SOA Exam FAM-F: Fundamentals of Actuarial Mathematics - Financial - Comprehensive Cheat Sheet

Table of Contents #

Exam Overview
Interest Rate Theory
Derivatives Pricing
Binomial Option Pricing
Advanced Option Strategies
Financial Risk Management
Investment Theory and Portfolio Management
Derivatives Markets and Applications
Numerical Methods
- Monte Carlo Simulation
- Finite Difference Methods
Exam Strategies and Tips

Exam Overview #

The Fundamentals of Actuarial Mathematics – Financial (FAM-F) exam is a comprehensive 3-hour examination consisting of approximately 30 multiple-choice questions. This exam tests candidates’ understanding of financial economics, derivatives pricing, risk management, and investment theory from both theoretical and practical perspectives.

Key Areas of Focus:

Financial mathematics and interest rate theory (25-30%)
Derivatives and option pricing (35-40%)
Risk management and measurement (20-25%)
Investment theory and portfolio management (15-20%)

Exam Format:

Duration: 3 hours
Questions: ~30 multiple choice
Calculator: SOA-approved financial calculator required
Reference materials: None allowed

Interest Rate Theory #

Short Rate Models #

The instantaneous short rate r(t) serves as the foundation for interest rate modeling and bond pricing. Understanding various short rate models is crucial for FAM-F success.

Ho-Lee Model #

The Ho-Lee model assumes the short rate follows:

$$r(t) = r(0) + \int_0^t \theta(s)ds + \sigma W(t)$$

Where:

$r(0)$ = initial short rate
$\theta(t)$ = deterministic drift function
$\sigma$ = constant volatility parameter
$W(t)$ = standard Brownian motion

Key Properties:

Normal distribution of rates (can go negative)
Perfect correlation between rates at different maturities
Volatility structure: $\text{Var}[r(t)] = \sigma^2 t$

Vasicek Model #

The Vasicek model incorporates mean reversion:

$$dr(t) = \kappa(\theta - r(t))dt + \sigma dW(t)$$

Where:

$\kappa$ = speed of mean reversion
$\theta$ = long-term mean rate
$\sigma$ = volatility parameter

Analytical Solution: $$r(t) = r(0)e^{-\kappa t} + \theta(1-e^{-\kappa t}) + \sigma\int_0^t e^{-\kappa(t-s)}dW(s)$$

Bond Price Formula: $$P(t,T) = A(t,T)e^{-B(t,T)r(t)}$$

Where:

$B(t,T) = \frac{1-e^{-\kappa(T-t)}}{\kappa}$
$A(t,T) = \exp\left[(\theta-\frac{\sigma^2}{2\kappa^2})(B(t,T)-(T-t))-\frac{\sigma^2}{4\kappa}B(t,T)^2\right]$

Cox-Ingersoll-Ross (CIR) Model #

The CIR model ensures non-negative interest rates:

$$dr(t) = \kappa(\theta - r(t))dt + \sigma\sqrt{r(t)}dW(t)$$

Condition for Non-negativity: $2\kappa\theta \geq \sigma^2$

Forward Rates and Yield Curves #

Forward Rate Relationship #

The instantaneous forward rate is defined as:

$$f(t,T) = -\frac{\partial}{\partial T}\ln P(t,T)$$

Spot-Forward Relationship #

$$P(t,T) = \exp\left(-\int_t^T f(t,s)ds\right)$$

Yield to Maturity #

The continuously compounded yield $y(t,T)$ satisfies: $$P(t,T) = e^{-y(t,T)(T-t)}$$

Therefore: $y(t,T) = -\frac{\ln P(t,T)}{T-t}$

Bond Pricing and Duration #

Duration Measures #

Macaulay Duration: $$D_{Mac} = \frac{\sum_{i=1}^n t_i \cdot PV(CF_i)}{P}$$

Modified Duration: $$D_{mod} = \frac{D_{Mac}}{1+y}$$

Dollar Duration: $$DD = D_{mod} \times P$$

Convexity #

$$C = \frac{1}{P}\frac{d^2P}{dy^2} = \frac{\sum_{i=1}^n t_i(t_i+1) \cdot PV(CF_i)}{P(1+y)^2}$$

Price Approximation #

$$\frac{\Delta P}{P} \approx -D_{mod} \cdot \Delta y + \frac{1}{2}C \cdot (\Delta y)^2$$

Derivatives Pricing #

Black-Scholes Model #

The Black-Scholes-Merton model provides closed-form solutions for European options under specific assumptions.

Model Assumptions #

Constant risk-free rate $r$
Constant volatility $\sigma$
No dividends (or constant dividend yield $q$)
Efficient markets (no transaction costs, unlimited borrowing/lending)
Log-normal stock price distribution

Stock Price Dynamics #

$$dS = \mu S dt + \sigma S dW$$

Under risk-neutral measure: $$dS = (r-q) S dt + \sigma S dW$$

Black-Scholes Formulas #

European Call Option: $$C = S_0 e^{-qT} N(d_1) - K e^{-rT} N(d_2)$$

European Put Option: $$P = K e^{-rT} N(-d_2) - S_0 e^{-qT} N(-d_1)$$

Where: $$d_1 = \frac{\ln(S_0/K) + (r-q+\sigma^2/2)T}{\sigma\sqrt{T}}$$ $$d_2 = d_1 - \sigma\sqrt{T}$$

Dividend Adjustments #

For stocks paying dividends with yield $q$:

Replace $S_0$ with $S_0 e^{-qT}$ in formulas
For discrete dividends $D$: Replace $S_0$ with $S_0 - De^{-rt_D}$

Option Greeks #

The Greeks measure sensitivities of option prices to various parameters.

Delta (Δ) - Price Sensitivity #

Call: $\Delta_c = e^{-qT} N(d_1)$ Put: $\Delta_p = -e^{-qT} N(-d_1) = \Delta_c - e^{-qT}$

Gamma (Γ) - Delta Sensitivity #

$$\Gamma = \frac{e^{-qT} \phi(d_1)}{S_0 \sigma \sqrt{T}}$$

Where $\phi(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}$ is the standard normal PDF.

Theta (Θ) - Time Decay #

Call: $$\Theta_c = -\frac{S_0 e^{-qT} \phi(d_1) \sigma}{2\sqrt{T}} + qS_0 e^{-qT} N(d_1) - rK e^{-rT} N(d_2)$$

Put: $$\Theta_p = -\frac{S_0 e^{-qT} \phi(d_1) \sigma}{2\sqrt{T}} - qS_0 e^{-qT} N(-d_1) + rK e^{-rT} N(-d_2)$$

Vega (ν) - Volatility Sensitivity #

$$\nu = S_0 e^{-qT} \sqrt{T} \phi(d_1)$$

Rho (ρ) - Interest Rate Sensitivity #

Call: $\rho_c = KT e^{-rT} N(d_2)$ Put: $\rho_p = -KT e^{-rT} N(-d_2)$

Put-Call Parity #

Without Dividends #

$$C - P = S_0 - K e^{-rT}$$

With Continuous Dividends #

$$C - P = S_0 e^{-qT} - K e^{-rT}$$

With Discrete Dividends #

$$C - P = S_0 - D e^{-rt_D} - K e^{-rT}$$

Binomial Option Pricing #

Single-Period Model #

Parameters #

Up factor: $u > 1$
Down factor: $d < 1$ (typically $d = 1/u$)
Risk-neutral probability: $p = \frac{e^{rΔt} - d}{u - d}$

Option Valuation #

$$V_0 = e^{-rΔt}[pV_u + (1-p)V_d]$$

Where $V_u$ and $V_d$ are option values in up and down states.

Multi-Period Models #

Cox-Ross-Rubinstein Parameters #

For $n$ periods with $Δt = T/n$:

$u = e^{\sigma\sqrt{Δt}}$
$d = e^{-\sigma\sqrt{Δt}}$
$p = \frac{e^{rΔt} - d}{u - d}$

Backward Induction #

At each node $(i,j)$ where $i$ is the time step and $j$ is the number of up moves:

$$V_{i,j} = e^{-rΔt}[pV_{i+1,j+1} + (1-p)V_{i+1,j}]$$

Stock Price at Node $(i,j)$ #

$$S_{i,j} = S_0 u^j d^{i-j}$$

American Options #

For American options, compare intrinsic value with continuation value:

$$V_{i,j} = \max(\text{Intrinsic Value}, e^{-rΔt}[pV_{i+1,j+1} + (1-p)V_{i+1,j}])$$

American Call: Intrinsic Value = $\max(S_{i,j} - K, 0)$ American Put: Intrinsic Value = $\max(K - S_{i,j}, 0)$

Advanced Option Strategies #

Basic Strategies #

Protective Put #

Long stock + Long put
Payoff: $\max(S_T, K) - P$
Maximum loss: $S_0 + P - K$

Covered Call #

Long stock + Short call
Payoff: $\min(S_T, K) + C$
Maximum gain: $K - S_0 + C$

Straddle (Long) #

Long call + Long put (same strike)
Payoff: $|S_T - K| - C - P$
Breakeven: $K ± (C + P)$

Strangle (Long) #

Long call ($K_2$) + Long put ($K_1$), where $K_2 > K_1$
Payoff: $\max(S_T - K_2, 0) + \max(K_1 - S_T, 0) - C - P$

Complex Combinations #

Butterfly Spread (Long) #

Long call ($K_1$) + Short 2 calls ($K_2$) + Long call ($K_3$)
Where $K_1 < K_2 < K_3$ and $K_2 - K_1 = K_3 - K_2$
Maximum payoff: $K_2 - K_1$ at $S_T = K_2$

Iron Condor #

Short call ($K_4$) + Long call ($K_3$) + Short put ($K_1$) + Long put ($K_2$)
Where $K_1 < K_2 < K_3 < K_4$
Profit range: $K_2 < S_T < K_3$

Exotic Options #

Asian Options #

Average Price Call: Payoff = $\max(\bar{S} - K, 0)$ Average Strike Call: Payoff = $\max(S_T - \bar{S}, 0)$

Where $\bar{S} = \frac{1}{n}\sum_{i=1}^n S_{t_i}$

Barrier Options #

Up-and-Out Call: Becomes worthless if $S_t > H$ for any $t$ Down-and-In Put: Activated only if $S_t < H$ for some $t$

Financial Risk Management #

Value at Risk (VaR) #

Parametric VaR (Normal Distribution) #

$$\text{VaR}_α = μ + σ Φ^{-1}(α)$$

Where:

$μ$ = expected return
$σ$ = standard deviation of returns
$Φ^{-1}(α)$ = α-quantile of standard normal distribution

Historical Simulation VaR #

Collect historical returns
Sort in ascending order
Select α-percentile as VaR estimate

Monte Carlo VaR #

Generate random scenarios using appropriate models
Calculate portfolio value for each scenario
Determine α-percentile of loss distribution

Portfolio Risk Measures #

Portfolio Variance #

For n-asset portfolio: $$σ_p^2 = \sum_{i=1}^n \sum_{j=1}^n w_i w_j σ_{ij}$$

Where $σ_{ij} = ρ_{ij}σ_i σ_j$ is the covariance between assets $i$ and $j$.

Expected Shortfall (Conditional VaR) #

$$\text{ES}_α = E[L | L ≥ \text{VaR}_α]$$

Coherent Risk Measures #

A risk measure $ρ$ is coherent if it satisfies:

Monotonicity: If $X ≤ Y$, then $ρ(X) ≥ ρ(Y)$
Translation Invariance: $ρ(X + c) = ρ(X) - c$
Positive Homogeneity: $ρ(λX) = λρ(X)$ for $λ > 0$
Subadditivity: $ρ(X + Y) ≤ ρ(X) + ρ(Y)$

Credit Risk #

Merton Model #

Company defaults when asset value $V_T < D$ (debt level)

Default probability: $$P(\text{default}) = N\left(\frac{\ln(D/V_0) - (r - σ_V^2/2)T}{σ_V\sqrt{T}}\right)$$

Credit Spread #

$$s = -\frac{1}{T}\ln(1-PD \cdot LGD)$$

Where:

PD = Probability of Default
LGD = Loss Given Default

Investment Theory and Portfolio Management #

Modern Portfolio Theory #

Efficient Frontier #

For two-asset portfolio: $$σ_p^2 = w_1^2σ_1^2 + w_2^2σ_2^2 + 2w_1w_2ρ_{12}σ_1σ_2$$

Minimum variance weight: $$w_1^* = \frac{σ_2^2 - ρ_{12}σ_1σ_2}{σ_1^2 + σ_2^2 - 2ρ_{12}σ_1σ_2}$$

Sharpe Ratio #

$$S = \frac{E[R_p] - R_f}{σ_p}$$

Capital Asset Pricing Model (CAPM) #

Security Market Line #

$$E[R_i] = R_f + β_i(E[R_m] - R_f)$$

Beta Calculation #

$$β_i = \frac{\text{Cov}(R_i, R_m)}{\text{Var}(R_m)} = ρ_{i,m}\frac{σ_i}{σ_m}$$

Arbitrage Pricing Theory (APT) #

Multi-factor model: $$E[R_i] = R_f + β_{i1}λ_1 + β_{i2}λ_2 + … + β_{ik}λ_k$$

Where:

$β_{ij}$ = sensitivity to factor $j$
$λ_j$ = risk premium for factor $j$

Derivatives Markets and Applications #

Forward and Futures Contracts #

Forward Price Formula #

No Income: $F_0 = S_0 e^{rT}$ With Income Yield: $F_0 = S_0 e^{(r-q)T}$ With Storage Costs: $F_0 = S_0 e^{(r+u)T}$

Value of Forward Contract #

At time $t < T$: $$V_f = S_t e^{-q(T-t)} - F_0 e^{-r(T-t)}$$

Futures vs. Forwards #

Convexity Adjustment: $$F_{\text{futures}} = F_{\text{forward}} \exp(ρ_{r,S}σ_rσ_ST)$$

Where $ρ_{r,S}$ is correlation between interest rates and underlying price.

Swaps #

Interest Rate Swap Valuation #

Fixed-for-floating swap value (from fixed payer’s perspective): $$V_{\text{swap}} = B_{\text{floating}} - B_{\text{fixed}}$$

Where:

$B_{\text{floating}}$ = value of floating rate bond
$B_{\text{fixed}}$ = value of fixed rate bond

Currency Swap #

Value in domestic currency: $$V_{\text{swap}} = S_0 B_f - B_d$$

Where:

$S_0$ = current exchange rate
$B_f$ = foreign bond value
$B_d$ = domestic bond value

Credit Derivatives #

Credit Default Swap (CDS) #

Premium Leg: Present value of premium payments Protection Leg: Present value of expected loss payments

CDS Spread: $$s = \frac{\text{Expected Annual Loss}}{\text{Risky Annuity}} = \frac{PD × LGD}{\text{Duration}}$$

Numerical Methods #

Monte Carlo Simulation #

Basic Algorithm #

Generate random variables
Transform to desired distribution
Calculate payoff for each path
Average and discount results

Variance Reduction Techniques #

Antithetic Variates: Use $-Z$ if $Z$ is standard normal
Control Variates: Use correlated variable with known expected value
Importance Sampling: Modify probability distribution

Asian Option Example #

For geometric average Asian call: $$\ln(\bar{S}{\text{geo}}) = \frac{1}{n}\sum{i=1}^n \ln(S_{t_i})$$

Finite Difference Methods #

Heat Equation Form #

Transform Black-Scholes PDE to: $$\frac{∂u}{∂τ} = \frac{∂^2u}{∂x^2}$$

Explicit Scheme #

$$u_{i,j+1} = αu_{i-1,j} + (1-2α)u_{i,j} + αu_{i+1,j}$$

Where $α = \frac{Δτ}{(Δx)^2}$

Stability Condition: $α ≤ 0.5$

Implicit Scheme #

$$-αu_{i-1,j+1} + (1+2α)u_{i,j+1} - αu_{i+1,j+1} = u_{i,j}$$

Always stable but requires matrix inversion.

Exam Strategies and Tips #

Time Management #

First Pass (45 minutes): Answer all questions you’re confident about
Second Pass (60 minutes): Work on challenging but solvable problems
Final Pass (75 minutes): Tackle remaining difficult questions and review

Formula Application Strategy #

Identify the Problem Type
- Option pricing vs. bond pricing
- European vs. American options
- Discrete vs. continuous time
Check for Special Cases
- At-the-money options
- Time to expiration near zero
- Very high or low volatility
Verify Units and Conventions
- Annual vs. semiannual rates
- Continuously compounded vs. effective rates
- Days vs. years for time calculations

Common Pitfalls to Avoid #

Interest Rate Conversions
- Mixing effective and continuous rates
- Incorrect compounding frequency adjustments
Option Pricing Errors
- Wrong sign on $d_1$ and $d_2$ calculations
- Forgetting dividend adjustments
- Misapplying put-call parity
Risk Measure Calculations
- Confusing VaR confidence levels
- Incorrect time scaling for volatility
Binomial Tree Mistakes
- Wrong risk-neutral probability calculation
- Incorrect backward induction procedure

Calculator Tips #

Essential Calculator Functions:

Normal distribution (N and N^-1)
Natural logarithm and exponential
Statistical functions (mean, standard deviation)
Cash flow analysis (NPV, IRR)

Quick Checks:

Put-call parity relationships
Delta hedge ratios
Basic arbitrage conditions

Final Preparation Checklist #

One Week Before:

Review all major formulas
Practice with exam-style questions
Time yourself on problem sets
Identify weak areas for final review

Day Before:

Light review of key concepts
Check calculator functionality
Prepare exam materials
Get adequate rest

Exam Day:

Arrive early and settle in
Read questions carefully
Show all work for partial credit
Double-check numerical calculations

This comprehensive cheat sheet covers all major topics tested on the SOA FAM-F exam. Focus your study efforts on understanding the underlying principles rather than just memorizing formulas, as the exam often tests conceptual understanding and application in novel contexts.

Remember that success on FAM-F requires both theoretical knowledge and practical problem-solving skills. Regular practice with varied problem types will build the intuition needed to tackle challenging exam questions efficiently.

Good luck with your exam preparation! The financial mathematics and derivatives knowledge gained through FAM-F forms the foundation for advanced actuarial practice in risk management, product development, and financial analysis.