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Financial Economic Theory and Engineering Formula Sheet
Morning and afternoon exam booklets will include a formula package identical to the one attached to this study note. The exam committee felt that by providing many key formulas, candidates would 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 syl- labus 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. For example, there are several versions of the Black-Scholes-Merton option pricing formula to differentiate between instruments paying dividends, tied to an index, etc. 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.
General Formulas
Lognormal distribution:
f (x) =
xσ
2 π
exp
(ln(x) − μ)^2 σ^2
E[X] = eμ+σ^2 /^2 V[X] = e^2 μ+σ^2
eσ^2 − 1
Pr[X ≤ x] = Φ
ln(x) − μ σ
x > 0
Normal distribution:
f (x) =
σ
2 π
exp
(x − μ)^2 σ^2
Capital Asset Pricing Model:
E[Ri] = Rf + βi (E[Rm] − Rf ) where βi = Cov(Ri, Rm) Var[Rm]
Weighted Average Cost of Capital:
WACC = (1 − τc)kb
B
B + S
S
B + S
where kb=return on debt, ks =return in equity.
GARCH(1,1) Model
Yt = μ + σtt σ^2 t = α 0 + α 1 (Yt− 1 − μ)^2 + βσ t^2 − 1
RSLN-2 Model
Yt|ρt ∼ N
μρt , σ ρ^2 t
where {ρt} , t = 1, 2 , ..., is a Markov process with 2-states.
pij = Pr[ρt+1 = j|ρt = i]
Conditional Tail Expectation For loss L, continuous at Vα = F (^) L− 1 (α):
CTEα(L) = E [L|L > Vα(L)]
If there is a probability mass at Vα, define β′^ = max{β : Vα = Vβ }, then
CTEα(L) = (1 − β′)E [X|X > Vα] + (β′^ − α)Vα 1 − α
pt = Ke−r(T^ −t)N (−d 2 ) − StN (−d 1 )
where d 1 = ln(St/K) + (r + σ^2 /2)(T − t) σ
T − t
and d 2 = d 1 − σ
T − t
Copeland, Weston, Shastri: Financial Theory and Corporate Policy
Chapter 2
(2.2) Revt + mtSt = Divt + (W &S)t + It
(2.5) N It = Revt − (W &S)t − dept
(2.7) S 0 =
∑^ ∞
t=
N It − ∆At (1 + ks)t (2.13) F CF = EBIT (1 − τc) + ∆dep − ∆I
Chapter 6
(6.34) Rjt = E(Rjt) + βj δmt + εjt
(6.36) Rjt − Rf t = (Rmt − Rf t)βj + εjt (6.36) R′ pt = γ 0 + γ 1 βp + εpt (6.37) Rit = αi + βiRmt + εit (6.38) E(Ri) = E(RZ ) + [E(Rm) − E(RZ )] βi (6.40) E(Ri) − Rf = bi [E(Rm) − Rf ] + siE(SMB) + hiE(HML) (6.41) λi = E(Rm) − E(RZ ) (6.46) Rit = ̂γ 0 t + ̂γ 1 tβit + εit (6.49) E(Ri) = E(Rz,I ) + [E(RI ) − E(Rz,I )] βi,I (6.50) R˜i = E( R˜i) + bi 1 F˜ 1 +... + bik F˜k + ˜εi (6.57) E( R˜i) = λ 0 + λ 1 bi 1 +... + λkbik (6.59) E(Ri) − Rf =
[¯
δ 1 − Rf
]
bi 1 +... +
[¯
δk − Rf
]
bik (6.60) bik = Cov(Ri, δk)/V ar(δk)
Chapter 9
(9.1) C(SA, SB , T ) = SAN (d 1 ) − SB N (d 2 )
where d 1 =
[
ln (SA/SB ) + V 2 T
]
/V
T ; d 2 = d 1 − V
T
and V 2 = V (^) A^2 − 2 ρAB VAVB + V (^) B^2
Chapter 10
(10.1) V (η) ≡
m
q(m) max a
e
p(e|m)U (a, e) − V (η 0 )
(12.18) V [D∗(t)] =
r
[
t 2 − τpD∗(t) − β [D∗(t)]^2 2 t
]
(12.20) V [D∗(t)] = (τp + βA)D∗(t)
(12.21) A = −
[
τp β
] [
1 + r 1 + 2r
]
[
τp β
] [
1 + r 1 + 2r
] √
β(1 + 2r) τ (^) p^2 (1 + r)^2
(12.22) I + D = C + N pe = C + Pe
(12.24) max D
L − τpD +
[
P + τpD − L P + τpD + I − C
]
X
(12.25) τp =
τp +
∂P
∂D
L + I − C
P + τpD + I − C
X
(12.26) D(X) =
τp max(I − C + L, 0) ln X
(12.30) V 0 old =
P ′
P ′^ + E
(E + S + a + b)
(12.36) W 0 (z) = αP (z) + βP − T (m, P )
(12.37) W s(n, z) = α P̂ (n, z) + βP − T (n, p)
(12.38) Wn = α P̂n − t 2 pγ−^1
(12.40) P̂ (n, z) = k[n + c(z)]^1 /γ^ where k = (t 2 γ/α)^1 /γ
(12.41) M (n, z) = P̂ (n, z) − T (n, P ) = k(1 − t 1 )[n + c(z)]^1 /γ^ − t 2 nk^1 −γ^ n + c(z)/γ
(12.45) V 0 = (1 − t)X 0
(12.46) E(X|n) = X̂ (n) = X 0 s 0 + Ŷ m(n)sm s 0 + sm
(12.47) V 1 (n) = X̂ (n) − T (n) − C
(12.50) V 2 (T, Y ) = X 0 s 0 + Ŷ msm + Y F T s s 0 + sm + F T s
− E
[
Xt(X/n)|T, Ŷ m, Y
]
− C
(12.51) E[V 2 (T )|Ym] =
X 0 s 0 + Ŷ msm +
X 0 s 0 +Ymsm s 0 +sm
F T s s 0 + sm + F T s
− T − C
dα dD
[V (n) − B(n, D)] − α(D)
∂B
∂D
d^2 α dD^2
[V (n) − B(n, D)] − 2
dα dD
∂B
∂D
− α(D)
∂^2 B
∂D^2
(12.56) α (D∗(n)) [V (n) − B (n, D∗(n))] − [V (n) − I] = 0
(12.57)
[
d^2 α dD^2
(V − B) − 2
dα dD
∂B
∂D
− α
∂^2 B
∂D^2
]
dD dn
dα dD
∂B
∂n
dV dn
∂^2 B
∂n∂D
(12.58) ̂ε(α, D) =
[
(a − D 0 )^2 + D^2 1 − α α^2
]
(12.59) D = X 1
[
α( − γ) (1 − α)(1 − )γ
]
(12.60) max c(s,p),a
U [s − c(s, p)] f (s, p|a)ds dp
V [c(s, p)] f (s, p|a)ds dp − G(a) ≥ V
U ′[s − c(s, p)] V ′[c(s, p)] = λ
(12.65) max a U (k) + λ
[∫ ∫
V [c(s, p)]f (s, p|a)ds dp − G(a) − V
]
(12.68) max c(s),a
U [s − c(s)]f (s|a) ds + λ
[∫
V [c(s)] f (s|a) ds − G(a) − V
]
+μ
[∫
V [c(s)] fa(s|a)ds − G′(a)
]
U ′[s − c(s)] V ′[c(s)] = λ + μ fa(s|a) f (s|a)
(12.73) max c(s,p),a
U [s − c(s, p)] f (s, p|a)ds dp+λ
[∫ ∫
V [c(s, p)]f (s, p|a) ds dp − G(a) − V
]
+μ
[∫ ∫
V [c(s, p)]fa(s, p|a) ds dp − G′(a)
]
U ′^ [s − c(s, p)] V ′[c(s, p)] = λ + μ fa(s, p|a) f (s, p|a)
(12.75) U ′^ [s − c(p)] V ′[c(p)] = λ + μ 1 fa 1 (p|a) f (p|a)
- μ 2 fa 1 (p|a) f (p|a) +... + μn fa 1 (p|a) f (p|a)
(12.81) max c(s,p,m),a(m),m(m) Es,p,m [U [s − c(s, p, m)] |a(m)]
Subject to (for all m) Es,p|m [[V [c(s, p, m)] − G[a(m)]] |a(m)] ≥ V
(12.82)
∂V (X∗)
∂X
∂P (X∗)
∂X
∂C(X∗)
∂X
(12.83) S =
(1 − β)E(s) − α − λ(1 − β)Cov(s, RM ) 1 + rf
(12.86) max α,β
(1 − β)E(s) − α − λ(1 − β)Cov(s, RM ) 1 + rf +μ
a [E(W ) + α + βE(s)] − b
[
Var(W ) + 2βCov(W, s) + β^2 Var(s)
]
− V
(12.90) β = λaCov(s, RM ) 2 bVar(s)
Cov(W, s) Var(s)
Note: This is a correction to the formula in the reading.
(15.55) dV V = μ(V, t)dt + σdW
(15.57)
σ^2 V 2 FV V (V ) + rV FV (V ) − rF (V ) + C = 0
(15.58) F (V ) = A 0 + A 1 V + A 2 V −(2r/σ (^2) )
(15.59) V = VB ⇒ B(V ) = (1 − α)VB V → ∞ ⇒ B(V ) → C/r (15.61) B(V ) = (1 − pB )C/r + pB [(1 − α)VB ] where pB = (V /VB )−^2 r/σ 2
(15.62) V = VB ⇒ DC(V ) = αVB V → ∞ ⇒ DC(V ) → 0 (15.63) DC(V ) = αVB (V /VB )−^2 r/σ
2
(15.67) VL(V ) = VU (V ) + TcB(V ) − DC(V ) = VU (V ) + TcB − pB TcB − αVB pB
(15.68) M = (1 + r)γ 0 V 0 + γ 1
V 1 if V 1 ≥ D V 1 − C if V 1 < D
(15.69) Ma =
γ 0 (1 + r) (^) 1+V^1 ar + γ 1 V 1 a if D∗^ < D ≤ V 1 a γ 0 (1 + r) (^) 1+V^1 br + γ 1 V 1 a if D < D∗
(15.70) Mb =
γ 0 (1 + r) (^) 1+V^1 ar + γ 1 (V 1 b − C) if D∗^ ≤ D ≤ V 1 a γ 0 (1 + r) (^) 1+V^1 br + γ 1 V 1 b if D < D∗
Chapter 16
(16.2) Vi(t) = Divi(t + 1) + ni(t)Pi(t + 1) 1 + ku(t + 1)
(16.8) V˜i(t) = EBIT˜ (^) i(t + 1) − I˜i(t + 1) + V˜i(t + 1) 1 + ku(t + 1) (16.9) Y˜di =
[(
EBIT˜ − rDc
(1 − τc) − rDpi
]
(1 − τpi)
(16.10) Y˜gi = ( EBIT˜ − rDc)(1 − τc)(1 − τgi) − rDpi(1 − τpi)
(16.26) S 1 − E(S 1 ) = ε 1
[
γ 1 + k
]
= [EBIT 1 − E 0 (EBIT 1 )]
[
γ 1 + k
]
(16.27) ∆Divit = ai + ci(Div it∗ − Divi,t− 1 ) + Uit (16.33) ∆Divt = β 1 Divt− 1 + β 2 N It + β 3 N It− 1 + Zt (16.35) Pit = a + bDivit + cREit + εit (16.38) R˜j = γ 0 +
[
R˜m − γ 0
]
βj + γ 1 [DYj − DYm] /DYm + εj
(16.43)
∆W
N 0 P 0
= (1 − FP )
PE − P 0
P 0
+ FP
PT − P 0
P 0
Hull: Options Futures and Other Derivatives
Chapter 12: Binomial Trees
(12.1) ∆ = fu − fd S 0 u − S 0 d
(12. 5 /6) f = e−r∆t^ [pfu + (1 − p)fd] where p = er∆t^ − d u − d (12. 13 /14) u = eσ
√∆t d = e−σ
√∆t
Chapter 13: Wiener Processes and Itˆos Lemma
(13.17) G = ln S dG =
μ − σ^2 / 2
dt + σ dz
Chapter 14: The Black-Scholes-Merton model
(14.9) df =
∂f ∂S μS + ∂f ∂t
∂^2 f ∂S^2 σ^2 S^2
dt + ∂f ∂S σSdz
∂f ∂t
σ^2 S^2 ∂^2 f ∂S^2 = rf
Chapter 18: Greek Letters
(18. ) ∆(call) = N (d 1 ) ∆(put) = N (d 1 ) − 1
(18.2) Θ(call) = − S 0 N ′(d 1 )σ 2
T
− rKe−rT^ N (d 2 )
(18.2) Θ(put) = − S 0 N ′(d 1 )σ 2
T
(18. ) Γ(call) = Γ(put) =
N ′(d 1 ) S 0 σ
T
(18.4) Θ + rS∆ +
σ^2 S^2 Γ = rΠ
(18. ) V(call) = V(put) = S 0
T N ′(d 1 ) (18. ) rho(call) = KT e−rT^ N (d 2 ) (18. ) rho(put) = −KT e−rT^ N (−d 2 )
Chapter 20: Basic Numerical Procedures
(20.8) ∆ = f 1 , 1 − f 1 , 0 S 0 u − S 0 d
Chapter 25: Exotic Options Call on a call
S 0 e−qT^2 M
a 1 , b 1 ;
T 1 /T 2
− K 2 e−rT^2 M
a 2 , b 2 ;
T 1 /T 2
− e−rT^1 K 1 N (a 2 )
where a 1 = ln(S 0 /S∗) + (r − q + σ^2 /2)T 1 σ
T 1
a 2 = a 1 − σ
T 1
b 1 = ln(S 0 /K 2 ) + (r − q + σ^2 /2)T 2 σ
T 2
b 2 = b 1 − σ
T 2
Put on a call
K 2 e−rT^2 M
−a 2 , b 2 ; −
T 1 /T 2
− S 0 e−qT^2 M
−a 1 , b 1 ; −
T 1 /T 2
Call on a put
K 2 e−rT^2 M
−a 2 , −b 2 ;
T 1 /T 2
− S 0 e−qT^2 M
−a 1 , −b 1 ;
T 1 /T 2
− e−rT^1 K 1 N (−a 2 )
Put on a put
S 0 e−qT^2 M
a 1 , −b 1 ; −
T 1 /T 2
− K 2 e−rT^2 M
a 2 , −b 2 ; −
T 1 /T 2
Chooser
max(c, p) = c + e−q(T^2 −T^1 )^ max
0 , Ke−(r−q)(T^2 −T^1 )^ − S 1
Barrier Options
λ =
r − q + σ^2 / 2 σ^2 y =
ln [H^2 /(S 0 K)] σ
T
T
x 1 = ln(S 0 /H) σ
T
T y 1 = ln(H/S 0 ) σ
T
T
If H ≤ K
cdi = S 0 e−qT^ (H/S 0 )^2 λN (y) − Ke−rT^ (H/S 0 )^2 λ−^2 N
y − σ
T
If H ≥ K
cdo = S 0 N (x 1 )e−qT^ − Ke−rT^ N
x 1 − σ
T
−S 0 e−qT^ (H/S 0 )^2 λN (y 1 ) + Ke−rT^ (H/S 0 )^2 λ−^2 N
y 1 − σ
T
If H > K
cui = S 0 N (x 1 )e−qT^ − Ke−rT^ N
x 1 − σ
T
−S 0 e−qT^ (H/S 0 )^2 λ^ [N (−y) − N (−y 1 )] +Ke−rT^ (H/S 0 )^2 λ−^2
[
N
−y + σ
T
− N
−y 1 + σ
T
)]
If H ≥ K
pui = −S 0 e−qT^ (H/S 0 )^2 λN (−y) + Ke−rT^ (H/S 0 )^2 λ−^2 N
−y + σ
T
If H ≤ K
puo = −S 0 N (−x 1 )e−qT^ + Ke−rT^ N
−x 1 + σ
T
+S 0 e−qT^ (H/S 0 )^2 λN (−y 1 ) − Ke−rT^ (H/S 0 )^2 λ−^2 N
−y 1 + σ
T
If H ≤ K
pdi = −S 0 N (−x 1 )e−qT^ + Ke−rT^ N
−x 1 + σ
T
- S 0 e−qT^ (H/S 0 )^2 λ^ [N (y) − N (y 1 )] −Ke−rT^ (H/S 0 )^2 λ−^2
[
N
y − σ
T
− N
y 1 − σ
T
)]
Lookback Options
cf l = S 0 e−qT^ N (a 1 )−S 0 e−qT^ σ^2 2(r − q)
N (−a 1 )−Smine−rT
N (a 2 ) − σ^2 2(r − q)
eY^1 N (−a 3 )
where a 1 = ln(S^0 /Smin)+(r−q+σ (^2) /2)T σ√T a^2 =^ a^1 −^ σ
T
a 3 = ln(S^0 /Smin)+(−r+q+σ (^2) /2)T σ√T Y 1 = 2(r−q−σ
(^2) /2) ln(S 0 /Smin) σ^2
pf l = Smaxe−rT
N (b 1 ) − σ^2 2(r − q) eY^2 N (−b 3 )
+S 0 e−qT^ σ^2 2(r − q) N (−b 2 )−S 0 e−qT^ N (b 2 )
where b 1 = ln(Smax/S^0 )+(−r+q+σ (^2) /2)T σ√T b 2 = b 1 − σ
T
b 3 = ln(Smax/S^0 )+(r−q−σ (^2) /2)T σ√T Y 2 = 2(r−q−σ
(^2) /2) ln(Smax/S 0 ) σ^2 Exchange Options
(25.5) V 0 e−qV^ T^ N (d 1 ) − U 0 e−qU^ T^ N (d 2 )
where d 1 = ln(V 0 /U 0 ) + (qU − qV + ˆσ^2 /2)T σ ˆ
T
d 2 = d 1 − σˆ
T
and σˆ =
σ U^2 + σ^2 V − 2 ρσU σV
Realized volatility: ¯σ =
n − 2
∑^ n−^1
i=
[
ln
Si+ Si
)] 2
(25.6) Eˆ( V¯ ) =
T
ln
F 0
S∗^
T
[
F 0
S∗^
]
T
[∫ S∗
K=
K^2
erT^ p(K)dK +
K=S∗
K^2
erT^ c(K)dK
]
Chapter 28: Interest Rate Derivatives: The Standard Market Models
(28.1) c = P (0, T ) [FB N (d 1 ) − KN (d 2 )]
(28.2) p = P (0, T ) [KN (−d 2 ) − FB N (−d 1 )]
where d 1 = ln (FB /K) + σ^2 B T / 2 σB
T
d 2 = d 1 − σB
T
(28.3) FB =
B 0 − I
P (0, T )
(28.4) σB = Dy 0 σy (28. 7 , Caplet) LδkP (0, tk+1) [FkN (d 1 ) − RK N (d 2 )]
where d 1 = ln(Fk/RK ) + σ^2 ktk/ 2 σk
tk
d 2 = d 1 − σk
tk
(28. 8 , Floorlet) LδkP (0, tk+1) [RK N (−d 2 ) − FkN (−d 1 )]
(28.10) LA [s 0 N (d 1 ) − skN (d 2 )] where A =
m
∑^ mn
i=
P (0, Ti)
Chapter 29: Convexity, Timing, and Quanto Adjustments
(29.1) ET (yT ) = y 0 −
y 02 σ y^2 T G′′(y 0 ) G′(y 0 )
(29.2) ET (RT ) = R 0 +
R^20 σ R^2 τ T 1 + R 0 τ (29.3) αV = ρV W σV σW
(29.4) ET ∗ (VT ) = ET (VT ) exp
[
ρV RσV σRR 0 (T ∗^ − T ) 1 + R 0 /m
T
]
Chapter 30: Interest Rate Derivatives: Models of the Short Rate
(30.20) c = LP (0, s)N (h) − KP (0, T )N (h − σp) p = KP (0, T )N (−h + σp) − LP (0, s)N (−h)
h =
σp ln LP (0, s) P (0, T )K
σp 2
σp = σ a
[
1 − e−a(s−T^ )
]
1 − e−^2 aT 2 a
(30.24) Pm+1 =
∑^ nm
j=−nm
Qm,j exp [−g (αm + j∆x) ∆t]
Chapter 32: Swaps Revisited - This Chapter is no longer on the syllabus
(32.1) Fi + F (^) i^2 σ i^2 τiti 1 + Fiτi
(31.2) yi −
y^2 i σ y,i^2 ti G′′ i (yi) G′ i(yi)
yiτiFiρiσy,iσF,iti 1 + Fiτi (32.3) Vi + ViρiσW,iσV,iti
Hardy: Investment Guarantees
Chapter 2 AR(1) (2.7):
(2.7) Yt = μ + a(Yt− 1 − μ) + σεt εt independent and identically distributed (iid), εt ∼ N (0, 1)
ARCH(1) (2.8-9):
(2.8) Yt = μ + σtεt
(2.9) σ^2 t = a 0 + a 1 (Yt− 1 − μ)^2
ARCH(1) with AR(1) stock price (2.10-11):
(2.10) Yt = μ + a(Yt− 1 − μ) + σtεt εt iid ∼ N (0, 1)
(2.11) σ^2 t = a 0 + α(Yt− 1 − μ)^2
GARCH(1,1) (2.12-13):
(2.12) Yt = μ + σtεt
(2.13) σ^2 t = a 0 + α 1 (Yt− 1 − μ)^2 + βσ^2 t− 1
GARCH(1,1) with AR(1) stock price (2.14-15):
(2.14) Yt = μ + a(Yt− 1 − μ) + σtεt εt iid ∼ N (0, 1)
(2.15) σ^2 t = α 0 + α 1 (Yt− 1 − μ)^2 + βσ t^2 − 1
Chapter 3
(3.1) L(θ) = f (X 1 , X 2 ,... , Xn; θ)
(3.2) L(θ) =
∏^ n
t=
f (xt; θ) l(θ) =
∑^ n
t=
logf (xt; θ)
GARCH(1,1) (3.31-33):
(3.31) Yt = μ + σtεt (3.32) σ^2 t = α 0 + α 1 (Yt− 1 − μ)^2 + βσ t^2 − 1 (3.33) Yt|Yt− 1 ∼ N (μ, σ^2 t ) t = 2, 3 ,... , n (3.35) μ˜ = ¯y ˜σ = sy
Chapter 8
(8. ) FT = F 0
ST
S 0
(1 − m)T
(8.3) P 0 = Ge−rT^ Φ(−d 2 ) − S 0 (1 − m)T^ Φ(−d 1 )
where d 1 = log
S 0 (1 − m)T^ /G
T )
d 2 = d 1 − σ
T
(8.8) H(0) =
∫ (^) n
0
Ge−rtΦ(−d 2 )
tpτx μ (d) x,t dt^ +
∫ (^) n
0
−S 0 (1 − m)tΦ(−d 1 )
tpτx μ (d) x,t dt
(8.9) H(0, t 3 ) =
PS (t 1 ) + S 0 (1 − m)t^1
×
1 + P (t 2 − t 1 ) (1 + P (t 3 − t 2 )) + (1 − m)t^2 −t^1 P (t 3 − t 2 )
−S 0 (1 − m)t^1
where PS (t 1 ) = BSP(S 0 (1 − m)t^1 , G, t 1 )
(8.15) α =
B
S 0 ¨aτx:nei′ (8.19) HEt = H(t) + (^) t− 1 |qxd
(G − Ft)+
− H(t−) (8.22) TCt = τ St |Ψt − Ψt− 1 |
FET-108-07 and FET-165-08: Doherty; Integrated Risk Management Chapter 13
(p. 465) V (E) = V (F ) − V (D) = V (F ) − DDF + P (V (F ), D) = C(V (F ), D)
(p. 481) V ∗(F ) = S + D
m n
(p. 494) V (^) R′(E) = −C + VR(F ) − D + P {Vr(F ), D} = −C + VR − D + PR V (^) N′ (E) = VN (F ) − D + P {VN (F ), D} = VN − D + PN
Chapter 16
(16.1) T = (E + P )(1 + ri) − L (16.2) E(T ) = (E + P − R(I; S)) (1 + E(ri)) − E(L(a)) + hC(I; S) − a
(16.3)
∂E(T )
∂a
−∂E(L(a)) ∂a − 1 + h
∂C
∂I
∂I
∂L
∂L
∂a
Manistre and Hancock; Variance of the CTE Estimator
(p. 130) C T Eˆ (α) =
k
∑^ k
j=
x(j)
(p. 131) V AR(C T Eˆ ) = E[V AR(C T Eˆ |X(k))] + V AR[E(C T Eˆ |X(k))]
(p. 131) V AR(C T Eˆ ) ≈ V AR(x(1),... , x(k)) + α · (C T Eˆ − x(k))^2 k
(p. 132) C T Eˆ (α) =
k
∑^ k
i=
x(i) V ˆaR(α) = x(k)
(p. 133) IFV aR(x) =
−(1−α) f (V aR) x < V aR 0 x = V aR α f (V aR) x > V aR
(p. 133) IFCT E (x) =
V aR − CT E x < V aR V aR − CT E + x− 1 −V aRα x > V aR
(p. 133) V AR(C T Eˆ n) ≈
E[IFCT E )^2 ]
n
V AR(X|X > V aR) + α · (CT E − V aR)^2 n · (1 − α)
(p. 133) V AR(V ˆaRn) ≈ E[IFV aR)^2 ] n
α · (1 − α) n · [f (V aR)]^2
(p. 133) Cov(C T Eˆ n, V ˆaRn) ≈ E[IFCT E · IFV aR] n
α · (CT E − V aR) n · f (V aR)
(p. 134) F SE(CT E) =
V AR(X(1),... , X(k)) + α · (C T Eˆ − X(k))^2 n · (1 − α)
