Quantitative Finance and Investments Advanced Formula Sheet. Fall 2014/Spring 2015

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1 Quaniaive Finance and Invemen Advanced Formula Shee Fall 2014/Spring 2015 Morning and afernoon exam bookle will include a formula package idenical o he one aached o hi udy noe. The exam commiee believe ha by providing many key formula, candidae will be able o focu more of heir exam preparaion ime on he applicaion of he formula and concep o demonrae heir underanding of he yllabu maerial and le ime on he memorizaion of he formula. The formula hee wa developed equenially by reviewing he yllabu maerial for each major yllabu opic. Candidae hould be able o follow he flow of he formula package eaily. We recommend ha candidae ue he formula package concurrenly wih he yllabu maerial. No every formula in he yllabu i in he formula package. Candidae are reponible for all formula on he yllabu, including hoe no on he formula hee. Candidae hould carefully oberve he omeime uble difference in formula and heir applicaion o lighly differen iuaion. Candidae will be expeced o recognize he correc formula o apply in a pecific iuaion of an exam queion. Candidae will noe ha he formula package doe no generally provide name or definiion of he formula or ymbol ued in he formula. Wih he wide variey of reference and auhor of he yllabu, candidae hould recognize ha he leer convenion and ue of ymbol may vary from one par of he yllabu o anoher and hu from one formula o anoher. Weruhayouwillfindheincluionofheformulapackageobeavaluableudyaide ha will allow for more of your preparaion ime o be pen on maering he learning objecive and learning oucome. Change from Spring 2014 are: Addiion of formula from Chaper 69 and 71 of Fabozzi Addiion of a formula from Chaper 6 of Bluhm Deleion of Chaper 3 and 8 of Tay 1

2 Inere Rae Model - Theory and Pracice, Brigo and Mercurio Chaper 3 Table 3.1 Summary of inananeou hor rae model Model Dynamic r > 0 r AB AO V dr = k[θ r ]d + σdw N N Y Y CIR dr = k[θ r ]d + σ r dw Y NCχ 2 Y Y D dr = ar d + σr dw Y LN Y N EV dr = r [η a ln r ]d + σr dw Y LN N N HW dr = k[θ r ]d + σdw N N Y Y BK dr = r [η ³ a ln r ]d + i σr dw Y LN N N MM dr = r hη λ ln r d + σr dw Y LN N N γ 1+γ CIR++ r = x + ϕ, dx = k[θ x ]d + σ x dw Y* SNCχ 2 Y Y EEV r = x + ϕ, dx = x [η a ln x ]d + σx dw Y* SLN N N *rae are poiive under uiable condiion for he deerminiic funcion ϕ. (3.5) dr() =k[θ r()]d + σdw(), r(0) = r 0 (3.6) r() =r()e k( ) + θ k( ) + σ R e k( u) dw (u) (3.7) E {r() F } = r()e k( ) + θ k( ) Var{r() F } = σ2 2k( ) 2k (3.8) B(,T )r() P (, T )=A(, T )e (3.9) dr() =[kθ B(, T )σ 2 kr()]d + σdw T () (3.11) dr() =[kθ (k + λσ)r()]d + σdw 0 (), r(0) = r 0 (3.12) dr() =[b ar()]d + σdw 0 () (3.13) r() =r()e a( ) + b a( ) + σ R a dw 0 (u) (3.14) ˆα = n P n r ir i 1 P n r P n i r i 1 n P n r2 i 1 (P n r i 1) 2 P n (3.15) ˆβ = [r i ˆαr i 1 ] n(1 ˆα) (3.16) V c 2 = 1 P h n r i ˆαr i 1 n ˆβ(1 i 2 ˆα) (3.19) E {r() F } = r()e a( ) and Var{r() F } = r 2 ()e ³e 2a( ) σ2 ( ) 1 (3.20) P (, T )= rp R in(2 r inh y) R f(z)in(yz)dzdy + 2 π Γ(2p) rp K 2p (2 r) (3.21) dr() =k(θ r())d + σ p r()dw (), r(0) = r 0 (3.22) dr() =[kθ (k + λσ)r()]d + σ p r()dw 0 (), r(0) = r 0 2

3 (3.23) E {r() F } = r()e k( ) + θ k( ) Var{r() F } = r() σ2 e k( ) e 2k( ) + θ σ2 k( ) 2 k 2k (3.24) B(,T )r() P (, T )=A(, T )e (3.25) 2kθ/σ 2 2h exp {(k + h)(t )/2} A(, T )= 2h +(k + h)(exp {(T )h} 1) B(, T )= 2(exp{(T )h} 1) 2h +(k + h)(exp {(T )h} 1), h = k 2 +2σ 2 (3.27) dr() =[kθ (k + B(, T )σ 2 )r()]d + σ p r()dw T () (3.28) p T r() r() (x) =p χ 2 (υ,δ(,))/q(,)(x) =q(, )p χ 2 (υ,δ(,))(q(, )x) q(, ) =2[ρ( )+ψ + B(, T )] and δ(, ) = 4ρ( )2 r()e h( ) q(, ) Page 68 R(, T )=α(, T )+β(, T )r(), B(,T )r() P(, T )=A(, T )e (3.29) σ f (, T )= B(, T ) σ(, r()) T Page 69 dr() = b(, r())d + σ(, r())dw () b(, x) =λ()x + η(), σ 2 (, x) =γ()x + δ() B(, T )+λ()b(, T ) 1 2 γ()b(, T )2 +1=0, B(T,T)=0 [ln A(, T )] η()b(, T )+1 2 δ()b(, T )2 =0, A(T,T)=1 Page 69/70 Vaicek λ() = k, η() =kθ, γ() =0, δ() =σ 2 Page 70 CIR λ() = k, η() =kθ, γ() =σ 2, δ() =0 b(x) =λx + η, σ 2 (x) =γx + δ µ θ Page 71 lim E{r() F } =exp a + σ2 4a µ µ 2θ (3.31) lim Var{r() F } =exp a + σ2 σ 2 exp 1 2a 2a (3.32) dr() =[ϑ() a()r()]d + σ()dw () (3.33) dr() =[ϑ() ar()]d + σdw() (3.34) ϑ() = fm (0,) + af M (0,)+ σ2 T 2a ( 2a ) (3.35) r() =r()e a( ) + R e a( u) ϑ(u)du + σ R e a( u) dw (u) = r()e a( ) + α() α()e a( ) + σ R e a( u) dw (u) (3.36) where α() =f M (0,)+ σ2 2a 2 ( a ) 2 3

4 (3.37) E{r() F } = r()e a( ) + α() α()e a( ) Var{r() F } = σ2 2a( ) 2a (3.38) dx() = ax()d + σdw(), x(0) = 0 Page 74 x() =x()e a( ) + σ R e a( u) dw (u) (3.47) E{x( i+1 ) x( i )=x i,j } = x i,j e a i =: M i,j Var{x( i+1 ) x( i )=x i,j } = σ2 2a i =: V 2 2a i r 3 (3.48) x i = V i 1 3=σ 2a [ 2a i 1 µ Mi,j (3.49) k =round x i+1 (3.50) p u = η2 j,k + η j,k 6Vi 2 2,p m = 2 3V i 3 η2 j,k,p 3Vi 2 d = η2 j,k 6Vi 2 2 3V i (3.64) dx α = μ(x α ; α)d + σ(x α ; α)dw x (3.65) P x (, T )=Π x (, T, x α ; α) (3.66) r = x + ϕ(; α), 0 h (3.67) P (, T )=exp R i T ϕ(; α)d Π x (, T, r ϕ(; α); α) (3.68) ϕ(; α) =ϕ (; α) :=f M (o, ) f x (0,; α) (3.69) h exp R i T ϕ(; α)d = Φ (, T, x 0 ; α) := P M (0,T) Π x (0,,x 0 ; α) Π x (0,T,x 0 ; α) P M (0,) (3.70) Π(, T, r ; α) =Φ (, T, x 0 ; α)π (, T, r ϕ (; α); α) (3.71) V x (, T, τ, K) =Ψ x (, T, τ, K, x α ; α) dϕ(; α) (3.74) dr = kθ + kϕ(; α)+ kr d + σdw d Page 100 ϕ VAS (; α) =f M (0,)+(e k 1) k2 θ σ 2 /2 k 2 η j,k σ2 2k 2 e k ( k ) x 0 e k Page 101 P (, T )= P M (0,T)A(0,)exp{ B(0,)x 0 } P M (0,)A(0,T)exp{ B(0,T)x 0 } A(, T )exp{ B(, T )[r ϕ VAS (; α)]} (3.76) dx() =k(θ x())d + σ p x()dw (), x(0) = x 0, r() =x()+ϕ() (3.77) ϕ CIR (; α) =f M (0,) f CIR (0,; α) f CIR 2kθ(exp{h} 1) (0,; α) = 2h +(k + h)(exp{h} 1) + x 4h 2 exp{h} 0 [2h +(k + h)(exp{h} 1)] 2 h = k 2 +2σ 2 4

5 Chaper 4 (4.4) r = x()+y()+ϕ(), r(0) = r 0 (4.5) dx() = ax()d + σdw 1 (), x(0) = 0 dy() = by()d + ηdw 2 (), y(0) = 0 (4.6) E{r() F } = x()e a( ) + y()e b( ) + ϕ() Var{r() F } = σ2 2a( ) + η2 2b( ) +2ρ ση (a+b)( ) 2a 2b a + b (4.7) r() =σ R 0 e a( u) dw 1 (u)+η R 0 e b( u) dw 2 (u)+ϕ() (4.8) dx() = ax()d + σdfw 1 () dy() = by()d + ηρdfw 1 ()+η p 1 ρ 2 dfw 2 () where dw 1 () =ddfw 1 () and dw 2 () =ρdfw 1 ()+ p 1 ρ 2 dfw 2 () ) ) a(t b(t (4.9) M(, T )= x()+ y() a b (4.10) V (, T )= σ2 T + 2 a 2 a e a(t ) 1 2a e 2a(T ) 3 2a + η2 T + 2 b 2 b e b(t ) 1 2b e 2b(T ) 3 2b +2ρ ση ab (4.11) P (, T )=exp T + e a(t ) 1 a ½ R T + e b(t ) 1 b ϕ(u)du a(t ) x() a e (a+b)(t ) 1 a + b ) b(t y()+ 1 ¾ b 2 V (, T ) (4.12) ϕ() =f M (0,T)+ σ2 at 2 + η2 bt 2 + ρ ση 2a 2 2b 2 ab ( at )( bt ) n (4.13) exp R o T ϕ(u)du = P M (0,T) ½ P M (0,) exp 12 ¾ [V (0,T) V (0,)] (4.14) P (, T )= P M (0,T) exp {A(, T )} P M (0,) A(, T ):= 1 ) ) a(t b(t [V (, T ) V (0,T)+V(0,)] x() y() 2 a b (4.15) P (, T )=A(, T )exp{ B(a,, T )x() B(b,, T )y()} (4.16) σ f (, T )= p σ 2 e 2a(T ) + η 2 e 2b(T ) +2ρσηe (a+b)(t ) 5

6 Page 152 Cov(df (, T 1 ),df(, T 2 )) d = σ 2 B T (a,, T 1) B T (a,, T 2)+η 2 B T (b,, T 1) B T (b,, T 2) B +ρση T (a,, T 1) B T (b,, T 2)+ B T (a,, T 2) B T (b,, T 1) = σ 2 e a(t 1+T 2 2) + η 2 e b(t 1+T 2 2) +ρση e at 1 bt 2 +(a+b) + e at 2 bt 1 +(a+b) Corr(df (, T 1 ),df(, T 2 )) = σ2 e a(t 1+T 2 2) + η 2 e b(t 1+T 2 2) σ f (, T 1 )σ f (, T 2 ) + ρση e at 1 bt 2 +(a+b) + e at 2 bt 1 +(a+b) σ f (, T 1 )σ f (, T 2 ) Page 153 f(, T 1 T 2 )= ln P (, T 1) ln P (, T 2 ) T 2 T 1 df (, T 1,T 2 )=...d + B(a,, T 2) B(a,, T 1 ) σdw 1 () T 2 T 1 + B(b,, T 2) B(b,, T 1 ) ηdw 2 () T 2 T 1 σ f (, T 1,T 2 )= p σ 2 β(a,, T 1,T 2 ) 2 + η 2 β(b,, T 1,T 2 ) 2 +2ρσηβ(a,, T 1,T 2 )β(b,, T 1,T 2 ) where β(z,, T 1,T 2 )= B(z,, T 2) B(z,,T 1 ) T 2 T 1 Cov(df (, T 1,T 2 ),df(, T 3,T 4 )) d σ 2 B(a,, T 2) B(a,, T 1 ) B(a,, T 4 ) B(a,, T 3 ) T 2 T 1 T 4 T 3 +η 2 B(b,, T 2) B(b,, T 1 ) B(b,, T 4 ) B(b,, T 3 ) T 2 T 1 T 4 T 3 B(a,, T2 ) B(a,, T 1 ) B(b,, T 4 ) B(b,, T 3 ) +ρση T 2 T 1 T 4 T 3 + B(a,, T 4) B(a,, T 3 ) B(b,, T 2 ) B(b,, T 1 ) T 4 T 3 T 2 T 1 Page 160 σ 3 = dz 3 () = σ σ2 2 (ā b) 2 +2 ρ σ 1σ 2 b ā σ 1 dz 1 () σ 2 ā b dz 2(), σ 4 = σ 2 σ 3 ā b Page 161 a =ā, b = b, σ = σ 3, η = σ 4, ρ = σ 1 ρ σ 4 σ 3 6

7 ϕ() =r 0 e ā + R 0 θ(v)e ā( v) dv ā = a, b = b, σ1 = p σ 2 + η 2 +2ρση, σ 2 = η(a b) ρ = σρ + η p σ2 + η 2 +2ρση, θ() =dϕ() + aϕ() d Managing Credi Rik: The Grea Challenge for Global Financial Marke, Caouee, e. al. Chaper 20 P (20.2) R p = N X i EAR (20.3) V p = N P j=1 (20.5) UAL p = N P Page 403 NP X i X j σ i σ j ρ ij j=1 NP X i X j σ i σ j ρ ij 1 CV ar(cl)=ead LGD µ µ ρφ 1 (CL)+Φ 1 (PD) Φ PD 1 ρ 1+(M 2.5) b(pd) 1 1.5b(PD) Liquidiy Rik Meauremen and Managemen: Guide o Global Be Pracice, Maz and Neu A Pracioner Chaper 2 Page 33 Page 33 log V () =α + β + σε log V q () =α + β σφ 1 (q) Bond-CDS Bai Handbook: Meauring, Trading and Analying Bai Trade, Elizalde, Docor, and Saluk Page 13, Equaion 1 S = PD (1 R) Page 15, Equaion 2 FR = U AI RA + FC Page 18, Equaion 3 PV[c + p] BP SS = RF A Page 25, Equaion 4 BTP1 =CN (100 R U CP F C)+BN (R+CR BP FC) Page 25, Equaion 5 BTP2 =BN (100 + CR BP FC) CN (U + CP + FC) Page 43, Equaion 7 CN = BP R 100 R U BN 7

8 A Survey of Behavioral Finance, Barberi and Thaler (1) (x, p : y,q) =π(p)v(x) +π(q)v(y) (2) P i π i v(x i ) where v = xα if x 0 λ( x) α if x<0 and π i = w(p i ) w(pi ), w(p )= P γ (P γ +(1 P ) γ ) 1/γ (3) D +1 D = e g D+σ D ε +1 (4) (5) C +1 = e g C+σ C η +1 C µ µµ µ ε 0 1 w N, η 0 w 1 P (6) E 0 ρ C1 γ =0 1 γ " µc+1 γ (7) 1 = ρe R +1# C, i.i.d.over ime (8) R +1 = D +1 + P +1 = 1+P +1/D +1 D +1 P P /D D (9) r +1 = d +1 +con. d +1 d +con. (10) E π v[(1 w)r f,+1 + wr +1 1] P (11) E 0 ρ C1 γ 1 γ + b 0C γ ˆv(X +1 ) =0 (13) R +1 = P +1 + D +1 P P P (14) p d = E ρ d +1+j E (15) E 0 P =0 j=0 j=0 ρ C1 γ 1 γ + b 0C γ ṽ(x +1,z ) ρ r +1+j + E lim ρ j (p +j d +j )+con. j (16) r i r f = β i.1 (F 1 r f )+...+ β i,k (F K r f ) (17) r i, r f, = α i + β i,1 (F 1, r f, )+...+ β i,k (F K, r f, )+ε i, (18) R f = 1 ρ eγg C+0.5γ 2 σ 2 C (19) 1 = ρ 1+f e g D γg C +0.5(σD 2 +γ2 σc 2 2γσ Cσ D w) f (20) R +1 = D +1 + P +1 = 1+P +1/D +1 D +1 = 1+f P P /D D f e g D+σ D ε +1 8

9 CAIA Level II: Advanced Core Topic in Alernaive Invemen, Black, Chamber, Kazemi Chaper 16 (16.1) P repored (16.2) P repored (16.3) P rue (16.4) P rue = α + β 0 P rue = αp rue =(1/α) P repored = P repored 1 + β 1 P rue 1 + β 2 P rue α(1 α)p rue 1 + α(1 α) 2 P rue 2 + [(1 α)/α] P repored 1 +[(1/α) (P repored P repored 1 )] (16.5) R,repored β 0 R,rue + β 1 R 1,rue + β 2 R 2,rue + (16.6) P repored (16.7) P repored =(1 ρ)p rue =(1 ρ) P rue + ρp repored 1 + ρ P repored 1 (16.8) R,repored (1 ρ)r,rue + ρr 1,repored (16.9) R,rue =(R,repored ρr 1,repored )/(1 ρ) (16.10) ˆρ = corr(r,repored R 1,repored ) (16.11) ρ i,j = σ ij /(σ i σ j ) (16.12) R repored Chaper 21 Page 262 = α + β 1 R repored 1 + β 2 R repored β k R repored k + ε Y = S I E H where Y = yield, S = oal olar radiaion over he area per period, I = fracion of olar radiaion capured by he crop canopy, E = phooynheic efficiency of he crop (oal plan dry maer per uni of olar radiaion), H = harve index (fracion of oal dry maer ha i harveable) Managing Invemen Porfolio: A Dynamic Proce, Maginn, Tule, Pino, McLeavey Chaper 8 Page 523 TRCI = CR + RR + SR Page 553 RR n, =(R + R 1 + R R n )/n Page 554 rp n DD = i r, 0)] 2 n 1 Page 555 ARR rf SR = SD Page 556 ARR rf SR = DD 9

10 The Secular and Cyclic Deerminan of Capializaion Rae: The Role of Propery Fundamenal, Macroeconic Facor, and "Srucural Change," Chervachidze, Coello, Wheaon (1) Log(C j, )=a 0 + a 1 log(c j, 1 )+a 2 log(c j, 4 )+a 3 log(rri j, )+a 4 RTB + a 7 Q2 (1.1) RRI j, = RR j, /M RR j +a 8 Q3 + a 9 Q4 + a 10 D j (2) Log(C j, )=a 0 + a 1 log(c j, 1 )+a 2 log(c j, 4 )+a 3 log(rri j, )+a 4 RTB (2.1) DEBTFLOW = TNBL /GDP +a 5 SPREAD + a 6 DEBTFLOW + a 7 Q2 + a 8 Q3 + a 9 Q4 + a 10 D j (3) Log(C j, )=a 0 + a 1 log(c j, 1 )+a 2 log(c j, 4 )+a 3 log(rri j, )+a 4 RTB +a 5 SPREAD + a 6 DEBTFLOW + a 7 Q2 + a 8 Q3 + a 9 Q4 (4) Log(C j, )=a 0 + a 1 yearq + a 2 log(c j, 1 )+a 3 log(c j, 4 )+a 4 log(rri j, )+a 5 RT B +a 6 SPREAD + a 7 DEBTFLOW + a 7 Q2 + a 8 Q3 + a 9 Q4 + a 10 D j Analyi of Financial Time Serie, Tay Chaper 9 (9.1) r i = α i + β i1 f β im f m + i, =1,...,T,,...,k (9.2) r = α + βf +, =1,...,T (9.3) R i = α i 1 T + Fβ 0 i + E i (9.4) R = Gξ 0 + E (9.5) r i = α i + β i r m + i, i =1,...,k =1,...,T (9.11) Var(y i )=wiσ 0 r w i, i =1,...,k (9.12) Cov(y i,y j )=wiσ 0 r w j, i, j =1,...,k (9.13) kp P Var(r i )=r(σ r )= k P λ i = k Var(y i ) (9.14) ˆΣ r [ˆσ ij,r ]= 1 T 1 (9.15) ˆρ r = Ŝ 1 ˆΣ r Ŝ 1 TP =1(r r)(r r) 0, r = 1 T TP r =1 (9.16) r μ = βf + (9.17) Σr = Cov(r )=E[(r μ)(r μ) 0 ]=E[(βf + )(βf + ) 0 ]=ββ 0 + D (9.18) Cov(r, f )=E[(r μ)f]=βe(f 0 f)+e( 0 f)=β 0 (9.19) ˆβ [ ˆβ i ij ]= hpˆλ1 ê 1 pˆλ2 ê 2 pˆλm ê m (9.20) LR(m) = T 1 16 (2k +5) 23 m ³ ln ˆΣ r ln ˆβ ˆβ 0 + ˆD 10

11 Handbook of Fixed Income Securiie, Fabozzi Chaper 69 (69 4) Ae Allocaion P (w P w B ) R B (69 5) Securiy Selecion P w P (R P R B ) (69 12) α P k f P k αb k f B k = P α P k, f P k, P α B k, f B k, Chaper 70 (70 1) Ae Allocaion w P P µ w P w wb (TR B P w B TR B ) (70 2) Secor Managemen P w P (TR P TR B ) (70 3) Top-Level Expoure (w P w B ) TR B (70 4) Ae Allocaion w P P µ w P w wb (ER B P w B ER B ) (70 5) Secor Managemen P w P (ER P ER B ) (70 6) Top-Level Expoure (w P w B ) ER B Chaper 71 Page 1737 R P R B = P A = β (R P R B ) (R P R B )/T (1 + R P ) 1/T (1 + R B ) 1/T C = RP R B A P T =1 (RP R B ) P T =1 (RP R B ) 2 β = A + C(R P R B ) Inroducion o Credi Rik Modeling, 2nd ed., Bluhm, Overbeck, Wagner Chaper 6 Page 237 M n = M1 n 11