2016 · frm® exam review 2016 frm ® part i covers all topics in part i formula sheets
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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
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
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
/= =
−
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
© 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
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
© 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
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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