{"id":97,"date":"2024-01-09T07:20:13","date_gmt":"2024-01-09T07:20:13","guid":{"rendered":"https:\/\/openbook.ums.edu.my\/financialmathematicsineconomics\/chapter\/5-3-present-value-of-an-ordinary-annuity\/"},"modified":"2024-09-25T08:37:10","modified_gmt":"2024-09-25T08:37:10","slug":"5-3-present-value-of-an-ordinary-annuity","status":"publish","type":"chapter","link":"https:\/\/openbook.ums.edu.my\/financialmathematicsineconomics\/chapter\/5-3-present-value-of-an-ordinary-annuity\/","title":{"raw":"5.3 Present Value of An Ordinary Annuity","rendered":"5.3 Present Value of An Ordinary Annuity"},"content":{"raw":"

The present value (or discounted value) of an annuity is the equivalent value of the set of payments due at the beginning of the term which is one period before the first payment and is equivalent to the sum of the present values of all the payments comprising the annuity.<\/p>\r\n

The relationship between the present value and future value can be written as:<\/p>\r\n

Present value, $a_{\\bar{n}\\mid{i}}$ = Future value \u00d7 <\/em>(1 + i<\/em>)\u2212n<\/em><\/sup><\/p>\r\n

Substitute the formula of future value in section 5.2 we have:<\/p>\r\n[latexpage]\r\n\r\n$a_{\\bar{n}\\mid{i}} = s_{\\bar{n}\\mid{i}} \\times (1 + i)^{-n}$\r\n\r\n$= \\frac{(1+{i})^{n}-1}{i} \\times (1 + i)^{-n}$\r\n\r\n$= \\frac{1-(1+{i})^{-n}}{i}$\r\n\r\n \r\n

Example 5.3<\/header>\r\n
\r\n

How much money is needed now to provide RM500 at the end of each year (first payment 1 year from now) for 15 years if the money earns interest at 12% p.a?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

Example 5.4<\/header>\r\n
\r\n

A student who borrowed some money to purchase a car was to repay the loan with monthly installments of RM150 for 3 years. Calculate the value of these repayments at the beginning of the loan if the interest rate was 9% convertible monthly.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n[h5p id=\"26\"]","rendered":"

The present value (or discounted value) of an annuity is the equivalent value of the set of payments due at the beginning of the term which is one period before the first payment and is equivalent to the sum of the present values of all the payments comprising the annuity.<\/p>\n

The relationship between the present value and future value can be written as:<\/p>\n

Present value, \"a_{\bar{n}\mid{i}}\" = Future value \u00d7 <\/em>(1 + i<\/em>)\u2212n<\/em><\/sup><\/p>\n

Substitute the formula of future value in section 5.2 we have:<\/p>\n

\"a_{\bar{n}\mid{i}} = s_{\bar{n}\mid{i}} \times (1 + i)^{-n}\"<\/p>\n

\"= \frac{(1+{i})^{n}-1}{i} \times (1 + i)^{-n}\"<\/p>\n

\"= \frac{1-(1+{i})^{-n}}{i}\"<\/p>\n

 <\/p>\n

\n
Example 5.3<\/header>\n
\n

How much money is needed now to provide RM500 at the end of each year (first payment 1 year from now) for 15 years if the money earns interest at 12% p.a?<\/p>\n<\/div>\n<\/div>\n

\n
Example 5.4<\/header>\n
\n

A student who borrowed some money to purchase a car was to repay the loan with monthly installments of RM150 for 3 years. Calculate the value of these repayments at the beginning of the loan if the interest rate was 9% convertible monthly.<\/p>\n<\/div>\n<\/div>\n

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