1,Chapter 13 Annuities Due,13.1 Future value of an annuity due,2,Annuity due: an annuity with payments at the beginning of the payment intervals is called an annuity due.,0,1,2,3,4,n-1,n,R,R,R,R,R,R,Term of the annuity,Payment interval,Interval number,3,Two types of annuity due:,4,Two types of ordinary annuity,5,Note: 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.,6,The 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.,7,0,1,2,3,4,n-1,n,R,R,R,R,R,R,Ordinary annuity,The date of thelast payment,The end of the annuity,Beginning of the annuity,8,0,1,2,3,4,n-1,n,R,R,R,R,R,R,Annuity due,The date of thelast payment,The end of the annuity,Beginning of the annuity,9,Future 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.,0,1,2,3,4,n-1,n,R,R,R,R,R,R,10,Future Value Using the Algebraic Method,0,1,2,3,4,n-1,n,R,R,R,R,R,R,0,1,2,3,4,n-1,n,R,R,R,R,R,R,FV,FV(due),11,Future value using the algebraic method,0,1,2,3,4,n-1,n,R,R,R,R,R,R,-1,FV,FV(due),FV(due)=FV(1+p),12,Future value of an annuity due,13,Future value using the financial calculator functions,Method 1,+/-,PV,PMT,n,1/Y,FV,R,0,n,P,FV,+/-,PV,PMT,n,FV,0,1,FV(due),CPT,CPT,14,Method 2,Texas Instruments BA Plus,+/-,PV,PMT,n,1/Y,FV,R,0,n,P,FV(due),P515,BGN,CPT,15,Example:,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.,16,j=8% compounded semiannually i=j/m=8%/2=4% term=60-27=33 yearspayment interval=half year n=33*2=66 p=i=4% R=$2000,Solution:,27,28,29,60,birthday,$2000,$2000,$2000,$2000,$2000,$2000,17,Algebraic method,18,Financial calculator method,+/-,PV,PMT,n,1/Y,FV,2000,0,66,4,BGN,mode,640,155.60,Stan will have $640,155.60 in his RRSP at age 60.,CPT,19,Example,To 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.,20,Solution:,n=66 R=$2000 j=8% compounded annually i=j/m=8% c=1/2 p=(1+i)c-1=3.923048% per half year,21,+/-,PV,PMT,n,1/Y,FV,2000,0,66,3.923048,BGN,mode,618,606,Stan will have $618,606 in his RRSP at age 60.,CPT,22,Example,Stephanie 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?,23,0,7,20,years,$2500 every 6 months,$2500 every 6 months,FV1(due),FV2(due),S,Future value=sum of FV2(due) and S,8% compoundedsemiannually,7% compoundedsemiannually,24,Step1: calculate FV1(due),R=$2500 term=7 years n=7*2=14j=8% compounded semiannually i=j/m=4%=p per half year,+/-,PV,PMT,n,1/Y,FV,2500,0,14,4,BGN,mode,47,558.97,FV1(due)= 47,558.97,CPT,25,Step2: calculate FV2(due),R=$2500 term=20-7=13 yearsn=13*2=26 j=7% compounded semiannually i=j/m=3.5%=p per half year,n,1/Y,FV,26,3.5,BGN,mode,106,897.65,FV2(due)= 106,897.65,CPT,26,Step3: calculate S,FV1(due)=$47,558.97 n=26 i=3.5% FV=PV(1+i)n,+/-,PV,PMT,n,1/Y,FV,47,558.97,0,26,3.5,116,327.27,S= 116,327.27,CPT,27,Step4: calculate the future value of all payments after 20 years,The future value =S+ FV2(due) =$223,224.92,Stephanie will have $223,224.92in her RRSP after 20 years.,