Chapter 13 Annuities Due13.1 Future value of an annuity due 1Annuity due: an annuity with payments at the beginning of the payment intervals is called an annuity due.01234n-1nRRRRRRTerm of the annuityPayment intervalInterval number2Two types of annuity due:3Two types of ordinary annuity4Note: What is meaning of “by the end of an annuity”?The end of an annuity means “the end of the annuitys term” or “the end of the last payment interval”. It occurs one payment interval after the last payment in an annuity due.5The beginning of an annuity refers to “the start of the annuitys term” or “the start of the first payment interval”.It does coincide with the first payment in an annuity due.601234n-1nRRRRRROrdinary annuityThe date of the last paymentThe end of theannuityBeginning of the annuity 701234n-1nRRRRRRAnnuity dueThe date of the last paymentThe end of the annuityBeginning of the annuity 8Future value of an annuity due -FV(due)The future value of an annuity is the single amount at the end of the annuity, that is economically equivalent to the annuity.01234n-1nRRRRRR9Future Value Using the Algebraic Method01234n-1nRRRRRR01234n-1nRRRRRRFVFV(due )10Future value using the algebraic method01234n-1nRRRRRR-1FVFV(due)FV(due)=FV(1+p)11Future value of an annuity due12Future value using the financial calculator functionsMethod 1+/-PVPMTn1/YFVR0nPFV+/-PVPMTnFV01FV(due)CPTCPT13Method 2Texas Instruments BA Plus+/-PVPMTn1/YFVR0nPFV(due)P515BGNCPT14Example:How much will Stan accumulate in his Registered Retirement Savings Plan (RRSP) by age 60 if he makes semiannual contributions of $2000 starting on his 27th birthday? Assume that the RRSP earns 8% compounded semiannually and that no contribution is made on Stans 60th birthday.15j=8% compounded semiannually i=j/m=8%/2=4% term=60-27=33 years payment interval=half year n=33*2=66p=i=4% R=$2000Solution:27282960birthday$2000$2000$2000$2000$2000$200016Algebraic method17Financial calculator method+/-PVPMTn1/YFV20000664BGNmode640,155.60Stan will have $640,155.60 in his RRSP at age 60.CPT18ExampleTo the nearest dollar, how much will Stan accumulate in his RRSP by age 60 if he makes semiannual contributions of $2000 starting on his 27th birthday? Assume that the RRSP earns 8% compounded annually and that no contribution is made on Stans 60th birthday. 19Solution:n=66 R=$2000 j=8% compounded annually i=j/m=8%c=1/2 p=(1+i)c-1=3.923048% per half year20+/-PVPMTn1/Y FV20000663.923048BGNmode618,606Stan will have $618,606 in his RRSP at age 60.CPT21ExampleStephanie intend to contribute $2500 to her RRSP at the beginning of every 6 months starting today. If the RRSP earns 8% compounded semiannually for the first 7 years and 7% compounded semiannually thereafter, what amount will she have in the plan after 20 years?220720 years$2500 every 6 months$2500 every 6 months FV1(due)FV2(due)SFuture value=sum of FV2(due) and S8%compounded semiannually7% compounded semiannually23Step1: calculate FV1(due) R=$2500 term=7 years n=7*2=14 j=8% compounded semiannuallyi=j/m=4%=p per half year+/-PVPMTn1/YFV25000144BGNmode47,558.97FV1(due)= 47,558.97CPT24Step2: calculate FV2(due)R=$2500 term=20-7=13 years n=13*2=26 j=7% compounded semiannuallyi=j/m=3.5%=p per half yearn1/YFV263.5BGNmode106,897.65FV2(due)= 106,897.65CPT25Step3: calculate SFV1(due)=$47,558.97 n=26 i=3.5%FV=PV(1+i)n+/-PVPMTn1/YFV47,558.970263.5116,327.27S= 116,327.27CPT26Step4: calculate the future value of all payments after 20 yearsThe future value =S+ FV2(due)=$223,224.92Stephanie will have $223,224.92 in her RRSP after 20 years.27