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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