![\"Timeline](\"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/04\/chap4-cashflow-300x77.png\")
Figure 4.1: Cash Flow[\/caption]\r\n\r\n \r\nExample 4.3.1<\/h2>\r\nTo see how the formula is developed, consider the $5,000 loan at 8% compounded semi-annually for two years.\r\n
First, [latex]i = \\frac{0.08}{2}=0.04[\/latex]<\/p>\r\nThe balances would be:\r\n\r\nAt 6 months:\r\n
[latex]$5,000(1+0.04) =$5,000(1.04)=$5,200[\/latex]<\/p>\r\nAt 1 year:\r\n
[latex]$5,200 (1.04)=$5,000(1.04)(1.04) =$5,000 (1.04)^2 = $5,408[\/latex]<\/p>\r\nAt 18 months:\r\n
[latex]$5,408(1.04)=$5,000(1.04)^2 (1.04) =$5,000(1.04)^3 = $5,624.32[\/latex]<\/p>\r\nAt 2 years:\r\n
[latex]$5,624.32(1.04)=$5,000(1.04)^3 (1.04) =$5,000(1.04)^4 = $5,849.29[\/latex]<\/p>\r\nThis last calculation for the two-year balance is the general formula for FV with:\r\n
- \r\n \t
- PV = $5,000<\/li>\r\n \t
- i<\/em> = 0.04<\/li>\r\n \t
- n<\/em> = 4 = 2\u00d7 2<\/li>\r\n<\/ul>\r\n
\r\n\r\n \r\n\r\n<\/div>\r\nIn general, the values for i<\/em> and n<\/em> are found by:\r\n[latex]i = \\frac{j_m}{m}[\/latex]<\/p>\r\nand\r\n
[latex]n = (\\text{number of years})\\times (\\text{number of periods per year})=t\\times m[\/latex]<\/p>\r\n \r\n
\r\n Knowledge Check 4.2<\/p>\r\n\r\n<\/header>\r\n
\r\n\r\nUse the compound-interest formula to find the following future values:\r\n- \r\n \t
- The future value of a deposit of $8,000 at 16% compounded qua1terly for nine months.<\/li>\r\n \t
- The future value of a loan of $1,000 for two years at 10% compounded annually.<\/li>\r\n \t
- The future value of a loan of $2,500 for four years at 8% compounded monthly.<\/li>\r\n<\/ol>\r\nSolutions at the end of the chapter<\/a>\r\n\r\n<\/div>\r\n<\/div>\r\n \r\n\r\n \r\n
Your Own Notes<\/h2>\r\n
- \r\n \t
- Are there any notes you want to take from this section? Is there anything you'd like to copy and paste below?<\/li>\r\n \t
- These notes are for you only (they will not be stored anywhere)<\/li>\r\n \t
- Make sure to download them at the end to use as a reference<\/li>\r\n<\/ul>\r\n[h5p id=\"1\"]","rendered":"
The procedure for adding interest each period can always be used to find the future value of a loan or deposit, but the following general formula gives the future value more directly.<\/p>\n
[latex]FV = PV(1+i)^n = PV \\left(1+\\frac{j_m}{m}\\right)^n[\/latex]<\/p>\n
where:<\/p>\n
- \n
- FV = future value of the loan<\/li>\n
- PV = present value of the loan (principal)<\/li>\n
- i<\/em> = periodic interest rate<\/li>\n
- n<\/em> = number of compounding periods<\/li>\n<\/ul>\n
Figure 4.1: Cash Flow<\/figcaption><\/figure>\n <\/p>\n
Example 4.3.1<\/h2>\n
To see how the formula is developed, consider the $5,000 loan at 8% compounded semi-annually for two years.<\/p>\n
First, [latex]i = \\frac{0.08}{2}=0.04[\/latex]<\/p>\n
The balances would be:<\/p>\n
At 6 months:<\/p>\n
[latex]$5,000(1+0.04) =$5,000(1.04)=$5,200[\/latex]<\/p>\n
At 1 year:<\/p>\n
[latex]$5,200 (1.04)=$5,000(1.04)(1.04) =$5,000 (1.04)^2 = $5,408[\/latex]<\/p>\n
At 18 months:<\/p>\n
[latex]$5,408(1.04)=$5,000(1.04)^2 (1.04) =$5,000(1.04)^3 = $5,624.32[\/latex]<\/p>\n
At 2 years:<\/p>\n
[latex]$5,624.32(1.04)=$5,000(1.04)^3 (1.04) =$5,000(1.04)^4 = $5,849.29[\/latex]<\/p>\n
This last calculation for the two-year balance is the general formula for FV with:<\/p>\n
- \n
- PV = $5,000<\/li>\n
- i<\/em> = 0.04<\/li>\n
- n<\/em> = 4 = 2\u00d7 2<\/li>\n<\/ul>\n
\n<\/p>\n<\/div>\n
In general, the values for i<\/em> and n<\/em> are found by:<\/p>\n
[latex]i = \\frac{j_m}{m}[\/latex]<\/p>\n
and<\/p>\n
[latex]n = (\\text{number of years})\\times (\\text{number of periods per year})=t\\times m[\/latex]<\/p>\n
<\/p>\n
\n\n Knowledge Check 4.2<\/p>\n<\/header>\n
\nUse the compound-interest formula to find the following future values:<\/p>\n
- \n
- The future value of a deposit of $8,000 at 16% compounded qua1terly for nine months.<\/li>\n
- The future value of a loan of $1,000 for two years at 10% compounded annually.<\/li>\n
- The future value of a loan of $2,500 for four years at 8% compounded monthly.<\/li>\n<\/ol>\n
- n<\/em> = 4 = 2\u00d7 2<\/li>\n<\/ul>\n
- n<\/em> = number of compounding periods<\/li>\n<\/ul>\n
- n<\/em> = 4 = 2\u00d7 2<\/li>\r\n<\/ul>\r\n