Exam FM Formula Summary

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收藏 2008-11-06

FM Formulas

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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:<ol style="list-style-type: decimal;"><li>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.</li><li>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.</li><li>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.</li><li>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
&amp; 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.</li></ol> 
no driver 10/08/2006 
 
Chapter 1: 
Basics: 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?a%28t%29" alt=""/> : accumulation function. Measures the amount in a fund with an investment of 1 at time 0 at the end of year <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?t" alt=""/>. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?a%28t%29-a%28t-1%29" alt=""/> : amount of growth in year <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?t" alt=""/>. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?i_t=%5Cfrac%20%7Ba%28t%29-a%28t-1%29%7D%7Ba%28t-1%29%7D" alt=""/> : rate of growth in year <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?t" alt=""/>, also known as the effective rate of interest in year <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?t" alt=""/>. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?A%28t%29=ka%28t%29" alt=""/>
: 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: 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?a%28t%29=1+it" alt=""/> : simple interest. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?a%28t%29=%5Cprod_%7Bj=1%7D%5Et%20%281+i_j%29" alt=""/> : variable interest. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?a%28t%29=%281+i%29%5Et" alt=""/> : compound interest. 
 
Present Value and Discounting: 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?PV=%5Cfrac%7B1%7D%7Ba%28t%29%7D=%5Cfrac%7B1%7D%7B%281+i%29%5Et%7D=%281+i%29%5E%7B-t%7D=v%5Et" alt=""/> : amount you must invest at time 0 to get 1 at time <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?t" alt=""/>. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?d_t=%5Cfrac%20%7Ba%28t%29-a%28t-1%29%7D%7Ba%28t%29%7D" alt=""/> : effective rate of discount in year <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?t" alt=""/>. 
 
Some Useful Relationships: 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?1-d=v" alt=""/> 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?d=%5Cfrac%7Bi%7D%7B1+i%7D=iv" alt=""/> 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?i=%5Cfrac%7Bd%7D%7B1-d%7D" alt=""/> 
 
Nominal Interest and Discount: 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?i%5E%7B%28m%29%7D" alt=""/> and <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?d%5E%7B%28m%29%7D" alt=""/> are the symbols for nominal rates of interest compounded m-thly. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?1+i=%281+%5Cfrac%7Bi%5E%7B%28m%29%7D%7D%7Bm%7D%29%5Em" alt=""/> 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?i%5E%7B%28m%29%7D=m%28%281+i%29%5E%7B%5Cfrac%7B1%7D%7Bm%7D%7D-1%29" alt=""/> 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?1-d=%281-%5Cfrac%7Bd%5E%7B%28m%29%7D%7D%7Bm%7D%29%5Em" alt=""/> 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?d%5E%7B%28m%29%7D=m%281-%281-d%29%5E%7B%5Cfrac%7B1%7D%7Bm%7D%7D%29" alt=""/> 
 
Force of Interest: 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%5Cdelta_t=%5Cfrac%7B1%7D%7Ba%28t%29%7D%20%5Cfrac%7Bd%7D%7Bdt%7D%20a%28t%29=%5Cfrac%7Bd%7D%7Bdt%7Dln%20a%28t%29" alt=""/> : definition of force of interest. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?a%28t%29=e%5E%7B%5Cint_0%5Et%20%5Cdelta_r%20dr%7D" alt=""/> 
 
If the Force of Interest is Constant: 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?a%28t%29=e%5E%7B%5Cdelta%20t%7D" alt=""/> 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?PV=e%5E%7B-%5Cdelta%20t%7D" alt=""/> 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%5Cdelta%20=%20ln%281+i%29" alt=""/> 
 
Chapter 3: 
Annuities: 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?a_%7B%5Coverline%7Bn%7C%7D%7D=%5Cfrac%7B1-v%5En%7D%7Bi%7D=v+v%5E2+..+v%5En" alt=""/> : PV of an annuity-immediate. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%5Cddot%7Ba%7D_%7B%5Coverline%7Bn%7C%7D%7D=%5Cfrac%7B1-v%5En%7D%7Bd%7D=1+v+v%5E2+..+v%5E%7Bn-1%7D" alt=""/> : PV of an annuity-due. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%5Cddot%7Ba%7D_%7B%5Coverline%7Bn%7C%7D%7D=%281+i%29a_%7B%5Coverline%7Bn%7C%7D%7D=1+%20%20a_%7B%5Coverline%7Bn-1%7C%7D%7D" alt=""/>
 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?s_%7B%5Coverline%7Bn%7C%7D%7D=%5Cfrac%7B%281+i%29%5En-1%7D%7Bi%7D=%281+i%29%5E%7Bn-1%7D+%281+i%29%5E%7Bn-2%7D+..+1" alt=""/> : AV of an annuity-immediate (on the date of the last deposit). 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%5Cddot%7Bs%7D_%7B%5Coverline%7Bn%7C%7D%7D=%5Cfrac%7B%281+i%29%5En-1%7D%7Bd%7D=%281+i%29%5En+%281+i%29%5E%7Bn-1%7D+..+%281+i%29" alt=""/> : AV of an annuity-due (one period after the date of the last deposit). 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%5Cddot%7Bs%7D_%7B%5Coverline%7Bn%7C%7D%7D=%281+i%29s_%7B%5Coverline%7Bn%7C%7D%7D=s_%20%20%7B%5Coverline%7Bn+1%7C%7D%7D-1" alt=""/> 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?a_%7B%5Coverline%7Bmn%7C%7D%7D=a_%7B%5Coverline%7Bn%7C%7D%7D+v%5En%20a_%7B%5Coverline%7Bn%7C%7D%7D+v%5E%7B2n%7D%20a_%7B%5Coverline%7Bn%7C%7D%7D+..+v%5E%7B%28m-1%29n%7D%20a_%7B%5Coverline%7Bn%7C%7D%7D" alt=""/> 
 
Perpetuities: 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%5Clim_%7Bn%5Cto%5Cinfty%7D%20a_%7B%5Coverline%7Bn%7C%7D%7D%20=%20%5Clim_%7Bn%5Cto%5Cinfty%7D%20%5Cfrac%7B1-v%5En%7D%7Bi%7D=%5Cfrac%7B1%7D%7Bi%7D=v+v%5E2+...=%20a_%7B%5Coverline%7B%5Cinfty%7C%7D%7D" alt=""/> : PV of a perpetuity-immediate. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%5Clim_%7Bn%5Cto%5Cinfty%7D%20%5Cddot%7Ba%7D_%7B%5Coverline%7Bn%7C%7D%7D%20=%20%5Clim_%7Bn%5Cto%5Cinfty%7D%20%5Cfrac%7B1-v%5En%7D%7Bd%7D=%5Cfrac%7B1%7D%7Bd%7D=1+v+v%5E2+...=%20%5Cddot%7Ba%7D_%7B%5Coverline%7B%5Cinfty%7C%7D%7D" alt=""/> : PV of a perpetuity-due. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%5Cddot%7Ba%7D_%7B%5Coverline%7B%5Cinfty%7C%7D%7D-a_%7B%5Coverline%7B%5Cinfty%7C%7D%7D=%5Cfrac%7B1%7D%7Bd%7D-%5Cfrac%7B1%7D%7Bi%7D=1" alt=""/> 
 
Chapter 4: 
m-thly Annuities &amp; Perpetuities: 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?a_%7B%5Coverline%7Bn%7C%7D%7D%5E%7B%28m%29%7D=%5Cfrac%7B1-v%5En%7D%7Bi%5E%7B%28m%29%7D%7D=%5Cfrac%7Bi%7D%7Bi%5E%7B%28m%29%7D%7Da_%7B%5Coverline%7Bn%7C%7D%7D=s%20%20_%7B%5Coverline%7B1%7C%7D%7D%5E%7B%28m%29%7Da_%7B%5Coverline%7Bn%7C%7D%7D" alt=""/> : PV of an n-year annuity-immediate of 1 per year payable in m-thly installments. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%5Cddot%7Ba%7D_%7B%5Coverline%7Bn%7C%7D%7D%5E%7B%28m%29%7D=%5Cfrac%7B1-v%5En%7D%7Bd%5E%7B%28m%29%7D%7D=%5Cfrac%7Bi%7D%7Bd%5E%7B%28m%29%7D%7Da_%7B%5Coverline%7Bn%7C%7D%7D=%20%5Cddot%7Bs%7D_%7B%5Coverline%7B1%7C%7D%7D%5E%7B%28m%29%7Da_%7B%5Coverline%7Bn%7C%7D%7D" alt=""/> : PV of an n-year annuity-due of 1 per year payable in m-thly installments. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?s_%7B%5Coverline%7Bn%7C%7D%7D%5E%7B%28m%29%7D=%5Cfrac%7B%281+i%29%5En-1%7D%7Bi%5E%7B%28m%29%7D%7D" alt=""/> : AV of an n-year annuity-immediate of 1 per year payable in m-thly installments. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%5Cddot%7Bs%7D_%7B%5Coverline%7Bn%7C%7D%7D%5E%7B%28m%29%7D=%5Cfrac%7B%281+i%29%5En-1%7D%7Bd%5E%7B%28m%29%7D%7D" alt=""/> : AV of an n-year annuity-due of 1 per year payable in m-thly installments. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%5Clim_%7Bn%5Cto%5Cinfty%7D%20a_%7B%5Coverline%7Bn%7C%7D%7D%5E%7B%28m%29%7D%20=%20%5Clim_%7Bn%5Cto%5Cinfty%7D%20%5Cfrac%7B1-v%5En%7D%7Bi%5E%7B%28m%29%7D%7D=%5Cfrac%7B1%7D%7Bi%5E%7B%28m%29%7D%7D=a_%7B%5Coverline%7B%20%5Cinfty%7C%7D%7D%5E%7B%28m%29%7D" alt=""/> : PV of a perpetuity-immediate of 1 per year payable in m-thly installments. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%5Clim_%7Bn%5Cto%5Cinfty%7D%20%5Cddot%7Ba%7D_%7B%5Coverline%7Bn%7C%7D%7D%5E%7B%28m%29%7D%20=%20%5Clim_%7Bn%5Cto%5Cinfty%7D%20%5Cfrac%7B1-v%5En%7D%7Bd%5E%7B%28m%29%7D%7D=%5Cfrac%7B1%7D%7Bd%5E%7B%28m%29%7D%7D=%20%5Cddot%7Ba%7D_%7B%5Coverline%7B%5Cinfty%7C%7D%7D%5E%7B%28m%29%7D" alt=""/> : PV of a perpetuity-due of 1 per year payable in m-thly installments. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%5Cddot%7Ba%7D_%7B%5Coverline%7B%5Cinfty%7C%7D%7D%5E%7B%28m%29%7D-a_%7B%5Coverline%7B%20%5Cinfty%7C%7D%7D%5E%7B%28m%29%7D=%5Cfrac%7B1%7D%7Bd%5E%7B%28m%29%7D%7D-%5Cfrac%7B1%7D%7Bi%5E%7B%28m%29%7D%7D=%5Cfrac%7B1%7D%7Bm%7D" alt=""/> 
 
Continuous Annuities: 
Since <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%5Clim_%7Bm%5Cto%5Cinfty%7D%20i%5E%7B%28m%29%7D=%5Clim_%7Bm%5Cto%5Cinfty%7D%20d%5E%7B%28m%29%7D=%5Cdelta" alt=""/>, 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%5Clim_%7Bm%5Cto%5Cinfty%7D%20a_%7B%5Coverline%7Bn%7C%7D%7D%5E%7B%28m%29%7D%20=%20%5Clim_%7Bm%5Cto%5Cinfty%7D%20%5Cfrac%7B1-v%5En%7D%7Bi%5E%7B%28m%29%7D%7D=%5Cfrac%7B1-v%5En%7D%7B%5Cdelta%7D=%20%5Coverline%7Ba%7D_%7B%5Coverline%7Bn%7C%7D%7D=%5Cfrac%7Bi%7D%7B%5Cdelta%7D%20a_%7B%5Coverline%7Bn%7C%7D%7D" alt=""/> : 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 <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?n" alt=""/> payments, where the first payment is <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?P" alt=""/> and each additional payment increases by <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?Q" alt=""/> can be represented by: 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?A=Pa_%7B%5Coverline%7Bn%7C%7D%7D+Q%5Cfrac%7Ba_%7B%5Coverline%7Bn%7C%7D%7D-nv%5En%7D%7Bi%7D=Pv+%28P+Q%29v%5E2+%28P+2Q%29v%5E3+..+%28P+%28n-1%29Q%29v%5En" alt=""/>
 
 
Similarly: 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%5Cddot%7BA%7D=P%20%5Cddot%7Ba%7D_%7B%5Coverline%7Bn%7C%7D%7D+Q%5Cfrac%7Ba_%7B%5Coverline%7Bn%7C%7D%7D-nv%5En%7D%7Bd%7D" alt=""/>
 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?S=Ps_%7B%5Coverline%7Bn%7C%7D%7D+Q%5Cfrac%7Bs_%7B%5Coverline%7Bn%7C%7D%7D-n%7D%7Bi%7D" alt=""/> : AV of a series of <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?n" alt=""/> payments, where the first payment is <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?P" alt=""/> and each additional payment increases by <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?Q" alt=""/>. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%5Cddot%7BS%7D=P%20%5Cddot%7Bs%7D_%7B%5Coverline%7Bn%7C%7D%7D+Q%5Cfrac%7Bs_%7B%5Coverline%7Bn%7C%7D%7D-n%7D%7Bd%7D" alt=""/>
 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%28Ia%29_%7B%5Coverline%7Bn%7C%7D%7D=%5Cfrac%7B%5Cddot%7Ba%7D_%7B%5Coverline%7Bn%7C%7D%20%20%7D-nv%5En%7D%7Bi%7D" alt=""/> : PV of an annuity-immediate with first payment 1 and each additional payment increasing by 1; substitute <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?d" alt=""/> for <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?i" alt=""/> in denominator to get due form. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%28Is%29_%7B%5Coverline%7Bn%7C%7D%7D=%5Cfrac%7B%5Cddot%7Bs%7D_%7B%5Coverline%7Bn%7C%7D%20%20%7D-n%7D%7Bi%7D" alt=""/> : AV of an annuity-immediate with first payment 1 and each additional payment increasing by 1; substitute <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?d" alt=""/> for <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?i" alt=""/> in denominator to get due form. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%28Da%29_%7B%5Coverline%7Bn%7C%7D%7D=%5Cfrac%7Bn-%7Ba%7D_%7B%5Coverline%7Bn%7C%7D%7D%7D%7Bi%7D" alt=""/> : PV of an annuity-immediate with first payment <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?n" alt=""/> and each additional payment decreasing by 1; substitute <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?d" alt=""/> for <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?i" alt=""/> in denominator to get due form. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%28Ds%29_%7B%5Coverline%7Bn%7C%7D%7D=%5Cfrac%7Bn%281+i%29%5En-%7Bs%7D_%7B%5Coverline%7Bn%7C%7D%7D%7D%7Bi%7D" alt=""/> : AV of an annuity-immediate with first payment <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?n" alt=""/> and each additional payment decreasing by 1; substitute <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?d" alt=""/> for <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?i" alt=""/> in denominator to get due form. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%28Ia%29_%7B%5Coverline%7B%5Cinfty%7C%7D%7D=%5Cfrac%7B1%7D%7Bid%7D=%5Cfrac%7B1%7D%7Bi%7D%20%20+%5Cfrac%7B1%7D%7Bi%5E2%7D" alt=""/> : PV of a perpetuity-immediate with first payment 1 and each additional payment increasing by 1. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%28I%5Cddot%7Ba%7D%29_%7B%5Coverline%7B%5Cinfty%7C%7D%7D=%5Cfrac%7B1%7D%7Bd%5E2%7D" alt=""/> : PV of a perpetuity-due with first payment 1 and each additional payment increasing by 1. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%28Ia%29_%7B%5Coverline%7Bn%7C%7D%7D%20+%20%28Da%29_%7B%5Coverline%7Bn%7C%7D%7D%20=%20%28n+1%29a_%7B%5Coverline%7Bn%7C%7D%7D" alt=""/> 
 
Additional Useful Results: 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%5Cfrac%7BP%7D%7Bi%7D+%5Cfrac%7BQ%7D%7Bi%5E2%7D" alt=""/> : PV of a perpetuity-immediate with first payment <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?P" alt=""/> and each additional payment increasing by <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?Q" alt=""/>. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%28Ia%29_%7B%5Coverline%7Bn%7C%7D%7D%5E%7B%28m%29%7D=%5Cfrac%7B%5Cddot%7Ba%7D_%7B%20%5Coverline%7Bn%7C%7D%7D-nv%5En%7D%7Bi%5E%7B%28m%29%7D%7D" alt=""/> : PV of an annuity-immediate with m-thly payments of <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%5Cfrac%7B1%7D%7Bm%7D" alt=""/> in the first year and each additional year increasing until there are m-thly payments of <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%5Cfrac%7Bn%7D%7Bm%7D" alt=""/> in the nth year. 
 
May God Have Mercy on Your Soul: 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%28I%5E%7B%28m%29%7Da%29_%7B%5Coverline%7Bn%7C%7D%7D%5E%7B%28m%29%7D=%5Cfrac%7B%5Cddot%7Ba%7D_%7B%20%5Coverline%7Bn%7C%7D%7D%5E%7B%28m%29%7D-nv%5En%7D%7Bi%5E%7B%28m%29%7D%7D" alt=""/> : PV of an annuity-immediate with payments of <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%5Cfrac%7B1%7D%7Bm%5E2%7D" alt=""/> at the end of the first mth of the first year, <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%5Cfrac%7B2%7D%7Bm%5E2%7D" alt=""/> at the end of the second mth of the first year, and each additional payment increasing until there is a payment of <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%5Cfrac%7Bmn%7D%7Bm%5E2%7D" alt=""/> at the end of the last mth of the nth year. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%28%5Coverline%7BI%7D%20%5Coverline%7Ba%7D%29_%7B%5Coverline%7Bn%7C%7D%7D=%5Cfrac%7B%20%5Coverline%7Ba%7D_%7B%5Coverline%7Bn%7C%7D%7D-nv%5En%7D%7B%5Cdelta%7D" alt=""/> : PV of an annuity with continuous payments that are continuously increasing. Annual rate of payment is <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?t" alt=""/> at time <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?t" alt=""/>. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%5Cint_0%5En%20f%28t%29v%5Et%20dt" alt=""/> : PV of an annuity with a continuously variable rate of payments and a constant interest rate. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%5Cint_0%5En%20f%28t%29e%5E%7B-%5Cint_0%5Et%20%5Cdelta_r%20dr%7D%20dt" alt=""/> : PV of an annuity with a continuously variable rate of payment and a continuously variable rate of interest. 
 
Payments in Geometric Progression: 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%5Cfrac%7B1-%28%5Cfrac%7B1+k%7D%7B1+i%7D%29%5En%7D%7Bi-k%7D" alt=""/> : PV of an annuity-immediate with an initial payment of 1 and each additional payment increasing by a factor of <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%281+k%29" alt=""/>. 
 
Chapter 5: 
Definitions: 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?R_t" alt=""/> : payment at time <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?t" alt=""/>. A negative value is an investment and a positive value is a return. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?P%28i%29=%5Csum%7Bv%5EtR_t%7D" alt=""/> : PV of a cash flow at interest rate <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?i" alt=""/>. 
 
Chapter 6: 
General Definitions: 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?R_t=I_t+P_t" alt=""/> : payment made at the end of year <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?t" alt=""/>, split into the interest <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?I_t" alt=""/> and the principle repaid <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?P_t" alt=""/>. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?I_t=iB_%7Bt-1%7D" alt=""/> : interest paid at the end of year <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?t" alt=""/>. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?P_t=R_t-I_t=%281+i%29P_%7Bt-1%7D+%28R_t-R_%7Bt-1%7D%29" alt=""/> : principle repaid at the end of year <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?t" alt=""/>. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?B_t=B_%7Bt-1%7D-P_t" alt=""/> : balance remaining at the end of year <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?t" alt=""/>, just after payment is made. 
 
On a Loan Being Paid with Level Payments: 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?I_t=1-v%5E%7Bn-t+1%7D" alt=""/> : interest paid at the end of year <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?t" alt=""/> on a loan of <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?a_%7B%5Coverline%7Bn%7C%7D%7D" alt=""/>. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?P_t=v%5E%7Bn-t+1%7D" alt=""/> : principle repaid at the end of year <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?t" alt=""/> on a loan of <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?a_%7B%5Coverline%7Bn%7C%7D%7D" alt=""/>. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?B_t=a_%7B%5Coverline%7Bn-t%7C%7D%7D" alt=""/> : balance remaining at the end of year <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?t" alt=""/> on a loan of <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?a_%7B%5Coverline%7Bn%7C%7D%7D" alt=""/>, just after payment is made. 
 
For a loan of <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?L" alt=""/>, level payments of <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%5Cfrac%7BL%7D%7Ba_%7B%5Coverline%7Bn%7C%7D%7D%7D" alt=""/> will pay off the loan in <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?n" alt=""/> years. In this case, multiply <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?I_t" alt=""/>, <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?P_t" alt=""/>, and <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?B_t" alt=""/> by <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%5Cfrac%7BL%7D%7Ba_%7B%5Coverline%7Bn%7C%7D%7D" alt=""/>, ie <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?B_t=%5Cfrac%7BL%7D%7Ba_%7B%5Coverline%7Bn%7C%7D%7D%7Da_%7B%5Coverline%7Bn-t%7C%7D%7D" alt=""/> etc. 
 
Sinking Funds: 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?PMT=Li+%5Cfrac%7BL%7D%7Bs_%7B%5Coverline%7Bn%7C%7Dj%7D%7D" alt=""/> : total yearly payment with the sinking fund method, where <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?Li" alt=""/> is the interest paid to the lender and <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%5Cfrac%7BL%7D%7Bs_%7B%5Coverline%7Bn%7C%7Dj%7D%7D" alt=""/> is the deposit into the sinking fund that will accumulate to <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?L" alt=""/> in <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?n" alt=""/> years. <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?i" alt=""/> is the interest rate for the loan and <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?j" alt=""/> is the interest rate that the sinking fund earns. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?L=%28PMT-Li%29s_%7B%5Coverline%7Bn%7C%7Dj%7D" alt=""/> 
 
Chapter 7: 
Definitions: 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?P" alt=""/> : Price paid for a bond. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?F" alt=""/> : Par/face value of a bond. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?C" alt=""/> : Redemption value of a bond. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?r" alt=""/> : coupon rate for a bond. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?g=%5Cfrac%7BFr%7D%7BC%7D" alt=""/> : modified coupon rate. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?i" alt=""/> : yield rate on a bond. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?K" alt=""/> : PV of <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?C" alt=""/>. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?n" alt=""/> : number of coupon payments. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?G=%5Cfrac%7BFr%7D%7Bi%7D" alt=""/> : base amount of a bond. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?Fr=Cg" alt=""/> 
 
Determination of Bond Prices: 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?P=Fra_%7B%5Coverline%7Bn%7C%7Di%7D+Cv%5En=Cga_%7B%5Coverline%7Bn%7C%7Di%7D+C%20%20v%5En" alt=""/> : price paid for a bond to yield <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?i" alt=""/>. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?P=C+%28Fr-Ci%29a_%7B%5Coverline%7Bn%7C%7Di%7D=C+%28Cg-Ci%29a_%7B%5Coverline%7Bn%7C%7Di%7D" alt=""/> : Premium/Discount formula for the price of a bond. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?P-C=%28Fr-Ci%29a_%7B%5Coverline%7Bn%7C%7Di%7D=%28Cg-Ci%29a_%7B%5Coverline%7Bn%7C%7Di%7D" alt=""/> : premium paid for a bond if <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?g%3Ei" alt=""/>. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?C-P=%28Ci-Fr%29a_%7B%5Coverline%7Bn%7C%7Di%7D=%28Ci-Cg%29a_%7B%5Coverline%7Bn%7C%7Di%7D" alt=""/> : discount paid for a bond if <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?g%3Ci" alt=""/>. 
 
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: 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?R_t" alt=""/> : coupon payment. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?I_t=iB_%7Bt-1%7D" alt=""/> : interest earned from the coupon payment. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?P_t=R_t-I_t=%28Fr-Ci%29v%5E%7Bn-t+1%7D=%28Cg-Ci%29v%5E%7Bn-t+1%7D" alt=""/> : adjustment amount for amortization of premium ("write down") or 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?P_t=I_t-R_t=%28Ci-Fr%29v%5E%7Bn-t+1%7D=%28Ci-Cg%29v%5E%7Bn-t+1%7D" alt=""/> : adjustment amount for accumulation of discount ("write up"). 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?B_t=B_%7Bt-1%7D-P_t" alt=""/> : book value of bond after adjustment from the most recent coupon paid. 
 
Price Between Coupon Dates: 
For a bond sold at time <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?k" alt=""/> after the coupon payment at time <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?t" alt=""/> and before the coupon payment at time <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?t+1" alt=""/>: 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?B_%7Bt+k%7D%5Ef=B_t%281+i%29%5Ek=%28B_%7Bt+1%7D+Fr%29v%5E%7B1-k%7D" alt=""/> : "flat price" of the bond, ie the money that actually exchanges hands on the sale of the bond. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?B_%7Bt+k%7D%5Em=B_%7Bt+k%7D%5Ef-kFr=B_t%281+i%29%5Ek-kFr" alt=""/> : "market price" of the bond, ie the price quoted in a financial newspaper. 
 
Approximations of Yield Rates on a Bond: 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?i%20%5Capprox%20%5Cfrac%7BnFr+C-P%7D%7B%5Cfrac%7Bn%7D%7B2%7D%28P+C%29%7D" alt=""/> : Bond Salesman's Method. 
 
Price of Other Securities: 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?P=%5Cfrac%7BFr%7D%7Bi%7D" alt=""/> : price of a perpetual bond or preferred stock. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?P=%5Cfrac%7BD%7D%7Bi-k%7D" alt=""/> : theoretical price of a stock that is expected to return a dividend of <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?D" alt=""/> with each subsequent dividend increasing by <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%281+k%29" alt=""/>, <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?k%3Ci" alt=""/>. 
 
Chapter 9: 
Recognition of Inflation: 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?i%27=%5Cfrac%7Bi-r%7D%7B1+r%7D" alt=""/> : real rate of interest, where <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?i" alt=""/> is the effective rate of interest and <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?r" alt=""/> is the rate of inflation. 
 
Method of Equated Time and (Macauley) Duration: 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%5Coverline%7Bt%7D=%20%5Cfrac%7B%5Csum_%7Bt=1%7D%5En%20tR_t%7D%7B%5Csum_%7Bt=1%7D%5En%20R_t%7D" alt=""/> : method of equated time. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%5Coverline%7Bd%7D=%20%5Cfrac%7B%5Csum_%7Bt=1%7D%5En%20tv%5EtR_t%7D%7B%5Csum_%7Bt=1%7D%5En%20v%5EtR_t%7D" alt=""/> : (Macauley) duration. 
 
Volatility and Modified Duration: 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?P%28i%29=%5Csum%7Bv%5EtR_t%7D" alt=""/> : PV of a cash flow at interest rate <img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?i" alt=""/>. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%5Coverline%7Bv%7D=%20-%20%5Cfrac%7BP%27%28i%29%7D%7BP%28i%29%7D=v%5Coverline%7Bd%7D=%5Cfrac%7B%5Coverline%7Bd%20%20%7D%7D%7B1+i%7D" alt=""/> : volatility/modified duration. 
 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%5Coverline%7Bd%7D=-%281+i%29%5Cfrac%7BP%27%28i%29%7D%7BP%28i%29%7D" alt=""/> : alternate definition of (Macauley) duration. 
 
Convexity and (Redington) Immunization: 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?%5Coverline%7Bc%7D=%5Cfrac%7BP%27%27%28i%29%7D%7BP%28i%29%7D" alt=""/> convexity 
 
To achieve Redington immunization we want: 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?P%27%28i%29=0" alt=""/> 
<img src="http://www.actuarialoutpost.com/cgi-bin/mimetex.cgi?P%27%27%28i%29%3E0" alt=""/>

[此贴子已经被作者于2008-11-6 5:13:33编辑过]