Manual for SOA Exam FM/CAS Exam 2. - Binghamton University

1 1/12 Chapter 1. Basic interest for SOA Exam FM/CAS Exam 1. Basic interest Force of 2008. Miguel A. Arcones. All rights from: Arcones Manual for the SOA Exam FM/CAS Exam 2,Financial Mathematics. Spring 2009 Edition ,available 2008. Miguel A. Arcones. All rights for SOA Exam FM/CAS Exam 1. Basic interest Force of of interestTheforce of interest tof an amount functionA(t) is defined by t=ddtlnA(t) =A (t)A(t).The force of interest is the fraction of the instantaneous rate ofchange of the accumulation function and the find the force of interest , we may use the accumulationfunction,ddtlnA(t) =ddtln(A(0)a(t)) =ddtln(A(0)) +ddtln(a(t))=ddtln(a(t)).c 2008. Miguel A. Arcones. All rights for SOA Exam FM/CAS Exam 1. Basic interest Force of of interestTheforce of interest tof an amount functionA(t) is defined by t=ddtlnA(t) =A (t)A(t).

2 The force of interest is the fraction of the instantaneous rate ofchange of the accumulation function and the find the force of interest , we may use the accumulationfunction,ddtlnA(t) =ddtln(A(0)a(t)) =ddtln(A(0)) +ddtln(a(t))=ddtln(a(t)).c 2008. Miguel A. Arcones. All rights for SOA Exam FM/CAS Exam 1. Basic interest Force of of interestTheforce of interest tof an amount functionA(t) is defined by t=ddtlnA(t) =A (t)A(t).The force of interest is the fraction of the instantaneous rate ofchange of the accumulation function and the find the force of interest , we may use the accumulationfunction,ddtlnA(t) =ddtln(A(0)a(t)) =ddtln(A(0)) +ddtln(a(t))=ddtln(a(t)).c 2008. Miguel A. Arcones. All rights for SOA Exam FM/CAS Exam 1. Basic interest Force of 1 Consider the amount functionA(t) = 25(1 +t4)3.

3 At what time isthe force of interest equal of :We have thatln(A(t)) = ln(25(1 +t4)3)= ln 25 + 3 ln(1 +t4).The force of interest is t=ddtln(A(t)) =ddt(ln 25 + 3 ln(1 +t4))= 3141 +t4=34 + the equation,34+t=12, we get thatt= 2008. Miguel A. Arcones. All rights for SOA Exam FM/CAS Exam 1. Basic interest Force of 1 Consider the amount functionA(t) = 25(1 +t4)3. At what time isthe force of interest equal of :We have thatln(A(t)) = ln(25(1 +t4)3)= ln 25 + 3 ln(1 +t4).The force of interest is t=ddtln(A(t)) =ddt(ln 25 + 3 ln(1 +t4))= 3141 +t4=34 + the equation,34+t=12, we get thatt= 2008. Miguel A. Arcones. All rights for SOA Exam FM/CAS Exam 1. Basic interest Force of force of interest is also called therate of interestcontinuously compoundedand thecontinuous interest have that t= limh 0A(t+h) A(t)A(t) h= limh 0interest earned over the nexthyearsinvestment at timet nominal annual rate earned in the next1myears compoundedmtimes a year at timetism(a(t+1m) a(t))a(t)=a(t+1m) a(t)a(t) have thatlimm a(t+1m) a(t)a(t)1m= 2008.

4 Miguel A. Arcones. All rights for SOA Exam FM/CAS Exam 1. Basic interest Force of compound interest ,a(t) = (1 +i)tand t=ddtlna(t) =ddtln(1 +i)t=ddttln(1 +i) = ln(1 +i)Under compound interest , the force of interest is a constant ,such that = ln(1 +i) = ln .Under compound interest ,limm i(m)= limm d(m)= .In the case of simple interest ,a(t) = 1 +itand t=ddtln(1 +it) =i1+it. The force of interest is decreasing 2008. Miguel A. Arcones. All rights for SOA Exam FM/CAS Exam 1. Basic interest Force of the force of interest t, we may find the accumulationfunctiona(t), usingTheorem 2 For each t 0, a(t) =eRt0 s=ddslna(s) anda(0) = 1, t0 sds= t0ddslna(s)ds= lna(s) t0= lna(t).So,a(t) =eRt0 2008. Miguel A. Arcones. All rights for SOA Exam FM/CAS Exam 1. Basic interest Force of the force of interest t, we may find the accumulationfunctiona(t), usingTheorem 2 For each t 0, a(t) =eRt0 s=ddslna(s) anda(0) = 1, t0 sds= t0ddslna(s)ds= lna(s) t0= lna(t).

5 So,a(t) =eRt0 2008. Miguel A. Arcones. All rights for SOA Exam FM/CAS Exam 1. Basic interest Force of 3A bank account credits interest using a force of interest t=3t2t3+ deposit of 100 is made in the account at timet= 0. Find theamount of interest earned by the account from the end of the 4 thyear until the end of the 8 th :First, we finda(t) =eRt0 sds. t0 sds= t03s2s3+ 2ds= ln(s3+ 2) t0= ln(t3+ 2) ln 2 = ln(t3+ 22)anda(t) =eRt0 sds=eln t3+22 =t3+ 22= 1 + amount of interest earned in the considered period is100(a(8) a(4)) = (100)(1 +832 (1 +432))= 2008. Miguel A. Arcones. All rights for SOA Exam FM/CAS Exam 1. Basic interest Force of 3A bank account credits interest using a force of interest t=3t2t3+ deposit of 100 is made in the account at timet= 0. Find theamount of interest earned by the account from the end of the 4 thyear until the end of the 8 th :First, we finda(t) =eRt0 sds.

6 T0 sds= t03s2s3+ 2ds= ln(s3+ 2) t0= ln(t3+ 2) ln 2 = ln(t3+ 22)anda(t) =eRt0 sds=eln t3+22 =t3+ 22= 1 + amount of interest earned in the considered period is100(a(8) a(4)) = (100)(1 +832 (1 +432))= 2008. Miguel A. Arcones. All rights for SOA Exam FM/CAS Exam 2.