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Rentenendwert ",[573,1146,1148,1197],{"className":1147},[576],[573,1149,1151],{"className":1150},[580],[582,1152,1153],{"xmlns":584},[586,1154,1155,1194],{},[589,1156,1157,1163,1165,1167,1170],{},[673,1158,1159,1161],{},[592,1160,677],{},[592,1162,594],{},[683,1164,685],{},[592,1166,633],{},[683,1168,1169],{},"⋅",[687,1171,1172,1186],{},[589,1173,1174,1180,1183],{},[702,1175,1176,1178],{},[592,1177,693],{},[592,1179,594],{},[683,1181,1182],{},"−",[679,1184,1185],{},"1",[589,1187,1188,1190,1192],{},[592,1189,693],{},[683,1191,1182],{},[679,1193,1185],{},[596,1195,1196],{"encoding":598},"R_n = r \\cdot \\frac{q^n - 1}{q - 1}",[573,1198,1200,1256,1275],{"className":1199,"ariaHidden":603},[602],[573,1201,1203,1206,1247,1250,1253],{"className":1202},[607],[573,1204],{"className":1205,"style":738},[611],[573,1207,1209,1212],{"className":1208},[616],[573,1210,677],{"className":1211,"style":745},[616,617],[573,1213,1215],{"className":1214},[749],[573,1216,1218,1239],{"className":1217},[753,754],[573,1219,1221,1236],{"className":1220},[758],[573,1222,1225],{"className":1223,"style":1224},[762],"height:0.1514em;",[573,1226,1227,1230],{"style":766},[573,1228],{"className":1229,"style":771},[770],[573,1231,1233],{"className":1232},[775,776,777,778],[573,1234,594],{"className":1235},[616,617,778],[573,1237,786],{"className":1238},[785],[573,1240,1242],{"className":1241},[758],[573,1243,1245],{"className":1244,"style":793},[762],[573,1246],{},[573,1248],{"className":1249,"style":800},[799],[573,1251,685],{"className":1252},[804],[573,1254],{"className":1255,"style":800},[799],[573,1257,1259,1263,1266,1269,1272],{"className":1258},[607],[573,1260],{"className":1261,"style":1262},[611],"height:0.4445em;",[573,1264,633],{"className":1265,"style":648},[616,617],[573,1267],{"className":1268,"style":898},[799],[573,1270,1169],{"className":1271},[902],[573,1273],{"className":1274,"style":898},[799],[573,1276,1278,1282],{"className":1277},[607],[573,1279],{"className":1280,"style":1281},[611],"height:1.4442em;vertical-align:-0.4811em;",[573,1283,1285,1288,1389],{"className":1284},[616],[573,1286],{"className":1287},[821,822],[573,1289,1291],{"className":1290},[687],[573,1292,1294,1381],{"className":1293},[753,754],[573,1295,1297,1378],{"className":1296},[758],[573,1298,1301,1321,1329],{"className":1299,"style":1300},[762],"height:0.9631em;",[573,1302,1303,1306],{"style":838},[573,1304],{"className":1305,"style":842},[770],[573,1307,1309],{"className":1308},[775,776,777,778],[573,1310,1312,1315,1318],{"className":1311},[616,778],[573,1313,693],{"className":1314,"style":852},[616,617,778],[573,1316,1182],{"className":1317},[902,778],[573,1319,1185],{"className":1320},[616,778],[573,1322,1323,1326],{"style":855},[573,1324],{"className":1325,"style":842},[770],[573,1327],{"className":1328,"style":863},[862],[573,1330,1332,1335],{"style":1331},"top:-3.4461em;",[573,1333],{"className":1334,"style":842},[770],[573,1336,1338],{"className":1337},[775,776,777,778],[573,1339,1341,1372,1375],{"className":1340},[616,778],[573,1342,1344,1347],{"className":1343},[616,778],[573,1345,693],{"className":1346,"style":852},[616,617,778],[573,1348,1350],{"className":1349},[749],[573,1351,1353],{"className":1352},[753],[573,1354,1356],{"className":1355},[758],[573,1357,1360],{"className":1358,"style":1359},[762],"height:0.7385em;",[573,1361,1363,1366],{"style":1362},"top:-2.931em;margin-right:0.0714em;",[573,1364],{"className":1365,"style":966},[770],[573,1367,1369],{"className":1368},[775,970,971,778],[573,1370,594],{"className":1371},[616,617,778],[573,1373,1182],{"className":1374},[902,778],[573,1376,1185],{"className":1377},[616,778],[573,1379,786],{"className":1380},[785],[573,1382,1384],{"className":1383},[758],[573,1385,1387],{"className":1386,"style":888},[762],[573,1388],{},[573,1390],{"className":1391},[894,822]," berechnen und um ",[573,1394,1396,1409],{"className":1395},[576],[573,1397,1399],{"className":1398},[580],[582,1400,1401],{"xmlns":584},[586,1402,1403,1407],{},[589,1404,1405],{},[592,1406,594],{},[596,1408,594],{"encoding":598},[573,1410,1412],{"className":1411,"ariaHidden":603},[602],[573,1413,1415,1418],{"className":1414},[607],[573,1416],{"className":1417,"style":612},[611],[573,1419,594],{"className":1420},[616,617]," Jahre abzinsen: ",[573,1423,1425,1459],{"className":1424},[576],[573,1426,1428],{"className":1427},[580],[582,1429,1430],{"xmlns":584},[586,1431,1432,1456],{},[589,1433,1434,1440,1442],{},[673,1435,1436,1438],{},[592,1437,677],{},[679,1439,681],{},[683,1441,685],{},[687,1443,1444,1450],{},[673,1445,1446,1448],{},[592,1447,677],{},[592,1449,594],{},[702,1451,1452,1454],{},[592,1453,693],{},[592,1455,594],{},[596,1457,1458],{"encoding":598},"R_0 = \\frac{R_n}{q^n}",[573,1460,1462,1517],{"className":1461,"ariaHidden":603},[602],[573,1463,1465,1468,1508,1511,1514],{"className":1464},[607],[573,1466],{"className":1467,"style":738},[611],[573,1469,1471,1474],{"className":1470},[616],[573,1472,677],{"className":1473,"style":745},[616,617],[573,1475,1477],{"className":1476},[749],[573,1478,1480,1500],{"className":1479},[753,754],[573,1481,1483,1497],{"className":1482},[758],[573,1484,1486],{"className":1485,"style":763},[762],[573,1487,1488,1491],{"style":766},[573,1489],{"className":1490,"style":771},[770],[573,1492,1494],{"className":1493},[775,776,777,778],[573,1495,681],{"className":1496},[616,778],[573,1498,786],{"className":1499},[785],[573,1501,1503],{"className":1502},[758],[573,1504,1506],{"className":1505,"style":793},[762],[573,1507],{},[573,1509],{"className":1510,"style":800},[799],[573,1512,685],{"className":1513},[804],[573,1515],{"className":1516,"style":800},[799],[573,1518,1520,1524],{"className":1519},[607],[573,1521],{"className":1522,"style":1523},[611],"height:1.3695em;vertical-align:-0.4811em;",[573,1525,1527,1530,1657],{"className":1526},[616],[573,1528],{"className":1529},[821,822],[573,1531,1533],{"className":1532},[687],[573,1534,1536,1649],{"className":1535},[753,754],[573,1537,1539,1646],{"className":1538},[758],[573,1540,1543,1583,1591],{"className":1541,"style":1542},[762],"height:0.8884em;",[573,1544,1545,1548],{"style":838},[573,1546],{"className":1547,"style":842},[770],[573,1549,1551],{"className":1550},[775,776,777,778],[573,1552,1554],{"className":1553},[616,778],[573,1555,1557,1560],{"className":1556},[616,778],[573,1558,693],{"className":1559,"style":852},[616,617,778],[573,1561,1563],{"className":1562},[749],[573,1564,1566],{"className":1565},[753],[573,1567,1569],{"className":1568},[758],[573,1570,1572],{"className":1571,"style":1093},[762],[573,1573,1574,1577],{"style":962},[573,1575],{"className":1576,"style":966},[770],[573,1578,1580],{"className":1579},[775,970,971,778],[573,1581,594],{"className":1582},[616,617,778],[573,1584,1585,1588],{"style":855},[573,1586],{"className":1587,"style":842},[770],[573,1589],{"className":1590,"style":863},[862],[573,1592,1594,1597],{"style":1593},"top:-3.4101em;",[573,1595],{"className":1596,"style":842},[770],[573,1598,1600],{"className":1599},[775,776,777,778],[573,1601,1603],{"className":1602},[616,778],[573,1604,1606,1609],{"className":1605},[616,778],[573,1607,677],{"className":1608,"style":745},[616,617,778],[573,1610,1612],{"className":1611},[749],[573,1613,1615,1637],{"className":1614},[753,754],[573,1616,1618,1634],{"className":1617},[758],[573,1619,1622],{"className":1620,"style":1621},[762],"height:0.1645em;",[573,1623,1625,1628],{"style":1624},"top:-2.357em;margin-left:-0.0077em;margin-right:0.0714em;",[573,1626],{"className":1627,"style":966},[770],[573,1629,1631],{"className":1630},[775,970,971,778],[573,1632,594],{"className":1633},[616,617,778],[573,1635,786],{"className":1636},[785],[573,1638,1640],{"className":1639},[758],[573,1641,1644],{"className":1642,"style":1643},[762],"height:0.143em;",[573,1645],{},[573,1647,786],{"className":1648},[785],[573,1650,1652],{"className":1651},[758],[573,1653,1655],{"className":1654,"style":888},[762],[573,1656],{},[573,1658],{"className":1659},[894,822],[565,1661,1662,1663,1714,1715,1141],{},"Der Rentenbarwert ist immer kleiner als die Summe ",[573,1664,1666,1684],{"className":1665},[576],[573,1667,1669],{"className":1668},[580],[582,1670,1671],{"xmlns":584},[586,1672,1673,1681],{},[589,1674,1675,1677,1679],{},[592,1676,594],{},[683,1678,1169],{},[592,1680,633],{},[596,1682,1683],{"encoding":598},"n \\cdot r",[573,1685,1687,1705],{"className":1686,"ariaHidden":603},[602],[573,1688,1690,1693,1696,1699,1702],{"className":1689},[607],[573,1691],{"className":1692,"style":1262},[611],[573,1694,594],{"className":1695},[616,617],[573,1697],{"className":1698,"style":898},[799],[573,1700,1169],{"className":1701},[902],[573,1703],{"className":1704,"style":898},[799],[573,1706,1708,1711],{"className":1707},[607],[573,1709],{"className":1710,"style":612},[611],[573,1712,633],{"className":1713,"style":648},[616,617]," aller Auszahlungen, weil das Kapital in der Zwischenzeit Zinsen bringt. Für Zahlungen am Jahresanfang (vorschüssig) multipliziert man zusätzlich mit ",[573,1716,1718,1731],{"className":1717},[576],[573,1719,1721],{"className":1720},[580],[582,1722,1723],{"xmlns":584},[586,1724,1725,1729],{},[589,1726,1727],{},[592,1728,693],{},[596,1730,693],{"encoding":598},[573,1732,1734],{"className":1733,"ariaHidden":603},[602],[573,1735,1737,1741],{"className":1736},[607],[573,1738],{"className":1739,"style":1740},[611],"height:0.625em;vertical-align:-0.1944em;",[573,1742,693],{"className":1743,"style":852},[616,617],[560,1745,1747],{"id":1746},"beispiele","Beispiele",[565,1749,1750,1751,1781,1782,1811,1812,1857],{},"Jemand möchte ",[573,1752,1754,1768],{"className":1753},[576],[573,1755,1757],{"className":1756},[580],[582,1758,1759],{"xmlns":584},[586,1760,1761,1766],{},[589,1762,1763],{},[679,1764,1765],{},"15",[596,1767,1765],{"encoding":598},[573,1769,1771],{"className":1770,"ariaHidden":603},[602],[573,1772,1774,1778],{"className":1773},[607],[573,1775],{"className":1776,"style":1777},[611],"height:0.6444em;",[573,1779,1765],{"className":1780},[616]," Jahre lang jeweils am Jahresende ",[573,1783,1785,1799],{"className":1784},[576],[573,1786,1788],{"className":1787},[580],[582,1789,1790],{"xmlns":584},[586,1791,1792,1797],{},[589,1793,1794],{},[679,1795,1796],{},"6000",[596,1798,1796],{"encoding":598},[573,1800,1802],{"className":1801,"ariaHidden":603},[602],[573,1803,1805,1808],{"className":1804},[607],[573,1806],{"className":1807,"style":1777},[611],[573,1809,1796],{"className":1810},[616]," € ausgezahlt bekommen. Das Kapital bringt ",[573,1813,1815,1837],{"className":1814},[576],[573,1816,1818],{"className":1817},[580],[582,1819,1820],{"xmlns":584},[586,1821,1822,1834],{},[589,1823,1824,1826,1830],{},[679,1825,708],{},[1827,1828,1829],"mtext",{}," ",[592,1831,1833],{"mathvariant":1832},"normal","%",[596,1835,1836],{"encoding":598},"2\\,\\%",[573,1838,1840],{"className":1839,"ariaHidden":603},[602],[573,1841,1843,1847,1850,1854],{"className":1842},[607],[573,1844],{"className":1845,"style":1846},[611],"height:0.8056em;vertical-align:-0.0556em;",[573,1848,708],{"className":1849},[616],[573,1851],{"className":1852,"style":1853},[799],"margin-right:0.1667em;",[573,1855,1833],{"className":1856},[616]," Zinsen. Benötigt werden heute",[573,1859,1862],{"className":1860},[1861],"katex-display",[573,1863,1865,1934],{"className":1864},[576],[573,1866,1868],{"className":1867},[580],[582,1869,1871],{"xmlns":584,"display":1870},"block",[586,1872,1873,1931],{},[589,1874,1875,1881,1883,1885,1887,1915,1918,1921,1923,1926,1928],{},[673,1876,1877,1879],{},[592,1878,677],{},[679,1880,681],{},[683,1882,685],{},[679,1884,1796],{},[683,1886,1169],{},[687,1888,1889,1902],{},[589,1890,1891,1898,1900],{},[702,1892,1893,1896],{},[679,1894,1895],{},"1,02",[679,1897,1765],{},[683,1899,1182],{},[679,1901,1185],{},[589,1903,1904,1910,1912],{},[702,1905,1906,1908],{},[679,1907,1895],{},[679,1909,1765],{},[683,1911,1169],{},[679,1913,1914],{},"0,02",[683,1916,1917],{},"≈",[679,1919,1920],{},"77",[1827,1922,1829],{},[679,1924,1925],{},"095,58",[1827,1927,1829],{},[1827,1929,1930],{},"Euro",[596,1932,1933],{"encoding":598},"R_0 = 6000 \\cdot \\frac{1{,}02^{15} - 1}{1{,}02^{15} \\cdot 0{,}02} \\approx 77\\,095{,}58\\,\\text{Euro}",[573,1935,1937,1992,2010,2214],{"className":1936,"ariaHidden":603},[602],[573,1938,1940,1943,1983,1986,1989],{"className":1939},[607],[573,1941],{"className":1942,"style":738},[611],[573,1944,1946,1949],{"className":1945},[616],[573,1947,677],{"className":1948,"style":745},[616,617],[573,1950,1952],{"className":1951},[749],[573,1953,1955,1975],{"className":1954},[753,754],[573,1956,1958,1972],{"className":1957},[758],[573,1959,1961],{"className":1960,"style":763},[762],[573,1962,1963,1966],{"style":766},[573,1964],{"className":1965,"style":771},[770],[573,1967,1969],{"className":1968},[775,776,777,778],[573,1970,681],{"className":1971},[616,778],[573,1973,786],{"className":1974},[785],[573,1976,1978],{"className":1977},[758],[573,1979,1981],{"className":1980,"style":793},[762],[573,1982],{},[573,1984],{"className":1985,"style":800},[799],[573,1987,685],{"className":1988},[804],[573,1990],{"className":1991,"style":800},[799],[573,1993,1995,1998,2001,2004,2007],{"className":1994},[607],[573,1996],{"className":1997,"style":1777},[611],[573,1999,1796],{"className":2000},[616],[573,2002],{"className":2003,"style":898},[799],[573,2005,1169],{"className":2006},[902],[573,2008],{"className":2009,"style":898},[799],[573,2011,2013,2017,2205,2208,2211],{"className":2012},[607],[573,2014],{"className":2015,"style":2016},[611],"height:2.3715em;vertical-align:-0.8804em;",[573,2018,2020,2023,2202],{"className":2019},[616],[573,2021],{"className":2022},[821,822],[573,2024,2026],{"className":2025},[687],[573,2027,2029,2193],{"className":2028},[753,754],[573,2030,2032,2190],{"className":2031},[758],[573,2033,2036,2115,2123],{"className":2034,"style":2035},[762],"height:1.4911em;",[573,2037,2039,2042],{"style":2038},"top:-2.314em;",[573,2040],{"className":2041,"style":842},[770],[573,2043,2045,2048,2056,2059,2093,2096,2099,2102,2105,2111],{"className":2044},[616],[573,2046,1185],{"className":2047},[616],[573,2049,2051],{"className":2050},[616],[573,2052,2055],{"className":2053},[2054],"mpunct",",",[573,2057,681],{"className":2058},[616],[573,2060,2062,2065],{"className":2061},[616],[573,2063,708],{"className":2064},[616],[573,2066,2068],{"className":2067},[749],[573,2069,2071],{"className":2070},[753],[573,2072,2074],{"className":2073},[758],[573,2075,2078],{"className":2076,"style":2077},[762],"height:0.7401em;",[573,2079,2081,2084],{"style":2080},"top:-2.989em;margin-right:0.05em;",[573,2082],{"className":2083,"style":771},[770],[573,2085,2087],{"className":2086},[775,776,777,778],[573,2088,2090],{"className":2089},[616,778],[573,2091,1765],{"className":2092},[616,778],[573,2094],{"className":2095,"style":898},[799],[573,2097,1169],{"className":2098},[902],[573,2100],{"className":2101,"style":898},[799],[573,2103,681],{"className":2104},[616],[573,2106,2108],{"className":2107},[616],[573,2109,2055],{"className":2110},[2054],[573,2112,2114],{"className":2113},[616],"02",[573,2116,2117,2120],{"style":855},[573,2118],{"className":2119,"style":842},[770],[573,2121],{"className":2122,"style":863},[862],[573,2124,2126,2129],{"style":2125},"top:-3.677em;",[573,2127],{"className":2128,"style":842},[770],[573,2130,2132,2135,2141,2144,2178,2181,2184,2187],{"className":2131},[616],[573,2133,1185],{"className":2134},[616],[573,2136,2138],{"className":2137},[616],[573,2139,2055],{"className":2140},[2054],[573,2142,681],{"className":2143},[616],[573,2145,2147,2150],{"className":2146},[616],[573,2148,708],{"className":2149},[616],[573,2151,2153],{"className":2152},[749],[573,2154,2156],{"className":2155},[753],[573,2157,2159],{"className":2158},[758],[573,2160,2163],{"className":2161,"style":2162},[762],"height:0.8141em;",[573,2164,2166,2169],{"style":2165},"top:-3.063em;margin-right:0.05em;",[573,2167],{"className":2168,"style":771},[770],[573,2170,2172],{"className":2171},[775,776,777,778],[573,2173,2175],{"className":2174},[616,778],[573,2176,1765],{"className":2177},[616,778],[573,2179],{"className":2180,"style":898},[799],[573,2182,1182],{"className":2183},[902],[573,2185],{"className":2186,"style":898},[799],[573,2188,1185],{"className":2189},[616],[573,2191,786],{"className":2192},[785],[573,2194,2196],{"className":2195},[758],[573,2197,2200],{"className":2198,"style":2199},[762],"height:0.8804em;",[573,2201],{},[573,2203],{"className":2204},[894,822],[573,2206],{"className":2207,"style":800},[799],[573,2209,1917],{"className":2210},[804],[573,2212],{"className":2213,"style":800},[799],[573,2215,2217,2221,2224,2227,2231,2237,2241,2244],{"className":2216},[607],[573,2218],{"className":2219,"style":2220},[611],"height:0.8778em;vertical-align:-0.1944em;",[573,2222,1920],{"className":2223},[616],[573,2225],{"className":2226,"style":1853},[799],[573,2228,2230],{"className":2229},[616],"095",[573,2232,2234],{"className":2233},[616],[573,2235,2055],{"className":2236},[2054],[573,2238,2240],{"className":2239},[616],"58",[573,2242],{"className":2243,"style":1853},[799],[573,2245,2248],{"className":2246},[616,2247],"text",[573,2249,1930],{"className":2250},[616],[565,2252,2253,2254,2339],{},"Ausgezahlt werden insgesamt ",[573,2255,2257,2285],{"className":2256},[576],[573,2258,2260],{"className":2259},[580],[582,2261,2262],{"xmlns":584},[586,2263,2264,2282],{},[589,2265,2266,2268,2270,2272,2274,2277,2279],{},[679,2267,1765],{},[683,2269,1169],{},[679,2271,1796],{},[683,2273,685],{},[679,2275,2276],{},"90",[1827,2278,1829],{},[679,2280,2281],{},"000",[596,2283,2284],{"encoding":598},"15 \\cdot 6000 = 90\\,000",[573,2286,2288,2306,2324],{"className":2287,"ariaHidden":603},[602],[573,2289,2291,2294,2297,2300,2303],{"className":2290},[607],[573,2292],{"className":2293,"style":1777},[611],[573,2295,1765],{"className":2296},[616],[573,2298],{"className":2299,"style":898},[799],[573,2301,1169],{"className":2302},[902],[573,2304],{"className":2305,"style":898},[799],[573,2307,2309,2312,2315,2318,2321],{"className":2308},[607],[573,2310],{"className":2311,"style":1777},[611],[573,2313,1796],{"className":2314},[616],[573,2316],{"className":2317,"style":800},[799],[573,2319,685],{"className":2320},[804],[573,2322],{"className":2323,"style":800},[799],[573,2325,2327,2330,2333,2336],{"className":2326},[607],[573,2328],{"className":2329,"style":1777},[611],[573,2331,2276],{"className":2332},[616],[573,2334],{"className":2335,"style":1853},[799],[573,2337,2281],{"className":2338},[616]," €, die Differenz stammt aus den Zinsen.",[565,2341,2342,2343,2383,2384,2424,2425,2453,2454,2482],{},"Wie lange reichen ",[573,2344,2346,2365],{"className":2345},[576],[573,2347,2349],{"className":2348},[580],[582,2350,2351],{"xmlns":584},[586,2352,2353,2362],{},[589,2354,2355,2358,2360],{},[679,2356,2357],{},"50",[1827,2359,1829],{},[679,2361,2281],{},[596,2363,2364],{"encoding":598},"50\\,000",[573,2366,2368],{"className":2367,"ariaHidden":603},[602],[573,2369,2371,2374,2377,2380],{"className":2370},[607],[573,2372],{"className":2373,"style":1777},[611],[573,2375,2357],{"className":2376},[616],[573,2378],{"className":2379,"style":1853},[799],[573,2381,2281],{"className":2382},[616]," € bei ",[573,2385,2387,2406],{"className":2386},[576],[573,2388,2390],{"className":2389},[580],[582,2391,2392],{"xmlns":584},[586,2393,2394,2403],{},[589,2395,2396,2399,2401],{},[679,2397,2398],{},"3",[1827,2400,1829],{},[592,2402,1833],{"mathvariant":1832},[596,2404,2405],{"encoding":598},"3\\,\\%",[573,2407,2409],{"className":2408,"ariaHidden":603},[602],[573,2410,2412,2415,2418,2421],{"className":2411},[607],[573,2413],{"className":2414,"style":1846},[611],[573,2416,2398],{"className":2417},[616],[573,2419],{"className":2420,"style":1853},[799],[573,2422,1833],{"className":2423},[616],", wenn jährlich ",[573,2426,2428,2441],{"className":2427},[576],[573,2429,2431],{"className":2430},[580],[582,2432,2433],{"xmlns":584},[586,2434,2435,2439],{},[589,2436,2437],{},[679,2438,1796],{},[596,2440,1796],{"encoding":598},[573,2442,2444],{"className":2443,"ariaHidden":603},[602],[573,2445,2447,2450],{"className":2446},[607],[573,2448],{"className":2449,"style":1777},[611],[573,2451,1796],{"className":2452},[616]," € entnommen werden? Löst man die Formel nach ",[573,2455,2457,2470],{"className":2456},[576],[573,2458,2460],{"className":2459},[580],[582,2461,2462],{"xmlns":584},[586,2463,2464,2468],{},[589,2465,2466],{},[592,2467,594],{},[596,2469,594],{"encoding":598},[573,2471,2473],{"className":2472,"ariaHidden":603},[602],[573,2474,2476,2479],{"className":2475},[607],[573,2477],{"className":2478,"style":612},[611],[573,2480,594],{"className":2481},[616,617]," auf, ergibt sich",[573,2484,2486],{"className":2485},[1861],[573,2487,2489,2615],{"className":2488},[576],[573,2490,2492],{"className":2491},[580],[582,2493,2494],{"xmlns":584,"display":1870},[586,2495,2496,2612],{},[589,2497,2498,2500,2502,2504,2559,2561,2563,2607,2609],{},[592,2499,594],{},[683,2501,685],{},[683,2503,1182],{},[687,2505,2506,2551],{},[589,2507,2508,2511,2514],{},[592,2509,2510],{},"ln",[683,2512,2513],{},"⁡",[589,2515,2516,2519,2521,2523,2549],{},[683,2517,2518],{"fence":603},"(",[679,2520,1185],{},[683,2522,1182],{},[687,2524,2525,2547],{},[589,2526,2527,2533,2535,2538,2540,2542,2544],{},[673,2528,2529,2531],{},[592,2530,677],{},[679,2532,681],{},[683,2534,1169],{},[683,2536,2518],{"stretchy":2537},"false",[592,2539,693],{},[683,2541,1182],{},[679,2543,1185],{},[683,2545,2546],{"stretchy":2537},")",[592,2548,633],{},[683,2550,2546],{"fence":603},[589,2552,2553,2555,2557],{},[592,2554,2510],{},[683,2556,2513],{},[592,2558,693],{},[683,2560,685],{},[683,2562,1182],{},[687,2564,2565,2598],{},[589,2566,2567,2569,2571],{},[592,2568,2510],{},[683,2570,2513],{},[589,2572,2573,2575,2577,2579,2596],{},[683,2574,2518],{"fence":603},[679,2576,1185],{},[683,2578,1182],{},[687,2580,2581,2594],{},[589,2582,2583,2585,2587,2589,2591],{},[679,2584,2357],{},[1827,2586,1829],{},[679,2588,2281],{},[683,2590,1169],{},[679,2592,2593],{},"0,03",[679,2595,1796],{},[683,2597,2546],{"fence":603},[589,2599,2600,2602,2604],{},[592,2601,2510],{},[683,2603,2513],{},[679,2605,2606],{},"1,03",[683,2608,1917],{},[679,2610,2611],{},"9,73",[596,2613,2614],{"encoding":598},"n = -\\frac{\\ln\\left(1 - \\frac{R_0 \\cdot (q - 1)}{r}\\right)}{\\ln q} = -\\frac{\\ln\\left(1 - \\frac{50\\,000 \\cdot 0{,}03}{6000}\\right)}{\\ln 1{,}03} \\approx 9{,}73",[573,2616,2618,2636,2890,3110],{"className":2617,"ariaHidden":603},[602],[573,2619,2621,2624,2627,2630,2633],{"className":2620},[607],[573,2622],{"className":2623,"style":612},[611],[573,2625,594],{"className":2626},[616,617],[573,2628],{"className":2629,"style":800},[799],[573,2631,685],{"className":2632},[804],[573,2634],{"className":2635,"style":800},[799],[573,2637,2639,2643,2646,2881,2884,2887],{"className":2638},[607],[573,2640],{"className":2641,"style":2642},[611],"height:3.0705em;vertical-align:-0.8804em;",[573,2644,1182],{"className":2645},[616],[573,2647,2649,2652,2878],{"className":2648},[616],[573,2650],{"className":2651},[821,822],[573,2653,2655],{"className":2654},[687],[573,2656,2658,2870],{"className":2657},[753,754],[573,2659,2661,2867],{"className":2660},[758],[573,2662,2665,2685,2694],{"className":2663,"style":2664},[762],"height:2.19em;",[573,2666,2668,2672],{"style":2667},"top:-2.464em;",[573,2669],{"className":2670,"style":2671},[770],"height:3.15em;",[573,2673,2675,2679,2682],{"className":2674},[616],[573,2676,2510],{"className":2677},[2678],"mop",[573,2680],{"className":2681,"style":1853},[799],[573,2683,693],{"className":2684,"style":852},[616,617],[573,2686,2688,2691],{"style":2687},"top:-3.38em;",[573,2689],{"className":2690,"style":2671},[770],[573,2692],{"className":2693,"style":863},[862],[573,2695,2697,2700],{"style":2696},"top:-4.19em;",[573,2698],{"className":2699,"style":2671},[770],[573,2701,2703,2706,2709],{"className":2702},[616],[573,2704,2510],{"className":2705},[2678],[573,2707],{"className":2708,"style":1853},[799],[573,2710,2712,2722,2725,2728,2731,2734,2861],{"className":2711},[1030],[573,2713,2717],{"className":2714,"style":2716},[821,2715],"delimcenter","top:0em;",[573,2718,2518],{"className":2719},[2720,2721],"delimsizing","size2",[573,2723,1185],{"className":2724},[616],[573,2726],{"className":2727,"style":898},[799],[573,2729,1182],{"className":2730},[902],[573,2732],{"className":2733,"style":898},[799],[573,2735,2737,2740,2858],{"className":2736},[616],[573,2738],{"className":2739},[821,822],[573,2741,2743],{"className":2742},[687],[573,2744,2746,2849],{"className":2745},[753,754],[573,2747,2749,2846],{"className":2748},[758],[573,2750,2753,2767,2775],{"className":2751,"style":2752},[762],"height:1.01em;",[573,2754,2755,2758],{"style":838},[573,2756],{"className":2757,"style":842},[770],[573,2759,2761],{"className":2760},[775,776,777,778],[573,2762,2764],{"className":2763},[616,778],[573,2765,633],{"className":2766,"style":648},[616,617,778],[573,2768,2769,2772],{"style":855},[573,2770],{"className":2771,"style":842},[770],[573,2773],{"className":2774,"style":863},[862],[573,2776,2778,2781],{"style":2777},"top:-3.485em;",[573,2779],{"className":2780,"style":842},[770],[573,2782,2784],{"className":2783},[775,776,777,778],[573,2785,2787,2828,2831,2834,2837,2840,2843],{"className":2786},[616,778],[573,2788,2790,2793],{"className":2789},[616,778],[573,2791,677],{"className":2792,"style":745},[616,617,778],[573,2794,2796],{"className":2795},[749],[573,2797,2799,2820],{"className":2798},[753,754],[573,2800,2802,2817],{"className":2801},[758],[573,2803,2806],{"className":2804,"style":2805},[762],"height:0.3173em;",[573,2807,2808,2811],{"style":1624},[573,2809],{"className":2810,"style":966},[770],[573,2812,2814],{"className":2813},[775,970,971,778],[573,2815,681],{"className":2816},[616,778],[573,2818,786],{"className":2819},[785],[573,2821,2823],{"className":2822},[758],[573,2824,2826],{"className":2825,"style":1643},[762],[573,2827],{},[573,2829,1169],{"className":2830},[902,778],[573,2832,2518],{"className":2833},[821,778],[573,2835,693],{"className":2836,"style":852},[616,617,778],[573,2838,1182],{"className":2839},[902,778],[573,2841,1185],{"className":2842},[616,778],[573,2844,2546],{"className":2845},[894,778],[573,2847,786],{"className":2848},[785],[573,2850,2852],{"className":2851},[758],[573,2853,2856],{"className":2854,"style":2855},[762],"height:0.345em;",[573,2857],{},[573,2859],{"className":2860},[894,822],[573,2862,2864],{"className":2863,"style":2716},[894,2715],[573,2865,2546],{"className":2866},[2720,2721],[573,2868,786],{"className":2869},[785],[573,2871,2873],{"className":2872},[758],[573,2874,2876],{"className":2875,"style":2199},[762],[573,2877],{},[573,2879],{"className":2880},[894,822],[573,2882],{"className":2883,"style":800},[799],[573,2885,685],{"className":2886},[804],[573,2888],{"className":2889,"style":800},[799],[573,2891,2893,2897,2900,3101,3104,3107],{"className":2892},[607],[573,2894],{"className":2895,"style":2896},[611],"height:2.5177em;vertical-align:-0.8804em;",[573,2898,1182],{"className":2899},[616],[573,2901,2903,2906,3098],{"className":2902},[616],[573,2904],{"className":2905},[821,822],[573,2907,2909],{"className":2908},[687],[573,2910,2912,3090],{"className":2911},[753,754],[573,2913,2915,3087],{"className":2914},[758],[573,2916,2919,2946,2954],{"className":2917,"style":2918},[762],"height:1.6372em;",[573,2920,2921,2924],{"style":2038},[573,2922],{"className":2923,"style":842},[770],[573,2925,2927,2930,2933,2936,2942],{"className":2926},[616],[573,2928,2510],{"className":2929},[2678],[573,2931],{"className":2932,"style":1853},[799],[573,2934,1185],{"className":2935},[616],[573,2937,2939],{"className":2938},[616],[573,2940,2055],{"className":2941},[2054],[573,2943,2945],{"className":2944},[616],"03",[573,2947,2948,2951],{"style":855},[573,2949],{"className":2950,"style":842},[770],[573,2952],{"className":2953,"style":863},[862],[573,2955,2957,2960],{"style":2956},"top:-3.74em;",[573,2958],{"className":2959,"style":842},[770],[573,2961,2963,2966,2969],{"className":2962},[616],[573,2964,2510],{"className":2965},[2678],[573,2967],{"className":2968,"style":1853},[799],[573,2970,2972,2978,2981,2984,2987,2990,3081],{"className":2971},[1030],[573,2973,2975],{"className":2974,"style":2716},[821,2715],[573,2976,2518],{"className":2977},[2720,971],[573,2979,1185],{"className":2980},[616],[573,2982],{"className":2983,"style":898},[799],[573,2985,1182],{"className":2986},[902],[573,2988],{"className":2989,"style":898},[799],[573,2991,2993,2996,3078],{"className":2992},[616],[573,2994],{"className":2995},[821,822],[573,2997,2999],{"className":2998},[687],[573,3000,3002,3070],{"className":3001},[753,754],[573,3003,3005,3067],{"className":3004},[758],[573,3006,3009,3023,3031],{"className":3007,"style":3008},[762],"height:0.8972em;",[573,3010,3011,3014],{"style":838},[573,3012],{"className":3013,"style":842},[770],[573,3015,3017],{"className":3016},[775,776,777,778],[573,3018,3020],{"className":3019},[616,778],[573,3021,1796],{"className":3022},[616,778],[573,3024,3025,3028],{"style":855},[573,3026],{"className":3027,"style":842},[770],[573,3029],{"className":3030,"style":863},[862],[573,3032,3033,3036],{"style":1331},[573,3034],{"className":3035,"style":842},[770],[573,3037,3039],{"className":3038},[775,776,777,778],[573,3040,3042,3045,3049,3052,3055,3058,3064],{"className":3041},[616,778],[573,3043,2357],{"className":3044},[616,778],[573,3046],{"className":3047,"style":3048},[799,778],"margin-right:0.1952em;",[573,3050,2281],{"className":3051},[616,778],[573,3053,1169],{"className":3054},[902,778],[573,3056,681],{"className":3057},[616,778],[573,3059,3061],{"className":3060},[616,778],[573,3062,2055],{"className":3063},[2054,778],[573,3065,2945],{"className":3066},[616,778],[573,3068,786],{"className":3069},[785],[573,3071,3073],{"className":3072},[758],[573,3074,3076],{"className":3075,"style":2855},[762],[573,3077],{},[573,3079],{"className":3080},[894,822],[573,3082,3084],{"className":3083,"style":2716},[894,2715],[573,3085,2546],{"className":3086},[2720,971],[573,3088,786],{"className":3089},[785],[573,3091,3093],{"className":3092},[758],[573,3094,3096],{"className":3095,"style":2199},[762],[573,3097],{},[573,3099],{"className":3100},[894,822],[573,3102],{"className":3103,"style":800},[799],[573,3105,1917],{"className":3106},[804],[573,3108],{"className":3109,"style":800},[799],[573,3111,3113,3117,3121,3127],{"className":3112},[607],[573,3114],{"className":3115,"style":3116},[611],"height:0.8389em;vertical-align:-0.1944em;",[573,3118,3120],{"className":3119},[616],"9",[573,3122,3124],{"className":3123},[616],[573,3125,2055],{"className":3126},[2054],[573,3128,3130],{"className":3129},[616],"73",[565,3132,3133,3134,3162,3163,3191,3192,3477],{},"Es gibt also ",[573,3135,3137,3150],{"className":3136},[576],[573,3138,3140],{"className":3139},[580],[582,3141,3142],{"xmlns":584},[586,3143,3144,3148],{},[589,3145,3146],{},[679,3147,3120],{},[596,3149,3120],{"encoding":598},[573,3151,3153],{"className":3152,"ariaHidden":603},[602],[573,3154,3156,3159],{"className":3155},[607],[573,3157],{"className":3158,"style":1777},[611],[573,3160,3120],{"className":3161},[616]," volle Auszahlungen; für ",[573,3164,3166,3179],{"className":3165},[576],[573,3167,3169],{"className":3168},[580],[582,3170,3171],{"xmlns":584},[586,3172,3173,3177],{},[589,3174,3175],{},[679,3176,3120],{},[596,3178,3120],{"encoding":598},[573,3180,3182],{"className":3181,"ariaHidden":603},[602],[573,3183,3185,3188],{"className":3184},[607],[573,3186],{"className":3187,"style":1777},[611],[573,3189,3120],{"className":3190},[616]," Raten hätten ",[573,3193,3195,3247],{"className":3194},[576],[573,3196,3198],{"className":3197},[580],[582,3199,3200],{"xmlns":584},[586,3201,3202,3244],{},[589,3203,3204,3206,3208,3234,3236,3239,3241],{},[679,3205,1796],{},[683,3207,1169],{},[687,3209,3210,3222],{},[589,3211,3212,3218,3220],{},[702,3213,3214,3216],{},[679,3215,2606],{},[679,3217,3120],{},[683,3219,1182],{},[679,3221,1185],{},[589,3223,3224,3230,3232],{},[702,3225,3226,3228],{},[679,3227,2606],{},[679,3229,3120],{},[683,3231,1169],{},[679,3233,2593],{},[683,3235,1917],{},[679,3237,3238],{},"46",[1827,3240,1829],{},[679,3242,3243],{},"716,65",[596,3245,3246],{"encoding":598},"6000 \\cdot \\frac{1{,}03^9 - 1}{1{,}03^9 \\cdot 0{,}03} \\approx 46\\,716{,}65",[573,3248,3250,3268,3451],{"className":3249,"ariaHidden":603},[602],[573,3251,3253,3256,3259,3262,3265],{"className":3252},[607],[573,3254],{"className":3255,"style":1777},[611],[573,3257,1796],{"className":3258},[616],[573,3260],{"className":3261,"style":898},[799],[573,3263,1169],{"className":3264},[902],[573,3266],{"className":3267,"style":898},[799],[573,3269,3271,3275,3442,3445,3448],{"className":3270},[607],[573,3272],{"className":3273,"style":3274},[611],"height:1.5511em;vertical-align:-0.4811em;",[573,3276,3278,3281,3439],{"className":3277},[616],[573,3279],{"className":3280},[821,822],[573,3282,3284],{"className":3283},[687],[573,3285,3287,3431],{"className":3286},[753,754],[573,3288,3290,3428],{"className":3289},[758],[573,3291,3294,3361,3369],{"className":3292,"style":3293},[762],"height:1.07em;",[573,3295,3296,3299],{"style":838},[573,3297],{"className":3298,"style":842},[770],[573,3300,3302],{"className":3301},[775,776,777,778],[573,3303,3305,3308,3314,3317,3346,3349,3352,3358],{"className":3304},[616,778],[573,3306,1185],{"className":3307},[616,778],[573,3309,3311],{"className":3310},[616,778],[573,3312,2055],{"className":3313},[2054,778],[573,3315,681],{"className":3316},[616,778],[573,3318,3320,3323],{"className":3319},[616,778],[573,3321,2398],{"className":3322},[616,778],[573,3324,3326],{"className":3325},[749],[573,3327,3329],{"className":3328},[753],[573,3330,3332],{"className":3331},[758],[573,3333,3335],{"className":3334,"style":959},[762],[573,3336,3337,3340],{"style":962},[573,3338],{"className":3339,"style":966},[770],[573,3341,3343],{"className":3342},[775,970,971,778],[573,3344,3120],{"className":3345},[616,778],[573,3347,1169],{"className":3348},[902,778],[573,3350,681],{"className":3351},[616,778],[573,3353,3355],{"className":3354},[616,778],[573,3356,2055],{"className":3357},[2054,778],[573,3359,2945],{"className":3360},[616,778],[573,3362,3363,3366],{"style":855},[573,3364],{"className":3365,"style":842},[770],[573,3367],{"className":3368,"style":863},[862],[573,3370,3371,3374],{"style":1331},[573,3372],{"className":3373,"style":842},[770],[573,3375,3377],{"className":3376},[775,776,777,778],[573,3378,3380,3383,3389,3392,3422,3425],{"className":3379},[616,778],[573,3381,1185],{"className":3382},[616,778],[573,3384,3386],{"className":3385},[616,778],[573,3387,2055],{"className":3388},[2054,778],[573,3390,681],{"className":3391},[616,778],[573,3393,3395,3398],{"className":3394},[616,778],[573,3396,2398],{"className":3397},[616,778],[573,3399,3401],{"className":3400},[749],[573,3402,3404],{"className":3403},[753],[573,3405,3407],{"className":3406},[758],[573,3408,3411],{"className":3409,"style":3410},[762],"height:0.8913em;",[573,3412,3413,3416],{"style":1362},[573,3414],{"className":3415,"style":966},[770],[573,3417,3419],{"className":3418},[775,970,971,778],[573,3420,3120],{"className":3421},[616,778],[573,3423,1182],{"className":3424},[902,778],[573,3426,1185],{"className":3427},[616,778],[573,3429,786],{"className":3430},[785],[573,3432,3434],{"className":3433},[758],[573,3435,3437],{"className":3436,"style":888},[762],[573,3438],{},[573,3440],{"className":3441},[894,822],[573,3443],{"className":3444,"style":800},[799],[573,3446,1917],{"className":3447},[804],[573,3449],{"className":3450,"style":800},[799],[573,3452,3454,3457,3460,3463,3467,3473],{"className":3453},[607],[573,3455],{"className":3456,"style":3116},[611],[573,3458,3238],{"className":3459},[616],[573,3461],{"className":3462,"style":1853},[799],[573,3464,3466],{"className":3465},[616],"716",[573,3468,3470],{"className":3469},[616],[573,3471,2055],{"className":3472},[2054],[573,3474,3476],{"className":3475},[616],"65"," € gereicht, der Rest reicht für eine kleinere zehnte Zahlung.",[560,3479,3481],{"id":3480},"typische-fehler","Typische Fehler",[654,3483,3484,3495,3560],{},[657,3485,3486,3487,3490,3491,3494],{},"Rentenbarwert und Rentenendwert verwechseln. Der Barwert ist der Betrag ",[569,3488,3489],{},"zu Beginn",", der Endwert das Guthaben ",[569,3492,3493],{},"am Ende"," eines Sparplans.",[657,3496,3497,3498,3559],{},"Den Faktor ",[573,3499,3501,3519],{"className":3500},[576],[573,3502,3504],{"className":3503},[580],[582,3505,3506],{"xmlns":584},[586,3507,3508,3516],{},[589,3509,3510],{},[702,3511,3512,3514],{},[592,3513,693],{},[592,3515,594],{},[596,3517,3518],{"encoding":598},"q^n",[573,3520,3522],{"className":3521,"ariaHidden":603},[602],[573,3523,3525,3529],{"className":3524},[607],[573,3526],{"className":3527,"style":3528},[611],"height:0.8588em;vertical-align:-0.1944em;",[573,3530,3532,3535],{"className":3531},[616],[573,3533,693],{"className":3534,"style":852},[616,617],[573,3536,3538],{"className":3537},[749],[573,3539,3541],{"className":3540},[753],[573,3542,3544],{"className":3543},[758],[573,3545,3548],{"className":3546,"style":3547},[762],"height:0.6644em;",[573,3549,3550,3553],{"style":2165},[573,3551],{"className":3552,"style":771},[770],[573,3554,3556],{"className":3555},[775,776,777,778],[573,3557,594],{"className":3558},[616,617,778]," im Nenner vergessen. Dann erhält man den Endwert statt des Barwerts.",[657,3561,3562,3563,3602],{},"Die Auszahlungen einfach addieren und die Zinsen ignorieren: ",[573,3564,3566,3584],{"className":3565},[576],[573,3567,3569],{"className":3568},[580],[582,3570,3571],{"xmlns":584},[586,3572,3573,3581],{},[589,3574,3575,3577,3579],{},[679,3576,2276],{},[1827,3578,1829],{},[679,3580,2281],{},[596,3582,3583],{"encoding":598},"90\\,000",[573,3585,3587],{"className":3586,"ariaHidden":603},[602],[573,3588,3590,3593,3596,3599],{"className":3589},[607],[573,3591],{"className":3592,"style":1777},[611],[573,3594,2276],{"className":3595},[616],[573,3597],{"className":3598,"style":1853},[799],[573,3600,2281],{"className":3601},[616]," € wären deutlich zu viel.",{"title":3604,"searchDepth":3605,"depth":3605,"links":3606},"",2,[3607,3608,3609],{"id":562,"depth":3605,"text":563},{"id":1746,"depth":3605,"text":1747},{"id":3480,"depth":3605,"text":3481},[3611,3614,3617,3620,3623,3626],{"lhs":3612,"rhs":3613},"r*(q^5 - 1)\u002F(q^5*(q - 1))","(r*(q^5 - 1)\u002F(q - 1)) \u002F q^5",{"lhs":3615,"rhs":3616},"r\u002Fq + r\u002Fq^2 + r\u002Fq^3","r*(q^3 - 1)\u002F(q^3*(q - 1))",{"lhs":3618,"rhs":3619},"round(6000 * (1.02^15 - 1) \u002F (1.02^15 * 0.02), 2)","77095.58",{"lhs":3621,"rhs":3622},"15 * 6000","90000",{"lhs":3624,"rhs":3625},"round(-log(1 - 50000 * 0.03 \u002F 6000) \u002F log(1.03), 2)","9.73",{"lhs":3627,"rhs":3628},"round(6000 * (1.03^9 - 1) \u002F (1.03^9 * 0.03), 2)","46716.65","p > 0,\\ n \\geq 1,\\ \\text{Zahlung jeweils am Ende des Jahres}","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>p\u003C\u002Fmi>\u003Cmo>>\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">p > 0,\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.7335em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">p\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">0\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>n\u003C\u002Fmi>\u003Cmo>≥\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">n \\geq 1,\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.7719em;vertical-align:-0.136em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">≥\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">Zahlung jeweils am Ende des Jahres\u003C\u002Fspan>",null,"md","annuity-present-value","R_0 = r \\cdot \\frac{q^n - 1}{q^n \\cdot (q - 1)}, \\qquad q = 1 + \\frac{p}{100}","\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>R\u003C\u002Fmi>\u003Cmn>0\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmo>=\u003C\u002Fmo>\u003Cmi>r\u003C\u002Fmi>\u003Cmo>⋅\u003C\u002Fmo>\u003Cmfrac>\u003Cmrow>\u003Cmsup>\u003Cmi>q\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsup>\u003Cmo>−\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003C\u002Fmrow>\u003Cmrow>\u003Cmsup>\u003Cmi>q\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsup>\u003Cmo>⋅\u003C\u002Fmo>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>q\u003C\u002Fmi>\u003Cmo>−\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmfrac>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmspace width=\"2em\"\u002F>\u003Cmi>q\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmo>+\u003C\u002Fmo>\u003Cmfrac>\u003Cmi>p\u003C\u002Fmi>\u003Cmn>100\u003C\u002Fmn>\u003C\u002Fmfrac>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">R_0 = r \\cdot \\frac{q^n - 1}{q^n \\cdot (q - 1)}, \\qquad q = 1 + \\frac{p}{100}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8333em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0077em;\">R\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">0\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.4445em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">r\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">⋅\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:2.2774em;vertical-align:-0.936em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mopen nulldelimiter\">\u003C\u002Fspan>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.3414em;\">\u003Cspan style=\"top:-2.314em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">q\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.5904em;\">\u003Cspan style=\"top:-2.989em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">⋅\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">q\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.23em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.677em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">q\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6644em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.936em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose nulldelimiter\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:2em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">q\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1.7936em;vertical-align:-0.686em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mopen nulldelimiter\">\u003C\u002Fspan>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.1076em;\">\u003Cspan style=\"top:-2.314em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">100\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.23em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.677em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">p\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.686em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose nulldelimiter\">\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","upper",{},true,"\u002Fde\u002Fformulas\u002Fmath\u002Farithmetic\u002Frentenbarwert",[],[3642,3643,3644,3645],"annuity-future-value","present-value","annuity-payment","geometric-sum",{"title":548,"description":3604},"de\u002Fformulas\u002Fmath\u002Farithmetic\u002Frentenbarwert","Der Rentenbarwert ist das Kapital, das heute nötig ist, um n Jahre lang jeweils am Jahresende den Betrag r auszuzahlen.",[3650,3651,3652,3653,3654],"rente","barwert","abzinsen","auszahlplan","finanzmathematik",[3656,3660,3663,3666,3669],{"symbol":3657,"meaning":3658,"symbolHtml":3659},"R_0","Rentenbarwert, das heute nötige Kapital","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>R\u003C\u002Fmi>\u003Cmn>0\u003C\u002Fmn>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">R_0\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8333em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0077em;\">R\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan 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style=\"height:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">r\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":693,"meaning":3664,"symbolHtml":3665},"Zinsfaktor","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>q\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">q\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">q\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":565,"meaning":3667,"symbolHtml":3668},"Zinssatz in Prozent pro 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