Exam_Formula_Sheet-Final.docx | Exams Management Fundamentals

Chapter 5 : Financial Instruments

Weather Derivatives:

Heating degree days:

HDD = Max(0, 65 – A)

Cooling degree days:

CDD = Max (0, A – 65)

where A = the average of the highest and lowest

temperature during a day in ºF

Optimal hedge ratio: Number of contracts:

h∗¿ρσ

N∗¿h∗N

F=future/ forward price and S=value of hedged asset

P=correlation between ΔS and ΔFS and ΔS and ΔFF

Standard deviation of portfolio returns

√

+2ρw

Chapter 6 : How Traders Manage Exposure

Delta: Rate of Change of the Option Price with

Respect to the Price of the underlying Asset

Protects against: small changes in S

Δ=ΔP

ΔS

Gamma Г: Rate of Change (derivative) in Delta with

Respect to the Price of the underlying Asset.

Protects against: large changes in S

Г of portfolio = wT* ГT+ Г

Г: gamma of delta neutral portfolio

ГT: gamma traded option

wT: number of traded options

Position in traded option to make P gamma

neutral: wT = - Г/ ГT

Theta Θ: Rate of Change in Portfolio Value with

Respect to the Passage of Time (not hedgable)

Vega: Rate of change in portfolio value with respect

to the volatility of the underlying Asset (=

uncertainty about future variable value)

ν=ΔS and ΔFΠ

Δσ

Rho ρ: Rate of Change in Portfolio Value with

Respect to the risk-free Rate of Interest.

Taylor Series Expansion (change in portfolio price):

Approximate non-linear with linear relationship

For a delta-neutral portfolio:

ΔP=Θ∗Δt +1/2Γ∗ΔS

Chapter 7 : Interest Rate Risk

Duration:

Measures the sensitivity of P value to a small parallel

shift in the zero-coupon yield curve (continuous comp.)

∑

i=1

[

−yt

]

With bond price:

∑

i=1

−yt

Key duration relationship:

ΔB=−DB Δy

Modified Duration (D*)

When yield y is expressed with compounding m times

per year:

D∗¿D

1+y/m

ΔB=− BD Δy

1+y/m

Dollar Duration

Product of Duration and bond price, input for D in key

duration relationship

Convexity

Measures curvature

C=1

∂2B

∂y2=

∑

i=1

ncit²ie−yt

The change in the bond

price (bond return):

ΔB/B=− DΔy+1

2C(Δy )

Portfolio

D=∑

i=1

nXi

P× Di

C=∑

i=1

nXi

P× Ci

Nonparallel yield curve shifts (partial duration)

Di=−1

∆ Pi

∆ yi

Principal Component Analysis:

to estimate delta exposure:

Delta exposure PC1= Factor loading PC1 3m*change in

portfolio value 3month + 6monthPC1*change in

portfolio value 6month+… (importance measured by

SD)

Chapter 8 : VaR

VaR=mean - σ*NN-1(X), where σ = standard deviation

of the portfolio change over the time horizon; X =

confidence level; N-1() = inverse cumulative normal

distribution; books assumes mean = 0

T-day VaR = 1-day VaR *N √T (requires zero mean!)

Impact of autocorrelation:

Variance of change in portfolio value over T days (sqrt

of formula used as new st. dev. in VaR formula):

σ²[T+2

(

T−1

)

ρ+2

(

T−2

)

(

T−3

)

+…]

Variance adjusted for autocorrelation

Convert VaR with a confidence level to VaR with a

different confidence level:

VaR(X∗)=VaR(X)∗N

−1

(X∗)/N

−1

(X)

Marginal VaR (sensitivity of VaR to the amount

invested in the ith component):

∂( VaR)

∂x

Incremental VaR (change in VaR resulting from a

particular trade) of the ith component:

Marginal VaR* xi

Component VaR for the ith component (The part of

VaR of the portfolio that can be attributed to this

component):

∂VaR

∂x

Back Testing:

One-tailed: The probability of the VaR limit being

exceeded on m or more is:

∑

k=m

k !(n−k)!p

(1−p)

n−k

When exceptions (m) are lower than expected number

of exceptions, the probability of m or less exceptions is:

∑

k=0

k !(n−k)!p

(1−p)

n−k

Two-tailed: If probability of exceptions under the VaR

model is p and m exceptions are observed in n trials,

then (Kupiec’s Test):

-2ln[(1-p)n-m *pm] + 2ln[(1- m/n)n-m*(m/n)m]

Should have chi-square distribution with 1 degree of

freed.

For 5%, reject the model, if the result > 3.84

Bunching:

If daily portfolio changes indep, should be spread

evenly through period

1) test for autocorrelation

2) Christofferson test statistic

−2 ln

[

(

1−π

)

]

+2 ln ⁡[(1−π

)

(1−π

)

]

: # of observations going from state i to j

State 0= no exception, state 1= exception

Χ2

distribution with 1df

π=u

π01=u01

u00+u01

; π 11=u11

u10+u11

Chapter 9: Volatility

σ(year) = σ(day) * √252

Estimating daily volatility from historical data:

Return: ui=⁡ln(Si/Si-1) ui: continuously compound.

Estimate s of standard deviation of ui:

√

n−1

∑

i=1

−´

u)²

√

n−1

∑

i=1

²−1

n(n−1)¿¿¿

Estimate of volatility:

σ=s/

√

with the standard error estimate:

σ/

√

Power

Law for extreme observations:

For variable v and when x is large

Prob(v⁡> x) = Kx-a K and a: constants

ln[Prob (υ> x) ]= ln K- α lnx

Monitoring daily volatility:

use: ui = (Si-Si-1)/Si-1 and

σn

2=1

∑

i=1

mun−i

When weighting:

σn²=∑

i=1

αiu²n−i

α: weight given to

observation

ARCH(m)

σn

2=γVV L+

∑

i=1

mαiun−i

γV+∑

i=1

αi=1

Exponentially Weighted Moving Average EWMA:

=λσ

n−1

+( 1−λ)u

n−1

Garch (1,1)

incl. mean reversion

With α + β < 1 to be stable

ω=γVV L

γ = 1- α – β

VL=ω

1−α−β

Maximum Likelihood Method – Estimating constant

variance

Likelihood of m observations occurring in the order in

which they are observed is:

∏

i=1

[

√

2πv exp

(

−ui

)

]

Best estimate of v is value that maximizes:

∑

i=1

[

−ln(v)− u

]

Maximum likelihood estimator of v :

v=1

∑

i=1

Estimating GARCH or EWMA parameters:

With use of max. likelihood

Maximize:

∑

i=1

(−ln

(

)

−u

)

By changing ω α and β.

Forecast future volatility with Garch (1,1):

E[σ

n+t

]=V

+( α+β)

(σ

−V

)

For large t, move towards VL

Volatility term structure:

Volatility (p. annum) that should be used for pricing an

option lasting T days (term structure):

σ(T)=

√

252

{

VL+1−e−aT

[

V(0)−VL

]

}

where:a=ln 1

α+β

Impact of volatility changes:

When σ(0) changes by ΔS and ΔF σ(0), σ(T) changes by:

1−e

−aT

σ(0)

σ(T)Δσ (0)

Chapter 10 : Correlation & Copulas

Coefficient of correlation (ρ) between V1 and V2:

E(V

)−E(V

)E(V

)

SD(V

)SD(V

)

cov(V

)=E(V

)−E(V

)E(V

)

ρ=cov (V

)

(

)

SD(V

)

V1 and V2 are independent, if the knowledge of one

does not affect the probability distribution of the other:

f(V

=x)=f(V

)

f() is the probability density function

Monitoring correlations:

Between 2 variables:

Returns: ⁡Xi=⁡(Xi-X⁡i-1)/Xi-1 Yi=⁡(Yi-Y⁡i-1)/Yi-1

covn=1

m∑

i=1

xn−iyn−i

varx , n=1

m∑

i=1

x²n−i

vary ,n=1

m∑

i=1

y²n−i

corr=covn

√

varx, n var y , n

EMWA

for updating covariance:

cov

=λcov

n−1

+(1−λ)x

n−1

Garch(1,1) for updating covariance:

cov

=ω+αx

n−1

+βcov

n−1

Long-term average cov. is ω/(1-α-β) where yVL= ω

xn-1 yn-1 : most recent observation on cov. (with x and y

being % changes)

A variance-covariance matrix, Ω, is internally

consistent (=positive semi-definite) if for all vectors w:

Ωw≥0

where wT is the transpose of w.

Multivariate normal distribution:

Generating random sample:

ε=

∑

i=1

−6

ε1=z1

and

ε2=ρz1+z2

√

1−ρ²

Cholesky decomposition:

Factor Models:

One-factor:

F+

√

1−a

F: common factor; Zi : specific to asset; ai is constant

(between -1 and +1); F and Zi have standard normal

distributions; Zi uncorrelated with F

Multi-factor:

+…a

√

1−a

−a

−…a

Correlation in multifactor

is:

∑

m=1

Copulas - Application to Loan Portfolios:

WCDR(T,X) is the default rate that we are X% certain

will not exceed in time T:

WCDR(T , X )=N

[

−1

[

Q(T)

]

√

ρ N

−1

(X)

√

1−ρ

]

Q(T): probability of default in time T

ρ: correlation parameter

N-1() = inverse cumulative normal distribution

VaR (T,X):

VaR for a loan portfolio with a time horizon T and a

confidence level of X:

L: Dollar size of loan portfolio

R: Proportional amount recovered

Chapter 12 : Market VaR – Historic. Sim. Approach

The ith scenario assumes that the value of the market

variable tomorrow (n+1) will be:

vi−1

Timo Hendriks

I423300

Vega neutrality: Take position –V/VT

Y: bond yield

t: time i

c: cash flow at time i

Δy= small change in yield

Δy=large change

Xi: bond i

P: price portfolio

Di: duration bond i

Xi: bond i

P: price portfolio

Di: convexity bond i

P: portfolio value

Δyi: small change to ith point

ΔPi: change in portfolio value

Sum of component VaR

should equal portfolio VaR

m=k: number exceptions

p: prob. of exception

n: total days (trials)

reject if < conf. level

k= 0

τ : length of time

M: days of observations

Gives⁡equal⁡weights⁡

VL: long-run variance

γVL = ω

Weights sum to 1

v: variance

m: observations

and V(0)= current variance per

day

σ(0)= current volatility p. day

(*√252)

σ(T)=volatility for T (yearly)

E( ) = expected value





kiki













jkik



vn: value of market variable today

ΔP=DELTA⋅ΔS+

GAMMA⋅

(

ΔS

)

+THETA⋅Δt+

VEGA⋅Δσ+RHO⋅Δr

=γVV

+αu

n−1

+βσ

n−1

VaR

(

T , X

)

=L×

(

1−R

)

×WCDR

(

T , X

)

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h∗¿ ρ

σ S

σ F

N∗¿ h∗ N Q^ A

ΔP=Θ∗Δt + 1 / 2 Γ∗ΔS^2

ti [

B ]

D∗¿ 1 +D y / m

D=∑

i= 1

i

C=∑

i= 1

i

∂( VaR)

∂ xi

n n!

k! (n−k )! p

k ( 1 − p )n−k

-2ln[(1-p)n-m^ pm] + 2ln[(1- m/n)n-m(m/n)m]

− 2 ln [ ( 1 −π )

u 00 +u 10

u 01 +u 11

]+ 2 ln [( 1 −π 01 )

u 00

u 01

u 10

u 11

2 distribution with 1df

σ(year) = σ(day) * √

Return: ui= ln(Si/Si- 1 ) ui: continuously compound.

i= 1

n

i= 1

n

σ^ ^ =s/√τ

with the standard error estimate: ^σ /√ 2 n Power

Prob(v > x) = Kx-a^ K and a: constants

ln[ Prob (υ> x )]= ln K- α lnx

use: ui = (Si-Si- 1 )/Si- 1 and

σ n ²=∑

i= 1

M

αi un^2 −i

α i= 1

ω=γVV L

V L= 1 −ωα −β

[

√ 2 πv exp(

−ui^2

2 v )]

[−ln(^ v^ )−^

ui^2

v ]

i= 1

m

σ ( T )=

252 {V L+^1 −e

aT [^ V^ (^0 )−V^ L]^ }

where :a=ln 1 α +β

cov (V 1 , V 2 )=E(V 1 V 2 )−E (V 1 ) E(V 2 )

f (V 2 |V 1 =x )=f (V 2 )

i = 1

m

i= 1

m

i= 1

m

√var^ x, n var^ y , n

EMWA

wT^ Ω w≥ 0 where wT^ is the transpose of w.

Ui=ai 1 F 1 +ai 2 F 2 +… aiM F M +

aim a jm

WCDR (T , X )=N [ N

√ 1 −ρ ]

1 ^2