CM2 formula sheet

S.MONK · CM2 formula sheet · 98 formulae · smonk.in/formula-sheet/cm2

Utility theory and risk aversion: non-satiation, A(w) and R(w)

A(w) = −U″(w) / U′(w)    (absolute risk aversion)
R(w) = −w · U″(w) / U′(w) = w · A(w)    (relative risk aversion)

U(certainty equivalent) = E[U(random wealth)]

Measures of investment risk: variance, semi-variance, VaR and Tail VaR

Var(R) = E[(R − μ)²] = E[R²] − μ²

semi-variance = E[((R − μ)⁻)²] = ∑ p_i(r_i − μ)² over the outcomes with r_i < μ

shortfall probability = P(R < L) for a chosen benchmark L

TVaR_α = E[loss | loss > VaR_α]
for a normal return, E[R | R < μ − z_ασ] = μ − σ · φ(z_α) / (1 − α)

CAPM and the security market line: beta and the market price of risk

β_i = Cov(R_i, R_M) / Var(R_M) = ρ_i,M · σ_i / σ_M

E[R_i] = r_f + β_i(E[R_M] − r_f)

market price of risk = (E[R_M] − r_f) / σ_M

σ_i² = β_i²σ_M² + σ_ε,i²
  (total) = (systematic) + (specific)

Ito's lemma explained term by term, and what it does to a share price

df = f′(x) dx + ½ f″(x)(dx)² + …

(dW)² = dt,    (dt)² = 0,    dW · dt = 0

dX_t = μ(X_t, t) dt + σ(X_t, t) dW_t

df = (∂f/∂x) dX + (∂f/∂t) dt + ½ (∂²f/∂x²) σ² dt

df = [ μ (∂f/∂x) + (∂f/∂t) + ½ σ² (∂²f/∂x²) ] dt + σ (∂f/∂x) dW

dS_t = μ S_t dt + σ S_t dW_t

d(ln S) = (1/S)(μS dt + σS dW) + ½(−1/S²)(σS)² dt
    = μ dt + σ dW − ½σ² dt
    = (μ − ½σ²) dt + σ dW

ln S_t = ln S₀ + (μ − ½σ²)t + σW_t
so ln S_t ~ N( ln S₀ + (μ − ½σ²)t , σ²t )

E[S_t] = exp( mean of ln S_t + ½ variance of ln S_t )
    = exp( ln S₀ + μt − ½σ²t + ½σ²t ) = S₀e^μt

The chain ladder method: estimating outstanding claims from a run-off triangle

f_j = ( ∑ C_i,j+1 ) / ( ∑ C_i,j )

ultimate = C_latest × f_j × f_j+1 × … × f_n−1
ultimate = C_latest × F_j, where F_j is the product of all the remaining factors

reserve for accident year i = C_latest × (F_j − 1)
total reserve = (total ultimate) − (total paid to date)

The Bornhuetter-Ferguson method: the chain ladder with a second opinion

expected ultimate = (expected loss ratio) × (earned premium)
reserve = expected ultimate × (1 − 1/F)
Bornhuetter-Ferguson ultimate = (claims paid to date) + reserve

BF ultimate = z × (chain ladder ultimate) + (1 − z) × (expected ultimate)

Binomial option pricing: where the risk-neutral probability comes from

Δ(uS) + B e^rh = f_u
Δ(dS) + B e^rh = f_d

Δ = (f_u − f_d) / (uS − dS)

option value = ΔS + B

f = e^−rh [ q f_u + (1 − q) f_d ]    where    q = (e^rh − d) / (u − d)

q(uS) + (1 − q)(dS) = S e^rh

f = e^−rT ∑ _nC_k q^k(1 − q)^n−k × (payoff after k up moves)

value at a node = max( e^−rh[q f_u + (1−q) f_d] , intrinsic value )

The Black-Scholes formula and put-call parity, read term by term

c = S · Φ(d₁) − K e^−rT · Φ(d₂)

d₁ = [ ln(S/K) + (r + ½σ²)T ] / (σ√T)
d₂ = d₁ − σ√T

p = K e^−rT · Φ(−d₂) − S · Φ(−d₁)

c − p = S − K e^−rT

Δ_call = Φ(d₁)     Δ_put = Φ(d₁) − 1
Γ = φ(d₁) / (Sσ√T)    (the same for a call and a put)

Ruin theory: the adjustment coefficient and Lundberg bound explained

U(t) = U + ct − S(t)

S(t) = X₁ + X₂ + … + X_N(t), where N(t) ~ Poisson(λt)

E[S(t)] = λt · E[X]
Var[S(t)] = λt · E[X²]

c = (1 + θ) λ μ

λ + cR = λ · M_X(R)

R = θ / ((1 + θ) μ)

reinsurance premium = (1 + θ_R) λ (1 − α) μ
net premium = c − reinsurance premium

Mean-variance portfolio theory: the efficient frontier and why diversification works

E[R_p] = w E[R_X] + (1 − w) E[R_Y]

Var[R_p] = w²σ_X² + (1 − w)²σ_Y² + 2w(1 − w)ρσ_Xσ_Y

Var[R_p] = ∑_i ∑_j w_i w_j σ_ij

w_min = (σ_Y² − ρσ_Xσ_Y) / (σ_X² + σ_Y² − 2ρσ_Xσ_Y)

Cov(R_min, R_X) = Cov(R_min, R_Y) = Var(R_min)

The term structure: spot rates, forward rates and par yields

v_n = 1 / (1 + s_n)ⁿ

(1 + s_n)ⁿ = (1 + s_m)^m × (1 + f(m, n))^n−m

1 + f(m, m+1) = (1 + s_m+1)^m+1 / (1 + s_m)^m = v_m / v_m+1

P = c(v₁ + v₂ + … + v_n) + 100v_n

y_n = 100(1 − v_n) / (v₁ + v₂ + … + v_n)

Short-rate models: Vasicek, Cox-Ingersoll-Ross and Hull-White compared

dr_t = α(μ − r_t) dt + (diffusion) dW_t

half-life = ln 2 / α

dr_t = α(μ − r_t) dt + σ dW_t

dr_t = α(μ − r_t) dt + σ√r_t dW_t

dr_t = α(μ(t) − r_t) dt + σ dW_t

E[r_t] = μ + (r₀ − μ)e^−αt
Var[r_t] = (σ²/2α)(1 − e^−2αt)

The Merton model: company equity as an option, and where a credit spread comes from

E₀ = V₀Φ(d₁) − De^−rTΦ(d₂)

d₁ = [ln(V₀/D) + (r + ½σ_V²)T] / (σ_V√T)
d₂ = d₁ − σ_V√T

B₀ = De^−rT − put(V₀, D, T)

risk-neutral P(default) = P(V_T < D) = Φ(−d₂)

yield = −(1/T) ln(B₀/D)
credit spread = yield − r

The Greeks: delta, gamma, vega, theta and rho, and how a hedge decays

Δ = ∂f/∂S
Δ_call = Φ(d₁)     Δ_put = Φ(d₁) − 1

Γ = ∂²f/∂S² = φ(d₁) / (Sσ√T)

ν = ∂f/∂σ = Sφ(d₁)√T

Θ = ∂f/∂t = −Sφ(d₁)σ/(2√T) − rKe^−rTΦ(d₂)

ρ = ∂f/∂r = KTe^−rTΦ(d₂)

The inflation-adjusted chain ladder: taking past inflation out and putting a future assumption in

payment year = accident year + development year

The average cost per claim method: developing numbers and severity separately

ultimate claims = (ultimate number of claims) × (ultimate average cost per claim)

average cost per claim = (cumulative claim amounts) / (cumulative claim numbers)

Valuing a maturity guarantee by simulation, and checking it against the formula

payout = max(F_T, G) = F_T + max(G − F_T, 0)

F_T = F₀ exp((r − ½σ²)T + σ√T · Z), with Z ~ N(0, 1)

standard error = s / √n, where s is the standard deviation of the payoffs

The Efficient Markets Hypothesis: the three forms, and how each one is tested

abnormal return = actual return − expected return given the market move

Single-index and multifactor models: getting a covariance matrix out of a few betas

R_i = α_i + β_iR_M + ε_i

σ_i² = β_i²σ_M² + σ_εi²
σ_ij = β_iβ_jσ_M²

R_i = α_i + b_i1F₁ + b_i2F₂ + … + ε_i

σ_i² = ∑_k b_ik²σ_Fk² + σ_εi²
σ_ij = ∑_k b_ikb_jkσ_Fk²

specific variance of the portfolio = (1/N²) ∑ σ_εi² = (average specific variance) / N

Excess-of-loss reinsurance: the retained claim, and choosing the retention

retained claim Y = min(X, M)
reinsurer pays Z = max(X − M, 0)

E[min(X, M)] = μ(1 − e^−M/μ)
E[max(X − M, 0)] = μ e^−M/μ

M_Y(r) = ∫₀^M e^rx(1/μ)e^−x/μ dx + e^rMP(X > M)
    = (1/μa)(1 − e^−aM) + e^−aM

reinsurance premium = (1 + θ_R)λ E[max(X − M, 0)] = (1 + θ_R)λμe^−M/μ
net premium c(M) = (1 + θ)λμ − (1 + θ_R)λμe^−M/μ

net expected profit = λμ(θ − θ_Re^−M/μ)

M_break-even = μ ln(θ_R/θ)

λ + c(M)R = λ M_Y(R)

State price deflators: pricing with real-world probabilities

price = e^−rT E_Q[payoff]

price = E_P[A × payoff]

e^−rT[q X_u + (1−q)X_d] = p A_uX_u + (1−p)A_dX_d

A_u = e^−rT q / p      A_d = e^−rT(1−q)/(1−p)

E_P[A] = p A_u + (1−p)A_d = e^−rT

Two-state and intensity credit models: backing a default rate out of a bond price

P(still healthy at time t) = e^−λt
P(default by time t) = 1 − e^−λt

B₀ = e^−rT[ e^−λTN + (1 − e^−λT)δN ]

spread = −(1/T)ln(B₀/N) − r

The martingale representation theorem, and why hedging is possible at all

E[M_t | F_s] = M_s for all s < t

e^−rtV_t = E_Q[e^−rTV_T | F_t], so V_t = e^−r(T−t)E_Q[payoff | F_t]

M_t = M₀ + ∫₀^t φ_s dW_s

dṼ_t = φ_t dS̃_t, where the tildes mean discounted

Free from S.MONK School of Actuaries · smonk.in · every formula here is explained, with a worked example, on the page it links to.