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

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.

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 $1$.

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 $\bar s_{\overline{n}|}=\dfrac{(1+i)^{n}-1}{\delta}$.

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

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

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.

Define the yield rate (internal rate of return, IRR) of a transaction.

The IRR is the rate $i$ at which $NPV=0$: $\sum_t C_t (1+i)^{-t}=0$. It equates present value of inflows and outflows.

Compare the dollar-weighted and time-weighted rates of return.

Dollar-weighted (money-weighted) is the IRR solving $\sum C_t(1+i)^{t}$ — sensitive to timing/size of deposits. Time-weighted chains sub-period returns: $1+i=\prod(1+r_k)$ — removes cash-flow timing effect, measuring manager performance.

Give the approximate simple-interest dollar-weighted return formula for a fund over a year.

$$i \approx \frac{I}{A+\sum_t C_t(1-t)},$$ where $I$ is interest earned, $A$ the beginning balance, and $C_t$ a deposit at time $t$ (fraction of year).

What is the reinvestment-rate consideration when comparing two projects by IRR?

IRR implicitly assumes interim cash flows are reinvested at the IRR. When that is unrealistic, use NPV at a stated reinvestment rate or compute a modified IRR; IRR can also be non-unique with sign changes in cash flows.

Describe the amortization method of loan repayment and how each payment is split.

Equal periodic payments are made; each payment first covers interest on the outstanding balance, with the remainder reducing principal. Outstanding balance is the present value of remaining payments.

For a loan of $L$ repaid by level payments over $n$ periods, give the payment and outstanding balance formulas.

Payment $R=\dfrac{L}{a_{\overline{n}|}}$. Outstanding balance after $t$ payments (prospective): $B_t=R\,a_{\overline{n-t}|}$; retrospective: $B_t=L(1+i)^{t}-R\,s_{\overline{t}|}$.

In an amortization schedule, give the interest and principal portions of the $t$-th payment $R$.

Interest paid $I_t=R\,(1-v^{n-t+1})$; principal repaid $P_t=R\,v^{n-t+1}$. Principal repayments form a geometric sequence: $P_{t+1}=P_t(1+i)$.

Describe the sinking-fund method of loan repayment.

The borrower pays periodic interest $Li$ on the full loan to the lender, and separately deposits into a sinking fund earning rate $j$ to accumulate the principal $L$ by maturity. Sinking-fund deposit $=L/s_{\overline{n}|j}$.

When sinking-fund rate $j$ equals loan rate $i$, how does the sinking-fund method compare to amortization?

If $j=i$, total periodic outlay equals the amortization payment, and the net amount of principal outstanding is identical. If $j<i$ the borrower's effective cost is higher than amortization.

How is an outstanding loan balance found by the retrospective vs prospective method?

Prospective: PV of remaining payments, $B_t=R\,a_{\overline{n-t}|}$. Retrospective: original loan accumulated minus accumulated payments, $B_t=L(1+i)^{t}-R\,s_{\overline{t}|}$. Both give the same value.

For a loan repaid with payments that vary, how do you handle a balloon or drop (final irregular) payment?

Find the integer number of full payments, then adjust the final payment: a balloon payment is a larger final payment added to the last regular payment; a drop payment is a smaller final payment. Solve so the outstanding balance is exactly cleared.

Give the basic price formula for a bond (general/Makeham not required) with face $F$, redemption $C$, coupon rate $r$, $n$ coupons, yield $i$.

$$P = F r\, a_{\overline{n}|i} + C\,v^{n},$$ where $Fr$ is the coupon amount and $C$ the redemption value, both discounted at yield $i$.

State the premium/discount formula for a bond price.

$$P = C + (Fr - Ci)\,a_{\overline{n}|i}.$$ The bond sells at a premium if $Fr>Ci$ and at a discount if $Fr<Ci$.

When does a bond sell at par, premium, or discount?

At par ($P=C$) when coupon rate equals yield ($Fr=Ci$, i.e. $g=i$ for $C=F$). At a premium when coupon rate $>$ yield; at a discount when coupon rate $<$ yield.

Define the modified coupon rate $g$ and Makeham's bond price formula.

$g=\dfrac{Fr}{C}$ (coupon per unit redemption). Makeham: $P=K+\dfrac{g}{i}(C-K)$, where $K=Cv^{n}$ is the PV of redemption.

Describe amortization of bond premium ('writing down') and accumulation of discount ('writing up').

For a premium bond, each coupon exceeds the yield-based interest; the excess writes the book value down toward $C$. For a discount bond, the yield interest exceeds the coupon; the shortfall writes book value up toward $C$. Book value reaches $C$ at maturity.

Give the book value of a bond between coupon dates and the write-up/down amount in period $t$.

Book value $B_t = Fr\,a_{\overline{n-t}|i}+Cv^{n-t}$. Write-down in premium = $(Fr-Ci)v^{n-t+1}=(Fr - B_{t-1}i)$ adjusting book value; equivalently the principal-adjustment column of the schedule.

Distinguish the flat (dirty) price from the market (clean) price of a bond between coupons.

Flat price = book value just after last coupon accumulated forward: $B^{flat}=B_{t}(1+i)^{f}$. Clean (market) price = flat price minus accrued coupon $f\cdot Fr$, where $f$ is the fraction of the period elapsed.

Define spot rates and the term structure of interest rates.

The spot rate $s_t$ is the annual effective yield on a zero-coupon bond maturing at time $t$; PV of $1$ at time $t$ is $(1+s_t)^{-t}$. The term structure (yield curve) is the set $\{s_t\}$ as a function of maturity.

Define the forward rate $f_{[t,t+k]}$ implied by spot rates.

The forward rate is the rate locked in today for a future period: $$(1+s_{t+k})^{t+k}=(1+s_t)^{t}\,(1+f_{[t,t+k]})^{k}.$$ A one-year forward starting at $t$: $1+f_t=\dfrac{(1+s_{t+1})^{t+1}}{(1+s_t)^{t}}$.

Define Macaulay duration and Macaulay convexity of a set of cash flows.

Macaulay duration $\bar D=\dfrac{\sum_t t\,C_t v^{t}}{\sum_t C_t v^{t}}$ (PV-weighted average time). Macaulay convexity $=\dfrac{\sum_t t^{2} C_t v^{t}}{\sum_t C_t v^{t}}$.

Define modified duration and its relationship to Macaulay duration and price sensitivity.

Modified duration $D_{mod}=-\dfrac{P'(i)}{P(i)}=\dfrac{\bar D}{1+i}$. It gives the first-order price change: $\dfrac{\Delta P}{P}\approx -D_{mod}\,\Delta i$.

Give the first/second-order (duration–convexity) approximation to a bond's price change.

$$\frac{\Delta P}{P}\approx -D_{mod}\,\Delta i+\tfrac{1}{2}\,C_{mod}\,(\Delta i)^{2},$$ where $C_{mod}=\dfrac{P''(i)}{P(i)}$ is modified convexity.

State the three Redington immunization conditions for protecting surplus against small rate changes.

1) $PV_{assets}=PV_{liabilities}$. 2) $D_{assets}=D_{liabilities}$ (matched durations / $P'_A=P'_L$). 3) Convexity of assets $>$ convexity of liabilities ($P''_A>P''_L$). This makes surplus a local minimum at the current rate.

How does full (absolute/cash-flow) immunization differ from Redington immunization?

Full immunization exactly matches asset and liability cash flows (or brackets each liability with asset flows before and after it) so surplus is protected against rate shocks of any size, not just small ones. Redington protects only against small parallel shifts.

Contrast a forward contract with a futures contract.

A forward is a customized OTC agreement to buy/sell an asset at a set price on a future date, settled at maturity with counterparty credit risk. A futures is exchange-traded, standardized, marked-to-market daily with margin, minimizing default risk.

Give the no-arbitrage forward price for an asset with spot $S_0$, risk-free rate $r$ (continuous), time $T$, no income.

$$F_{0,T}=S_0\,e^{rT}.$$ With continuous dividend yield $\delta$: $F_{0,T}=S_0 e^{(r-\delta)T}$. With discrete income PV $I$: $F_{0,T}=(S_0-I)e^{rT}$.

Define a call option and a put option, and state each holder's payoff at expiration with strike $K$, price $S_T$.

A call gives the right to buy at $K$: payoff $\max(S_T-K,0)$. A put gives the right to sell at $K$: payoff $\max(K-S_T,0)$. The buyer pays a premium for this right.

State put-call parity for European options on a non-dividend stock.

$$C - P = S_0 - K e^{-rT}.$$ A long call plus short put replicates a forward; rearranged, $C+Ke^{-rT}=P+S_0$ (a fiduciary call equals a protective put).