frm formula sheets: part 2 - wiley el · pdf filede servigny and renault, chapter 3 sv tt nk v...
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FRM® EXAM REVIEW2016
FRM PART II®
COVERS ALL TOPICS
IN PART II
FORMULA SHEETS
Cover image: Loewy DesignCover design: Loewy Design
Copyright © 2016 by John Wiley & Sons, Inc. All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada.
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Library of Congress Cataloging-in-Publication Data:
ISBN 978-1-119-34824-5
Market Risk Management and Measurement (MR)
DowD, Chapter 3
© 2016 wiley 2
Dowd, Chapter 3
VaR Z= − +µ σ α
VaR Z P) t= − + −( µ σ α 1
VaR P exp Z)past= − −[ ( ]1 µ σ
SFRatioE R RP L
P
= −( )
σ
Meissner, Chapter 1
© 2016 wiley 3
Meissner, Chapter 1
ρ ρrealized =−
∑22n n
i j,
2
3 33 2 4 3
2 −+ + =(. . . ) .
Meissner, Chapter 2
© 2016 wiley 4
Meissner, Chapter 2
S S a St t S t− = −− −1 1( )µ
ACCov
( , )( , )
( ) ( )ρ ρ ρ ρ
σ ρ σ ρt tt t
t t−
−
−=1
1
1
Meissner, Chapter 3
© 2016 wiley 5
Meissner, Chapter 3
Correlation = cov( , )X Y
X Yσ σ
16
1
2
2−
−∑d
n n
i
( ).
τ = −−
n n
n nc d
( ) /1 2
Meissner, Chapter 4
© 2016 wiley 6
Meissner, Chapter 4
p Z= < =P( ) ( )ζ ζΦ
where phi is the standard expression of the cumulative normal distribution and ζ is the log of the strike we subjectively set where we assume an asset will default.
tuCkMan, Chapter 6
© 2016 wiley 7
Tuckman, Chapter 6
F FDV
DVR N
N
R= − × ×01
01β̂
FaceValue FaceValueDVFirstHedge = − What Im hedging
What Im He01 ddging
DV
FaceValue Face
FirstHedgeFirstHedge
SecondHedge
01⋅
= −
β
VValueDV
DV SecondHedgeSeWhat Im hedging
What Im Hedging01
01⋅ β ccondHedge
tuCkMan, Chapter 7
© 2016 wiley 8
Tuckman, Chapter 7
1 000 0002
, ,r kcmt −
tuCkMan, Chapter 9
© 2016 wiley 9
Tuckman, Chapter 9
dr dW= σ
dr dt dW= +µ σ
dr k r)dt dW= − +(θ σ
rk r
t dt0
0= − + −(/
)θ σ
dr k r r dt dt dw
k rk
r dt dw
= − + +
= +
−
+
∞
∞
( ) λ σ
λ σ
θ λ≡ +∞rk
Credit Risk Measurement and Management (CR)
De servigny anD renault, Chapter 3
© 2016 wiley 11
de Servigny and Renault, Chapter 3
S V N k T t Ke N kt t vr T t= + − − − −( ) ( )( )σ
stulz, Chapter 18
© 2016 wiley 12
Stulz, Chapter 18
D T T Kif A
K A if AK K AT
T TT( , ) max[ , ]= −
≥− <
= − −
00
K
K
Value of risky debt = Value of risk-free debt – Value of put option on company assets
S A N d Ke N dt t 1r T t
2= − − −( )( ) ( ) |
D t T) A N d Ke N dt 1T t)
2( , ( () )(= − + − −r
Credit spread = yD (t,T) – yP(t,T) = λϒ
Malz, Chapter 6
© 2016 wiley 13
Malz, Chapter 6
• Probability of default = Prob(A T < K) = 1 − Prob(A T ≥ K) = 1 − N(e2)
e
lnAK
u T t) T t)
T t
e e T t
t
1
2
2 1
12
=
+ − + −
−= − −
( (σ
σσ
• Expected loss = KN(−e2) − Ateu(T−t) N(−e1)• Present value of expected loss = KP(t,T) − D(t,T)
This simplifies to Ke−r(T−t) N(−d2)AtN(−d1)
Malz, Chapter 7
© 2016 wiley 14
Malz, Chapter 7
P t t F t e t[ ] ( )* < ≡ = − −1 λ
′ = −F t e t( ) λ λ
Malz, Chapter 8
© 2016 wiley 15
Malz, Chapter 8
ai i i im i= + − ∈ =β β1 1 22 , ,...
gregory, Chapter 8
© 2016 wiley 16
Gregory, Chapter 8
Netting factor =n n n+ −( )1 ρ
n
gregory, Chapter 12
© 2016 wiley 17
Gregory, Chapter 12
CVA (1 Rec) DF(t )EE t PD t ti ii
m
i 1 i≈ −=
−∑ ( ) ( ).,1
F( ) exp( ),u hu= − −1
h ≈−
Spread
(1 Recovery),
F(u)Spread
(1 Recovery)u= − −
−
1 exp .
CVA 1 Rec DF(t )EE t t P,iIncremental
j iIncremental
j
m
j 1 j= −=
−∑( ) ( )1
DD(t t ),j 1 j− .
ChouDhry, Chapter 12
© 2016 wiley 18
Choudhry, Chapter 12
CPR SM= − −1 1 12( )M
rt t
t=
× ≤>
0 2 30
6 0 30
. % ,
. %,
for months
for months
Average lifePrincipal received at
Total principal= ×
×=∑ t t
t
N
121 ( )
Operational and Integrated Risk Management (OR)
GirlinG, Chapter 12
© 2016 Wiley 20
Girling, Chapter 12
f ne
n
N d
( )!
=−λ
Crouhy, Chapter 17
© 2016 Wiley 21
Crouhy, Chapter 17
RAROC = Risk Adjusted Return
Risk Adjusted Capital
ARAROCRAROC
=− Rf
βe
DoWD, Chapter 14
© 2016 Wiley 22
Dowd, Chapter 14
LVaR 1 spread 1 645 spread /2 VaR= + +[ ( . ) ]µ σ
dS dt dWL
dN= + +µ σ 1
Malz, Chapter 12
© 2016 Wiley 23
Malz, Chapter 12
r Lr L re a d= − −( )1
1
22 33( . )s s+ σ
( )( )1
6
+ T 1 + 2T
T
hull, Chapter 15
© 2016 Wiley 24
Hull, Chapter 15
Max aLV, 0( ) +
w L w Ci i j ji=1
M
i=1
N
+ ∑∑
max( , * )Var m Var SRCt 1 c average− +
WCDR NN PD N
1
1i
1
=+
−
− −( ) ( . )ρρ
0 9999
EAD * LGD WCDR∑ *
EAD * LGD PD∑ *
EAD * LCG * WCDR PD * MA( )−
12 5. * ( )EAD * LCG * WCDR PD * MA−
hull, Chapter 16
© 2016 Wiley 25
Hull, Chapter 16
m * VaR + m * VaRc ave s stressed
Risk Management and Investment Management (IM)
GrinolD anD Kahn, Chapter 14
© 2016 Wiley 27
Grinold and Kahn, Chapter 14
λψAP
= IR
2.
− ≤ ≤SC MCVA PCn n n
Jorion, Chapter 7
© 2016 Wiley 28
Jorion, Chapter 7
Portfolio VaR VaR= = = ′∑p pW x xασ α
∆VaRVaR VaR cov(R )
i = ∂∂
= ∂∂
=∂∂
=x w W w
R
i i
p
i
p
p
ασ
ασ
,
Jorion, Chapter 17
© 2016 Wiley 29
Jorion, Chapter 17
R R R w R w R W Rasset polic ymix active mgt ib
ib
ii i i
bib
i
= + = + −∑ ∑. ( )
BoDie, Kane, anD MarCus, Chapter 24
© 2016 Wiley 30
Bodie, Kane, and Marcus, Chapter 24
Sharpe ratioR Rp f
p
=−
σ
Treynor ratioR Rp f
p
=−
β
α βp p f p m fR R R R= − + −[ ( )]
τ α ασ
( )( )
= N
e
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