CP351 Sec C

Equations from CP 351 ALM Sec C

Annual portfolio growth required to match annual liability growth

r(L)/(1-d)

Put Call Parity

Call(K,T1)-Put(K,T1)=Z(0,T1)*(Pfwd(0,T1,T2)-K)

C(u1, u2) with rho = -1

max(0,u1+u2-1)

C(u1, u2) with rho = 0

u1*u2

C(u1, u2) with rho = 1

min(u1,u2)

BS Call Price

S0*e^(-delta*T)*N(d1)-K*e^(-r*t)*N(d2)

BS d1

(ln(s0/k)+(r-delta+sigma^2/2)*T)/(sigma*sqrt(T))

BS d2

d1-sigma*sqrt(T)

BS Put Price

K*e^(-r*t)*N(-d2)-S0*e^(-delta*T)*N(-d1)

Delta Call

e^(-delta*t)*N(d1)

Delta Put

-e^(-delta*T)*N(-d1)

Gamma Call & Put

(e^(-delta*T)*phi(d1))/(s0*sigma*sqrt(T))

1-day % VaR Delta Normal method for portfolio

delta portfolio*S*sigma*z%*(1/sqrt(250))

Swap bootstrapping yield curve T1 Z(t)

1/(1+c(t,T1)/n)

Swap bootstrapping yield curve > T2 Z(t)

(1-c(t,Ti)/n*sumof(Z(t,Tj)))/(1+c(t,Ti)/n)

swap rate

n*(1-Z(0,Tm))/sumof(Z(0,Tj))

Price of Eurodollars Futures Contract that expires at time T (P Fut(t,T))

100-f4Fut(t,T)

Payoff of a Cap

N*0.25*max(r4(Ti-1)-K,0)

Index of upper tail dependence

lim q→1 (1-2q+C(q,q))/(1-q)

Index of lower tail dependence

lim q→0 C(q,q)/q

Kendalls Tau Clayton

theta/(theta+2)

Clayton lower tail lamda

2^(-1/theta)

Gumbel Kendalls Tau

1-1/theta

Gumbell upper tail lamba

2-2^(1/theta)

Probability A given B

Pr[A&B]/Pr[B]

Joint survival probability

1-u-v+C(u,v)

Binomial PMF

(n choose x)*p^x*(1-p)^(n-x)

Clayton Copula

(u^-theta+v^-theta-1)^(-1/theta)

Gumbel Copula

e^(-((-ln(u))^theta+((-ln(v))^theta))^(1/theta))

Gaussian z2 bivariate pair

z_1*rho+sqrt(1-rho²)*z_2

TAA Objective Function

Max(sumof(alpha*Ps*Ri,s*xi-(1-alpha)*Ps*(A*zs+B*zs²)))

General fund policy mathematical reserve

PM(t)=PM(t-1)+Annual Benefits+Complementary Premiums-Withdrawals-Annual Management Fees

Proportional Insurance Premium

(1-a)*E(S)+xi*sqrt(Var[(1-a)*S])

Proportional Insurance Expected Outgo

P+a*E[S]+.10*95%*ES

1st moment Pareto distribution Proportional

theta/(alpha-1)

2nd moment Pareto distribution Proportional

2*theta²/((alpha-1)*(alpha-2))

1st moment Pareto distribution with deductible

(theta+D)/(alpha-1)*(theta/(theta+D))^alpha

2nd moment Pareto distribution with deductible

2*(theta+D)²/((alpha-1)*(alpha-2))*(theta/(theta+D))^alpha

Cat Bond rho (Break Even Premium Spread)

(P-A)/(C*annuity factor)

Cat bond annuity factor

(1-e^(-r*T))/(2*(e^(.5*r)-1))

Conditional PLL GPD (Lambda)

(Beta/(Beta+Xi(exhaustion point-attachment point)))^(1/Xi)

Conditional EL GPD

Xi*lambda^Xi*(1-lambda^(1-Xi))/((1-Xi)*(1-lambda^Xi))

Dollar duration payer swap

Duration(floating)-Duration(fixed)

delta approximation portfolio

Change in value = -Portfolio delta*change in Stock Price

Covered Int Rate Parity

E(0) (USD/foreign)*exp((r_USD-r_foreign)*t)

Value of Forward Rate Agreement (FRA)

N*Z(t,T2)*delta*[fn(0,T1,T2)-fn(t,T1,T2)]