2)Annuity due O Future value of an Annuity Due FVAND=PMT(FVIFA)(1+i) Table III =PMT(FVIFAi1-1) O Present Value of an Annuity Due PVANDo=PMT(PVIFA)(1+i) Table IV PMT =PMT(PVIFA.n+1） 渊外经价货多方是 YO年N0事0E0003
2) Annuity due Annuity due z Future Value of an Annuity Due FVANDn = PMT(FVIFAi, n)(1 + i ) Table III = PMT =PMT(FVIFAi, n+1 -1 ) z Present Value of an Annuity Due PVAND0 = PMT(PVIFAi, n)(1 + i ) Table IV = PMT = PMT(PVIFAi, n-1+1） ⎥⎦⎤ ⎢⎣⎡ − + − + 1 (1 ) 1 1 i i n ⎥⎦⎤ ⎢⎣⎡ + − + − − 1 1 (1 ) ( 1) ii n
Iigure 4.6 Timeline of the Future Value of an Annuity Due (PMT=$1,000;i=6%;n=3) t(Year) 0 1 2 3 L PMT,($1.000) PMT($1.000) PMT3($1,000) Compounded 1 Year $1.060 Compounded 2 Years 1.124 Compounded 3 Years 1,191 $3,375 Future Value of an Annuity Due (FVAND) DuD 剥补楂价货多方是 U日U0年0LB月50E00MN
22 Figure 4.6 Timeline of the Future Value of an Annuity Due (PMT = $1,000; i = 6%; n = 3))
Iigare 4.7 Timeline of a Present Value of an Annuity Due (PMT=$1,000:i=6%;n=5) t(Year) 0 2 3 4 5 L PMT,($1.000) PMT2($1.000) PMT2($1.000) PMT,($1.000) PMT-($1000) Undiscounted $1.000 Discounted 1 Year $943 Discounted 2 Years 890 Discounted 3 Years 840 Discounted 4 Years 792 $4,465=Present Value of an Annuity Due (PVAND) 渊补经价货多方居 T0年0LB月50E00以
Figure 4.7 Timeline of a Present Value of an Annuity Due (PMT = $1,000; i = 6%; n = 5)
3)Deferred annuity O Deferred annuity:an annuity begins more than one year in the future PVAN-PMT(D) =PMT(PVIFAD)(PVIFim) Formula m deferred period OPVAN=PMT 1-(1+i)m =PMII (PVIFAim)-(PVIFAim)] Formula 剥外经价贫多方号 YO年N0l到8E事0E00h03
3) Deferred annuity 3) Deferred annuity z Deferred annuity: an annuity begins more than one year in the future z PVAN 0= PMT = PMT (PVIFAi,n)(PVIFi,m) Formula m = deferred period z PVAN 0= PMT = PMT[ (PVIFAi,m+n ) －(PVIFAi,m)] Formula ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − + − i i n 1 (1 ) m i − ( 1 + ) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − + − − + − + − i i i i m n m 1 (1 ) 1 (1 ) ( )
Iigure 4.8 Timeline of a Deferred Four-Year Annuity (1=12%) t(Year) 3 5 6 8 9 $15.000 $15.000 $15.000 $15,000 (PVAN1,000(PVIFA =$15,0003.037 (PV)PVAN (PVIF =$45,555 =$45.5550.636 =S28,973=Present Value of a Deferred Annuity 正”=1T。 YO年NlE事eO003
25 Figure 4.8 Timeline of a Deferred Four-Year Annuity (i = 12%)