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

Financial Mathematics and Actuarial Valuation (CM1, CB1) syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Financial Mathematics and Actuarial Valuation (CM1, CB1) in Institute and Faculty of Actuaries (IFoA) Exams, not a summary of it.

1. 1

   Theory of Interest

   3 topics

   • Interest rates and discounting
     • Simple, compound and continuously compounded interest
     • Nominal and effective rates, force of interest
     • Real and money interest rates and inflation adjustment
   • Annuities and cash flow valuation
     • Level and increasing annuities certain
     • Deferred and continuously payable annuities
     • Equation of value and unknown time/rate problems
   • Loan schedules and capital projects
     • Loan repayment schedules and outstanding capital
     • Net present value and internal rate of return
     • Payback period and discounted payback appraisal
2. 2

   Life Contingencies

   3 topics

   • Life tables and assurance functions
     • Survival probabilities and curtate future lifetime
     • Whole life, term and endowment assurances
     • Life annuities and their relationships
   • Premiums and reserves
     • Net and gross premium calculation
     • Prospective and retrospective reserves
     • Thiele's differential equation and recursive reserving
   • Multiple state and multiple life models
     • Joint life and last survivor functions
     • Multiple decrement tables
     • Profit testing and unit-linked contracts
3. 3

   Bonds, Equities and Term Structure

   3 topics

   • Fixed-income securities
     • Bond pricing, yields and accrued interest
     • Duration, convexity and immunisation theory
   • Term structure of interest rates
     • Spot, forward and par yields
     • Theories of the yield curve
   • Equity and project cash flow analysis
4. 4

   Corporate Finance and Financial Reporting (CB1)

   3 topics

   • Financial statements
     • Income statement, balance sheet and cash flow statement
     • Accounting concepts under IFRS
     • Ratio analysis and interpretation of accounts
   • Corporate finance principles
     • Capital structure and the cost of capital (WACC)
     • Dividend policy and capital project appraisal
     • Sources of finance: equity, debt and hybrid instruments
   • Taxation and the business environment

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

25 of 57 cards from the Financial Mathematics and Actuarial Valuation (CM1, CB1) deck — real questions with worked answers.

1. Define the accumulation factor $A(t_1, t_2)$ and the discount factor $v(t_1,t_2)$ under a constant effective rate $i$ per period.

   $A(t_1,t_2)=(1+i)^{t_2-t_1}$ accumulates a unit invested at $t_1$ to time $t_2$. The discount factor is its reciprocal, $v(t_1,t_2)=(1+i)^{-(t_2-t_1)}$, with $v=\frac{1}{1+i}$ for one period.

2. State the relationship between the effective annual rate $i$ and the nominal rate $i^{(p)}$ convertible (compounded) $p$ times per year.

   $$1+i=\left(1+\frac{i^{(p)}}{p}\right)^{p}\quad\Rightarrow\quad i^{(p)}=p\left[(1+i)^{1/p}-1\right].$$

3. How is the force of interest $\delta$ related to the effective annual rate $i$, and what is the accumulation factor over time $t$ for constant force?

   $\delta=\ln(1+i)$, equivalently $1+i=e^{\delta}$. With constant force, a unit accumulates as $e^{\delta t}$ over $t$ years.

4. For a time-varying force of interest $\delta(t)$, write the accumulation factor from time $0$ to $t$ and the present value at time $0$ of a payment at time $t$.

   Accumulation: $A(0,t)=\exp\left(\int_{0}^{t}\delta(s)\,ds\right)$. Present value of $1$ paid at $t$: $\exp\left(-\int_{0}^{t}\delta(s)\,ds\right)$.

5. State the relationship between the effective rate of interest $i$ and the effective rate of discount $d$.

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

6. Give the present value of a level annuity-immediate of $1$ per period for $n$ periods, $a_{\overline{n}|}$.

   $$a_{\overline{n}|}=\frac{1-v^{n}}{i}=v+v^{2}+\dots+v^{n}.$$ Payments are made at the end of each period.

7. Give the present value of a level annuity-due of $1$ per period for $n$ periods, $\ddot{a}_{\overline{n}|}$, and relate it to $a_{\overline{n}|}$.

   $$\ddot{a}_{\overline{n}|}=\frac{1-v^{n}}{d}=1+v+\dots+v^{n-1}.$$ Relationship: $\ddot{a}_{\overline{n}|}=(1+i)\,a_{\overline{n}|}=1+a_{\overline{n-1}|}$.

8. Give the accumulated value of a level annuity-immediate of $1$ for $n$ periods, $s_{\overline{n}|}$.

   $$s_{\overline{n}|}=\frac{(1+i)^{n}-1}{i}=(1+i)^{n}\,a_{\overline{n}|}.$$ It is the value at time $n$ of $n$ unit end-of-period payments.

9. Write the present value of a level perpetuity-immediate and a perpetuity-due of $1$ per period.

   Perpetuity-immediate: $a_{\overline{\infty}|}=\frac{1}{i}$. Perpetuity-due: $\ddot{a}_{\overline{\infty}|}=\frac{1}{d}$. Both require $i>0$ for convergence.

10. State the present value of an increasing annuity-immediate $(Ia)_{\overline{n}|}$ paying $1,2,\dots,n$ at the ends of periods $1$ to $n$.

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

11. Give the present value of a level annuity-immediate of $1$ per year payable $p$ times per year (each instalment $\tfrac{1}{p}$), $a_{\overline{n}|}^{(p)}$.

    $$a_{\overline{n}|}^{(p)}=\frac{1-v^{n}}{i^{(p)}}.$$

12. Give the present value of a continuously payable level annuity of rate $1$ per year for $n$ years, $\bar{a}_{\overline{n}|}$.

    $$\bar{a}_{\overline{n}|}=\int_{0}^{n}v^{t}\,dt=\frac{1-v^{n}}{\delta}.$$

13. For a loan repaid by level instalments, how are the interest and capital portions of each repayment determined, and how does the split change over time?

    Each instalment's interest portion is $i$ times the outstanding capital at the start of the period; the remainder repays capital. Over time the interest portion decreases and the capital portion increases.

14. State the two methods for finding the outstanding loan balance (capital) immediately after the $t$-th repayment of a level-instalment loan.

    Prospective method: present value of remaining repayments, $X a_{\overline{n-t}|}$. Retrospective method: original loan accumulated less repayments accumulated, $L(1+i)^{t}-X s_{\overline{t}|}$. Both give the same value.

15. How is the level annual repayment $X$ on a loan of amount $L$ over $n$ years at rate $i$ calculated?

    $X=\dfrac{L}{a_{\overline{n}|}}$, since the present value of repayments must equal the loan: $L=X a_{\overline{n}|}$.

16. Define the net present value (NPV) of a project and the decision rule based on it.

    $\text{NPV}=\sum_{t}C_{t}\,v^{t}$ (or $\int C(t)v^{t}dt$), the discounted value of all net cash flows at the risk discount rate. Rule: accept a project if $\text{NPV}>0$; rank competing projects by highest NPV.

17. Define the internal rate of return (IRR) of a project and state a limitation of the measure.

    The IRR is the rate $i$ solving $\text{NPV}(i)=0$. Limitations: it may not exist or may not be unique when net cash flows change sign more than once; it can give a different ranking from NPV.

18. Define the discounted payback period of a project.

    The discounted payback period is the first time $t$ at which the accumulated NPV of cash flows (discounted at the risk rate) first becomes non-negative — i.e. the project has recouped its outlay in present-value terms.

19. Define $l_{x}$, $d_{x}$, and the survival/death probabilities $p_{x}$ and $q_{x}$ in a life table.

    $l_{x}$ = expected number alive at exact age $x$; $d_{x}=l_{x}-l_{x+1}$ = expected deaths between $x$ and $x+1$. $p_{x}=\frac{l_{x+1}}{l_{x}}$ (survival), $q_{x}=\frac{d_{x}}{l_{x}}=1-p_{x}$ (death within one year).

20. Express $_{t}p_{x}$ and $_{t}q_{x}$ in terms of $l_{x}$, and define $_{t|}q_{x}$ (deferred mortality).

    $_{t}p_{x}=\frac{l_{x+t}}{l_{x}}$, $_{t}q_{x}=1-{}_{t}p_{x}=\frac{l_{x}-l_{x+t}}{l_{x}}$. Deferred: $_{t|}q_{x}={}_{t}p_{x}\,q_{x+t}=\frac{l_{x+t}-l_{x+t+1}}{l_{x}}$, the probability of dying between ages $x+t$ and $x+t+1$.

21. 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 $_{t}p_{x}=\exp\left(-\int_{0}^{t}\mu_{x+s}\,ds\right)$.

22. 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 $\mathring{e}_{x}\approx e_{x}+\tfrac{1}{2}$.

23. 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.

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

25. 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 $A_{x:\overline{n}|}$.

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