Casualty Actuarial Society Credentials (ACAS/FCAS) Probability, Financial Mathematics, and VEE Foundations Syllabus

State the Central Limit Theorem for the sample mean.

For i.i.d. $X_i$ with mean $\mu$ and variance $\sigma^2$, as $n \to \infty$ the sample mean $\bar{X}$ satisfies $$\frac{\bar{X} - \mu}{\sigma/\sqrt{n}} \xrightarrow{d} N(0,1)$$

Distinguish the accumulated value and the present value of $1$ under compound interest at effective rate $i$ over $n$ periods.

Accumulated value: $(1+i)^n$. Present value (discount factor): $v^n = (1+i)^{-n}$ where $v = \frac{1}{1+i}$.

Define the nominal rate of interest $i^{(m)}$ compounded $m$ times per year and relate it to the effective annual rate $i$.

$$1 + i = \left(1 + \frac{i^{(m)}}{m}\right)^{m}$$ The effective rate per subperiod is $\frac{i^{(m)}}{m}$.

Relate the force of interest $\delta$ to the effective annual rate $i$ and the discount factor $v$.

$\delta = \ln(1+i)$, equivalently $1 + i = e^{\delta}$ and $v = e^{-\delta}$. The accumulated value over $n$ years is $e^{\delta n}$.

Define the effective rate of discount $d$ and give its relationships to $i$ and $v$.

$d = \frac{i}{1+i} = iv = 1 - v$. Also $1 - d = v$ and $\frac{1}{1-d} = 1 + i$.

Give the present value and accumulated value of an annuity-immediate of $1$ per period for $n$ periods at rate $i$.

Present value: $a_{\overline{n}|} = \frac{1 - v^n}{i}$. Accumulated value: $s_{\overline{n}|} = \frac{(1+i)^n - 1}{i}$.

Give the present value and accumulated value of an annuity-due of $1$ per period for $n$ periods at rate $i$.

Present value: $\ddot{a}_{\overline{n}|} = \frac{1 - v^n}{d}$. Accumulated value: $\ddot{s}_{\overline{n}|} = \frac{(1+i)^n - 1}{d}$. Each equals the annuity-immediate value times $(1+i)$.

How are annuity-immediate and annuity-due values related?

$\ddot{a}_{\overline{n}|} = (1+i)\,a_{\overline{n}|}$ and $\ddot{s}_{\overline{n}|} = (1+i)\,s_{\overline{n}|}$. Equivalently $\ddot{a}_{\overline{n}|} = a_{\overline{n}|} + 1 - v^n$. The difference reflects payments at the beginning vs. end of each period.

What is the present value of a perpetuity-immediate and a perpetuity-due paying $1$ per period?

Perpetuity-immediate: $a_{\overline{\infty}|} = \frac{1}{i}$. Perpetuity-due: $\ddot{a}_{\overline{\infty}|} = \frac{1}{d}$.

Give the present value of an increasing annuity-immediate $(Ia)_{\overline{n}|}$ with payments $1, 2, \dots, n$.

$$(Ia)_{\overline{n}|} = \frac{\ddot{a}_{\overline{n}|} - n v^n}{i}$$

What is the present value of a level continuous annuity paying at rate $1$ per year for $n$ years?

$$\bar{a}_{\overline{n}|} = \frac{1 - v^n}{\delta}$$ where $\delta$ is the force of interest.

Under the amortization method, how do you split a loan payment into interest and principal, and how does the principal portion change over time?

Interest portion = (outstanding balance) $\times i$; principal portion = payment $-$ interest. As the balance declines, the interest portion decreases and the principal portion increases geometrically by factor $(1+i)$.

For a loan of $L$ repaid by $n$ level payments at rate $i$, give the level payment and the outstanding balance after $t$ payments (prospective method).

Level payment $P = \frac{L}{a_{\overline{n}|}}$. Prospective outstanding balance: $B_t = P\,a_{\overline{n-t}|}$ (present value of remaining payments).

Contrast the amortization method and the sinking fund method of loan repayment.

Amortization: each payment covers interest plus part of principal, directly reducing the balance. Sinking fund: the borrower pays periodic interest on the full loan to the lender and separately deposits into a sinking fund (often at a different rate $j$) to accumulate the principal repaid at the end.

Give the basic price formula for a bond with face $F$, redemption $C$, coupon rate $r$ per period, $n$ periods, and yield $i$.

$$P = Fr\,a_{\overline{n}|i} + C v^n$$ The coupon is $Fr$ per period and $Cv^n$ is the present value of redemption.

State the premium/discount relationship for a bond (with $C = F$).

If coupon rate $r >$ yield $i$, the bond sells at a premium ($P > F$); if $r < i$, at a discount ($P < F$); if $r = i$, at par ($P = F$). Premium amortizes down (and discount accretes up) toward $F$ at maturity.

State the Makeham bond price formula.

$$P = K + \frac{g}{i}(C - K)$$ where $K = C v^n$ is the present value of redemption and $g = \frac{Fr}{C}$ is the modified coupon rate.

Define the (Macaulay) duration of a series of cash flows.

$$D = \frac{\sum_t t\, v^t\, CF_t}{\sum_t v^t\, CF_t}$$ It is the present-value-weighted average time to receipt of the cash flows, measured in periods.

Relate modified duration to Macaulay duration and to price sensitivity.

Modified duration $D_{mod} = \frac{D_{Mac}}{1+i}$. Price sensitivity: $\frac{dP}{di} \approx -D_{mod}\,P$, so a small yield change $\Delta i$ gives $\frac{\Delta P}{P} \approx -D_{mod}\,\Delta i$.

Define convexity and give the second-order price-change approximation.

Convexity $C = \frac{1}{P}\frac{d^2P}{di^2}$. Second-order approximation: $$\frac{\Delta P}{P} \approx -D_{mod}\,\Delta i + \tfrac{1}{2}C(\Delta i)^2$$

State the three conditions for Redington immunization of a portfolio of assets against liabilities.

(1) Present value of assets equals PV of liabilities: $PV_A = PV_L$. (2) Duration of assets equals duration of liabilities (equal first derivatives of surplus w.r.t. $i$). (3) Convexity of assets exceeds convexity of liabilities (asset second derivative greater). This protects against small interest rate changes.

How does full (exact / dedication) immunization differ from Redington immunization?

Full immunization structures assets so the surplus is protected against interest rate movements of any size (not just small ones), typically with one asset cash flow before and one after each liability. Redington only guarantees protection against small changes in $i$.

Define a forward contract and give the payoff to the long position at expiration.

A forward is an agreement to buy/sell an asset at a fixed price $K$ at a future date $T$. Long payoff at expiration: $S_T - K$, where $S_T$ is the spot price at $T$. It is an obligation, not an option, with zero cost to enter.

Give the payoffs at expiration for a long call and a long put with strike $K$.

Long call: $\max(S_T - K, 0)$. Long put: $\max(K - S_T, 0)$. Profit subtracts the accumulated premium paid.

State the put-call parity relationship for European options on a non-dividend asset.

$$C - P = S_0 - K e^{-rT}$$ where $C$ and $P$ are call and put prices, $S_0$ the spot, $K$ the strike, $r$ the continuously compounded risk-free rate, and $T$ the time to expiry.

In economics, define price elasticity of demand and state what makes demand elastic vs. inelastic.

$E_d = \left|\frac{\%\,\Delta Q_d}{\%\,\Delta P}\right|$. Demand is elastic if $E_d > 1$ (quantity responds strongly), inelastic if $E_d < 1$, and unit elastic if $E_d = 1$.

State the GDP expenditure identity used in VEE Economics.

$$Y = C + I + G + (X - M)$$ where $C$ is consumption, $I$ investment, $G$ government spending, and $X - M$ net exports.

In accounting, state the fundamental accounting equation and how it underlies the balance sheet.

$$\text{Assets} = \text{Liabilities} + \text{Equity}$$ The balance sheet must always balance because every transaction affects both sides equally (double-entry bookkeeping).

In corporate finance, give the Capital Asset Pricing Model (CAPM) formula for expected return.

$$E[R_i] = R_f + \beta_i\big(E[R_m] - R_f\big)$$ where $R_f$ is the risk-free rate, $\beta_i$ the asset's systematic risk, and $E[R_m] - R_f$ the market risk premium.

In VEE Mathematical Statistics, define an unbiased estimator and state the formula for mean squared error.

An estimator $\hat{\theta}$ is unbiased if $E[\hat{\theta}] = \theta$. Mean squared error: $$\text{MSE}(\hat{\theta}) = \operatorname{Var}(\hat{\theta}) + \big(\text{Bias}(\hat{\theta})\big)^2$$ For unbiased estimators, MSE equals the variance.

In hypothesis testing, define Type I and Type II errors and relate them to significance and power.

Type I error: rejecting a true null hypothesis; its probability is $\alpha$ (significance level). Type II error: failing to reject a false null; its probability is $\beta$. Power $= 1 - \beta$ is the probability of correctly rejecting a false null.