FRM® EXAM REVIEW2016

FRM PART I®

COVERS ALL TOPICS

IN PART I

FORMULA SHEETS

Cover image: Loewy Design Cover 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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ISBN 978-1-119-34823-8 (ebk)

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

Foundations of Risk Management (FRM)

Elton, ChaptEr 13

© 2016 Wiley 2

Elton, Chapter 13

E R RE R R

f fm f

mX( )

( )= +

−σ

σ

E R R E R Ri F i M F( ) ( ( ) )= + −β

Where:

E R i

Rp

F

( ) ==

expected return of asset (of portfolio)

risk-freee rate of returnexpected rate of return of the markE RM( ) = eet portfolioCov

Var1β =( , )

( )

R R

Ri M

M

Equation of CML:

E R RE R R

pp fm f

m

( ) = + ( )

σσ

−×

βσ

ρ σ σσ

ρ σσi

i m

m

i m i m

m

i m i

m

R R= = =

Cov ,

2

,( , ) ,2

© 2016 Wiley 3

amEnC, ChaptEr 4

Amenc, Chapter 4

Sharpe ratio = R Rp f

p

−σ

Treynor ratio = R Rp f

p

−β

α βp p f p m fR R R R= − + −[ ( )]

TrackingError = σ (ActiveReturn − BenchmarkReturn)

IR( )

=−−

R R

s R RP B

P B

SR T

DR= −

BodiE, ChaptEr 10

© 2016 Wiley 4

Bodie, Chapter 10

E R Rp F p k p K( ) ..., ,= + +λ β λ β1 1

Required return = Risk-free rate + (Risk premium)1 + (Risk premium)2 + . . . + (Risk premium)k

Risk premiumi = Factor sensitivityi × Factor risk premiumi

Quantitative Analysis (QA)

millEr, ChaptEr 2

© 2016 Wiley 6

Miller, Chapter 2

P A or B) P(A) P(B) P(AB)

P A and B) P(A) P(B)

(

(

= + −= ×

© 2016 Wiley 7

millEr, ChaptEr 3

Miller, Chapter 3

μ = X

N

ii

N

=∑

1

σµ

=−

=∑( )X

N

ii

N2

1

Cov(XY) = E{[X − E(X)][Y − E(Y)]}

Cov(R ,R ) P(R R R ER R ERA B A,i B,J A,i A B,j Bji

= − −∑∑ , )( )( )

Corr(R , R ) R RCov R , R

A B A BA B

A B

= =( , )( )

( )( )σ σ

E R w E R w E R w E R w E Rp i ii

N

N N( ) ( ) ( ) ( ) ... ( )= = + + +=∑

11 1 2 2

Var R w w Cov R Rp i j i jj

N

i

N

( ) ( , )===∑∑

11

Var R w R w R w w Cov R Rp A A B B A B A B( ) ( ) ( ) ( , )= + +2 2 2 2 2σ σ

millEr, ChaptEr 4

© 2016 Wiley 8

Miller, Chapter 4

P rn

r n rp qr n r( )

!

!( )!=

−−

µ σµ σ µ σ σ

L Le e e= = −

+( ) +( )0 5 22 2 2 2

1.

© 2016 Wiley 9

millEr, ChaptEr 6

Miller, Chapter 6

P(Event |Information) =P(Information|Event) P(Event)

P(Inform

×aation)

millEr, ChaptEr 7

© 2016 Wiley 10

Miller, Chapter 7

sX X

n2

2

=−

−∑( )

1

FoRMulA oF StAndARd ERRoR

90% confidence interval: Xs

n±1 645.

95% confidence interval: Xs

n±1 960.

99% confidence interval: Xs

n± 2 575.

Test statisticSample statistic Hypothesized value

Standard err= −

oor of sample statistic

© 2016 Wiley 11

hull, ChaptEr 11

Hull, Chapter 11

COV COV 1n n n nX Y= + −− − −λ λ( )1 1 1

COV 1n n n nx Y= + + +− − −ω α β1 1cov

∈ =

∈ = + −1 1

2 1 221

Z

Z Zρ ρ

StoCk, ChaptEr 4

© 2016 Wiley 12

Stock, Chapter 4

Regression model equation = Yi = b0 + b1 Xi + εi, i = 1,...., n

Regression line equation = ˆ ˆ ˆ , , ...,Y b b X i ni i= + =0 1 1

ESS Y Yii

n

= −( )=∑ ˆ 2

1

SEE =Y b b X

n n

i ii

n

ii

n

− −( )−

=−

= =∑ ∑ˆ ˆ (ˆ )

/

0 1

2

1

1 2

2

1

2 2

ε

=−

1 2

1 2

2

/

/SSE

n

R2 = = −Explained variation

Total variation

Total variation Unexplaiined variation

Total variation

Unexplained variation

Total va= −1

rriation

© 2016 Wiley 13

StoCk, ChaptEr 5

Stock, Chapter 5

ˆ ( )

(

ˆb t s

t

j c bj± ×

±estimated regression coefficient critical -valuue)(coefficient standard error)

StoCk, ChaptEr 6

© 2016 Wiley 14

Stock, Chapter 6

Var Slopex x

( )( )

=−∑

σ2

2

Fk

n k +-

MSR

MSE

SS/

SSE [ ( 1)]stat

R

/= =

−

© 2016 Wiley 15

StoCk, ChaptEr 7

Stock, Chapter 7

Adjusted 2R R R= = − −− −

−2 211

11

n

n k( )

diEBold, ChaptEr 5

© 2016 Wiley 16

diebold, Chapter 5

se

T k

tt

T

2

2

1=−

=∑

,

AIC =

=∑

ee

T

k

Tt

t

T

22

1

SIC =

=∑

Te

T

k

Tt

t

T2

1 .

© 2016 Wiley 17

hull, ChaptEr 23

Hull, Chapter 23

σ γ α βσn L n nU21

21

2= + +− −V

σ ω α βσn n nU21

21

2= + +− −

xb

bt =−

0

11

Financial Markets and Products (FMP)

© 2016 Wiley 19

Hull, CHapter 1

Hull, Chapter 1

V T S F TT T( , ) ( , )0 0= −

F 0 T = S 1+ r0T( , ) ( )

V 0 T = St F 0 T / 1+rtT t( , ) [ ( , ) ( ) ]− −

Hull, CHapter 3

© 2016 Wiley 20

Hull, Chapter 3

MinimumVarianceHedgeRa o s

r

ti = ρσσ

# of Futures =−

×MD MD

MD

MV

MVTarget Portfolio

Futures

Portfolio

Futurres ContractYield× β

# of Futures =−

×β β

βTarget Portfolio

Futures

Portfolio

Futures

MV

MV CContract

© 2016 Wiley 21

Hull, CHapter 4

Hull, Chapter 4

PV =PMT

1 + Z

PMT

1 + Z

PMT + FV

1 + Z11

2 NN( ) ( )

...( )

+ + +2

relationship between multiperiod spot rates and forward rates:

1 1 11 1 2

2+( ) +( ) = +( )s f s0 1 0

1 1 12

2

1 3

3+( ) +( ) = +( )s f s0 2 0

∆ ∆ ∆B

BD y C y= − + 1

22( )

Convexity adjustment = Convexity estimate r 1002× ×( )∆

Hull, CHapter 5

© 2016 Wiley 22

Hull, Chapter 5

F Si

iFC/DC FC/DCFC

DC

= ×+( )+( )

1

1

F Si

iFC/BC FC/BCFC

BC

= ×+( )+( )

1

1

F S Si i Actual

i ActualFC/DC FC/DC FC/DC

FC DC

DC

− =−( ) ×

+ ×( )

360

1 360

− =−( ) ×

+ ×F S S

i i Actual

i ActuFC/BC FC/BC FC/BC

FC BC

BC

360

1 aal360( )

F S eContinuously compounded risk-free rate

0rT=

=0

r

F S e r+U T0 0= ( )

F S e r+U Y T0 0= −( )

© 2016 Wiley 23

Hull, CHapter 6

Hull, Chapter 6

( )days between dates/days in period * during the perinterest earned iiod

f TB T + Y r T FV CI 0 T

CF T

CF T Conversion

0C T

0

01( ) =

( ) + ( ) − ( )( )

( ) =

, ,

ffactor on CTD bond

Forward Rate Futures Rate T T= − 1

22

1 2σ

AI = the boxed formula

Hull, CHapter 7

© 2016 Wiley 24

Hull, Chapter 7

Swap fixed rate =1 B N

B B B B N0

− ( )( ) + ( ) + + + ( )

×0

0 0 01 2 31

( ) ...000

© 2016 Wiley 25

Hull, CHapter 11

Hull, Chapter 11

c S≤

C S≤

p K≤

p K≤

c S Ke rT− ≥ − −max 0 0( , )

C S KT= −max( , )0

p Ke rT S≥ − −max( , )0 0

C X/ 1+r S Pt0+ = +( )

C c0 0≥

P p0 0≥

Hull, CHapter 12

© 2016 Wiley 26

Hull, Chapter 12

Bear spread valueT = MAX(0, StrikeH − AssetT) − MAX(0, StrikeL − AssetT)

PayoffBear spread = Bear spread valueT − PutStrikeH + PutStrikeL

Bull spread value MAX 0 Asset Strike MAX 0 Asset StrikeT T L T= − − −( , ) ( , HH )

Payoff Bull spread value Call CallBullspread StrikeStrike= − +T L H

Butterfly spread valueT = MAX(0, AssetT − StrikeL) − 2MAX(0, AssetT − StrikeM) + MAX(0, AssetT − StrikeH)

PayoffButterfly spread = Butterfly spread valueT − CallStrikeL + 2CallStrikeM − CallStrikeH

Box strategy valueT = StrikeH − StrikeL

PayoffBox strategy = StrikeH − StrikeL − CallStrikeH − PutStrikeH + PutStrikeL

Breakeven Asset Strike ( )Straddle 0 0T = ± +Call Put

Payoff Straddle value Call PutStraddle 0 0= − −T

© 2016 Wiley 27

MCDonalD, CHapter 6

McDonald, Chapter 6

F S eo Tr T

,( )= −

0δ

F S e r U T0 0= +( )

SaunDerS, CHapter 13

© 2016 Wiley 28

Saunders, Chapter 13

( ) ( )

(

Forward rate Spot rate

Spot rate 1

− =−

+IR IR

IRDomestic Foreign

FForeign

Domestic

Foreign

and

IR

IR

)

( )

( )

Forward

Spot

1

1=

++

Real exchange rate = S P PDC/FC DC/FC FC DC× ( )/

© 2016 Wiley 29

tuCkMan, CHapter 20

Tuckman, Chapter 20

SMMepayment in month

Beginning mortgage balance for month St

t

t=

−Pr

ccheduled principal payment in month t

Valuation and Risk Models (VRM)

© 2016 Wiley 31

allen, CHapter 3

Allen, Chapter 3

PortfolioVaR = ΔVaR underlying

DoWD, CHapter 2

© 2016 Wiley 32

Dowd, Chapter 2

ES =− ∑1

1 α( ) * ( )greatest loss pr loss

© 2016 Wiley 33

Hull, CHapter 13

Hull, Chapter 13

n = −−

+ −

+ −c c

S S

cc c

r= + −

+

+ −π π( )

( )

1

1

π = + −−

( )

( )

1 r d

u d

Hull, CHapter 15

© 2016 Wiley 34

Hull, Chapter 15

RP P

PNi

i i 1

i 1

= , i = 1 to− −

−

R R Nic

i i to= + =ln( ),1 1

σ2

2

1

1=

−

−=∑( )R R

N

ic

ic

i

N

σ σ= 2

c = S0N(d1) − Ke−rT N(d2)

and

p = Ke−rT N(−d2) − S0N(−d1)

where

dS K r T

T

dS K r T

Td T

10

2

20

2

1

2

2

=+ +

=+ −

= −

ln( / ) ( / )

ln( / ) ( / )

σσ

σσ

σ

© 2016 Wiley 35

Hull, CHapter 19

Hull, Chapter 19

Delta =Change in option price

Change in underlying price

tuCkMan, CHapter 1

© 2016 Wiley 36

Tuckman, Chapter 1

PV FVDays

YearDR= × − ×

1

DRYear

Days

FV PV

FV=

× −

PVFull = PVFull + AI

AI = t / T × PMT

© 2016 Wiley 37

tuCkMan, CHapter 2

Tuckman, Chapter 2

d tr t t

( )(

( ))

=+

1

12

2

tuCkMan, CHapter 3

© 2016 Wiley 38

Tuckman, Chapter 3

RP c P

Pt t

t

=+ −+1

P Ti y

T

( ) = −+

C1

1

12

2

© 2016 Wiley 39

tuCkMan, CHapter 4

Tuckman, Chapter 4

Effective duration = −− +P P

P y2 0 ( )∆

ΔPrice = –Δy × Duration × Price

FaceFace DV

DVBA A

B

=− 01

01

PV PV

PV yy y− +−∆ ∆

∆2 0( )

tuCkMan, CHapter 5

© 2016 Wiley 40

Tuckman, Chapter 5

DVP

yk

k01

1

10 000= − ∂

∂,

DP

P

yk

k= − ∂

∂1

© 2016 Wiley 41

SCHroeCk, CHapter 5

Schroeck, Chapter 5

UL EA PD LRLR PD= ⋅ ⋅ + ⋅σ σ2 2 2

UL UL ULP i j ij i jj

n

i

n

===

∑∑ ω ω ρ11

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