Institute and Faculty of Actuaries (IFoA) Exams Financial Mathematics and Actuarial Valuation (CM1, CB1) Syllabus

Define the force of mortality $\mu_{x}$ and relate it to $_{t}p_{x}$.

$\mu_{x}=-\frac{1}{l_{x}}\frac{d l_{x}}{dx}=-\frac{d}{dx}\ln l_{x}$. Then $_{t}p_{x}=\exp\left(-\int_{0}^{t}\mu_{x+s}\,ds\right)$.

Define the complete expectation of life $\mathring{e}_{x}$ and the curtate expectation $e_{x}$.

Complete: $\mathring{e}_{x}=\int_{0}^{\infty}{}_{t}p_{x}\,dt$ (mean future lifetime). Curtate: $e_{x}=\sum_{t=1}^{\infty}{}_{t}p_{x}$ (mean number of complete future years). Approximately $\mathring{e}_{x}\approx e_{x}+\tfrac{1}{2}$.

Define the whole-life assurance EPV $A_{x}$ (unit benefit paid at end of year of death).

$$A_{x}=\sum_{k=0}^{\infty}v^{k+1}\,{}_{k|}q_{x}=\sum_{k=0}^{\infty}v^{k+1}\,{}_{k}p_{x}\,q_{x+k}.$$ It is the expected present value of $1$ paid at the end of the year of death.

State the EPV of a whole-life annuity-due $\ddot{a}_{x}$ and its relationship with $A_{x}$.

$\ddot{a}_{x}=\sum_{k=0}^{\infty}v^{k}\,{}_{k}p_{x}$. Relationship: $A_{x}=1-d\,\ddot{a}_{x}$, equivalently $\ddot{a}_{x}=\frac{1-A_{x}}{d}$.

Write the EPV of an $n$-year term assurance $A^{1}_{x:\overline{n}|}$ and an $n$-year pure endowment $A_{x:\overline{n}|}^{\ \ 1}$.

Term: $A^{1}_{x:\overline{n}|}=\sum_{k=0}^{n-1}v^{k+1}\,{}_{k}p_{x}\,q_{x+k}$. Pure endowment: $A_{x:\overline{n}|}^{\ \ 1}=v^{n}\,{}_{n}p_{x}$. Their sum is the endowment assurance $A_{x:\overline{n}|}$.

State the endowment assurance relationship linking $A_{x:\overline{n}|}$ and $\ddot{a}_{x:\overline{n}|}$.

$$A_{x:\overline{n}|}=1-d\,\ddot{a}_{x:\overline{n}|},\qquad \ddot{a}_{x:\overline{n}|}=\frac{1-A_{x:\overline{n}|}}{d}.$$

State the equivalence (equation of value) principle for setting a net premium.

At the policy outset, the EPV of premium income equals the EPV of benefit outgo: $\text{EPV(premiums)}=\text{EPV(benefits)}$. The net premium ignores expenses; the gross premium includes expenses.

Give the net annual premium $P_{x}$ for a whole-life assurance of $1$ payable by level annual premiums.

$$P_{x}=\frac{A_{x}}{\ddot{a}_{x}}=\frac{d\,A_{x}}{1-A_{x}}=\frac{1}{\ddot{a}_{x}}-d.$$

Define the net premium reserve and state the two methods of calculating it.

The reserve is the EPV of future benefits minus the EPV of future net premiums (prospective method). Equivalently, the retrospective method accumulates past premiums less benefits. For net premium reserves on the original basis both agree.

Write the prospective net premium reserve $_{t}V_{x}$ for a whole-life assurance of $1$.

$$_{t}V_{x}=A_{x+t}-P_{x}\,\ddot{a}_{x+t}=1-\frac{\ddot{a}_{x+t}}{\ddot{a}_{x}}.$$

State the general purpose of a reserve and why reserves are needed for a level-premium policy.

A reserve is the amount the insurer holds to meet future liabilities. For level premiums, early premiums exceed the cost of cover (risk rises with age), so the surplus is held as a reserve to fund later years when premiums fall short of the cost.

State Thiele's differential equation for the reserve $_{t}V$ of a policy under a continuous-time model.

$$\frac{d}{dt}\,_{t}V=\delta\,_{t}V+P_{t}-\mu_{x+t}\left(S_{t}-{}_{t}V\right)-(\text{expenses}),$$ where $P_t$ is premium rate, $S_t$ the death benefit, and $S_t-{}_tV$ the death strain at risk.

Define the death strain at risk (DSAR) and the expected death strain (EDS) for a policy in a given year.

DSAR $=S-{}_{t}V$ (sum at risk: death benefit payable less reserve held). Expected death strain $=q_{x+t}\times\text{DSAR}$, the expected cost above the reserve of paying death claims that year.

In a multiple-state model, write the Kolmogorov forward equation form for the transition probability and define a transition intensity $\mu^{ij}_{x}$.

$\mu^{ij}_{x}$ is the instantaneous rate of transition from state $i$ to state $j$ at age $x$ ($j\neq i$). For an occupancy probability $_{t}p^{ii}_{x}=\exp\left(-\int_{0}^{t}\sum_{j\neq i}\mu^{ij}_{x+s}\,ds\right)$ (probability of remaining in state $i$).

In the alive–dead two-state model, relate the transition intensity to ordinary mortality.

The single intensity $\mu^{AD}_{x}=\mu_{x}$ (force of mortality). Then $_{t}p^{AA}_{x}={}_{t}p_{x}=\exp(-\int_{0}^{t}\mu_{x+s}ds)$ and $_{t}p^{AD}_{x}={}_{t}q_{x}$.

Define $_{t}p_{xy}$ (joint-life survival) and $_{t}p_{\overline{xy}}$ (last-survivor survival) assuming independence.

Joint life (both survive): $_{t}p_{xy}={}_{t}p_{x}\cdot{}_{t}p_{y}$. Last survivor (at least one survives): $_{t}p_{\overline{xy}}={}_{t}p_{x}+{}_{t}p_{y}-{}_{t}p_{x}\,{}_{t}p_{y}$.

State the identity linking joint-life and last-survivor annuities/assurances for two lives $x$ and $y$.

$a_{x}+a_{y}=a_{xy}+a_{\overline{xy}}$ and $A_{x}+A_{y}=A_{xy}+A_{\overline{xy}}$. (The values for the two individuals split into joint-life and last-survivor components.)

For a bond redeemable at $R$ in $n$ years paying coupons of $D$ per year (annually), write the price $P$ at yield $i$.

$$P=D\,a_{\overline{n}|}+R\,v^{n},$$ the present value of the coupon annuity plus the redemption payment, all at the required yield $i$.

State the relationship between a bond's coupon rate and its yield to maturity that determines whether it trades at a premium or discount.

If coupon rate $>$ yield, the bond trades at a premium (price $>$ redemption value). If coupon rate $<$ yield, it trades at a discount. If equal, it trades at par.

Define the running (income) yield and the gross redemption yield of a fixed-interest security.

Running yield $=\frac{\text{annual coupon}}{\text{price}}$, measuring income only. Gross redemption yield is the IRR equating price to the present value of all gross coupons and the redemption proceeds (allows for capital gain/loss).

Define the Macaulay duration of a series of cash flows and state its interpretation.

$$\text{Macaulay duration}=\frac{\sum_{t}t\,C_{t}v^{t}}{\sum_{t}C_{t}v^{t}}.$$ It is the present-value-weighted mean term of the cash flows and measures sensitivity of price to interest-rate changes.

Define the volatility (modified duration) and convexity of a cash-flow series.

Effective/modified duration (volatility) $\nu=-\frac{1}{P}\frac{dP}{di}=\frac{\sum t\,C_{t}v^{t+1}}{P}=\frac{\text{Macaulay duration}}{1+i}$. Convexity $c=\frac{1}{P}\frac{d^{2}P}{di^{2}}=\frac{\sum t(t+1)C_{t}v^{t+2}}{P}$, measuring curvature of price w.r.t. yield.

State Redington's three conditions for immunisation of a portfolio of assets against small interest-rate changes.

(1) PV of assets $=$ PV of liabilities; (2) the durations (volatilities) are equal, $\frac{dV_A}{di}=\frac{dV_L}{di}$; (3) the convexity of assets exceeds that of liabilities, $\frac{d^{2}V_A}{di^{2}}>\frac{d^{2}V_L}{di^{2}}$.

Define the spot rate $y_{t}$ and the forward rate $f_{t,r}$ in the term structure of interest rates.

The spot rate $y_{t}$ is the annual effective yield on a zero-coupon bond maturing at time $t$: price $=(1+y_{t})^{-t}$. The forward rate $f_{t,r}$ is the rate agreed now for borrowing/lending over $[t,t+r]$, satisfying $(1+y_{t+r})^{t+r}=(1+y_{t})^{t}(1+f_{t,r})^{r}$.

Name and briefly describe three theories explaining the shape of the yield curve.

Expectations theory: long rates reflect expected future short rates. Liquidity preference: investors demand a premium for longer terms, raising long yields. Market segmentation (preferred habitat): supply/demand in distinct maturity segments set rates, allowing non-smooth curves.

State the dividend discount (Gordon growth) model for the value of an equity with dividends growing at constant rate $g$.

$$P_{0}=\frac{D_{1}}{i-g}=\frac{D_{0}(1+g)}{i-g},\quad i>g,$$ where $D_{1}$ is next year's dividend and $i$ the required return; valid only when the discount rate exceeds the growth rate.

List the three primary financial statements and what each shows.

Balance sheet (statement of financial position): assets, liabilities and equity at a point in time. Income statement (profit and loss): revenues and expenses over a period, giving profit. Cash flow statement: cash inflows/outflows over a period (operating, investing, financing).

State the fundamental accounting equation and the distinction between accruals (profit) accounting and cash accounting.

Accounting equation: Assets $=$ Liabilities $+$ Shareholders' equity. Accruals accounting recognises revenue/expense when earned/incurred (matching), so profit differs from cash; cash accounting records transactions only when cash moves.

Distinguish depreciation methods: straight-line vs reducing-balance.

Straight-line: equal annual charge $\frac{\text{cost}-\text{residual}}{\text{useful life}}$. Reducing (declining) balance: a fixed percentage of the asset's written-down value each year, giving higher charges early and lower later. Both spread an asset's cost over its useful life.

Define gearing (leverage) and explain its effect on a company's risk and returns.

Gearing measures debt relative to equity (e.g. $\frac{\text{debt}}{\text{debt}+\text{equity}}$). Higher gearing magnifies returns to equity when profits are good but increases financial risk and the volatility of shareholder returns, since interest must be paid before dividends.

Compare debt finance and equity finance from a company's perspective.

Debt: fixed interest (tax-deductible), ranks ahead of equity, no ownership dilution, but obligatory payments raise insolvency risk. Equity: no obligation to pay dividends, permanent capital, but dividends are not tax-deductible, ownership/control is diluted, and equity holders demand a higher return for greater risk.

Distinguish a company's systematic (market) risk from its specific (diversifiable) risk.

Systematic risk affects the whole market and cannot be removed by diversification (compensated by higher expected return). Specific (unsystematic) risk is unique to a firm/asset and can be diversified away in a well-spread portfolio, so it earns no risk premium.

Explain how corporation tax and capital allowances affect the appraisal of a capital project's cash flows.

Project profits are reduced by corporation tax, lowering after-tax cash flows used in NPV. Capital allowances (tax depreciation) reduce taxable profit, generating tax savings (a positive cash flow), so appraisal uses after-tax cash flows discounted at an after-tax required return.

Explain the difference between an investor's real rate of return and nominal rate of return, and the link via inflation.

Nominal return includes inflation; real return is adjusted for it. With inflation rate $e$: $1+i_{\text{real}}=\frac{1+i_{\text{nominal}}}{1+e}$, so $i_{\text{real}}\approx i_{\text{nominal}}-e$ for small rates.

Describe the purpose and basic mechanism of a discounted cash flow approach to comparing two investment projects with different cash-flow timings.

Discount each project's net cash flows at a common risk discount rate to obtain NPVs on a comparable present-value basis. Choose the higher NPV. This properly accounts for the timing and magnitude of cash flows, unlike undiscounted comparisons.

State how to value an annuity whose payments increase geometrically at rate $g$ per year (first payment $1$ at end of year 1, $n$ payments).

$$PV=\sum_{t=1}^{n}(1+g)^{t-1}v^{t}=\frac{1-\left(\frac{1+g}{1+i}\right)^{n}}{i-g}\quad(i\neq g).$$ This is a level annuity evaluated at the modified rate $i'=\frac{1+i}{1+g}-1$.

Define working capital and explain its significance for a business.

Working capital $=$ current assets $-$ current liabilities; it funds day-to-day operations (inventory, receivables less payables). Adequate working capital ensures short-term liquidity to meet obligations; too little risks insolvency, too much ties up capital inefficiently.