Algebra

Interest and Annuities

Interest

If 𝑟 be the Interest on £1 for 1 year, 𝑛 the number of years 𝑃 the Principal 𝐴 the amount in 𝑛 years. Then At Simple Interest 𝐴=𝑃(1+𝑛𝑟) 296 At Compound Interest 𝐴=𝑃(1+𝑟)^𝑛 By (84) 297 But if the payments of Interest be made 𝑞 times a year ⋯ 𝐴=𝑃(1+𝑟𝑞)^𝑛𝑞 298 If 𝐴 be an amount due in 𝑛 years' time, and 𝑃 the present worth of 𝐴. Then At Simple Interest 𝑃=𝐴1+𝑛𝑟 By (296) 299 At Compound Interest 𝑃=𝐴(1+𝑟)^𝑛 By (297) 300 Discount=𝐴−𝑃 301

Annuities

The amount of an Annuity of £1 in 𝑛 years, at Simple Interest ⋯: =𝑛+𝑛(𝑛−1)2𝑟. By (82) 302 Present value of same =𝑛+12𝑛(𝑛−1)𝑟1+𝑛𝑟. By (299) 303 Amount at Compound Interst ⋯: =(1+𝑟)^𝑛−1(1+𝑟)−1 By (85) Present worth of same 1−(1+𝑟)−𝑛(1+𝑟)−1 By (300) 304 Amount when the payments of Interest are made 𝑞 times per annum ⋯: =1+𝑟𝑞^𝑛𝑞−11+𝑟𝑞^𝑞−1 By (298) Present value of same =1−1+𝑟𝑞^−𝑛𝑞1+𝑟𝑞^𝑞−1 305 Amount when the payments of the Annuity are made 𝑚 times per annum ⋯: =(1+𝑟)^𝑛−1𝑚{(1+𝑟)^1𝑚−1} Present value of same =1−(1+𝑟)^−𝑛𝑚{(1+𝑟)^1𝑚−1} 306 Amount when the Interest is paid 𝑞 times and the Annuity 𝑚 times per annum ⋯ =1+𝑟𝑞^𝑛𝑞−1𝑚1+𝑟𝑞^𝑞𝑚−1 Present value of same =1−1+𝑟𝑞^−𝑛𝑞𝑚1+𝑟𝑞^𝑞𝑚−1 307

Sources and References

https://archive.org/details/synopsis-of-elementary-results-in-pure-and-applied-mathematics-pdfdrive