Compound Interest
2.5 Number of Compounding Periods and Time: Calculator Approach
If the duration of a loan or investment is not specified, we can determine the number of compounding periods, [asciimath]N[/asciimath], by using the present value, the future value, and the interest rate. Once we have the number of compounding periods, we can then calculate the total length of time for the term.
We can use the financial calculator to compute [asciimath]N[/asciimath]the same way you calculate other variables. Subsequently, in both formula and calculator approaches, the time period of the term [asciimath]t[/asciimath] is determined by Formula 2.2b, which is the rearranged form of Formula 2.2a.
[asciimath]t=N/(C//Y)[/asciimath] Formula 2.2b
How long will an investment of $24,000 grow to at least $96,000 at 5.8% compounded semi-annually? Express your answer in a) years, rounded to two decimal places, b) years and months, and c) years and days.
Show/Hide Solution
Given information:
- Present value: [asciimath]PV = -$24,000[/asciimath]
- Future value: [asciimath]FV = $96,000[/asciimath]
- Interest was compounded semi-annually so [asciimath]C//Y = 2[/asciimath]
- Nominal interest rate: [asciimath]I//Y = 5.8%[/asciimath]
Note that for investments, PV is considered a cash outflow and entered as a negative value, while FV is a cash inflow and should be entered as a positive value.
Finding N
[asciimath]N = 48.4930985[/asciimath] Keep at least six decimal places for intermediate values
Finding time (t)
Therefore, the investment term time period is obtained by Formula 2.2b:
[asciimath]t=N/(C//Y)[/asciimath]
[asciimath]t=(48.493098... )/2[/asciimath]
[asciimath]=24.246549...[/asciimath] years
a) The time in years rounded to two decimal places is [asciimath]t=24.25[/asciimath] years.
b) To express time in terms of years and months, take the integer portion as the number of years. Then, multiply the decimal part by 12 to convert it into months.
[asciimath]t=24.246549...[/asciimath] years
[asciimath]=24[/asciimath] years [asciimath]+(0.246549...xx12)[/asciimath] months
[asciimath]=24[/asciimath] years [asciimath]+(2.958588...)[/asciimath] months
[asciimath]=24[/asciimath] years and [asciimath]3[/asciimath] months
c) To express time in terms of years and days, take the integer portion as the number of years. Then, multiply the decimal part by 365 to convert it into days.
[asciimath]t=24.246549...[/asciimath] years
[asciimath]=24[/asciimath] years [asciimath]+(0.246549...xx365)[/asciimath] days
[asciimath]=24[/asciimath] years [asciimath]+(89.990385...)[/asciimath] days
[asciimath]=24[/asciimath] years and [asciimath]90[/asciimath] days
See Section 1.1 to recall how to convert the number of days and months to the number of years and vice versa.
Note: In time calculations for financial problems, the resulting figure often includes a decimal, such as a fraction of a day or month. In such cases, unless instructed otherwise, always round up the value to the nearest whole number, regardless of the decimal value. This rounding-up approach ensures that the required future value is fully accumulated.
Try an Example
Section 2.5 Exercises
- Consider an investment of $13,000.00 that grows to at least $28,541.83 at 7.97% compounded monthly. a) What is the number of compounding periods in the term rounded to two decimal places? b) How long does the investment take? Express the time in years and months.
Show/Hide Answer
a) N = 118.8
b) t = 9 years and 11 months
- How long will it take for an investment of $18,200 to grow to $24,103.80 at an interest rate of 3% compounded quarterly? Express the time in years and days.
Show/Hide Answer
t = 9 years and 146 days
