of the portfolio (P), and the value of stocks

underlying one futures contract (A), the number of stock index futures contracts (i.e., which

minimizes the portfolio variance) is given by:

*P

NA

By extension, when the goal is to shift portfolio beta from () to a target beta (*), the number of

contracts required is given by:

( * )P

NA

Compounding

Assuming:

R c: rate of interest with continuous compounding

R m

: rate of interest with discrete compounding (m per annum)

n: number of years

1

1

mnR n mc

mR mc

RAe A

m

Re

m

/ 1

ln 1

R mcm

mc

R m e

RR m

m

27 / 55

Convert rates based on different compounding frequencies

The present value is discretely discounted at (m) periods per year (e.g., m=2 for semi-annual

compounding) over n years by using the formula on the left. The continuous equivalent is the

right. Note that if the future value is one dollar (FV = $1), then the PV is the discount factor (DF).

Discrete Continuous

1

$1

1

m n

m n

FVPV

r

m

PVr

m

$1

r n

r n

PV FV e

PV e

Calculate forward interest rates from a set of spot rates

Hull assumes a continuous compound/discount frequency. For example, given a two-year spot

rate of 4% and a one-year spot rate of 3%, the one-year implied forward rate = [(4%*2) –

(3%)(1)]/[2-1] = 5%.

2 2 1 1

2 1

R T R T

T T

Par Yield

The par yield for a certain maturity is the coupon rate that causes the bond price to equal its face

value. For example, assume a 2-year bond that pays semi-annual coupons. Further, assume the

zero rate term structure is given by {0.5 years = 5.0%, 1.0 years = 5.8%, 1.5 years = 6.4%, and 2.0

years = 6.8%}. Then solve for the coupon rate (c) that solves for a price (present value) equal to

the par (100):

0.05 0.5 0.058 1.0 0.064 1.5 0.068 2.0100 1002 2 2 2

to get 6 87 (with s.a. compounding)

c c c ce e e e

c = .

28 / 55

Yield

The bond yield is also known as the yield to maturity (YTM). The yield (YTM) is the discount

rate that makes the present value of the cash flows on the bond equal to the market price of the

bond.

(Macaulay) Duration

Duration of a bond that provides cash flow ci at time t

i is

1

ytn ii

ii

c eD t

B

, where B is its price

and y is its yield (continuously compounded). This leads to:

BD y

B

Modified Duration

When the yield y is expressed with compounding m times per year 1

BD yB

y m

Modified duration (D*) is related to (Macaulay) duration (D) by the following:

*1

DD

y m

Such that the estimated change in bond price is a function of the modified duration:

*B BD y

Dollar duration

Dollar duration (DD**), also known as value duration, is the slope of the tangent line (a first

partial derivative)

*

* * *

* *

B BD y

D BD

B D y

29 / 55

Convexity

Convexity is the weighted average of maturity-squares of a bond, where weights are the present

values of the bond’s cash flows, given as proportions of bond’s price. Convexity can be

mathematically expressed

221

2

1n yti

i iic t ed B

CB Bdy

Cost of Carry Model

For a non-dividend-paying investment asset (i.e., an asset which has no storage cost) the cost

of carry model says the futures price is given by:

0 0 0 0cT rTF S e F S e

The equations for forward prices are essentially similar to futures prices. The generalized forward

price (F0) is either case (futures or forwards) is therefore given by:

0 0rTF S e

If the asset provides interim cash flows (e.g., a stock that pays dividends), then let (I) equal

the present value of the cash flows received and the cost-of-carry model is then given by:

0 0( ) rTF S I e

If the asset provides income (e.g., a stock that pays dividends), where the income can be

expressed as a constant percentage of the spot price (given by q), then the model is given by:

( )0 0

r q TF S e

If the asset has a storage cost and produces a convenience yield (where the convenience yield is a

constant percentage of the spot price, denoted by ‘y’), the cost-of-carry model expands to:

( )0 0

r u y TF S e

Where r is the risk-free rate, u is the storage cost as a constant percentage, and y is the

convenience yield.

30 / 55

Value of a forward contract

The value of a forward contract (f) is given by either equation below:

0

0

( ) rT

qT rT

f F K e

f S e Ke

Interest rate parity: Discrete annual compounding

In discrete terms, the equality is given by:

11 [1 ]D F

domestic foreign tt

r r FS

1

1

Ddomestic t

Ftforeign

r F

Sr

Where:

1 1 + the domestic interest rate at time exchange rate ( /foreign) at time

1 1 + the foreign interest rate tspot

ime exchange rate at time

Ddomestic

tFforeign

t

r tS domestic t

r tF tforward

Interest rate parity: continuous

Alternatively, interest rate parity (IRP) can be given in continuously compounded terms:

( )

0 0

r r TfF S e

Where r is the domestic interest rate and rf is the foreign interest rate.

Convenience Yield

For a consumption asset—where (y) is the convenience yield and (c) is the cost of carry—the

futures price is given by:

( )0 0

c y TF S e

If a non dividend-paying stock offered a “convenience yield” then its forward price calculation

would mirror the above formula:

( )0 0

r y TF S e

31 / 55

Cost of carry with storage costs

The futures price for a commodity can be given by two formulae:

0 0( ) rTF S U e

Where U is the present value of storage costs

( )0 0

r u y TF S e

Where

u is the storage costs as a proportion of the spot price y is the convenience yield

Day Count

Day count conventions are important for computing accrued interest:

Actual/actual: U.S. Treasury bonds 30/360: U.S. corporate and municipal bonds Actual/360: U.S. Treasury bills and other money market instruments

Calculate the cost of delivering a bond into a Treasury bond futures contract

The cost to deliver is the dirty price, which is the bond quoted price plus accrued interest (AI).

The short position will receive the settlement multiplied by the conversion factor plus accrued

interest (AI). The cheapest to deliver (CTD) is:

The bond that minimizes MIN: Quoted Bond Price - (Settlement)(CF), or similarly The bond that maximizes MAX: (Settlement)(CF) - Quoted Bond Price

Describe and compute the Eurodollar futures contract convexity adjustment

The convexity adjustment assumes continuous compounding. Given that () is the standard

deviation of the change in the short-term interest rate in one year, t1 is the time to maturity of the

futures contract and t2 is the time to maturity of the rate underlying the futures contract.

21 2

1Forward = Futures

2t t

Duration‐based hedge ratio

The number of contracts required to hedge against an uncertain change in the yield, given by y,

is given by:

32 / 55

* P

C F

PDN

F D

Note: FC = contract price for the interest rate futures contract. DF = duration of asset underlying

futures contract at maturity. P = forward value of the portfolio being hedged at the maturity of

the hedge (typically assumed to be today’s portfolio value). DP = duration of portfolio at maturity

of the hedge

Duration of a Bond

The duration of a bond (D) is given by a formula that says “the percentage change of a bond’s

price (B) is a function of its duration (D) and the change in the yield:”

1

BD y

BB

DB y

If we recast the same equation with deltas, we get: the duration multiplied by the change in yield

(D

BD y

B

And solving this equation for the duration (D) gives us:

1BD

B y

Modified Duration of a Bond (D*)

* where k = compound periods per year1

DD

y k

Black-Scholes-Merton

0 1 2( ) ( )rTc S N d Ke N d

33 / 55

Identify, interpret and compute upper and lower bounds for option prices

0 0

0

0

Upper Bounds

Lower bound for European CALL non-dividend paying stock:

max( ,0)

Lower bound for European PUT on non-dividend paying stock:

max( ,0)

rT

rT

c S C S

p X P X

c S Ke

p Ke S

Put‐call parity

Put–call parity is based on a no-arbitrage argument; it can be shown that arbitrage opportunities

exist if put–call parity does not hold. Put–call parity is given by:

0

0

rT

rT

c Ke p S

c p Ke S

Explain the early exercise features of American call and put options on a

non‐dividend‐paying stock and the price effect early exercise may have

The difference between an American call and an American put (C–P) is bounded by the following:

0 0rTS K C P S Ke

Discuss the effects dividends have on the put‐call parity, the bounds of put and call

option prices, and on the early exercise feature of American options

The ex-dividend date is specified when a dividend is declared. Investors who own shares of the

stock as of the ex-dividend date receive the dividend.

An American option should never be exercised early in the absence of dividends. In the case of

a dividend-paying stock, it would only be optimal to exercise immediately before the stock

goes ex-dividend. Specifically, early exercise would remain sub-optimal if the following

inequality applied:

( )1(1 )r t ti iiD K e

34 / 55

Derive the basic equilibrium formula for pricing commodity forwards and futures

The forward price is equal to the expected spot price in the future, but discounted to the present.

( )0, 0( ) r T

T TF E S e

Where:

0

0,T

( ) Spot price of S at time T, as expected at time 0F Forward pricer Risk-free rate

Discount rate for commodity S

TE S

And, as McDonald says, the forward price [F0] is a biased estimate of expected spot price

[E(St)], where the bias is due to the risk premium on the commodity (risk premium = α – r).

Explain the implication basic equilibrium has for different types of commodities

The forward price is a biased estimate of the expected spot price.

0, 0( )rT TT Te F E S e

For commodities on which forward prices are available, the forward price can be discounted; i.e.,

Forward Price * EXP[(-rate)(time)]. This give the present value of the commodity received at

future time (T).

The forward price when there is a storage (carry) cost is given by:

( )0, 0

r TTF S e

Where:

: lease rate

: commodity discount rate

: commodity growth rate

: storage cost

g

g

35 / 55

Define the lease rate

If the lease rate is given by , then the forward price is given by:

( )0, 0

r TTF S e

The lease rate = commodity discount rate – growth rate:

g

The lease rate is economically like (~) a dividend yield.

Define carry markets

A commodity that is stored is in a carry market. Storage is carry. Storage permits

consumption throughout the year

( )0, 0

r c TTF S e

Compare the lease rate with the convenience yield

If we are given the forward price, we only need to re-arrange the above formula to solve for the

implicit lease rate. We re-arrange as follows:

0,( ) ( )0, 0

0

0, 0,

0 0

0,

0

1ln ( ) n ( )

1n

Tr T r TT

T T

T

FF S e e

SF F

r T l rS T S

Fr l

T S

Define basis risk and the variance of the basis

, Pr ( )Tt T tBasis Spot ice F t

2 2 2( ( )) ( ) ( ( )) 2 ( ) ( ( ))T T Tt t tS F t S F t S F t

This equation shows that basis risk is zero when

Variances between the Futures and spot prices are identical, and The correlation coefficient between spot and futures prices is equal to one.

36 / 55

Effectiveness of hedging a spot position with a futures contract

The classical measure of the effectiveness of hedging a spot position with Futures contracts is

given by:

2

2

( )1

( )t

basish

S

The nearer (h) is to one, the better (more perfect) the hedge.

Historical > standard deviation (simple parametric)

A moving average forecast requires a window of fixed length; e.g., 30 or 60 trading days. If we

observe returns (rt) over M days, this volatility estimate is constructed from a moving average

(MA):

2 2

1

(1 / )M

t t ii

M r

The moving average (MA) series is simple but has two drawbacks

The MA series ignores the order of the observations. Older observations may no longer be relevant, but they receive the same weight.

The MA series has a so-called ghosting feature: data points are dropped arbitrarily due to length of the window.

GARCH (p, q) and in particular GARCH (1, 1)

GARCH (p, q) is General Autoregressive Conditional Heteroskedastic model. There are three key

aspects to the GARCH moniker:

Autoregressive (AR): tomorrow’s variance (or volatility) is a regressed function of today’s variance—it regresses on itself

Conditional (C): tomorrow’s variance depends—is conditional on—the most recent variance. An unconditional variance would not depend on today’s variance

Heteroskedastic (H): variances are not constant, they flux over time

GARCH regresses on “lagged” or historical terms. The lagged terms are either variance or squared

returns. The generic GARCH (p, q) model regresses on (p) squared returns and (q) variances.

Therefore, GARCH (1, 1) “lags” or regresses on last period’s squared return (i.e., just 1 return) and

last period’s variance (i.e., just 1 variance). GARCH (1, 1) given by the following equation.

2 2 21, 1t t t ta br c

37 / 55

Hull writes the same GARCH equation as: 2 2 2

1 1n L n nV u . The first term (VL) is

important because VL is the long run average variance. Therefore, (VL) is a product: it is the

weighted long-run average variance.

The same GARCH (1, 1) formula can be given with Greek parameters. The GARCH (1, 1) model

solves for the conditional variance as a function of three variables (previous variance, previous

return^2, and long-run variance):

20 1 1 1t t th r h

2t

21 t-1

2 2t-1 t-1,t

or conditional variance (i.e., we're solving for it)

or weighted long-run (average) variance

or previous variance

r or r previous squared return

t

t

h

a

h

EWMA

EWMA is a special case of GARCH (1,1). Here is how we get from GARCH (1,1) to EWMA:

2 2 21, 1GARCH(1,1) t t t ta br c

Then we let a = 0 and (b + c) =1, such that the above equation simplifies to:

2 2 21, 1GARCH(1,1) = (1 )t t t tbr b

This is now equivalent to the formula for exponentially weighted moving average (EWMA):

2 2 21, 1

2 2 21 1,

(1 )

(1 )

t t t t

t t t t

EWMA br b

r

In EWMA, the lambda parameter now determines the “decay:” a lambda that is close to one

(high lambda) exhibits slow decay.

EWMA is recursive solution to infinite series

The exponentially weighted moving average (EWMA) is given by:

2 2 2

1 1(1 )n n nu

38 / 55

The above formula is a recursive simplification of the “true” EWMA series which is given by:

2 0 21

1 22

2 23

(1 )

(1 )

(1 ) ...

n n

n

n

u

u

u

In the EWMA series, each weight assigned to the squared returns is a constant ratio of the

preceding weight. Specifically, lambda () is the ratio of between neighboring weights. In this

way, older data is systematically discounted. The systematic discount can be gradual (slow) or

abrupt, depending on lambda. If lambda is high (e.g., 0.99), then the discounting is very gradual.

If lambda is low (e.g., 0.7), the discounting is more abrupt.

Explain how persistence is related to the reversion to the mean.

Given the GARCH (1, 1) equation:

20 1 1 1t t th r h

Persistence is given by:

1Persistence

GARCH (1, 1) is unstable if the persistence > 1. A persistence of 1.0 indicates no mean reversion. A

low persistence (e.g., 0.6) indicates rapid decay and high reversion to the mean. The average,

unconditional variance in the GARCH (1, 1) model is given by:

0

11VL

Using GARCH (1, 1) to forecast volatility

The expected future variance rate, in (t) periods forward, is given by:

2 2[ ] ( ) ( )tn t L n LE V V

Multivariate Density Estimation (MDE)

2 2

1

( )K

t t ii

t i u

Instead of weighting returns^2 by time,

Weighting by proximity to current state

Kernel Function Vector describing Economic state at time t-i

39 / 55

Square root rule (i.e., variance is linear with time) only applies under restrictive i.i.d.

The simplest approach to extending the horizon is to use the “square root rule”

, , 1( ) ( )t t J t tr r J J-period VAR = J 1-period VAR

For example, if the 1-period VAR is $10, then the 2-period VAR is $14.14 ($10 x square root of 2)

and the 5-period VAR is $22.36 ($10 x square root of 5).

The square-root-rule: under the two assumptions below, VaR scales with the square root of time. Extend one-period VaR to J-period VAR by multiplying by the square root of J.

The square-root rule for extending the time horizon requires i.i.d., that’s two assumptions:

Random-walk (acceptable) Constant volatility (unlikely)

Auto Regression, AR(1)

The analysis of auto-regression AR (1) model describes tomorrow’s return as a function of

(dependent on) today’s return:

1 1t t tX a bX e

If the expected value of the error term is zero, then the expected value of Xt simplifies to the

equation below where the parameter (b) is called the “speed of reversion.” If (b=1) then the

formula is a random walk:

1[ ]t tE X a bX

Explain how to calculate VaR for linear derivatives.

By definition, the transmission parameter is constant. Therefore, in the case of a linear derivative,

VaR scales directly with the underlying risk factor.

Linear Derivative Underlying Risk FactorVaR VaR

Taylor Series Approximation

20 0 0 0 0( ) ( ) ( )( ) 1 2 ( )( )f x f x f x x x f x x x

40 / 55

Explain the full revaluation method for computing VaR.

Full revaluation considers values for a range of price levels. New values can be generated by:

Historical simulation, Bootstrap (simulation) Monte Carlo simulation

1 0( ) ( )dV V S V S

… Multifactor Models

Stock returns (as the dependent variable) are regressed against multiple factors. This is a multiple

regression where Iit are the external risk factors and the betas are the sensitivity (of each firm) to

the external risk factors:

1 1 1 1it it i t i t itR I I

The risk factors are external to the firm; e.g., interest rates, GDP. Also, note the multi-factor

model cannot help model low-frequency, high-severity loss (LFHS) events.

… Income Based Models

These are also called Earning at Risk (EaR) models. Income or revenue (as the dependent

variable) is regressed against credit risk factor(s) and market risk factor(s). The residual, or

unexplained, volatility component is deemed to be the measure of operational risk.

Extract market & credit risk from historical income volatility

Residual volatility (volatility of ε) is operational risk measure

1 1 2 2it it t t t t itE C M 1

1

Credit Risk

Market Risk

Residual. Volatility of residual is operational riskit

C

M

41 / 55

Binomial

We need the following notation:

0

price/value of option = proportional "up" jump ( 1) stock price proportional "down" jump ( 1)

number of shares of stock option payoff if stock jumps up= option payoff if stock jumps

u

d

f u uS d d

ff

down

The probability of an “up jump” (or up movement) is denoted by (p) and given by:

rTe dp

u d

This probability (p) then plugs into the equation that solves for the option price:

[ (1 ) ]rTu df e pf p f

and t tu e d e

r te dp

u d

[ (1 ) ]r tu df e pf p f

Describe the impact dividends have on the binomial model

If the stock pays a known dividend yield at rate (q), the probability (p) of an up movement is

adjusted:

( )r q te dp

u d

Up movement (u) and down movement (d)

Probability of up (p)

If u = 1.15, then stock moves up +15% to S0*u

p = probability of up movement,

So (1-p) = probability of down movement

42 / 55

Options on Currencies

Analogous to the adaptation of the cost of carry model to foreign exchange forwards, if (rf) is the

foreign risk-free rate, we can use:

( )

option on currency

r rf te dp

u d

Options on Futures

Since it costs nothing to take a long or short position in a futures contract, in a risk-neutral world

the futures price has an expected growth rate of zero. In this case, we can use:

futures

1 dp

u d

Lognormal property of stock prices

Under GBM (a Weiner process), Periodic returns are normally distributed

~ ( , )S

t tS

Price levels are log-normally distributed

2

0ln ~ ln , )2

TS S T

An Ito process is a generalized Weiner process (a stochastic process) where the change in the

variable during a short interval is normally distributed. The mean and variance of the distribution

are proportional to t. In an Ito process, the parameters are a function of the variables x and t.

2

0

2

0

ln ~ ,2

ln ~ ln ,2

T

T

ST T and

S

S S T T

Let ST equal the stock price at future time T. The expected value of ST [i.e., E(ST) is given by:

0( ) TTE S S e

22 20( ) ( 1)T T

Tvar S S e e

43 / 55

Distribution of the Rate of Return

The continuously compounded rate of return per annum is normally distributed. The distribution

of this rate of return is given by the following:

2

~ ,2 T

The Expected Return: Arithmetic vs. Geometric

The phrase “expected return” has two common meanings: arithmetic and geometric.

( )ArithmeticE 2

( )2

GeometricE

The continuously compounded return realized over T years is given by:

0

1ln( )TS

T S

Compute the realized return and historical volatility of a stock

Start with the variable (ui) which is the natural log of the ratio between a stock price at time (i)

and the previous stock price at time (i-1):

1

ln ii

i

Su

S

An unbiased estimate of the variance is given by:

2 21

1

1( )

1

m

n n ni

u um

Important: the equation above is the variance. The volatility is the standard deviation and,

therefore, is given by:

2 21

1

1( )

1

m

n n ni

u um

44 / 55

For purposes of calculating VAR—and often for volatility calculations in general—a few

simplifying assumptions are applied to this volatility formula. Specifically:

Instead of the natural log of the ratio [Si/Si-1], we can substitute a simple percentage

change in price: %S = [(Si-Si-1)/Si-1]

Assume the average price change is zero

Replace the denominator (m-1) with (m)

With these three simplifications, an alternative volatility calculation is based on the following

simplified variance:

2 21

1

1 m

n ni

um

The stock price process is described by the following formula:

dS Sdt Sdz

dSdt dz

S

The Black–Scholes–Merton Differential Equation is given by:

22 2

2

1

2

f f frS S rf

t S S

European option using Black‐Scholes‐Merton model on a non‐dividend‐paying stock

The Black–Scholes model gives the following values for a call (c) and a put (p) in the case of a

European option:

0 1 2( ) ( )rTc S N d Ke N d 2 0 1( ) ( )rTp Ke N d S N d

Where d1 and d2 are given by:

20

1

ln( ) ( )2

Sr T

KdT

20

2 1

ln( ) ( )2

Sr T

Kd d TT

45 / 55

European option using the Black‐Scholes‐Merton model on a dividend‐paying stock

A European option on a dividend-paying stock can be analyzed as the sum of two components:

A riskless component = known dividends during the life of the option, plus

A risk component

A dividend yield effectively reduces the stock price (the option holder forgoes dividends).

Operationally, the amounts to reducing the stock price by the present value of all the

dividends during the life of the option. If (q) represents the annual continuous dividend yield

on a stock (or stock index), the adjusted Black-Scholes-Merton for a European call option is given

by:

* *

0 1 2( ) ( )qT rTc S e N d Ke N d

20

*1

ln( ) ( )2

Sr q T

KdT

* *2 1d d T

Identify the complications involving the valuation of warrants

Assume that VT equals the value of the company’s equity and N equals the number of

outstanding shares. Further, assume that a company will issue (M) number of warrants with a

strike price equal to K. ST equals the stock price at time T. The (adjusted) stock price, after we

account for the dilution effect of the issued warrants, is:

Tadjusted

V MKS

N M

The Black–Scholes can be used to value a warrant; however, three adjustments are required

The stock price (S0) is replaced by an “adjusted” stock price The volatility input is calculated on equity (i.e., common equity plus warrants) not stock price The calculation is reduced by a multiplier. The multiplier captures dilution and is also called a

“haircut.” The haircut is given by:

N

N M

Warrant Optionvalue =Value , N=shares & M=warrantsN

N M

46 / 55

Define delta hedging for an option, forward, and futures contracts

Delta is the rate of change of the option price with respect to the price of the underlying asset:

= c

deltaS

change in the price of the call option

S = change in the price of the stock price

c

Delta of European Stock Options

1

1

1

1

( ) European call on stock

( ) European call on dividend stoc

non-dividend

non-d

k

( ) 1 European put on stock

( ) 1 European put on dividend

ivide

stoc

nd

k

qT

qT

N d

e N d

N d

e N d

Delta of Forward Contracts

The delta of a forward contract on one share of stock is 1.0.

If the stock pays a dividend, the delta = EXP(-qT).

Delta of Futures Contract

The delta of a futures contract is erT

If the asset pays a dividend, the delta = EXP[(r-q)*T]

Gamma

Gamma is the rate of change of the portfolio’s delta with respect to the underlying asset; it is

therefore a second partial derivative of the portfolio:

2

2 = Gamma

S

2

2 2

the second partial derivative of the call price

the second partial derivative of the stock priceS

Vega

Vega is the rate of change of the value of a portfolio (of derivatives) with respect to the

volatility of the underlying asset:

Vega =

47 / 55

Rho

Rho is the rate of change of the value of a portfolio (of derivatives) with respect to the interest

rate (or, as in the Black–-Scholes, the risk-free interest rate):

= Rhor

Relationship between delta, theta, and gamma

The risk-free rate multiplied by the portfolio (i.e., a fractional share of the portfolio) is directly

related to a linear function of theta, delta and gamma:

2 21

2r rS S

2

risk-free interest rate value of the portfolio option stock price option variance of underlying stock option

r

thetaS

delta

gamma

If theta is large and positive then gamma tends to be large and negative. Delta is zero by

definition in a “delta-neutral” portfolio, in which case the formula simplifies to:

2 21

2r S

Discount factor and discount function

The discount factor, d(t), for a term of (t) years, gives the present value of one unit of currency

($1) to be received at the end of that term.

If d(.5)=.97557, the present value of $1 to be received in six months is 97.557 cents

Assume A pays $105 in six months. Given the same discount factor of 0.97557, $105 to be received

in six months is worth .97557 x $105 = $102.43

$1 $1$1.025 in six months

(.5) 0.97557d

The discount function is simply the series of discount factors that correspond to a series of times

to maturity (t). For example, a discount function is the series of discount factors: d(0.5), d(1.0),

d(1.5), d(2.0).

48 / 55

Impact of different compounding frequencies on a bond’s value.

Investing (x) at an annual rate of (r) compounded semiannually for (T) years produces a terminal

wealth (w) of:

2

12

Tr

w x

Discount factor

Let d(t) equal the discounted value of one unit of currency. Assuming the one unit of currency is

discounted for (t) years at the semiannual compound rate r(t), then the discount rate d(t) is given

by:

2

1( )

(̂ )1

2

td t

r t

The relationship between continuous compounding and discrete compounding (semi-annual

compounding is discrete compounding where the number of periods per year is equal to 2) is

given by:

1mn

R n mcR

Ae Am

The continuous rate of return as function of

the discrete rate of return (where m is the

number of periods per year) is given by:

ln 1 mc

RR m

m

The discrete rate of return as a function of

the continuous rate of return is given by:

1( )R mcmR m e

Compute semi‐annual compounded rate of return for a C‐Strip

If the price of one unit of currency maturing in t years is given by d(t), the semiannual

compounded return, is given by:

1

21(̂ ) 2 1

( )

tr t

d t

49 / 55

Derive spot rates from discount factors.

Given a t-period discount factor d(t), the semiannual compounded return is given by:

1

21(̂ ) 2 1

( )

tr t

d t

The relationship between spot rates and maturity/term is called the term structure of spot

rates. When spot rates increase with maturity, the term structure is said to be upward-sloping.

When spot rates decrease with maturity, the term structure is said to be downward-sloping or

inverted.

Define, interpret, and apply a bond’s yield‐to‐maturity (YTM) to bond pricing.

Yield-to-maturity (YTM), sometimes just yield, is the single rate that, when used to discount a

bond’s cash flows, produces the bond’s market price. Given an annual coupon of c (and therefore

a semi-annual coupon of c/2), a final principal payment of F, a market price of P(T) with T years

to maturity, the yield to maturity (YTM) is given by (y) is the following equation:

2

21

2( )(1 ) (1 )

2 2

T

t Tt

cF

P Ty y

Price of an annuity and a perpetuity

An annuity with semiannual payments is a security that makes a payment c/2 every six months

for T years but never makes a final “principal” payment (i.e., FV=0). The price of an annuity, A(T),

is given by:

2

1( ) 1

12

T

cA T

yy

A perpetuity bond is a bond that pays coupons forever. The price of a perpetuity is simply the

coupon divided by the yield (i.e., the price of a perpetuity = c/y).

50 / 55

Calculate the price of an annuity.

Annuity: makes semiannual payments of c/2 ever six months for T years but never makes a final

payment. Price is given by:

21

( ) 11 2

Tc

A ty y

One-factor measures of sensitivity

DV01 = dollar value of an ’01

a.k.a., PV01, price value of an ’01

Gives the dollar value change of a fixed income security for a one-basis point decline in rates.

Modified duration

Percentage change in value of security for a one unit change (10,000 basis points)

Key relationship:

0110,000

ModP DDV

DV01

DV01 is an acronym for “dollar value of an 01” (.01%). DV01 gives the change in the value of a fixed

income security for a one-basis point decline:

DV01 = 10,000

P

y

Importantly, the DV01 is related to modified duration:

Duration PriceDuration PriceDV01 = =

10,000 10,000

DV01Duration = (10,000)

Price

MacaulayMod

Mod

51 / 55

Duration

Duration (D) is given by:

1 PD

P y

If we multiply both sides of equation, then we get the following key equation:

PD y

P

The above equation says: the percentage change in the price is equal to the modified duration

multiplied by the change in the rate (the minus sign indicates they move in opposite directions;

i.e., a positive yield change corresponds to a negative price change).

Duration can be calculated with the following formula:

price if yields decline - price if yields riseDuration =

2 (initial price) (yield change in decimal)

0

D = 2( )( )

V V

V y

Convexity

Convexity also measures interest rate sensitivity. Mathematically, convexity is given by the

formula below where the term (d2P/dy2) is the second derivative of the price-rate function:

2

2

1 d PC

P dy

The common convexity formula is given by:

02

0

2convexity measure =

( )

V V V

V y

Where:

V0 is the initial price of the bond V+ is the price of the bond if yields increase by Δy V- is the price of the bond if yields decrease by Δy Δy is a change in the yield (in decimal terms)

52 / 55

Applying the Convexity Measure

In order to estimate the percentage price change due to a bond’s convexity (i.e., the percentage

price change not explained by duration), the convexity measure must by “translated” into a

convexity adjustment:

21Convexity adjustment = convexity measure ( )

2y

The (1/2) in the formula above is called the “scaling factor.”

Debt service ratio

Exports generate dollars and hard currencies; the greater the exports, the easier it is to service

debt

interest + amortization on debtDSR =

export

DSR Likely to reschedule

Debt Service Ratio (DSR) has a positive (+) relationship to the likelihood of debt rescheduling:

Exports are its primary way of generating hard currencies for an LDC. Larger debt repayments

(i.e., in relation to export revenues) imply a greater probability that the country will need to

reschedule.

Import ratio

To pay for imports, LDC must run down its stock of hard currencies

total importsImport Ratio (IR) =

total foreign exchange reserves

IR Likely to reschedule

Import Ratio (IR) has a positive (+) relationship to the likelihood of debt rescheduling: To pay

for imports, the LDC must run down its stock of hard currencies. The greater the need for

imports, the quicker a country can be expected to deplete its foreign exchange reserves.

53 / 55

Investment ratio

Higher investment implies greater future productivity (i.e., negative relationship: less likely to

reschedule) but also greater bargaining power with creditors (positive relationship)

real investmentInvestment Ratio (IRVR) =

gross national product

IR Likely to reschedule

Likely to reschedule

Variance of export ratio

LDC export revenue variability impacted by (i) Quantity risk [how much sold?] and (ii) Price risk

[exchange rate]

2VAREX = ER

V Likely to reschedule

Variance of Export Revenue (VAREX) has a positive Relationship (+) to the likelihood of debt

rescheduling.

Domestic money supply growth

Faster growth in money supply → higher domestic inflation rate → weaker currency

money supplyDomestic Money Supply Growth (MG) =

money supply

MG Likely to reschedule

Domestic Money Supply Growth (MG) has a positive relationship (+) to likelihood of debt

rescheduling: a higher rate of growth in domestic money supply should cause a higher domestic

inflation rate and, consequently, a weaker currency.

Both views

54 / 55

Define, calculate and interpret the expected loss for an individual credit instrument.

Expected loss = Assured payment at maturity time T x Loss Given Default (LGD) x Probability

that default occurs before maturity T (PD)

However, “Assure payment at maturity time T” should be replaced with “Exposure.” Therefore,

the key formula is given by:

Expected loss = Exposure (at default, EAD) x Loss Given Default (LGD) x Probability of default

(PD)

Or, equivalently and ultimately:

Expected loss = Exposure at default (EAD) x Loss Given Default (LGD) x Expected Default

Frequency (EDF)

EL AE LGD EDF

EL AE LGD PD

Define exposures, adjusted exposures, commitments, covenants, and outstandings.

Assume

Value of bank asset = V Outstandings = OS Commitments = COM

Then V = OS + COM

Outstandings: generic term referring to the portion of the bank asset which has already been

extended to the borrowers and also to other receivables in the form of contractual payments

which are due from customers. Examples of outstandings include term loans, credit cards, and

receivables.

Commitments: An amount the bank has committed to lend, at the borrower’s request, up to the

full amount of the commitment. An example of a commitment is a line of credit (LOC). A

commitment consists of two portions:

Drawn, or Undrawn

But the drawn commitment should be treated as part of the outstanding (i.e., the amount

currently borrowed).

55 / 55

Define, calculate and interpret the unexpected loss of an asset.

Unexpected loss (UL) = Standard Deviation of unconditional value of the asset at horizon.

Unexpected loss (UL) is given by:

2 2 2LGD EDFUL AE EDF LGD

Where the variance of the default frequency (EDF) is given by:

2 (1 )EDF EDF EDF

Note: the variance of loss given default (LGD), unlike the variance of EDF, is non-trivial.

Unexpected loss (UL) is average loss bank can expect (to lose on its asset) over the specified

horizon.

The standard deviation of EDF = SQRT[(EDF)(1-EDF)]

The standard deviation of LGD is given as an input (not solved, being non-trivial)

Unexpected loss = SQRT[(EDF)(variance of LGD) + (LGD^2)(Variance of EDF)