Financial Economic Theory and Engineering Formula Sheet

Transcript

1 Financial Economic heory and Engineering Formula Sheet 2010 Morning and afternoon exam booklets will include a formula package identical to the one attached to this study note. he 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. he 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. 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. 1

2 General Formulas Lognormal distribution: { 1 fx = xσ 2π exp 1 } lnx µ 2 2 σ 2 EX = e µ+σ2 /2 VX = e 2µ+σ2 e σ2 1 lnx µ PrX x = Φ σ x > 0 Normal distribution: fx = 1 { σ 2π exp 1 } x µ 2 2 σ 2 Capital Asset Pricing Model: ER i = R f + β i ER m R f where β i = CovR i, R m VarR m Weighted Average Cost of Capital: B WACC = 1 τ c k b B + S + k S s B + S where k b =return on debt, k s =return in equity. GARCH1,1 Model Y t = µ + σ t ɛ t σ 2 t = α 0 + α 1 Y t 1 µ 2 + βσ 2 t 1 RSLN-2 Model Y t ρ t N µ ρt, σ 2 ρ t where {ρt }, t = 1, 2,..., is a Markov process with 2-states. p ij = Prρ t+1 = j ρ t = i Conditional ail Expectation For loss L, continuous at V α = F 1 L α: CE α L = E L L > V α L If there is a probability mass at V α, define β = max{β : V α = V β }, then CE α L = 1 β E X X > V α + β αv α 1 α 2

3 Vasicek Model Bt; rt dr = ab rdt + dz P t; = At; e a t 1 e h Bt; + t a 2 b 2 =2 2 Bt; 2 i Bt; = At; = exp a a 2 4a Cox Ingersoll Ross Model Bt; rt dr = ab rdt + p rdz P t; = At; e 2e t 1 2e a+ t=2 Bt; = At; = + ae t ae t Ho-Lee Model dr = tdt + dz t = F t 0; t + 2 t P t; = At; e rt t ln At; = ln P 0; P 0; t + tf 0; t t t 2 Hull-White Extended Vasicek Model 2ab= 2 dr = t ardt + dz t = F t 0; t + af 0; t + 2 2a 1 Bt; rt P t; = At; e a t 1 e Bt; = a ln At; = ln P 0; P 0; t + Bt; F 0; t 1 4a 3 2 e a e at 2 e 2at 1 e 2at Itô s Lemma Let X be an Itô process such that dx t = ut; X t dt + vt; X t dw t ; and let gt; x denote a twice di erentiable function. hen, for Y t = gt; X t, dy t; X t t; X t dx t Geometric Brownian 2 t; X t dx 2 t 2 : ds t S t = dt + dz t Black-Scholes Pricing Formulas c t = S t Nd 1 Ke r t Nd 2 3

4 p t = Ke r t N d 2 S t N d 1 where d 1 = lns t/k + r + σ 2 /2 t σ t and d 2 = d 1 σ t 4

5 Copeland, Weston, Shastri: Financial heory and Corporate Policy Chapter Rev t + m t S t = Div t + W &S t + I t 2.5 NI t = Rev t W &S t dep t 2.7 NI t A t S 0 = 1 + k s t t= F CF = EBI 1 τ c + dep I Chapter R jt = ER jt + β j δ mt + ε jt 6.36 R jt R ft = R mt R ft β j + ε jt 6.36 R pt = γ 0 + γ 1 β p + ε pt 6.37 R it = α i + β i R mt + ε it 6.38 ER i = ER Z + ER m ER Z β i 6.40 ER i R f = b i ER m R f + s i ESMB + h i EHML 6.41 λ i = ER m ER Z 6.46 R it = γ 0t + γ 1t β it + ε it 6.49 ER i = ER z,i + ER I ER z,i β i,i 6.50 Ri = E R i + b i1 F b ik Fk + ε i 6.57 E R i = λ 0 + λ 1 b i λ k b ik 6.59 ER i R f = δ1 R f bi δk R f bik 6.60 b ik = CovR i, δ k /V arδ k Chapter CS A, S B, = S A Nd 1 S B Nd 2 where d 1 = ln S A /S B + V 2 /V ; d 2 = d 1 V and V 2 = V 2 A 2ρ AB V A V B + V 2 B Chapter V η m qm max a pe mua, e V η 0 e 5

6 10.4 pr c pdr c 2 = p r/d c pr c Fair Game: ε j,t+1 = P j,t+1 P jt P jt EP j,t+1 η t P jt P jt = a Submartingale: EP j,t+1 η t P jt P jt = Er j,t+1 η t > b Martingale: EP j,t+1 η t P jt = Er j,t+1 η t = 0 P jt ER jt β jt = R jt + ER mt β mt R ft βjt Chapter R jt = α j + β 1j R mt R ft + β 2j RLE t RSE t + β 3j HB M t LB M t + ε jt 11.3 NI jt = â + ˆb j m t + ε jt 11.4 NI j,t+1 = â + ˆb j m t+1 Chapter V α = 1 µα λ 1 + r 12.3 W 0 = X + βv M + Y 1 αv α 12.4 W1 = αµ + ε µα + λ + β M 1 + rv M rw 0 X + µα λ E U W = E U W1 µ + ε µα + λ + 1 αµ α = 0 α E U W = E U W1 M 1 + rvm = 0 where µ α = µ β α E U W1 ε + λ αµ α = E U W ED = 1 V D + µ τ p D β 1 + r ED = r D X V D + t2 τ pd β D2 2t V D = τ p + β D t V D t = 1 t r 2 τ pd t β D t 2 2t 6 X DfXdX

7 12.20 V D t = τ p + βad t τp 1 + r τp A = + β 1 + 2r β I + D = C + Np e = C + P e L τ p D max D τ p = τ p + P D 1 + r r P + τp D L P + τ p D + I C L + I C P + τ p D + I C X DX = 1 τ p maxi C + L, 0 ln X V0 old = P E + S + a + b P + E W 0 z = αp z + βp m, P W s n, z = α P n, z + βp n, p W n = α P n t 2 p γ 1 = P n, z = kn + cz 1/γ X where k = t 2 γ/α 1/γ β1 + 2r τ 2 p 1 + r Mn, z = P n, z n, P = k1 t 1 n + cz 1/γ t 2 nk 1 γ n + cz 1 γ/γ V 0 = 1 tx EX n = Xn = X 0s 0 + Ŷmns m s 0 + s m V 1 n = Xn n C V 2, Y = X 0s 0 + Ŷms m + Y F s E s 0 + s m + F s XtX/n, Ŷm, Y C X 0 s 0 +Y ms m s 0 +s m F s X 0 s 0 + Ŷms m EV 2 Y m = C s 0 + s m + F s dα B V n Bn, D αd = 0 dd D d 2 α dα B 2 B V n Bn, D 2 αd < 0 dd 2 dd D D α D n V n B n, D n V n I = 0 d 2 α dα B V B 2 dd2 dd D α 2 B dd D 2 dn = dα dd εα, D = a D D 2 1 α α 2 7 B n dv dn 2 B + α n D

8 αɛ γ D = X 1 1 α1 ɛγ max U s cs, p fs, p ads dp cs,p,a V cs, p fs, p ads dp Ga V U s cs, p V cs, p max Uk + λ a max cs,a max cs,p,a = λ V cs, pfs, p ads dp Ga V Us csfs a ds + λ +µ V cs f a s ads G a U s cs = λ + µ f as a V cs fs a U s cs, p fs, p ads dp+λ +µ U s cs, p V cs, p U s cp V cp V cs, pf a s, p a ds dp G a = λ + µ f as, p a fs, p a V cs fs a ds Ga V f a1 p a = λ + µ 1 fp a + µ f a1 p a 2 fp a µ f a1 p a n fp a max cs,p,m,am,mm E s,p,m U s cs, p, m am Subject to for all m E s,p m V cs, p, m Gam am V V X X = P X X CX X = S = 1 βes α λ1 βcovs, R M 1 + r f max α,β V cs, pfs, p a ds dp Ga V 1 βes α λ1 βcovs, R M 1 + r f +µ a EW + α + βes b VarW + 2βCovW, s + β 2 Vars V β = λacovs, R M CovW, s 2bVars Vars { V + db B new B = D P D + dd, 1,, r f, σ V + P 8 } V D, 1,, r f, σ V = D P X +P Y

9 Chapter V U = E F CF or V U = EẼBI 1 τ c ρ ρ 15.3 ÑI + k d D = Rev Ṽ C F CC dep 1 τ c + k d Dτ c 15.4 V L = EẼBI 1 τ c ρ 15.6 V L = V U + τ c B V L I = S0 I + Sn + B 0 I 1 τ c EEBI I > ρ + k ddτ c k b = S0 I τ c B I k s = NI/ S = ρ + 1 τ c ρ k b B S G = V L V U = τ c B V U = EEBI 1 τ c1 τ ps ρ V L = V U τ c1 τ ps 1 τ pb β L = τ c B S β U B where B = k d D1 τ pb /k b R f S L + λ 1 τ c CovEBI, R m λ 1 τ c B CovR bj, R m = EEBI 1 τ c ER bj B1 τ c ds = S S dv + V t dt S 2 2 V 2 σ2 V 2 dt r s = S V V S r V S = V Nd 1 e r f DNd β S = 1 1 D/V e r f Nd 2 /Nd 1 β V 15.47/8 k s = R f + R m R f Nd 1 V S β V = R f + Nd 1 R v R f V S k b = R f + ρ R f N d 1 V B Note: his is a correction to the formula in the reading dv V = µv, tdt + σdw 9

10 σ2 V 2 F V V V + rv F V V rf V + C = F V = A 0 + A 1 V + A 2 V 2r/σ V = V B BV = 1 αv B V BV C/r BV = 1 p B C/r + p B 1 αv B where p B = V/V B 2r/σ V = V B DCV = αv B V DCV DCV = αv B V/V B 2r/σ V L V = V U V + c BV DCV = V U V + c B p B c B αv B p B { V1 if V M = 1 + rγ 0 V 0 + γ 1 D 1 V 1 C if V 1 < D { γ0 1 + r V 1a M a = 1+r 1V 1a if D < D V 1a γ r V 1b 1+r 1V 1a if D < D { γ0 1 + r V 1a M b = 1+r 1V 1b C if D D V 1a γ r V 1b 1+r 1V 1b if D < D Chapter V i t = Div it n i tp i t k u t Ṽ i t = ẼBI it + 1 Ĩit Ṽit k u t Ỹ di = ẼBI rdc 1 τ c rd pi 1 τ pi Ỹ gi = ẼBI rd c 1 τ c 1 τ gi rd pi 1 τ pi S 1 ES 1 = ε γ = EBI 1 E 0 EBI γ 1 + k 1 + k Div it = a i + c i Div it Div i,t 1 + U it Div t = β 1 Div t 1 + β 2 NI t + β 3 NI t 1 + Z t P it = a + bdiv it + cre it + ε it Rj = γ 0 + Rm γ 0 β j + γ 1 DY j DY m /DY m + ε j W PE P 0 P P = 1 F P + F P N 0 P 0 P 0 P 0 10

11 Hull: Options Futures and Other Derivatives Chapter 11: Binomial rees 11.1 = f u f d S 0 u S 0 d 11.5/6 f = e r t pf u + 1 pf d where p = er t d u d 11.13/14 u = e σ t d = e σ t Chapter 12: Wiener Processes and Itôs Lemma G = ln S dg = µ σ 2 /2 dt + σ dz Chapter 13: he Black-Scholes-Merton model f f 13.9 df = µs + S t f 2 S 2 σ2 S 2 dt + f S σsdz f t + rs f S σ2 S 2 2 f S 2 = rf Chapter 17: Greek Letters 17. call = Nd 1 put = Nd Θcall = S 0N d 1 σ Θput = S 0N d 1 σ Γcall = Γput = N d 1 S 0 σ 17.4 Θ + rs σ2 S 2 Γ = rπ rke r Nd Vcall = Vput = S 0 N d rhocall = K e r Nd rhoput = K e r N d 2 + rke r N d 2 Chapter 19: Basic Numerical Procedures 19.8 = f 1,1 f 1,0 S 0 u S 0 d 11

12 19.9 Γ = f 2,2 f 2,1 /S 0 u 2 S 0 f 2,1 f 2,0 /S 0 S 0 d 2 h Θ = f 2,1 f 0,0 2 t p = eft gt t d u d a j f i,j 1 + b j f i,j + c j f i,j+1 = f i+1,j where a j = 1 2 r qj t 1 2 σ2 j 2 t, b j = 1 + σ 2 j 2 t + r t and c j = 1 2 r qj t 1 2 σ2 j 2 t f i,j = a jf i+1,j 1 + b jf i+1,j + c jf i+1,j+1 where a 1 j = r t 2 r qj t σ2 j 2 t, b j = and c 1 1 j = 1 + r t 2 r qj t σ2 j 2 t 1 1 σ 2 j 2 t 1 + r t α j f i,j 1 + β j f i,j + γ j f i,j+1 = f i+1,j where α j = t r q σ2 2 Z 2 and γ j = t r q σ2 2 Z 2 t 2 Z 2 σ2 t 2 Z 2 σ2 β j = 1 + t Z 2 σ2 + r t αjf i+1,j 1 + βj f i+1,j + γj f i+1,j+1 = f i,j where αj 1 = t r q σ2 1 + r t 2 Z 2 and βj 1 = 1 t 1 + r t Z 2 σ2 and γ j = r t t 2 Z r q σ2 2 + t 2 Z 2 σ2 + t 2 Z 2 σ2 Chapter 21: Estimating Volatilities and Correlations 21.7 σ 2 n = λσ 2 n λu 2 n σn 2 = γv L + αu 2 n 1 + βσn 1 2 m lnv i u2 i i= E σn+t 2 = VL + α + β t σn 2 V L v i 12

13 Chapter 24: Exotic Options Call on a call S 0 e q 2 M a 1, b 1 ; 1 / 2 K 2 e r 2 M a 2, b 2 ; 1 / 2 e r 1 K 1 Na 2 where a 1 = lns 0/S + r q + σ 2 /2 1 σ 1 a 2 = a 1 σ 1 b 1 = lns 0/K 2 + r q + σ 2 /2 2 σ 2 b 2 = b 1 σ 2 Put on a call K 2 e r 2 M a 2, b 2 ; 1 / 2 S 0 e q 2 M a 1, b 1 ; 1 / 2 + e r 1 K 1 N a 2 Call on a put K 2 e r 2 M a 2, b 2 ; 1 / 2 S 0 e q 2 M a 1, b 1 ; 1 / 2 e r 1 K 1 N a 2 Put on a put S 0 e q 2 M a 1, b 1 ; 1 / 2 K 2 e r 2 M a 2, b 2 ; 1 / 2 + e r 1 K 1 Na 2 Chooser maxc, p = c + e q 2 1 max 0, Ke r q 2 1 S 1 Barrier Options λ = r q + σ2 /2 σ 2 x 1 = lns 0/H σ If H K y = ln H2 /S 0 K σ + λσ y 1 = lnh/s 0 σ + λσ + λσ c di = S 0 e q H/S 0 2λ Ny Ke r H/S 0 2λ 2 N If H K c do = S 0 Nx 1 e q Ke r N x 1 σ S 0 e q H/S 0 2λ Ny 1 + Ke r H/S 0 2λ 2 N If H > K c ui = S 0 Nx 1 e q Ke r N x 1 σ S 0 e q H/S 0 2λ N y N y 1 +Ke r H/S 0 2λ 2 y + σ N 13 y σ y 1 σ y 1 + σ

14 If H K p ui = S 0 e q H/S 0 2λ N y + Ke r H/S 0 2λ 2 N If H K p uo = S 0 N x 1 e q + Ke r N x 1 + σ +S 0 e q H/S 0 2λ N y 1 Ke r H/S 0 2λ 2 N y + σ y 1 + σ If H K p di = S 0 N x 1 e q + Ke r N x 1 + σ + S 0 e q H/S 0 2λ Ny Ny 1 Ke r H/S 0 N 2λ 2 y σ N y 1 σ Lookback Options c fl = S 0 e q Na 1 S 0 e q σ 2 where a 1 = lns 0/S min +r q+σ 2 /2 σ a 3 = lns 0/S min + r+q+σ 2 /2 σ Y 1 = 2r q σ2 /2 lns 0 /S min σ 2 p fl = S max e r Nb 1 where b 1 = lnsmax/s 0+ r+q+σ 2 /2 σ b 2 = b 1 σ b 3 = lnsmax/s 0+r q σ 2 /2 σ Y 2 = 2r q σ2 /2 lns max/s 0 σ 2 Exchange Options 24.3 V 0 e qv Nd 1 U 0 e qu Nd 2 2r q N a 1 S min e r a 2 = a 1 σ Na 2 σ2 2r q ey 1 N a 3 σ2 2r q ey 2 N b 3 +S 0 e q σ 2 2r q N b 2 S 0 e q Nb 2 where d 1 = lnv 0/U 0 + q U q V + ˆσ 2 /2 ˆσ and ˆσ = σu 2 + σ2 V 2ρσ Uσ V d 2 = d 1 ˆσ Chapter 26: More on Models and Numerical Procedures 26.2, 3 ds S = r qdt + V dz S dv = av L V dt + ξv α dz V 14

15 Chapter 27: Martingales and Measures 27.4 Π = σ 2 f 2 f 1 σ 1 f 1 f Π = µ 1 σ 2 f 1 f 2 µ 2 σ 1 f 1 f 2 t 27.7 df = µdt + σdz f 27.8 µ r = λ σ n µ r = λ i σ i d i=1 f = σ f σ g f g g dz f 0 = P 0, E f st = P t, 0 P t, N At f f 0 = A0E A A c = P 0, E maxs K, 0 V f 0 = U 0 E U max 1, 0 U f 0 = V 0 Nd 1 U 0 Nd α v = ρσ v σ w Chapter 28: Interest Rate Derivatives: he Standard Market Models 28.1 c = P 0, F B Nd 1 KNd p = P 0, KN d 2 F B N d 1 where d 1 = ln F B/K + σ 2 B /2 σ B 28.3 F B = B 0 I P 0, 28.4 σ B = D y0 σ y d 2 = d 1 σ B 28.7, Caplet Lδ k P 0, t k+1 F k Nd 1 R K Nd 2 where d 1 = lnf k/r K + σ 2 k t k/2 σ k tk d 2 = d 1 σ k tk 28.8, Floorlet Lδ k P 0, t k+1 R K N d 2 F k N d 1 15

16 28.10 LA s 0 Nd 1 s k Nd 2 Chapter 29: Convexity, iming, and Quanto Adjustments 29.1 E y = y y2 0σ 2 y G y 0 G y E R = R 0 + R2 0σ 2 R τ 1 + R 0 τ 29.3 α V = ρ V W σ V σ W 29.4 E V = E V exp ρ V Rσ V σ R R R 0 /m Chapter 30: Interest Rate Derivatives: Models of the Short Rate 30.20, 21 c = LP 0, snh KP 0, Nh σ p p = KP 0, N h + σ p LP 0, sn h h = 1 σ p ln σ p = σ a P m+1 = n m LP 0, s P 0, K + σ p 2 1 e as 1 e 2a 2a j= n m Q m,j exp g α m + j x t Chapter 32: Swaps Revisited 32.1 F i + F 2 i σ 2 i τ i t i 1 + F i τ i 31.2 y i 1 2 y2 i σ 2 y,it i G i y i G i y i y iτ i F i ρ i σ y,i σ F,i t i 1 + F i τ i 32.3 V i + V i ρ i σ W,i σ V,i t i 16

17 Hardy: Investment Guarantees Chapter F t + = F t 1 m = F t 1 1 m S t S t 1 s t 1 m t M t = F t m c = m c F 0 S C n = n p τ xg F n + errata sheet 6.13 C t = t p τ xm t + t 1 1 q d xg F t C t = t p τ xf 0 S t 1 m t m d + t 1 1 q d x errata sheet G F0 S t 1 m t + errata sheet Chapter 8 8. F = F 0 S S 0 1 m 8.3 P 0 = Ge r Φ d 2 S 0 1 m Φ d H0 = where d 1 = log S 0 1 m /G + r + σ 2 /2 σ n 0 Ge rt Φ d 2 tp τ x µ d x,t dt + n 0 d 2 = d 1 σ S0 1 m t Φ d 1 tp τ x µ d x,t dt 8.9 H0, t 3 = P S t 1 + S 0 1 m t 1 { 1 + P t2 t P t 3 t m t 2 t 1 P t 3 t 2 } 8.15 α = S 0 1 m t 1 where P S t 1 = BSPS 0 1 m t 1, G, t 1 B S 0 ä τ x:n i 8.19 HE t = Ht + t 1 q d x G Ft + Ht 8.22 C t = τ S t Ψ t Ψ t 1 Chapter β% Confidence Interval for α-quantile: 1 + β Nα1 L 0Nα A, L 0Nα+A ; A = Φ 1 α 2 log G/S0 nµ + log1 m 9.4 ξ = 1 Φ nσ 17

18 9.7 V α = G F 0 exp z α nσ + nµ + log1 m e rn { 9.17 CE α L = e rn /2 G enµ+log1 m+σ2 Φ z α nσ } 1 α Chapter H 0 = B0, ne Q Fn ga 65 n H 0 = F 0 E Q ga d 65 0, n B0, n H t = F t {ga 65 tφd 1 t Φd 2 t} where d 1 t = log ga 65t + σ 2 yn t/2 σ y n t, d 2 t = d 1 t σ y n t Chapter PP: H = = αp S 0 + Sn P 1 + α 1 G S 0 { S n S } + 0 G 1 α α P 13.7 H 0 = αp e dn Φd 1 G P 1 α e rn Φd 2 αp ln + r d + σ2 n G P 1 α 2 where d 1 = σ, d 2 = d 1 σ n n CAR: H = P { e r + α e d Φd 1 e r Φd 2 } n FE and FE : Doherty; Integrated Risk Management Chapter 13 p. 465 V E = V F V D = V F D DF + P V F, D = CV F, D p. 481 V F = S + D 1 + m n p. 494 V R E = C + V RF D + P {V r F, D} = C + V R D + P R V N E = V NF D + P {V N F, D} = V N D + P N 18

19 Chapter E + P 1 + r i L 16.2 E = E P RI; S 1 + ER i ELa + hci; S a 16.3 E a = ELa a 1 + h C I I L L a = 0 FE : Babbel and Fabozzi, Investment Management for Insurers Chapter 13 8 Ct = P t, Bχ 2r 2 ϕ + ψ + G, B, 4c, σ 2 X P t, χ 2r 2 φ + ψ, 4c, 2φ2 re γ t σ 2 φ+ψ 2ϕ 2 reγ t ϕ+ψ+g, B FE : Ho and Lee, he Oxford Guide to Financial Modeling Chapter 5: Interest Rate Derivatives: Interest Rate Models 5.10 dr = a 1 + b 1 l rdt + rσ 1 dz dl = a 2 + b 2 r + c 2 l dt + lσ 2 dw 5.12 dv = Mt, rdt + Ωt, rdz Mt, r = V t + µt, rv r σt, r2 V rr Ωt, r = σt, rv r Chapter 6: Implied Volatility Surface: Calibrating the Models 6.10 Lk, j + 1 = Lk, j exp k i=j P, i ; = P + P Li,j Λ 1+Li,j i j 1Λ k j 1 Λ2 k j t= 1 t=1 ht ht δ i where ht = δ t + k j 1 Z FE : he Cost of Capital for Financial Firms p.27 E 0 + F 0 = t=1 D t 1 + COE t 19

20 = F 0 = t=1 t=1 D t + E t E t 1 COE E t COE t ROE t COE 1 + COE t E t 1 t=1 E t 1 + COE + E t 1 t 1 + COE t 1 p R f {A 0 L 0 + F 0 } = A 0 L k {A 0 R f + m A k A L 0 R f m L + k L } + F R f F 0 = 1 k {A 0 m A k A + L 0 m L k L } k R f A 0 L 0 + F 1 p R f F 0 = A 0 1 k m A k A + L 0 1 k m L k L + 1 sf 1 R f + sk A 0 L 0 p R f A 0 L 0 + F 0 = A 0 {1 + 1 k R f m A k A } L 0 {1 + 1 k R f m L k L } + 1 sf 1 sk A 0 L 0 p.41 R f + s g + sgf 0 = A 0 1 k m A k A + L 0 1 k m L k L R f + sk A 0 L 0 t=1 FE : Solvency Measurement for Property-Liability Risk-Based Capital Applications 4a d L = D L c c L = kφ cφ k k d A = D A L = 1 c ca k A φ c A Φ 1 c A k A k A 5a d L = Φa 1 + cφa k 5b d A = Φb Φb k A 1 c A where a = k/2 ln1 + c/k b = k A /2 + ln1 + c A /k A 6 d 1 = 0 zpzdz 7 c 1 = c1 + p c1 + p r + pc b g n 1/2 n 8 C = c 2 i + ρ ij c i c j A1 i=1 i j µ µ D = σφ µφ σ σ 20

21 A2 A3 A4 A5 d = D c c L = k φ c Φ d L = kφ k c c Φ k d A = D A L = 1 1 c A F = SΦa Ee it Φ k c k ca ca c A Φ k A φ k A a σ t where a = lns/e + i + σ2 /2t σ t A6 D L = LΦa 1 + clφa σ L A7 d L = Φa 1 + cφa k A8 D L = AΦa LΦa σ A A9 d A = Φb Φb k A 1 c A where b = k A /2 + ln1 c A /k A k A FE : Capital Allocation in Financial Firms p.116 NP V = 1 dv {S + } C µ 1 + mv {S } = µ dv {S + } + mv {S } V {S + } V {S } z = σnz+znz 1+r = σnz zn z 1+r = C1 + r/σ FE : Babbel and Merrill, Real and Illusory Value Creation by Insurance Companies 2 V E = F + V A P V L + O FE : A Comparative Analysis of U.S. Canadian and Solvency II Capital Adequacy Requirements in Life Insurance, Sharara, Hardy and Saunders p 8 CCM = i ccr SCRt 1e rtt t=1 p 9 SCR = ρi, j SCR i SCR j i j 21

22 Hardy, Freeland and ill, Valuation of Long-erm Equity Return Models for Equity-Linked Guarantees MARCH Y t F t 1 = { Q1 w.p. q w.p. 1 q Q 2 where Q 1 F t 1 = µ 1 + σ t z t σ 2 t = α 1,0 + α 1,1 Y t 1 µ α 1,2 Y t 2 µ 1 2 Q 2 F t 1 = µ 2 + α 2,0 z t z t N0, 1, i.i.d. RSDD Y t ρ t = κ ρt + ϕ ρt D t 1 + σ ρt z t where D t 1 = min 0, D t 2 + Y t 1 RSGARCH Y t ρ t = µ ρt + σ ρt,tz t where σρ 2 t,t = α ρt,0 + α ρt,1ε 2 t 1 + β ρt σt 1 2 ε t = Y t p 1 tµ p 1 tµ 2 σt 2 = p 1 tµ σ1,t p 1 tµ σ2,t 2 p 1 tµ p 1 tµ 2 2 AAA Model Y t = µ t /12 + σ t / 12z y,t where µ t = A + Bσ t + Cσ 2 t log σ t = ν t = 1 ϕν t 1 + ϕ log τ + σ ν z ν,t 4.1 r t,1 = r t ρ t = 1 = y t µ 1 σ r t,2 = r t ρ t = 2 = y t µ 2 σ r t = I {ρt>0.5} r t,1 + 1 I {ρt>0.5} rt,2 Smith: Investor & Management Expectations of the Return on Equity Measure vs. Some Basic ruths of Financial Accounting E t EV t = t ROEx IRRE x IRR t x x=1 22