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#1
09-27-2006, 02:54 PM
 Tom Actuarial Outpost Administrator Contact me: Send me a PMor email me tom.troceen@dwsimpson.com CAS SOA COPA Join Date: Jan 1987 Location: Sitting in front of a red button w/ your name on it College: FSU ActSci Alumni Favorite beer: Root Posts: 11,152 Blog Entries: 7
Done with the exam? Share your Exam 2/FM study notes here!

You can post your notes online using the upload button after clicking post reply, or you can always email them to me at tom(at)actuarialoutpost.com. If that isn't easy enough, just PM me here and I can send you my fax number or we can work something else out!

Thanks!

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Questions? Contact me at tom@actuarialoutpost.com or send me a PM here.
#2
09-27-2006, 03:17 PM
 MyKenk Note Contributor CAS Join Date: Nov 2005 Location: twitter.com/mykenk College: Drake '06 Posts: 8,595

Attached find my notes for the Kellison book. Focusing on MY weak spots, there's very little in the way of formula development, i tried to make it theory heavy:
Attached Images
 Kellison Reading Notes.pdf (229.1 KB, 28240 views)
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#3
10-04-2006, 04:10 PM
 tbug Note Contributor SOA Join Date: Jul 2006 Studying for EA Exams Favorite beer: Yuengling Posts: 10,799
wow!

Thank you so much for posting these notes! I've been using them 1) to look up formulas I've forgotten when I'm studying at work (since my books are at home) and 2) to get specific information that differs from Broverman (such as GICs), since I don't have the Kellison book. You rock
#4
10-08-2006, 07:09 PM
 no driver Note Contributor SOA Join Date: Jan 2006 Studying for nothing! Posts: 2,352
FM Formulas

NEW PRINTABLE VERSION!
Scroll down to the bottom for a PDF that prints nicely. I will leave the original post up here so that folks can use it as a quick reference.

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.

no driver 10/08/2006

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.

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

$\frac{P}{i}+\frac{Q}{i^2}$ : 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:
$P=Fra_{\overline{n|}i}+Cv^n=Cga_{\overline{n|}i}+C v^n$ : 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.

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.

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:
$P'(i)=0$
$P''(i)>0$

Prints nicer than this post too.
Attached Images
 FM Formulas 2.01.pdf (110.6 KB, 19838 views)

Last edited by no driver; 11-14-2006 at 03:02 PM..
#5
10-13-2006, 07:02 PM
 ebony Member SOA Join Date: Mar 2006 Studying for CSP - GH Favorite beer: +Hb!l pnq Posts: 985

Sweet No Driver. Your awesome for this!
#6
10-13-2006, 07:35 PM
 MyKenk Note Contributor CAS Join Date: Nov 2005 Location: twitter.com/mykenk College: Drake '06 Posts: 8,595

That's a lot of formulas... I'm gonna stick to memorizing the 6 or 7 that I plan to memorize... but very impressive, none-the-less! You have the time to put this together... you're probably ready.
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#7
10-19-2006, 09:50 PM
 hao_sarah Join Date: Aug 2006 Posts: 4

Thank you for sharing.
#8
10-28-2006, 07:17 PM
 beck Member Join Date: Oct 2006 Posts: 349

thx for this nice and long summary, no driver... amazingly i recognize and remember most of those formulas, doing tons of questinos really do help : D
#9
10-28-2006, 07:18 PM
 Buzz Lightyear Notes Contributor CAS Join Date: Aug 2006 Favorite beer: Excuse me, I think the word you're searching for is "alcoholic malt beverage". Posts: 337
Using the BAII Plus workbook to calculate duration and convexity

The following is an outline of how the BA II plus calculator workbook can be used to calculate the Macaully Duration, Modified Duration and Convexity of a bond for short duration bonds quite easily. I will use the question below to help illustrate: -

Quote:
 A 2-year bond has an annual coupon rate of 10% and makes semiannual coupon payments. At the end of the 2 years, the bond's par value of \$100 is repaid. The yield on the bond is 10% compounded semiannually. Calculate the convexity of the bond.
First create a table like the screenshot below.

The following letters are assigned for calculations: -
m = coupon frequency/year
r = annual effective interest rate
Calculating the price
• Enter the original coupon and redemption value data into the fields of the CF worksheet on your calculator.
• Press the NPV key, enter the interest rate (i.e. 5% for this question), and CPT NPV.
In this question it is obviously 100 as the coupon rate and the yield rate are equal.
• Save this value into the memory.
Calculating the Mac Duration
• Change C values to "Cash Flow * t" values.
• CPT new NPV (call NPV'), and save to different memory.
Quote:
 D(MAC) = (1/m)*(NPV'/NPV)
For example D(MAC) = 1/2 * 372.3/100 = 1.86

Modified Duration
Quote:
 D(MOD) = D(MAC)/(1+r)
Calculating the Convexity
• Change C values to "Cash Flow * t^2" values.
• CPT new NPV (call NPV''), and save to different memory.
Quote:
 Convexity = (1/(1+r)^2) * [(1/m^2) * (NPV''/NPV) + D(MAC)]
For example Convexity = (1/(1.05)^4) * [1/4 * (1443.9/100) + 1.86] = 4.50

Credit: Credit where credit is due. I learned this method using Yufeng Guo's awesome manual for FM.
Attached Images

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Last edited by Buzz Lightyear; 10-28-2006 at 07:22 PM..
#10
10-28-2006, 11:21 PM
 hokensuri Member Join Date: May 2006 Posts: 106

Have you tried the question below using the formula shown above?

http://www.math.ilstu.edu/krzysio/KO-FM-Exercise4.pdf