ALM Sec B Formulas

Formulas for exam CP351 Sec B

Bond Equivalent Yield

2*(1-Z(t))/sumof(Z(t))

Dollar Duration

Duration*Bond Price

Total Change in Bond Price

-DD* ir change+1/2*DConv*(ir change)^2

Convexity Change

1/2*Convexity*Price*(ir change)^2

Equity Duration

D(A)+L/E*(D(A)-D(L))

Effective Duration

(L- - L+)/(2*ir change*L0)

Effective Convexity

(L+-2*L0+L-)/((ir change)^2*L0)

Vasicek Model dr

dr = k*(theta-r)*dt+sigma*dW

Akaike Information Criterion (AIC)

MLL-k

Bayesian Information Criterion

MLL-(k*ln(n))/2

DV01

(-1/10000)*(price change/ir change)

Convexity of Cash Flow - used for annually effective discounting

t*(t+1)*CF*v^(t+2)

RSLN pi1

p21/(p12+p21)

RSLN pi2

p12/(p12+p21)

RSLN E(Var)

pi1*var(1)+pi2*var(2)

RSLN Var(exp value)

pi1*mu^2(1)+pi2*mu^2(2) - (pi1*mu1+pi2*mu2)^2

Yield based DV01 ZCB

T/(100*(1+y/2)^(2T-1))

Yield based Duration ZCB

T/(1+y/2)

Yield based DV01 Par Bond

(1/100y)*(1-1/((1+y/2)^2T))

Yield based Duration Par Bond

(1/y)*(1-1/((1+y/2)^2T))

Standard Deviation of Hedge P&L Under Regression Hedging

abs(FA Bond*DV01 Bond/100)*sigma(residuals)

YBD Fixed Bond

1/P*[c/y²(1-1/(1+y/2)^2T)+T(1-c/100y)*(100/(1+y/2)^(2T+1))]

Bermuda Assets

Assets backing Technical Provision + Surplus Assets

Bermuda Liabilities + Surplus

EBS Technical Provision + ECR + Free Surplus

Bermuda Enhanced Capital Requirement (ECR)

max[BSCR, MSM]

Bermuda Minimum Margin for Solvency

max(25%*ECR, $1BMD for Class 3A/3B - $100BMD for Class 4)

NAIC RBC C2

C2 = sqrt(C2mort² + C2long² + 2 * C2mort * C2long * CorrFactor)

LICAT Total Ratio

(Available Capital + Surplus Allowance + Eligible Deposits) / Base Solvency Buffer

LICAT Core Ratio

(Tier 1 Capital + 70% Surplus Allowance + 70% of Eligible Deposits) / Base Solvency Buffer

ACL RBC

C0 + C4a + sqrt((C1o + C3a)² + (C1cs + C3c)² + C2² + C3b² + C4b²)

Solvency II Risk Margin

CoC * sumof(SCR(t) / ((1+r(t+1))^(t+1)))

Effective Dollar Duration

(L- - L+) / (2*ir change)

Effective Dollar Convexity

(L+ - 2*L0 + L-)/((ir change)²)

Equity dollar std dev

Equity MV * Equity Volatility

Fixed Income dollar std dev

FI MV*FI mod dur*IR implied volatility

Carve Out CoC

(Target Cap Ratio * Cap Factor * (Required Return on Cap - Surplus Return))/(1-Tax Rate)

Carve Out Net Expected Return

Gross Expected Return - Asset Default Provisions - CoC

Carve Out Sharpe Ratio

Net Risk Premium/Tracking Error

Cox-Ingersoll-Ross (CIR) dr

dr = k*(theta-r)*dt+sigma*sqrt(t)*dW

Black-Karasinski dr

d*ln(r)=k*(ln(theta)-ln(r))*dt+sigma*dW

Market Price of Risk

(Expected Average Return - r) / Std Dev of Price

Vasicek RN theta

theta + sigma*lambda/k

CIR RN k

k + lambda

CIR RN theta

k*theta/(k+lambda)