Exam 2 FM

Financial Mathematics

The examination for this material consists of two hours of multiple-choice questions and is identical to CAS Exam 2.

The goal of the Financial Mathematics course of reading is to provide an understanding of the fundamental concepts of financial mathematics, and how those concepts are applied in calculating present and accumulated values for various streams of cash flows as a basis for future use in: reserving, valuation, pricing, duration calculation, asset/liability management, investment income, capital budgeting and valuing contingent cash flows.

The following learning outcomes are presented with the understanding that candidates are allowed to use specified calculators on the exam. The education and examination of candidates should reflect that fact. In particular, such calculators eliminate the need for candidates to learn and be examined on certain mathematical methods of approximation.

## Study Notes

Notes By: no driver 11/12/2006

### Introduction

Since ASM does not have a formula summary, I decided to compile one to use as I started working on old test questions. In the interest of other actuarial students, I thought I would share the results.

A few notes:

1. This set of formulas is mostly derived from the 3rd edition of the ASM manual for Exam FM/2. As a reference, it does not attempt to recreate the methods presented in the ASM manual and skips many of the necessary techniques for using these formulas to solve certain types of problems. In particular you will notice that there are no formulas from chapters 2 and 8, and very little from chapter 5.
2. Since the syllabus for the exam will change after the November 2006 sitting, this compilation will not be complete for exams given in 2007 and beyond, but it can probably be used as a starting point for future exam takers.
3. I may have misstated some of the explanations of the formulas either through lack of understanding or inadequate keyboard/Tex skills. Please let me know if you find errors in this document and I will attempt to correct them. Also note that some formulas have no explanation, and are intended to show identities and useful relationships between terms that have been defined previously.
4. This summary is meant as a reference. You don’t need to memorize all of these formulas to do well on the exam. In fact, most of them can be easily derived from one another. As you work problems, some of these formulas will become second nature. For some of the problems where these formulas may work, you may prefer working from first principles or an intermediate derivation. Mykenk has suggested that you only need to know five formulas for the 2006 exam: Arithmetically increasing & decreasing annuity, geometrically increasing annuity, principle repaid at time t, and the price of a bond. As you learn the material you will figure out what works for you.

### Chapter 1

Basics: $a(t)$ : accumulation function. Measures the amount in a fund with an investment of 1 at time 0 at the end of year $t$.

$a(t)-a(t-1)$ : amount of growth in year $t$.

$i_t=\frac {a(t)-a(t-1)}{a(t-1)}$ : rate of growth in year $t$, also known as the effective rate of interest in year $t$.

$A(t)=ka(t)$ : any accumulation function can be multiplied by a constant (usually the principal amount invested) to obtain a result specific to the amount invested.

Common Accumulation Functions: $a(t)=1+it$ : simple interest.

$a(t)=\prod_{j=1}^t (1+i_j)$ : variable interest.

$a(t)=(1+i)^t$ : compound interest.

Present Value and Discounting: $PV=\frac{1}{a(t)}=\frac{1}{(1+i)^t}=(1+i)^{-t}=v^t$ : amount you must invest at time 0 to get 1 at time $t$.

$d_t=\frac {a(t)-a(t-1)}{a(t)}$ : effective rate of discount in year $t$.

Some Useful Relationships: $1-d=v$

$d=\frac{i}{1+i}=iv$

$i=\frac{d}{1-d}$

Nominal Interest and Discount: $i^{(m)}$ and $d^{(m)}$ are the symbols for nominal rates of interest compounded m-thly.

$1+i=(1+\frac{i^{(m)}}{m})^m$

$i^{(m)}=m((1+i)^{\frac{1}{m}}-1)$

$1-d=(1-\frac{d^{(m)}}{m})^m$

$d^{(m)}=m(1-(1-d)^{\frac{1}{m}})$

Force of Interest: $\delta_t=\frac{1}{a(t)} \frac{d}{dt} a(t)=\frac{d}{dt}ln a(t)$ : definition of force of interest.

$a(t)=e^{\int_0^t \delta_r dr}$

If the Force of Interest is Constant: $a(t)=e^{\delta t}$

$PV=e^{-\delta t}$

$\delta = ln(1+i)$

### Chapter 3

Annuities: $a_{\overline{n|}}=\frac{1-v^n}{i}=v+v^2+..+v^n$ : PV of an annuity-immediate.

$\ddot{a}_{\overline{n|}}=\frac{1-v^n}{d}=1+v+v^2+..+v^{n-1}$ : PV of an annuity-due.

$\ddot{a}_{\overline{n|}}=(1+i)a_{\overline{n|}}=1+a_{\overline{n-1|}}$

$s_{\overline{n|}}=\frac{(1+i)^n-1}{i}=(1+i)^{n-1}+(1+i)^{n-2}+..+1$ : AV of an annuity-immediate (on the date of the last deposit).

$\ddot{s}_{\overline{n|}}=\frac{(1+i)^n-1}{d}=(1+i)^n+(1+i)^{n-1}+..+(1+i)$ : AV of an annuity-due (one period after the date of the last deposit).

$\ddot{s}_{\overline{n|}}=(1+i)s_{\overline{n|}}=s_{\overline{n+1|}}-1$

$a_{\overline{mn|}}=a_{\overline{n|}}+v^n a_{\overline{n|}}+v^{2n} a_{\overline{n|}}+..+v^{(m-1)n} a_{\overline{n|}}$

Perpetuities: $\lim_{n\to\infty} a_{\overline{n|}} = \lim_{n\to\infty} \frac{1-v^n}{i}=\frac{1}{i}=v+v^2+...= a_{\overline{\infty|}}$ : PV of a perpetuity-immediate.

$\lim_{n\to\infty} \ddot{a}_{\overline{n|}} = \lim_{n\to\infty} \frac{1-v^n}{d}=\frac{1}{d}=1+v+v^2+...= \ddot{a}_{\overline{\infty|}}$ : PV of a perpetuity-due.

$\ddot{a}_{\overline{\infty|}}-a_{\overline{\infty|}}=\frac{1}{d}-\frac{1}{i}=1$

### Chapter 4

m-thly Annuities & Perpetuities: $a_{\overline{n|}}^{(m)}=\frac{1-v^n}{i^{(m)}}=\frac{i}{i^{(m)}}a_{\overline{n|}}=s_{\overline{1|}}^{(m)}a_{\overline{n|}}$ : PV of an n-year annuity-immediate of 1 per year payable in m-thly installments.

$\ddot{a}_{\overline{n|}}^{(m)}=\frac{1-v^n}{d^{(m)}}=\frac{i}{d^{(m)}}a_{\overline{n|}}= \ddot{s}_{\overline{1|}}^{(m)}a_{\overline{n|}}$ : PV of an n-year annuity-due of 1 per year payable in m-thly installments.

$s_{\overline{n|}}^{(m)}=\frac{(1+i)^n-1}{i^{(m)}}$ : AV of an n-year annuity-immediate of 1 per year payable in m-thly installments.

$\ddot{s}_{\overline{n|}}^{(m)}=\frac{(1+i)^n-1}{d^{(m)}}$ : AV of an n-year annuity-due of 1 per year payable in m-thly installments.

$\lim_{n\to\infty} a_{\overline{n|}}^{(m)} = \lim_{n\to\infty} \frac{1-v^n}{i^{(m)}}=\frac{1}{i^{(m)}}=a_{\overline{ \infty|}}^{(m)}$ : PV of a perpetuity-immediate of 1 per year payable in m-thly installments.

$\lim_{n\to\infty} \ddot{a}_{\overline{n|}}^{(m)} = \lim_{n\to\infty} \frac{1-v^n}{d^{(m)}}=\frac{1}{d^{(m)}}= \ddot{a}_{\overline{\infty|}}^{(m)}$ : PV of a perpetuity-due of 1 per year payable in m-thly installments.

$\ddot{a}_{\overline{\infty|}}^{(m)}-a_{\overline{ \infty|}}^{(m)}=\frac{1}{d^{(m)}}-\frac{1}{i^{(m)}}=\frac{1}{m}$

Continuous Annuities: Since $\lim_{m\to\infty} i^{(m)}=\lim_{m\to\infty} d^{(m)}=\delta$,

$\lim_{m\to\infty} a_{\overline{n|}}^{(m)} = \lim_{m\to\infty} \frac{1-v^n}{i^{(m)}}=\frac{1-v^n}{\delta}= \overline{a}_{\overline{n|}}=\frac{i}{\delta} a_{\overline{n|}}$ : PV of an annuity (immediate or due) of 1 per year paid continuously.

Payments in Arithmetic Progression: In general, the PV of a series of $n$ payments, where the first payment is $P$ and each additional payment increases by $Q$ can be represented by: $A=Pa_{\overline{n|}}+Q\frac{a_{\overline{n|}}-nv^n}{i}=Pv+(P+Q)v^2+(P+2Q)v^3+..+(P+(n-1)Q)v^n$

Similarly: $\ddot{A}=P \ddot{a}_{\overline{n|}}+Q\frac{a_{\overline{n|}}-nv^n}{d}$

$S=Ps_{\overline{n|}}+Q\frac{s_{\overline{n|}}-n}{i}$ : AV of a series of $n$ payments, where the first payment is $P$ and each additional payment increases by $Q$.

$\ddot{S}=P \ddot{s}_{\overline{n|}}+Q\frac{s_{\overline{n|}}-n}{d}$

$(Ia)_{\overline{n|}}=\frac{\ddot{a}_{\overline{n|}}-nv^n}{i}$ : PV of an annuity-immediate with first payment 1 and each additional payment increasing by 1; substitute $d$ for $i$ in denominator to get due form.

$(Is)_{\overline{n|}}=\frac{\ddot{s}_{\overline{n|}}-n}{i}$ : AV of an annuity-immediate with first payment 1 and each additional payment increasing by 1; substitute $d$ for $i$ in denominator to get due form.

$(Da)_{\overline{n|}}=\frac{n-{a}_{\overline{n|}}}{i}$ : PV of an annuity-immediate with first payment $n$ and each additional payment decreasing by 1; substitute $d$ for $i$ in denominator to get due form.

$(Ds)_{\overline{n|}}=\frac{n(1+i)^n-{s}_{\overline{n|}}}{i}$ : AV of an annuity-immediate with first payment $n$ and each additional payment decreasing by 1; substitute $d$ for $i$ in denominator to get due form.

$(Ia)_{\overline{\infty|}}=\frac{1}{id}=\frac{1}{i}+\frac{1}{i^2}$ : PV of a perpetuity-immediate with first payment 1 and each additional payment increasing by 1.

$(I\ddot{a})_{\overline{\infty|}}=\frac{1}{d^2}$ : PV of a perpetuity-due with first payment 1 and each additional payment increasing by 1.

$(Ia)_{\overline{n|}} + (Da)_{\overline{n|}} = (n+1)a_{\overline{n|}}$

Additional Useful Results: $\frac{P}{i}+\frac{Q}{i^2}$ : PV of a perpetuity-immediate with first payment $P$ and each additional payment increasing by $Q$.

$(Ia)_{\overline{n|}}^{(m)}=\frac{\ddot{a}_{ \overline{n|}}-nv^n}{i^{(m)}}$ : PV of an annuity-immediate with m-thly payments of $\frac{1}{m}$ in the first year and each additional year increasing until there are m-thly payments of $\frac{n}{m}$ in the nth year.

May God Have Mercy on Your Soul: $(I^{(m)}a)_{\overline{n|}}^{(m)}=\frac{\ddot{a}_{ \overline{n|}}^{(m)}-nv^n}{i^{(m)}}$ : PV of an annuity-immediate with payments of $\frac{1}{m^2}$ at the end of the first mth of the first year, $\frac{2}{m^2}$ at the end of the second mth of the first year, and each additional payment increasing until there is a payment of $\frac{mn}{m^2}$ at the end of the last mth of the nth year.

$(\overline{I} \overline{a})_{\overline{n|}}=\frac{ \overline{a}_{\overline{n|}}-nv^n}{\delta}$ : PV of an annuity with continuous payments that are continuously increasing. Annual rate of payment is $t$ at time $t$.

$\int_0^n f(t)v^t dt$ : PV of an annuity with a continuously variable rate of payments and a constant interest rate.

$\int_0^n f(t)e^{-\int_0^t \delta_r dr} dt$ : PV of an annuity with a continuously variable rate of payment and a continuously variable rate of interest.

Payments in Geometric Progression: $\frac{1-(\frac{1+k}{1+i})^n}{i-k}$ : PV of an annuity-immediate with an initial payment of 1 and each additional payment increasing by a factor of $(1+k)$.

### Chapter 5

Definitions: $R_t$ : payment at time $t$. A negative value is an investment and a positive value is a return.

$P(i)=\sum{v^tR_t}$ : PV of a cash flow at interest rate $i$.

### Chapter 6

General Definitions: $R_t=I_t+P_t$ : payment made at the end of year $t$, split into the interest $I_t$ and the principle repaid $P_t$.

$I_t=iB_{t-1}$ : interest paid at the end of year $t$.

$P_t=R_t-I_t=(1+i)P_{t-1}+(R_t-R_{t-1})$ : principle repaid at the end of year $t$.

$B_t=B_{t-1}-P_t$ : balance remaining at the end of year $t$, just after payment is made.

On a Loan Being Paid with Level Payments: $I_t=1-v^{n-t+1}$ : interest paid at the end of year $t$ on a loan of $a_{\overline{n|}}$.

$P_t=v^{n-t+1}$ : principle repaid at the end of year $t$ on a loan of $a_{\overline{n|}}$.

$B_t=a_{\overline{n-t|}}$ : balance remaining at the end of year $t$ on a loan of $a_{\overline{n|}}$, just after payment is made.

For a loan of $L$, level payments of $\frac{L}{a_{\overline{n|}}}$ will pay off the loan in $n$ years. In this case, multiply $I_t$, $P_t$, and $B_t$ by $\frac{L}{a_{\overline{n|}}$, ie $B_t=\frac{L}{a_{\overline{n|}}}a_{\overline{n-t|}}$ etc.

Sinking Funds:

$PMT=Li+\frac{L}{s_{\overline{n|}j}}$ : total yearly payment with the sinking fund method, where $Li$ is the interest paid to the lender and $\frac{L}{s_{\overline{n|}j}}$ is the deposit into the sinking fund that will accumulate to $L$ in $n$ years. $i$ is the interest rate for the loan and $j$ is the interest rate that the sinking fund earns.

$L=(PMT-Li)s_{\overline{n|}j}$

### Chapter 7

Definitions: $P$ : Price paid for a bond.

$F$ : Par/face value of a bond.

$C$ : Redemption value of a bond.

$r$ : coupon rate for a bond.

$g=\frac{Fr}{C}$ : modified coupon rate.

$i$ : yield rate on a bond.

$K$ : PV of $C$.

$n$ : number of coupon payments.

$G=\frac{Fr}{i}$ : base amount of a bond.

$Fr=Cg$

Determination of Bond Prices: $P=Fra_{\overline{n|}i}+Cv^n=Cga_{\overline{n|}i}+Cv^n$ : price paid for a bond to yield $i$.

$P=C+(Fr-Ci)a_{\overline{n|}i}=C+(Cg-Ci)a_{\overline{n|}i}$ : Premium/Discount formula for the price of a bond.

$P-C=(Fr-Ci)a_{\overline{n|}i}=(Cg-Ci)a_{\overline{n|}i}$ : premium paid for a bond if $g>i$.

$C-P=(Ci-Fr)a_{\overline{n|}i}=(Ci-Cg)a_{\overline{n|}i}$ : discount paid for a bond if $g<i$.

Bond Amortization: When a bond is purchased at a premium or discount the difference between the price paid and the redemption value can be amortized over the remaining term of the bond. Using the terms from chapter 6: $R_t$ : coupon payment.

$I_t=iB_{t-1}$ : interest earned from the coupon payment.

$P_t=R_t-I_t=(Fr-Ci)v^{n-t+1}=(Cg-Ci)v^{n-t+1}$ : adjustment amount for amortization of premium ("write down") or

$P_t=I_t-R_t=(Ci-Fr)v^{n-t+1}=(Ci-Cg)v^{n-t+1}$ : adjustment amount for accumulation of discount ("write up").

$B_t=B_{t-1}-P_t$ : book value of bond after adjustment from the most recent coupon paid.

Price Between Coupon Dates: For a bond sold at time $k$ after the coupon payment at time $t$ and before the coupon payment at time $t+1$: $B_{t+k}^f=B_t(1+i)^k=(B_{t+1}+Fr)v^{1-k}$ : "flat price" of the bond, ie the money that actually exchanges hands on the sale of the bond.

$B_{t+k}^m=B_{t+k}^f-kFr=B_t(1+i)^k-kFr$ : "market price" of the bond, ie the price quoted in a financial newspaper.

Approximations of Yield Rates on a Bond: $i \approx \frac{nFr+C-P}{\frac{n}{2}(P+C)}$ : Bond Salesman's Method.

Price of Other Securities: $P=\frac{Fr}{i}$ : price of a perpetual bond or preferred stock.

$P=\frac{D}{i-k}$ : theoretical price of a stock that is expected to return a dividend of $D$ with each subsequent dividend increasing by $(1+k)$, $k<i$.

### Chapter 9

Recognition of Inflation: $i'=\frac{i-r}{1+r}$ : real rate of interest, where $i$ is the effective rate of interest and $r$ is the rate of inflation.

Method of Equated Time and (Macauley) Duration: $\overline{t}= \frac{\sum_{t=1}^n tR_t}{\sum_{t=1}^n R_t}$ : method of equated time.

$\overline{d}= \frac{\sum_{t=1}^n tv^tR_t}{\sum_{t=1}^n v^tR_t}$ : (Macauley) duration.

Volatility and Modified Duration: $P(i)=\sum{v^tR_t}$ : PV of a cash flow at interest rate $i$.

$\overline{v}= - \frac{P'(i)}{P(i)}=v\overline{d}=\frac{\overline{d}}{1+i}$ : volatility/modified duration.

$\overline{d}=-(1+i)\frac{P'(i)}{P(i)}$ : alternate definition of (Macauley) duration.

Convexity and (Redington) Immunization: $\overline{c}=\frac{P(i)}{P(i)}$ convexity

To achieve Redington immunization we want: $P'(i)=0$ $P(i)>0$