Předpokládejme 33801

Kolik bude stát nákup dvouúrovňové důchodové renty, která bude platit 2000 $ na konci každého měsíce po dobu prvních 10 let a 3000 $ měsíčně po dobu dalších 15 let? Předpokládejme, že platba představuje návratnost pro osobu, která platbu přijímá, ve výši 7,6 % sčítanou čtvrtletně.

Správná odpověď:

x =  430245,06 USD

Postup správného řešení:

$$\begin{gathered} a=2000\cdot 4=8000 \\ b=3000\cdot 4=12000 \\ p=7,6\%=\frac{7,6}{100}=0,076 \\ r=1+p/4=1+0,076/4=1,019 \\ {n}_1=10\cdot 4=40 \\ {n}_2=15\cdot 4=60 \\ s=a\cdot n_1+b\cdot n_2=8000\cdot 40+12000\cdot 60=1040000 \\ {a}_1=a/r^0=8000/1,019^0=8000 \\ {a}_2=a/r^1=8000/1,019^1\doteq 7850,8342 \\ {a}_3=a/r^2=8000/1,019^2\doteq 7704,4496 \\ {a}_4=a/r^3=8000/1,019^3\doteq 7560,7945 \\ {a}_5=a/r^4=8000/1,019^4\doteq 7419,818 \\ {a}_6=a/r^5=8000/1,019^5\doteq 7281,47 \\ {a}_7=a/r^6=8000/1,019^6\doteq 7145,7017 \\ {a}_8=a/r^7=8000/1,019^7\doteq 7012,4649 \\ {a}_9=a/r^8=8000/1,019^8\doteq 6881,7123 \\ {a}_{10}=a/r^9=8000/1,019^9\doteq 6753,3978 \\ {a}_{11}=a/r^{10}=8000/1,019^{10}\doteq 6627,4757 \\ {a}_{12}=a/r^{11}=8000/1,019^{11}\doteq 6503,9016 \\ {a}_{13}=a/r^{12}=8000/1,019^{12}\doteq 6382,6316 \\ {a}_{14}=a/r^{13}=8000/1,019^{13}\doteq 6263,6228 \\ {a}_{15}=a/r^{14}=8000/1,019^{14}\doteq 6146,833 \\ {a}_{16}=a/r^{15}=8000/1,019^{15}\doteq 6032,2208 \\ {a}_{17}=a/r^{16}=8000/1,019^{16}\doteq 5919,7456 \\ {a}_{18}=a/r^{17}=8000/1,019^{17}\doteq 5809,3676 \\ {a}_{19}=a/r^{18}=8000/1,019^{18}\doteq 5701,0477 \\ {a}_{20}=a/r^{19}=8000/1,019^{19}\doteq 5594,7475 \\ {a}_{21}=a/r^{20}=8000/1,019^{20}\doteq 5490,4293 \\ {a}_{22}=a/r^{21}=8000/1,019^{21}\doteq 5388,0563 \\ {a}_{23}=a/r^{22}=8000/1,019^{22}\doteq 5287,592 \\ {a}_{24}=a/r^{23}=8000/1,019^{23}\doteq 5189,001 \\ {a}_{25}=a/r^{24}=8000/1,019^{24}\doteq 5092,2483 \\ {a}_{26}=a/r^{25}=8000/1,019^{25}\doteq 4997,2996 \\ {a}_{27}=a/r^{26}=8000/1,019^{26}\doteq 4904,1213 \\ {a}_{28}=a/r^{27}=8000/1,019^{27}\doteq 4812,6804 \\ {a}_{29}=a/r^{28}=8000/1,019^{28}\doteq 4722,9444 \\ {a}_{30}=a/r^{29}=8000/1,019^{29}\doteq 4634,8817 \\ {a}_{31}=a/r^{30}=8000/1,019^{30}\doteq 4548,4609 \\ {a}_{32}=a/r^{31}=8000/1,019^{31}\doteq 4463,6515 \\ {a}_{33}=a/r^{32}=8000/1,019^{32}\doteq 4380,4235 \\ {a}_{34}=a/r^{33}=8000/1,019^{33}\doteq 4298,7473 \\ {a}_{35}=a/r^{34}=8000/1,019^{34}\doteq 4218,594 \\ {a}_{36}=a/r^{35}=8000/1,019^{35}\doteq 4139,9352 \\ {a}_{37}=a/r^{36}=8000/1,019^{36}\doteq 4062,7431 \\ {a}_{38}=a/r^{37}=8000/1,019^{37}\doteq 3986,9903 \\ {a}_{39}=a/r^{38}=8000/1,019^{38}\doteq 3912,6499 \\ {a}_{40}=a/r^{39}=8000/1,019^{39}\doteq 3839,6957 \\ {b}_1=b/r^{n_1+0}=12000/1,019^{40+0}\doteq 5652,1527 \\ {b}_2=b/r^{n_1+1}=12000/1,019^{40+1}\doteq 5546,7642 \\ {b}_3=b/r^{n_1+2}=12000/1,019^{40+2}\doteq 5443,3407 \\ {b}_4=b/r^{n_1+3}=12000/1,019^{40+3}\doteq 5341,8456 \\ {b}_5=b/r^{n_1+4}=12000/1,019^{40+4}\doteq 5242,243 \\ {b}_6=b/r^{n_1+5}=12000/1,019^{40+5}\doteq 5144,4976 \\ {b}_7=b/r^{n_1+6}=12000/1,019^{40+6}\doteq 5048,5746 \\ {b}_8=b/r^{n_1+7}=12000/1,019^{40+7}\doteq 4954,4403 \\ {b}_9=b/r^{n_1+8}=12000/1,019^{40+8}\doteq 4862,0611 \\ {b}_{10}=b/r^{n_1+9}=12000/1,019^{40+9}\doteq 4771,4044 \\ {b}_{11}=b/r^{n_1+10}=12000/1,019^{40+10}\doteq 4682,4381 \\ {b}_{12}=b/r^{n_1+11}=12000/1,019^{40+11}\doteq 4595,1306 \\ {b}_{13}=b/r^{n_1+12}=12000/1,019^{40+12}\doteq 4509,4511 \\ {b}_{14}=b/r^{n_1+13}=12000/1,019^{40+13}\doteq 4425,369 \\ {b}_{15}=b/r^{n_1+14}=12000/1,019^{40+14}\doteq 4342,8548 \\ {b}_{16}=b/r^{n_1+15}=12000/1,019^{40+15}\doteq 4261,8791 \\ {b}_{17}=b/r^{n_1+16}=12000/1,019^{40+16}\doteq 4182,4132 \\ {b}_{18}=b/r^{n_1+17}=12000/1,019^{40+17}\doteq 4104,4291 \\ {b}_{19}=b/r^{n_1+18}=12000/1,019^{40+18}\doteq 4027,899 \\ {b}_{20}=b/r^{n_1+19}=12000/1,019^{40+19}\doteq 3952,7959 \\ {b}_{21}=b/r^{n_1+20}=12000/1,019^{40+20}\doteq 3879,0931 \\ {b}_{22}=b/r^{n_1+21}=12000/1,019^{40+21}\doteq 3806,7646 \\ {b}_{23}=b/r^{n_1+22}=12000/1,019^{40+22}\doteq 3735,7847 \\ {b}_{24}=b/r^{n_1+23}=12000/1,019^{40+23}\doteq 3666,1283 \\ {b}_{25}=b/r^{n_1+24}=12000/1,019^{40+24}\doteq 3597,7706 \\ {b}_{26}=b/r^{n_1+25}=12000/1,019^{40+25}\doteq 3530,6875 \\ {b}_{27}=b/r^{n_1+26}=12000/1,019^{40+26}\doteq 3464,8553 \\ {b}_{28}=b/r^{n_1+27}=12000/1,019^{40+27}\doteq 3400,2505 \\ {b}_{29}=b/r^{n_1+28}=12000/1,019^{40+28}\doteq 3336,8504 \\ {b}_{30}=b/r^{n_1+29}=12000/1,019^{40+29}\doteq 3274,6324 \\ {b}_{31}=b/r^{n_1+30}=12000/1,019^{40+30}\doteq 3213,5744 \\ {b}_{32}=b/r^{n_1+31}=12000/1,019^{40+31}\doteq 3153,655 \\ {b}_{33}=b/r^{n_1+32}=12000/1,019^{40+32}\doteq 3094,8528 \\ {b}_{34}=b/r^{n_1+33}=12000/1,019^{40+33}\doteq 3037,147 \\ {b}_{35}=b/r^{n_1+34}=12000/1,019^{40+34}\doteq 2980,5172 \\ {b}_{36}=b/r^{n_1+35}=12000/1,019^{40+35}\doteq 2924,9433 \\ {b}_{37}=b/r^{n_1+36}=12000/1,019^{40+36}\doteq 2870,4056 \\ {b}_{38}=b/r^{n_1+37}=12000/1,019^{40+37}\doteq 2816,8847 \\ {b}_{39}=b/r^{n_1+38}=12000/1,019^{40+38}\doteq 2764,3619 \\ {b}_{40}=b/r^{n_1+39}=12000/1,019^{40+39}\doteq 2712,8183 \\ {b}_{41}=b/r^{n_1+40}=12000/1,019^{40+40}\doteq 2662,2358 \\ {b}_{42}=b/r^{n_1+41}=12000/1,019^{40+41}\doteq 2612,5965 \\ {b}_{43}=b/r^{n_1+42}=12000/1,019^{40+42}\doteq 2563,8827 \\ {b}_{44}=b/r^{n_1+43}=12000/1,019^{40+43}\doteq 2516,0773 \\ {b}_{45}=b/r^{n_1+44}=12000/1,019^{40+44}\doteq 2469,1632 \\ {b}_{46}=b/r^{n_1+45}=12000/1,019^{40+45}\doteq 2423,1238 \\ {b}_{47}=b/r^{n_1+46}=12000/1,019^{40+46}\doteq 2377,9429 \\ {b}_{48}=b/r^{n_1+47}=12000/1,019^{40+47}\doteq 2333,6044 \\ {b}_{49}=b/r^{n_1+48}=12000/1,019^{40+48}\doteq 2290,0927 \\ {b}_{50}=b/r^{n_1+49}=12000/1,019^{40+49}\doteq 2247,3922 \\ {b}_{51}=b/r^{n_1+50}=12000/1,019^{40+50}\doteq 2205,4879 \\ {b}_{52}=b/r^{n_1+51}=12000/1,019^{40+51}\doteq 2164,365 \\ {b}_{53}=b/r^{n_1+52}=12000/1,019^{40+52}\doteq 2124,0088 \\ {b}_{54}=b/r^{n_1+53}=12000/1,019^{40+53}\doteq 2084,4051 \\ {b}_{55}=b/r^{n_1+54}=12000/1,019^{40+54}\doteq 2045,5399 \\ {b}_{56}=b/r^{n_1+55}=12000/1,019^{40+55}\doteq 2007,3993 \\ {b}_{57}=b/r^{n_1+56}=12000/1,019^{40+56}\doteq 1969,9699 \\ {b}_{58}=b/r^{n_1+57}=12000/1,019^{40+57}\doteq 1933,2383 \\ {b}_{59}=b/r^{n_1+58}=12000/1,019^{40+58}\doteq 1897,1917 \\ x=a_1+a_2+a_3+a_4+a_5+a_6+a_7+a_8+a_9+a_{10}+a_{11}+a_{12}+a_{13}+a_{14}+a_{15}+a_{16}+a_{17}+a_{18}+a_{19}+a_{20}+a_{21}+a_{22}+a_{23}+a_{24}+a_{25}+a_{26}+a_{27}+a_{28}+a_{29}+a_{30}+a_{31}+a_{32}+a_{33}+a_{34}+a_{35}+a_{36}+a_{37}+a_{38}+a_{39}+a_{40}+b_1+b_2+b_3+b_4+b_5+b_6+b_7+b_8+b_9+b_{10}+b_{11}+b_{12}+b_{13}+b_{14}+b_{15}+b_{16}+b_{17}+b_{18}+b_{19}+b_{20}+b_{21}+b_{22}+b_{23}+b_{24}+b_{25}+b_{26}+b_{27}+b_{28}+b_{29}+b_{30}+b_{31}+b_{32}+b_{33}+b_{34}+b_{35}+b_{36}+b_{37}+b_{38}+b_{39}+b_{40}+b_{41}+b_{42}+b_{43}+b_{44}+b_{45}+b_{46}+b_{47}+b_{48}+b_{49}+b_{50}+b_{51}+b_{52}+b_{53}+b_{54}+b_{55}+b_{56}+b_{57}+b_{58}+b_{59}=8000+7850,8342+7704,4496+7560,7945+7419,818+7281,47+7145,7017+7012,4649+6881,7123+6753,3978+6627,4757+6503,9016+6382,6316+6263,6228+6146,833+6032,2208+5919,7456+5809,3676+5701,0477+5594,7475+5490,4293+5388,0563+5287,592+5189,001+5092,2483+4997,2996+4904,1213+4812,6804+4722,9444+4634,8817+4548,4609+4463,6515+4380,4235+4298,7473+4218,594+4139,9352+4062,7431+3986,9903+3912,6499+3839,6957+5652,1527+5546,7642+5443,3407+5341,8456+5242,243+5144,4976+5048,5746+4954,4403+4862,0611+4771,4044+4682,4381+4595,1306+4509,4511+4425,369+4342,8548+4261,8791+4182,4132+4104,4291+4027,899+3952,7959+3879,0931+3806,7646+3735,7847+3666,1283+3597,7706+3530,6875+3464,8553+3400,2505+3336,8504+3274,6324+3213,5744+3153,655+3094,8528+3037,147+2980,5172+2924,9433+2870,4056+2816,8847+2764,3619+2712,8183+2662,2358+2612,5965+2563,8827+2516,0773+2469,1632+2423,1238+2377,9429+2333,6044+2290,0927+2247,3922+2205,4879+2164,365+2124,0088+2084,4051+2045,5399+2007,3993+1969,9699+1933,2383+1897,1917=430245,06\text{USD}\doteq 4,3\cdot 10^5\text{USD} \end{gathered}$$

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