Associate of the Society of Actuaries (ASA/FSA) Exam FM — Financial Mathematics Syllabus

Exam FM — Financial Mathematics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Exam FM — Financial Mathematics in Associate of the Society of Actuaries (ASA/FSA), not a summary of it.

1. 1

   Time Value of Money

   3 topics

   • Interest and Discount Measures
     • Simple vs. compound interest
     • Effective and nominal rates of interest
     • Effective and nominal rates of discount
     • Force of interest
   • Accumulation and Present Value
     • Accumulation and discount functions
     • Equation of value and unknown time/rate
   • Rate Conversions and Equivalence
     • Converting between i, d, and the force of interest
     • Inflation-adjusted (real) interest rates
2. 2

   Annuities and Cash Flow Valuation

   4 topics

   • Level Annuities
     • Annuity-immediate and annuity-due
     • Deferred annuities and perpetuities
   • Varying Annuities
     • Arithmetic increasing/decreasing annuities
     • Geometric (compound) increasing annuities
     • Continuously varying payment streams
   • Non-Annual Payments
     • Payments more/less frequent than interest conversion
     • Continuously payable annuities
   • Yield Rates and Project Appraisal
     • Net present value and internal rate of return
     • Dollar-weighted vs. time-weighted returns
     • Reinvestment rate considerations
3. 3

   Loans and Amortization

   3 topics

   • Loan Repayment Methods
     • Amortization method
     • Sinking fund method
   • Amortization Schedules
     • Outstanding balance (prospective and retrospective)
     • Principal and interest split per payment
   • Loan Variations
     • Varying payment loans
     • Refinancing and rate changes
4. 4

   Bonds and Fixed-Income Securities

   3 topics

   • Bond Pricing
     • Basic price formula and premium/discount
     • Makeham's formula
     • Pricing between coupon dates
   • Bond Yields and Amortization
     • Yield to maturity
     • Premium amortization and discount accumulation (book value)
     • Callable bonds
   • Term Structure of Interest Rates
     • Spot rates and forward rates
     • Yield curve interpretation
5. 5

   Interest Rate Risk and Immunization

   3 topics

   • Duration and Convexity
     • Macaulay and modified duration
     • Convexity and price sensitivity
   • Immunization Strategies
     • Redington immunization conditions
     • Full immunization
     • Cash-flow matching (dedication)
   • Asset-Liability Management Basics
     • Matching asset and liability cash flows
     • Surplus and reinvestment risk
6. 6

   Financial Derivatives and Risk Management

   3 topics

   • Forwards and Futures
     • Long and short positions
     • Forward pricing and payoff diagrams
   • Options
     • Call and put payoffs
     • Put-call parity
   • Hedging and Spreads
     • Swaps (interest rate)
     • Spreads, straddles, and collars

Exam FM — Financial Mathematics flashcards for Associate of the Society of Actuaries (ASA/FSA)

25 of 54 cards from the Exam FM — Financial Mathematics deck — real questions with worked answers.

1. What is the effective rate of interest $i$ over a period, expressed in terms of accumulated values $A(t)$?

   $i = \dfrac{A(t+1)-A(t)}{A(t)}$, the ratio of interest earned in the period to the amount at the start of the period.

2. Define the effective rate of discount $d$ for a period and give its relationship to the effective interest rate $i$.

   $d$ is interest as a fraction of the end-of-period value: $d = \dfrac{i}{1+i}$. Equivalently $i = \dfrac{d}{1-d}$ and $d = iv$, where $v=\frac{1}{1+i}$.

3. What is the discount factor $v$ and what does $v^{n}$ represent?

   $v = \dfrac{1}{1+i} = 1-d$. The factor $v^{n}=(1+i)^{-n}$ is the present value of $1$ payable $n$ periods in the future.

4. State the fundamental identity linking $i$, $d$, and $v$.

   $$d = i v = \frac{i}{1+i}, \qquad i - d = i d, \qquad v = 1-d.$$

5. Define the force of interest $\delta_t$ and give it for a compound-interest fund.

   $\delta_t = \dfrac{A'(t)}{A(t)} = \dfrac{d}{dt}\ln A(t)$. Under constant compound interest, $\delta_t=\delta=\ln(1+i)$, a constant.

6. Express the accumulation factor over $[t_1,t_2]$ in terms of the force of interest $\delta_t$.

   $$\frac{A(t_2)}{A(t_1)} = \exp\!\left(\int_{t_1}^{t_2}\delta_s\,ds\right).$$

7. Give the equivalence relationships between $\delta$, $i$, $d$, $v$.

   $e^{\delta}=1+i=\dfrac{1}{v}=\dfrac{1}{1-d}$, so $\delta=\ln(1+i)=-\ln v=-\ln(1-d)$, and $v=e^{-\delta}$.

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

   $i^{(m)}$ is an annual rate applied as $\frac{i^{(m)}}{m}$ each of $m$ subperiods: $\left(1+\dfrac{i^{(m)}}{m}\right)^{m}=1+i$.

9. Define the nominal discount rate $d^{(m)}$ and relate it to effective $d$.

   $\left(1-\dfrac{d^{(m)}}{m}\right)^{m}=1-d=v$, so the per-period discount is $\frac{d^{(m)}}{m}$ applied $m$ times.

10. Rank for a fixed effective annual rate: $d$, $d^{(m)}$, $\delta$, $i^{(m)}$, $i$.

    $$d < d^{(m)} < \delta < i^{(m)} < i,$$ with all nominal/effective rates converging to $\delta$ as $m\to\infty$.

11. What is the limiting relationship $\lim_{m\to\infty} i^{(m)}$ and $\lim_{m\to\infty} d^{(m)}$?

    Both converge to the force of interest: $\lim_{m\to\infty} i^{(m)} = \lim_{m\to\infty} d^{(m)} = \delta = \ln(1+i)$.

12. Give the present value and accumulated value of an ordinary annuity-immediate of $1$ for $n$ periods.

    $$a_{\overline{n}|}=\frac{1-v^{n}}{i}, \qquad s_{\overline{n}|}=\frac{(1+i)^{n}-1}{i}.$$ Payments occur at the end of each period.

13. Give the present value and accumulated value of an annuity-due of $1$ for $n$ periods.

    $$\ddot a_{\overline{n}|}=\frac{1-v^{n}}{d}, \qquad \ddot s_{\overline{n}|}=\frac{(1+i)^{n}-1}{d}.$$ Payments occur at the beginning of each period.

14. How are annuity-due values related to annuity-immediate values?

    $\ddot a_{\overline{n}|}=(1+i)\,a_{\overline{n}|}=a_{\overline{n}|}+1-v^{n}$ and $\ddot s_{\overline{n}|}=(1+i)\,s_{\overline{n}|}$. Also $\ddot a_{\overline{n}|}=1+a_{\overline{n-1}|}$.

15. Give the present value of a perpetuity-immediate and a perpetuity-due of $1$.

    $$a_{\overline{\infty}|}=\frac{1}{i}, \qquad \ddot a_{\overline{\infty}|}=\frac{1}{d}.$$ No accumulated value exists since payments never end.

16. What is the present value of an $n$-period deferred annuity-immediate, deferred $k$ periods?

    $_{k|}a_{\overline{n}|}=v^{k}\,a_{\overline{n}|}=a_{\overline{k+n}|}-a_{\overline{k}|}$. The first payment is at time $k+1$.

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

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

18. State the present value of a decreasing annuity-immediate $(Da)_{\overline{n}|}$ (payments $n,n-1,\dots,1$).

    $$(Da)_{\overline{n}|}=\frac{n-a_{\overline{n}|}}{i}.$$ Note $(Ia)_{\overline{n}|}+(Da)_{\overline{n}|}=(n+1)\,a_{\overline{n}|}$.

19. What is the present value of a perpetuity with payments increasing $1,2,3,\dots$ (increasing perpetuity-immediate)?

    $$(Ia)_{\overline{\infty}|}=\frac{1}{i}+\frac{1}{i^{2}}=\frac{1}{id}.$$

20. Give the present value of a geometric (compound-increasing) annuity-immediate: first payment $1$, growing at rate $g$ for $n$ payments.

    $$PV=\frac{1-\left(\frac{1+g}{1+i}\right)^{n}}{i-g}\quad(i\neq g);\qquad PV=\frac{n}{1+i}\ \text{if } i=g.$$

21. For a geometric perpetuity with payments growing at rate $g<i$, what is the present value (first payment $1$ at time 1)?

    $$PV=\frac{1}{i-g}.$$ This is the level-perpetuity formula generalized to growth.

22. How do you value an annuity paying $1$ per year via $m$thly payments of $\frac{1}{m}$ each, present value?

    $a_{\overline{n}|}^{(m)}=\dfrac{1-v^{n}}{i^{(m)}}$ and $\ddot a_{\overline{n}|}^{(m)}=\dfrac{1-v^{n}}{d^{(m)}}$, where total annual payment is $1$.

23. Give the present value of a continuously-paid level annuity of $1$ per year for $n$ years.

    $$\bar a_{\overline{n}|}=\int_0^{n} v^{t}\,dt=\frac{1-v^{n}}{\delta}.$$ Accumulated value $\bar s_{\overline{n}|}=\dfrac{(1+i)^{n}-1}{\delta}$.

24. What is the present value of a continuously increasing continuous annuity $(\bar I \bar a)_{\overline{n}|}$?

    $$(\bar I\bar a)_{\overline{n}|}=\frac{\bar a_{\overline{n}|}-n v^{n}}{\delta}.$$

25. Define the net present value (NPV) of a series of cash flows $C_t$ at rate $i$.

    $$NPV=\sum_{t} C_t\, v^{t}=\sum_t \frac{C_t}{(1+i)^{t}}.$$ Accept a project if $NPV>0$ at the required rate.

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