Institute and Faculty of Actuaries (IFoA) Exams Economics, Business and Financial Engineering (CB2, CM2) Syllabus

What does the short-run Phillips curve describe?

The short-run Phillips curve shows an inverse relationship between the unemployment rate and the rate of inflation. In the long run it is vertical at the natural rate of unemployment, so there is no permanent inflation-unemployment trade-off.

Distinguish expansionary from contractionary fiscal policy and give the main tools.

Fiscal policy uses government spending $G$ and taxation $T$. Expansionary policy raises $G$ and/or cuts $T$ to boost aggregate demand (typically increasing the deficit); contractionary policy cuts $G$ and/or raises $T$ to reduce aggregate demand.

State the principle of comparative advantage in international trade.

A country should specialise in and export goods in which it has a comparative advantage, i.e. a lower opportunity cost of production, even if it has no absolute advantage. Mutually beneficial trade is possible whenever opportunity-cost ratios differ between countries.

Define the expected utility (von Neumann-Morgenstern) of a random wealth $X$ and state how risk attitude relates to the utility function.

$E[U(X)] = \sum_{i} p_{i}\, U(x_{i})$ (or $\int U(x) f(x)\,dx$). A risk-averse investor has $U'(x) > 0$ and $U''(x) < 0$ (concave $U$); risk-neutral has $U'' = 0$; risk-seeking has $U'' > 0$.

Define the absolute and relative risk aversion coefficients.

Arrow-Pratt absolute risk aversion: $A(w) = -\dfrac{U''(w)}{U'(w)}$. Relative risk aversion: $R(w) = -\dfrac{w\,U''(w)}{U'(w)} = w\,A(w)$.

State the definitions of declining, constant and increasing absolute risk aversion and give an example utility.

Declining (DARA): $A'(w) < 0$ — risky assets are normal goods (e.g. $U(w)=\ln w$). Constant (CARA): $A'(w)=0$ — e.g. exponential $U(w) = -e^{-aw}$. Increasing (IARA): $A'(w) > 0$ — e.g. quadratic utility.

Define first-order stochastic dominance (FSD).

Portfolio $A$ first-order stochastically dominates $B$ if $F_{A}(x) \leq F_{B}(x)$ for all $x$ (with strict inequality somewhere), where $F$ is the cumulative distribution function. Then every investor preferring more to less ($U' > 0$) prefers $A$.

Define second-order stochastic dominance (SSD) and state the relevant investor class.

$A$ second-order stochastically dominates $B$ if $\int_{-\infty}^{x} \big(F_{B}(t) - F_{A}(t)\big)\,dt \geq 0$ for all $x$. Then every risk-averse investor ($U' > 0$, $U'' < 0$) prefers $A$ to $B$.

Give the formulas for the expected return and variance of a two-asset portfolio.

$E[R_{p}] = w_{A}E[R_{A}] + w_{B}E[R_{B}]$ and $\sigma_{p}^{2} = w_{A}^{2}\sigma_{A}^{2} + w_{B}^{2}\sigma_{B}^{2} + 2 w_{A} w_{B}\,\rho_{AB}\,\sigma_{A}\sigma_{B}$, where $\rho_{AB}$ is the correlation between the two assets' returns.

What is the efficient frontier in mean-variance portfolio theory?

The efficient frontier is the set of portfolios that offer the maximum expected return for each level of risk (variance), equivalently the minimum risk for each level of expected return. Rational mean-variance investors hold only portfolios on this frontier.

State the Capital Asset Pricing Model (CAPM) equation and define beta.

$E[R_{i}] = R_{f} + \beta_{i}\big(E[R_{m}] - R_{f}\big)$, where $\beta_{i} = \dfrac{\mathrm{Cov}(R_{i}, R_{m})}{\mathrm{Var}(R_{m})}$ measures the sensitivity of asset $i$'s return to the market return. $(E[R_m]-R_f)$ is the market risk premium.

What is the Capital Market Line (CML) and how does it differ from the Security Market Line (SML)?

The CML plots expected return against total risk $\sigma$ for efficient portfolios combining the risk-free asset and the market portfolio. The SML plots expected return against systematic risk $\beta$ and applies to all assets, efficient or not.

Distinguish systematic (market) risk from specific (diversifiable) risk.

Systematic risk affects all assets and cannot be removed by diversification; it is the only risk priced by the CAPM (via $\beta$). Specific (idiosyncratic/diversifiable) risk is unique to an individual asset and can be eliminated by holding a well-diversified portfolio.

State the single-index (market) model for security returns.

$R_{i} = \alpha_{i} + \beta_{i} R_{m} + \varepsilon_{i}$, where $\alpha_{i}$ is the asset-specific intercept, $\beta_{i}$ the market sensitivity, $R_{m}$ the market return and $\varepsilon_{i}$ a mean-zero specific term uncorrelated with $R_{m}$ and across securities.

State the assumed return-generating equation of Arbitrage Pricing Theory (APT).

$R_{i} = E[R_{i}] + \sum_{k=1}^{K} b_{ik} F_{k} + \varepsilon_{i}$, where $F_{k}$ are common (systematic) factors, $b_{ik}$ the factor sensitivities, and $\varepsilon_{i}$ specific risk. Expected return is linear in the factor risk premia: $E[R_{i}] = R_{f} + \sum_{k} b_{ik}\lambda_{k}$.

State the three forms of the Efficient Markets Hypothesis (EMH) and the information set each reflects.

Weak form: prices reflect all past price/trading information (technical analysis cannot give excess returns). Semi-strong form: prices reflect all publicly available information. Strong form: prices reflect all information, public and private (even insiders cannot beat the market).

List four behavioural finance biases that can cause prices to deviate from fundamental value.

Common biases: overconfidence, anchoring (over-reliance on an initial value), loss aversion / prospect theory (losses loom larger than gains), herding, framing, and representativeness. They challenge the rational-investor assumption of the EMH.

Outline the key features of prospect theory.

Investors evaluate outcomes as gains/losses relative to a reference point, not final wealth. The value function is concave for gains, convex for losses (risk-seeking over losses), and steeper for losses (loss aversion). Probabilities are distorted by a probability-weighting function that overweights small probabilities.

State the standard model for the price of a non-dividend-paying share, geometric Brownian motion (GBM).

$dS_{t} = \mu S_{t}\,dt + \sigma S_{t}\,dW_{t}$, where $\mu$ is the drift, $\sigma$ the volatility and $W_{t}$ a standard Brownian motion. Equivalently $S_{t} = S_{0}\exp\!\big((\mu - \tfrac{1}{2}\sigma^{2})t + \sigma W_{t}\big)$, so $\ln S_{t}$ is normally distributed.

Distinguish the continuous-time lognormal model from the Wilkie stochastic investment model.

The continuous-time lognormal model assumes log-returns are i.i.d. normal (random walk, suitable for short-term derivative pricing). The Wilkie model is a discrete-time, multi-series model with autoregressive structure linking inflation, share dividends/yields and bond yields, designed for long-term actuarial projections.

Define a martingale and state the martingale property of a stochastic process $X_t$.

A process $X_{t}$ is a martingale with respect to a filtration $\mathcal{F}_{t}$ if it is integrable and $E[X_{t} \mid \mathcal{F}_{s}] = X_{s}$ for all $s \leq t$. The best forecast of a future value is the current value; discounted asset prices are martingales under the risk-neutral measure.

State the Black-Scholes formula for a European call option on a non-dividend-paying stock.

$c = S_{0}\,N(d_{1}) - K e^{-rT} N(d_{2})$, with $d_{1} = \dfrac{\ln(S_{0}/K) + (r + \tfrac{1}{2}\sigma^{2})T}{\sigma\sqrt{T}}$ and $d_{2} = d_{1} - \sigma\sqrt{T}$, where $N(\cdot)$ is the standard normal CDF.

State the put-call parity relationship for European options.

$c - p = S_{0} - K e^{-rT}$, where $c$ and $p$ are the prices of European call and put options with the same strike $K$ and maturity $T$ on the same non-dividend-paying underlying $S_{0}$, and $r$ is the continuously compounded risk-free rate.

Give the one-step risk-neutral valuation formula in the binomial option-pricing model.

The option value is $f = e^{-r\,\delta t}\big(q\,f_{u} + (1-q)\,f_{d}\big)$, where the risk-neutral up-probability is $q = \dfrac{e^{r\,\delta t} - d}{u - d}$, and $u$, $d$ are the up/down multipliers with payoffs $f_{u}$, $f_{d}$.

Define the option Greeks delta and gamma.

Delta $\Delta = \dfrac{\partial f}{\partial S}$ is the rate of change of the option price with respect to the underlying price (the hedge ratio). Gamma $\Gamma = \dfrac{\partial^{2} f}{\partial S^{2}}$ is the rate of change of delta with respect to the underlying price.

Distinguish one-factor short-rate models: the Vasicek model versus the Cox-Ingersoll-Ross (CIR) model.

Vasicek: $dr_{t} = a(b - r_{t})\,dt + \sigma\,dW_{t}$ — mean-reverting, constant volatility, but rates can go negative. CIR: $dr_{t} = a(b - r_{t})\,dt + \sigma\sqrt{r_{t}}\,dW_{t}$ — mean-reverting with volatility scaling by $\sqrt{r_t}$, keeping rates non-negative.

Distinguish structural from reduced-form (intensity-based) credit risk models.

Structural models (e.g. Merton) treat default as occurring when the firm's asset value falls below its debt, modelling the firm's balance sheet directly. Reduced-form models treat default as an exogenous event governed by a hazard rate / intensity process $\lambda_{t}$, calibrated to market credit spreads.

In the Merton model, how is a company's equity and debt interpreted in option terms?

Equity is a European call option on the firm's assets $V$ with strike equal to the debt face value $L$ at maturity $T$: equityholders receive $\max(V_{T} - L, 0)$. The debt equals risk-free debt minus a put option, so default risk equals a short put on the firm's assets.

Define Value at Risk (VaR) and state its main shortcoming.

VaR at confidence level $p$ over horizon $t$ is the loss $L$ such that $P(\text{loss} > \text{VaR}) = 1 - p$, i.e. $\mathrm{VaR}_{p} = \inf\{x : P(L \leq x) \geq p\}$. Its main shortcoming is that it is not sub-additive (not coherent) and gives no information about the size of losses beyond the threshold.

Define Expected Shortfall (TailVaR) and state why it is preferred to VaR.

Expected Shortfall (conditional tail expectation) is the expected loss given that the loss exceeds the VaR threshold: $\mathrm{ES}_{p} = E[L \mid L > \mathrm{VaR}_{p}]$. It is preferred because it is a coherent (sub-additive) risk measure and captures the magnitude of tail losses.

Define semi-variance and shortfall probability as downside risk measures.

Semi-variance measures dispersion below the mean only: $\text{semi-var} = E\big[(\min(X - \mu, 0))^{2}\big]$. Shortfall probability is the probability that the return falls below a chosen benchmark level $L$: $P(X < L)$. Both focus on downside risk rather than total variability.