Quantitative Finance and Investments Advanced Formula Sheet
Fall 2015/Spring 2016
Morning and afternoon exam booklets will include a formula package identical to the one
attached to this study note. The exam committee believes that by providing many key
formulas, candidates will be able to focus more of their exam preparation time on the application of the formulas and concepts to demonstrate their understanding of the syllabus
material and less time on the memorization of the formulas.
The formula sheet was developed sequentially by reviewing the syllabus material for each
major syllabus topic. Candidates should be able to follow the flow of the formula package
easily. We recommend that candidates use the formula package concurrently with the syllabus material. Not every formula in the syllabus is in the formula package. Candidates are
responsible for all formulas on the syllabus, including those not on the formula
sheet.
Candidates should carefully observe the sometimes subtle differences in formulas and their
application to slightly different situations. Candidates will be expected to recognize the
correct formula to apply in a specific situation of an exam question.
Candidates will note that the formula package does not generally provide names or definitions
of the formula or symbols used in the formula. With the wide variety of references and
authors of the syllabus, candidates should recognize that the letter conventions and use of
symbols may vary from one part of the syllabus to another and thus from one formula to
another.
We trust that you will find the inclusion of the formula package to be a valuable study aide
that will allow for more of your preparation time to be spent on mastering the learning
objectives and learning outcomes.
1
Interest Rate Models - Theory and Practice, Brigo and Mercurio
Chapter 3
Table 3.1 Summary of instantaneous short rate models
Model
V
CIR
D
EV
HW
BK
MM
Dynamics
drt = k[θ − rt ]dt + σdWt
√
drt = k[θ − rt ]dt + σ rt dWt
drt = art dt + σrt dWt
drt = rt [η − a ln rt ]dt + σrt dWt
drt = k[θt − rt ]dt + σdWt
drth = rt [η
t ]dt +iσrt dWt
³t − a ln r´
drt = rt ηt − λ −
γ
1+γt
ln rt dt + σrt dWt
√
CIR++
rt = xt + ϕt , dxt = k[θ − xt ]dt + σ xt dWt
EEV
rt = xt + ϕt , dxt = xt [η − a ln xt ]dt + σxt dWt
r>0
N
Y
Y
Y
N
Y
r∼
N
NCχ2
LN
LN
N
LN
Y
LN
Y*
Y*
2
SNCχ
SLN
AB AO
Y
Y
Y
Y
Y
N
N
N
Y
Y
N
N
N
N
Y
N
Y
N
*rates are positive under suitable conditions for the deterministic function ϕ.
(3.5)
(3.6)
(3.7)
dr(t) = k[θ − r(t)]dt + σdW (t),
r(0) = r0
¡
¢
Rt
r(t) = r(s)e−k(t−s) + θ 1 − e−k(t−s) + σ s e−k(t−u) dW (u)
¡
¢
E {r(t)|Fs } = r(s)e−k(t−s) + θ 1 − e−k(t−s)
¤
σ2 £
1 − e−2k(t−s)
2k
P (t, T ) = A(t, T )e−B(t,T )r(t)
Var{r(t)|Fs } =
(3.8)
(3.9)
(3.11)
(3.12)
(3.13)
(3.14)
(3.15)
(3.16)
(3.19)
(3.20)
(3.21)
(3.22)
dr(t) = [kθ − B(t, T )σ 2 − kr(t)]dt + σdW T (t)
dr(t) = [kθ − (k + λσ)r(t)]dt + σdW 0 (t),
r(0) = r0
dr(t) = [b − ar(t)]dt + σdW 0 (t)
¢
Rt
b¡
r(t) = r(s)e−a(t−s) +
1 − e−a(t−s) + σ s e−a(t−u) dW 0 (u)
a
Pn
Pn
P
n i=1 ri ri−1 − i=1 ri ni=1 ri−1
α̂ =
P
P
2
2
n ni=1 ri−1
− ( ni=1 ri−1 )
Pn
[ri − α̂ri−1 ]
β̂ = i=1
n(1 − α̂)
i2
1 Pn h
−
α̂r
−
β̂(1
−
α̂)
r
Vc2 =
i
i−1
n i=1
³
´
2
E {r(t)|Fs } = r(s)ea(t−s) and Var{r(t)|Fs } = r2 (s)e2a(t−s) eσ (t−s) − 1
R∞
√
√
2 p
r̄p R ∞
r̄
sin(2
r̄
sinh
y)
f(z)
sin(yz)dzdy
+
K
(2
r̄)
2p
0
0
π2
Γ(2p)
p
r(0) = r0
dr(t) = k(θ − r(t))dt + σ r(t)dW (t),
p
dr(t) = [kθ − (k + λσ)r(t)]dt + σ r(t)dW 0 (t),
r(0) = r0
P (t, T ) =
2
(3.23)
¡
¢
E {r(t)|Fs } = r(s)e−k(t−s) + θ 1 − e−k(t−s)
Var{r(t)|Fs } = r(s)
(3.24)
(3.25)
¢
¢2
σ 2 ¡ −k(t−s)
σ2 ¡
− e−2k(t−s) + θ
e
1 − e−k(t−s)
k
2k
P (t, T ) = A(t, T )e−B(t,T )r(t)
∙
¸2kθ/σ2
2h exp {(k + h)(T − t)/2}
A(t, T ) =
2h + (k + h)(exp {(T − t)h} − 1)
√
2(exp{(T − t)h} − 1)
,
h = k2 + 2σ 2
2h + (k + h)(exp {(T − t)h} − 1)
p
dr(t) = [kθ − (k + B(t, T )σ 2 )r(t)]dt + σ r(t)dW T (t)
B(t, T ) =
(3.27)
(3.28)
pTr(t)|r(s) (x) = pχ2 (υ,δ(t,s))/q(t,s) (x) = q(t, s)pχ2 (υ,δ(t,s)) (q(t, s)x)
q(t, s) = 2[ρ(t − s) + ψ + B(t, T )] and δ(t, s) =
Page 68
(3.29)
P (t, T ) = A(t, T )e−B(t,T )r(t)
R(t, T ) = α(t, T ) + β(t, T )r(t),
∂B(t, T )
σ(t, r(t))
∂T
dr(t) = b(t, r(t))dt + σ(t, r(t))dW (t)
σf (t, T ) =
Page 69
Page 69/70
Page 70
4ρ(t − s)2 r(s)eh(t−s)
q(t, s)
b(t, x) = λ(t)x + η(t),
σ 2 (t, x) = γ(t)x + δ(t)
1
∂
B(t, T ) + λ(t)B(t, T ) − γ(t)B(t, T )2 + 1 = 0,
B(T, T ) = 0
∂t
2
∂
1
A(T, T ) = 1
[ln A(t, T )] − η(t)B(t, T ) + δ(t)B(t, T )2 = 0,
∂t
2
Vasicek λ(t) = −k, η(t) = kθ, γ(t) = 0, δ(t) = σ 2
CIR λ(t) = −k,
η(t) = kθ,
γ(t) = σ 2 ,
δ(t) = 0
σ 2 (x) = γx + δ
µ
¶
θ σ2
lim E{r(t)|Fs } = exp
+
t→∞
a 4a
¶∙
µ 2¶
¸
µ
σ
2θ σ 2
+
exp
−1
lim Var{r(t)|Fs } = exp
t→∞
a
2a
2a
b(x) = λx + η,
Page 71
(3.31)
(3.32)
dr(t) = [ϑ(t) − a(t)r(t)]dt + σ(t)dW (t)
(3.33)
dr(t) = [ϑ(t) − ar(t)]dt + σdW (t)
(3.34)
ϑ(t) =
(3.35)
(3.36)
∂f M (0, t)
σ2
+ af M (0, t) + (1 − e−2at )
∂T
2a
R t −a(t−u)
Rt
−a(t−s)
+ se
ϑ(u)du + σ s e−a(t−u) dW (u)
r(t) = r(s)e
Rt
= r(s)e−a(t−s) + α(t) − α(s)e−a(t−s) + σ s e−a(t−u) dW (u)
where α(t) = f M (0, t) +
σ2
(1 − e−at )2
2a2
3
(3.37)
(3.38)
Page 74
(3.47)
(3.48)
(3.49)
E{r(t)|Fs } = r(s)e−a(t−s) + α(t) − α(s)e−a(t−s)
¤
σ2 £
1 − e−2a(t−s)
Var{r(t)|Fs } =
2a
dx(t) = −ax(t)dt + σdW (t),
x(0) = 0
R t −a(t−u)
−a(t−s)
+σ s e
dW (u)
x(t) = x(s)e
E{x(ti+1 )|x(ti ) = xi,j } = xi,j e−a∆ti =: Mi,j
¤
σ2 £
1 − e−2a∆ti =: Vi2
Var{x(ti+1 )|x(ti ) = xi,j } =
2a
r
√
3
∆xi = Vi−1 3 = σ
[1 − e−2a∆ti−1
2a
¶
µ
Mi,j
k =round
∆xi+1
(3.64)
2
2
2
ηj,k
ηj,k
ηj,k
1
ηj,k
2
1
ηj,k
√
+
,
p
=
,
p
=
− √
+
−
+
m
d
2
2
2
6 6Vi
3 3Vi
6 6Vi
2 3Vi
2 3Vi
α
α
α
x
dxt = μ(xt ; α)dt + σ(xt ; α)dWt
(3.65)
P x (t, T ) = Πx (t, T, xαt ; α)
(3.66)
t≥0
rt = xt + ϕ(t; α),
i
h R
T
P (t, T ) = exp − t ϕ(s; α)ds Πx (t, T, rt − ϕ(t; α); α)
(3.50)
(3.67)
(3.68)
(3.69)
(3.70)
(3.71)
(3.74)
Page 100
Page 101
(3.76)
(3.77)
pu =
ϕ(t; α) = ϕ∗ (t; α) := f M (o, t) − f x (0, t; α)
h R
i
T
exp − t ϕ(s; α)ds = Φ∗ (t, T, x0 ; α) :=
P M (0, T ) Πx (0, t, x0 ; α)
Πx (0, T, x0 ; α) P M (0, t)
Π(t, T, rt ; α) = Φ∗ (t, T, x0 ; α)Π∗ (t, T, rt − ϕ∗ (t; α); α)
V x (t, T, τ, K) = Ψx (t, T, τ, K, xαt ; α)
¸
∙
dϕ(t; α)
drt = kθ + kϕ(t; α) +
− krt dt + σdWt
dt
k2 θ − σ 2 /2
σ 2 −kt
−
e (1 − e−kt ) − x0 e−kt
k2
2k2
P M (0, T )A(0, t) exp{−B(0, t)x0 }
P (t, T ) = M
P (0, t)A(0, T ) exp{−B(0, T )x0 }
ϕV AS (t; α) = f M (0, t) + (e−kt − 1)
×A(t, T ) exp{−B(t, T )[rt − ϕV AS (t; α)]}
p
x(0) = x0 ,
r(t) = x(t) + ϕ(t)
dx(t) = k(θ − x(t))dt + σ x(t)dW (t),
ϕCIR (t; α) = f M (0, t) − f CIR (0, t; α)
f CIR (0, t; α) =
h=
√
k2 + 2σ 2
4h2 exp{th}
2kθ(exp{th} − 1)
+ x0
2h + (k + h)(exp{th} − 1)
[2h + (k + h)(exp{th} − 1)]2
4
Chapter 4
(4.4)
rt = x(t) + y(t) + ϕ(t),
r(0) = r0
(4.5)
dx(t) = −ax(t)dt + σdW1 (t),
dy(t) = −by(t)dt + ηdW2 (t),
−a(t−s)
x(0) = 0
y(0) = 0
−b(t−s)
+ y(s)e
+ ϕ(t)
E{r(t)|Fs } = x(s)e
2 £
2
¤ η £
¤
¤
σ
ση £
Var{r(t)|Fs } =
1 − e−2a(t−s) +
1 − e−2b(t−s) + 2ρ
1 − e−(a+b)(t−s)
2a
2b
a+b
R t −a(t−u)
R t −b(t−u)
(4.7)
r(t) = σ 0 e
dW1 (u) + η 0 e
dW2 (u) + ϕ(t)
p
f1 (t) + η 1 − ρ2 dW
f2 (t)
f1 (t)
dy(t) = −by(t)dt + ηρdW
(4.8)
dx(t) = −ax(t)dt + σdW
p
f1 (t) and dW2 (t) = ρdW
f1 (t) + 1 − ρ2 dW
f2 (t)
where dW1 (t) = ddW
(4.6)
(4.9)
(4.10)
(4.11)
(4.12)
(4.13)
1 − e−b(T −t)
1 − e−a(T −t)
x(t) +
y(t)
a
b
∙
¸
2 −a(T −t)
1 −2a(T −t)
3
σ2
− e
−
V (t, T ) = 2 T − t + e
a
a
2a
2a
∙
¸
2
1
3
η2
+ 2 T − t + e−b(T −t) − e−2b(T −t) −
b
b
2b
2b
∙
¸
e−a(T −t) − 1 e−b(T −t) − 1 e−(a+b)(T −t) − 1
ση
T −t+
+
−
+2ρ
ab
a
b
a+b
½
¾
−a(T
−t)
RT
1−e
1 − e−b(T −t)
1
P (t, T ) = exp − t ϕ(u)du −
x(t) −
y(t) + V (t, T )
a
b
2
2 ¡
2
¢2
¢2
σ
η ¡
ση
ϕ(t) = f M (0, T ) + 2 1 − e−aT + 2 1 − e−bT + ρ (1 − e−aT )(1 − e−bT )
2a
2b
ab
½
¾
n R
o P M (0, T )
1
T
exp − t ϕ(u)du = M
exp − [V (0, T ) − V (0, t)]
P (0, t)
2
M(t, T ) =
P M (0, T )
exp {A(t, T )}
P M (0, t)
(4.14)
P (t, T ) =
(4.15)
1 − e−a(T −t)
1 − e−b(T −t)
1
x(t) −
y(t)
A(t, T ) := [V (t, T ) − V (0, T ) + V (0, t)] −
2
a
b
P (t, T ) = A(t, T ) exp{−B(a, t, T )x(t) − B(b, t, T )y(t)}
p
σf (t, T ) = σ 2 e−2a(T −t) + η2 e−2b(T −t) + 2ρσηe−(a+b)(T −t)
(4.16)
5
Page 152
Cov(df(t, T1 ), df (t, T2 ))
dt
∂B
∂B
∂B
∂B
(a, t, T1 )
(a, t, T2 ) + η2
(b, t, T1 )
(b, t, T2 )
= σ2
∂T
∂T
∂T
∂T
¸
∙
∂B
∂B
∂B
∂B
(a, t, T1 )
(b, t, T2 ) +
(a, t, T2 )
(b, t, T1 )
+ρση
∂T
∂T
∂T
∂T
= σ 2 e−a(T1 +T2 −2t) + η2 e−b(T1 +T2 −2t)
£
¤
+ρση e−aT1 −bT2 +(a+b)t + e−aT2 −bT1 +(a+b)t
σ 2 e−a(T1 +T2 −2t) + η2 e−b(T1 +T2 −2t)
σf (t, T1 )σf (t, T2 )
¤
£ −aT −bT +(a+b)t
+ e−aT2 −bT1 +(a+b)t
ρση e 1 2
+
σf (t, T1 )σf (t, T2 )
Corr(df (t, T1 ), df(t, T2 )) =
Page 153
ln P (t, T1 ) − ln P (t, T2 )
T2 − T1
B(a, t, T2 ) − B(a, t, T1 )
df (t, T1 , T2 ) = ...dt +
σdW1 (t)
T2 − T1
B(b, t, T2 ) − B(b, t, T1 )
+
ηdW2 (t)
T2 − T1
p
σf (t, T1 , T2 ) = σ 2 β(a, t, T1 , T2 )2 + η2 β(b, t, T1 , T2 )2 + 2ρσηβ(a, t, T1 , T2 )β(b, t, T1 , T2 )
f(t, T1 T2 ) =
B(z, t, T2 ) − B(z, t, T1 )
T2 − T1
Cov(df(t, T1 , T2 ), df (t, T3 , T4 ))
dt
B(a, t, T2 ) − B(a, t, T1 ) B(a, t, T4 ) − B(a, t, T3 )
σ2
T2 − T1
T4 − T3
B(b, t, T2 ) − B(b, t, T1 ) B(b, t, T4 ) − B(b, t, T3 )
+η2
T2 − T1
T4 − T3
∙
B(a, t, T2 ) − B(a, t, T1 ) B(b, t, T4 ) − B(b, t, T3 )
+ρση
T2 − T1
T4 − T3
¸
B(a, t, T4 ) − B(a, t, T3 ) B(b, t, T2 ) − B(b, t, T1 )
+
T4 − T3
T2 − T1
s
σ22
σ1 σ2
+ 2ρ̄
Page 160
σ3 = σ12 +
2
b̄ − ā
(ā − b̄)
σ2
σ1 dZ1 (t) −
dZ2 (t)
σ2
ā
−
b̄
dZ3 (t) =
,
σ4 =
σ3
ā − b̄
σ1 ρ̄ − σ4
Page 161
a = ā, b = b̄, σ = σ3 , η = σ4 , ρ =
σ3
where
β(z, t, T1 , T2 ) =
6
ϕ(t) = r0 e−āt +
ā = a,
b̄ = b,
Rt
θ(v)e−ā(t−v) dv
p
σ1 = σ 2 + η2 + 2ρση,
0
σρ + η
ρ̄ = p
,
σ 2 + η2 + 2ρση
θ(t) =
σ2 = η(a − b)
dϕ(t)
+ aϕ(t)
dt
Managing Credit Risk: The Great Challenge for Global Financial
Markets, Caouette, et. al.
Chapter 20
(20.2)
Rp =
N
P
Xi EAR
i=1
(20.3)
Vp =
N
N P
P
Xi Xj σi σj ρij
i=1 j=1
(20.5)
UALp =
N
N P
P
Xi Xj σi σj ρij 1
i=1 j=1
Page 403
µ µ √ −1
¶
¶
ρΦ (CL) + Φ−1 (P D)
√
CV aR(CL) = EAD • LGD • Φ
− PD
1−ρ
1 + (M − 2.5) • b(P D)
×
1 − 1.5b(P D)
Liquidity Risk Measurement and Management: A Practioner’s
Guide to Global Best Practices, Matz and Neu
Chapter 2
Page 33
Page 33
log V (t) = α + βt + σεt
√
log Vq (t) = α + βt − σΦ−1 (q) t
Bond-CDS Basis Handbook: Measuring, Trading and Analysing
Basis Trades, Elizalde, Doctor, and Saltuk
Page 13, Equation 1
Page 15, Equation 2
Page 18, Equation 3
Page 25, Equation 4
Page 25, Equation 5
Page 43, Equation 7
S = P D × (1 − R)
U − AI
+ FC
FR =
RA
P V [c + p] − BP
SS =
RF A
BT P 1 = CN ×(100−R−U −CP −F C)+BN ×(R+CR−BP −F C)
BT P 2 = BN × (100 + CR − BP − F C) − CN × (U + CP + F C)
BP − R
× BN
CN =
100 − R − U
7
A Survey of Behavioral Finance, Barberis and Thaler
(1)
(2)
(x, p : y, q) = π(p)v(x) + π(q)v(y)
P
xα
if x ≥ 0
πi v(xi ) where v =
and πi = w(Pi ) − w(Pi∗ ),
α
−λ(−x)
if
x
<
0
i
Pγ
w(P ) =
(P γ + (1 − P )γ )1/γ
(3)
(4)
(5)
(6)
(7)
Dt+1
= egD +σD εt+1
Dt
Ct+1
= egC +σC ηt+1
Ct
¶
µµ ¶ µ
¶¶
µ
0
1 w
εt
˜N
,
, i.i.d.over time
ηt
0
w 1
Ct1−γ
ρ
E0
1−γ
t=0
"µ
#
¶−γ
Ct+1
Rt+1
1 = ρEt
Ct
∞
P
t
Dt+1 + Pt+1
1 + Pt+1 /Dt+1 Dt+1
=
Pt
Pt /Dt
Dt
(8)
Rt+1 =
(9)
rt+1 = ∆dt+1 +const.≡ dt+1 − dt +const.
(10)
(11)
(13)
(14)
Eπ v[(1 − w)Rf,t+1 + wRt+1 − 1]
∙
¸
1−γ
∞
P
−γ
t Ct
E0
ρ
+ b0 C t v̂(Xt+1 )
1−γ
t=0
Pt+1 + Dt+1
Rt+1 =
Pt
∞
∞
P
P
pt − dt = Et
ρt ∆dt+1+j − Et
ρt rt+1+j + Et lim ρj (pt+j − dt+j )+const.
j=0
(15)
(16)
(17)
(18)
(19)
(20)
j→∞
j=0
∙
¸
1−γ
−γ
t Ct
E0
ρ
+ b0 C t ṽ(Xt+1 , zt )
1−γ
t=0
∞
P
ri − rf = βi.1 (F 1 − rf ) + . . . + βi,K (F K − rf )
ri,t − rf,t = αi + βi,1 (F1,t − rf,t ) + . . . + βi,K (FK,t − rf,t ) + εi,t
1
2 2
Rf = eγgC +0.5γ σC
ρ
1 + f gD −γGC +0.5(σ2 +γ 2 σ2 −2γσC σD w)
D
C
1=ρ
e
f
Dt+1 + Pt+1
1 + Pt+1 /Dt+1 Dt+1
1 + f gD +σD εt+1
Rt+1 =
=
=
e
Pt
Pt /Dt
Dt
f
8
CAIA Level II: Advanced Core Topics in Alternative Investments,
Black, Chambers, Kazemi
Chapter 16
(16.1)
(16.2)
(16.3)
(16.4)
(16.5)
(16.6)
(16.7)
(16.8)
(16.9)
(16.10)
true
true
Ptreported = α + β0 Pttrue + β1 Pt−1
+ β2 Pt−2
+ ···
true
true
Ptreported = αPttrue + α(1 − α)Pt−1
+ α(1 − α)2 Pt−2
+···
reported
Pttrue = (1/α) × Ptreported − [(1 − α)/α] × Pt−1
reported
reported
Pttrue = Pt−1
+ [(1/α) × (Ptreported − Pt−1
)]
Rt,reported ≈ β0 Rt,true + β1 Rt−1,true + β2 Rt−2,true + · · ·
reported
Ptreported = (1 − ρ)Pttrue + ρPt−1
reported
∆Ptreported = (1 − ρ)∆Pttrue + ρ∆Pt−1
Rt,reported ≈ (1 − ρ)Rt,true + ρRt−1,reported
Rt,true = (Rt,reported − ρRt−1,reported )/(1 − ρ)
ρ̂ = corr(Rt,reported − Rt−1,reported )
(16.11)
ρi,j = σij /(σi σj )
(16.12)
reported
reported
reported
Rtreported = α + β1 Rt−1
+ β2 Rt−2
+ · · · + βk Rt−k
+ εt
Chapter 21
Page 262
Y = S × I × E × H where Y = yield, S = total solar radiation over the area
per period, I = fraction of solar radiation captured by the crop canopy, E = photosynthetic
efficiency of the crop (total plant dry matter per unit of solar radiation), H = harvest index
(fraction of total dry matter that is harvestable)
Managing Investment Portfolio: A Dynamic Process, Maginn, Tuttle, Pinto, McLeavey
Chapter 8
Page 523
T RCI = CR + RR + SR
Page 553
RRn,t = (Rt + Rt−1 + Rt−2 + . . . + Rt−n )/n
r Pn
∗
2
i=1 [min(ri − r , 0)]
DD =
n−1
ARR − rf
Sharpe Ratio =
SD
ARR − rf
Sortino Ratio =
DD
Page 554
Page 555
Page 556
9
The Secular and Cyclic Determinants of Capitalization Rates: The
Role of Property Fundamentals, Macroeconic Factors, and "Structural Changes," Chervachidze, Costello, Wheaton
(1)
Log(Cj,t ) = a0 + a1 log(Cj,t−1 ) + a2 log(Cj,t−4 ) + a3 log(RRIj,t ) + a4 RT Bt + a7 Q2t
+a8 Q3t + a9 Q4t + a10 Dj
(1.1)
(2)
RRIj,t−s = RRj,t /MRRj
Log(Cj,t ) = a0 + a1 log(Cj,t−1 ) + a2 log(Cj,t−4 ) + a3 log(RRIj,t−s ) + a4 RT Bt
+a5 SP READt + a6 DEBT F LOWt + a7 Q2t + a8 Q3t + a9 Q4t + a10 Dj
(2.1)
(3)
DEBT F LOWt = T NBLt /GDPt
Log(Cj,t ) = a0 + a1 log(Cj,t−1 ) + a2 log(Cj,t−4 ) + a3 log(RRIj,t−s ) + a4 RT Bt
+a5 SP READt + a6 DEBT F LOWt + a7 Q2t + a8 Q3t + a9 Q4t
(4)
Log(Cj,t ) = a0 + a1 yearq + a2 log(Cj,t−1 ) + a3 log(Cj,t−4 ) + a4 log(RRIj,t−s ) + a5 RT Bt
+a6 SP READt + a7 DEBT F LOWt + a7 Q2t + a8 Q3t + a9 Q4t + a10 Dj
Analysis of Financial Time Series, Tsay
Chapter 9
(9.1)
rit = αi + βi1 f1t + · · · + βim fmt +
(9.2)
rt = α + βf t + t , t = 1, . . . , T
(9.3)
Ri = αi 1T + Fβ 0i + Ei
(9.4)
R = Gξ 0 + E
(9.5)
rit = αi + βi rmt +
(9.11)
(9.12)
(9.13)
Var(yi ) =
Cov(yi , yj ) =
k
P
k
P
λi =
k
P
Var(yi )
i=1
T
T
1 P
1 P
(rt − r̄)(rt − r̄)0 , r̄ =
rt
T − 1 t=1
T t=1
(9.15)
ρ̂r = Ŝ−1 Σ̂r Ŝ−1
(9.16)
rt − μ = βf t +
(9.20)
i, j = 1, . . . , k
i=1
Σ̂r ≡ [σ̂ij,r ] =
(9.19)
i = 1, . . . , k
Var(ri ) = tr(Σr ) =
(9.14)
(9.18)
t = 1, . . . , T, i = 1, . . . , k
i = 1, . . . , k t = 1, . . . , T
wi0 Σr wj ,
i=1
(9.17)
it ,
wi0 Σr wi ,
it ,
t
Σr = Cov(rt ) = E[(rt − μ)(rt − μ)0 ] = E[(βf t + t )(βf t + t )0 ] = ββ 0 + D
Cov(rt , ft ) = E[(rt − μ)ft0 ] = βE(ft ft0 ) + E( t ft0 ) = β
hp
i
p
p
β̂ ≡ [β̂ij ] =
λ̂1 ê1 | λ̂2 ê2 | · · · | λ̂m êm
∙
¸³
´
0
1
2
LR(m) = − T − 1 − (2k + 5) − m ln |Σ̂r | − ln |β̂ β̂ + D̂|
6
3
10
Handbook of Fixed Income Securities, Fabozzi
Chapter 69
P P
(ws − wsB ) · RsB
(69 − 4)
Asset Allocation
(69 − 5)
Security Selection
(69 − 12)
s
αkP fkP
Chapter 70
(70 − 1)
(70 − 2)
(70 − 3)
(70 − 4)
(70 − 5)
(70 − 6)
(70 − 7)
(70 − 8)
(70 − 9)
(70 − 10)
(70 − 11)
−
αkB fkB
=
P
s
wsP · (RsP − RsB )
s
P
P
αk,s
fk,s
−
P
P
s
B B
αk,s
fk,s
µ
¶
P wsP
wsB
Asset Allocation w ·
− B · (T RsB − T RB )
P
w
w
s
P P
P
Sector Management
ws · (T Rs − T RsB )
P
s
Top-Level Exposure (wP − wB ) · T RB
µ
¶
P wsP
wsB
P
Asset Allocation w ·
− B · (ERsB − ERB )
wP
w
s
P P
ws · (ERsP − ERsB )
Sector Management
s
Top-Level Exposure (wP − wB ) · ERB
¡ P
¢
B
Outperformance from average carry yavg
− yavg
· ∆t
¢
¡
¢¢
P¡ P ¡
P
B
ωj yj − yavg
− ωjB yj − yavg
· ∆t
Key rate contributions
j
¡
¢
Outperformance from avg. parallel shifts − OADP − OADB · ∆yavg
¢
P¡
Outperformance from reshaping −
KRDjP − KRDjB · (∆yj − ∆yavg )
j
Asset Allocation
µ P
¶
P
B
B
¡
¢
P
OASD
OASD
w
w
s
s
s
s
B
B
·
∆OAS
−
−
∆OAS
−OASDP ·
s
OASDP
OASDB
s
¡
¢
P
(70 − 12)
Security Selection − wsP OASDsP · ∆OASsP − ∆OASsB
s
(70 − 13)
(70 − 14)
(70 − 15)
Spread Duration Mismatch −(OASDP − OASDB ) · ∆OAS B
¢
P¡ P
Asset Allocation −
ws OASDsP − wsB OASDsB · ∆OASsB
s
¢
¡
P
Security Selection − wsP OASDsP · ∆OASsP − ∆OASsB
s
11
Chapter 71
Page 1737
RP − RB =
P
t
βt (RtP − RtB )
(RP − RB )/T
(1 + RP )1/T − (1 + RB )1/T
P
RP − RB − A Tt=1 (RtP − RtB )
C=
PT
P
B 2
t=1 (Rt − Rt )
A=
βt = A + C(RtP − RtB )
Introduction to Credit Risk Modeling, 2nd ed., Bluhm, Overbeck,
Wagner
Chapter 6
Page 237
Mn = M1n
Guarantees and Target Volatility Funds, Morrison and Tadrowski
Page 4
wtequity
µ
¶
σtarget
= min
, 100%
σ̂tequity
¶¶2
µ µ
¡ equity ¢2
¡ equity ¢2
St
1
σ̂t
ln
= λ σ̂t−∆t + (1 − λ)
∆t
St−∆t
12