9.7: Determining the Number of Compounds
- Page ID
- 28558
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)How long will it take to reach a financial goal? At a casual get-together at your house, a close friend discusses saving for a 14-day vacation to the Blue Bay Grand Esmeralda Resort in the Mayan Riviera of Mexico upon graduation. The estimated cost from Travelocity.ca is $1,998.94 including fares and taxes. He has already saved $1,775 into a fund earning 8% compounded quarterly. Assuming the costs remain the same and he makes no further contributions, can you tell him how soon he will be basking in the sun on the beaches of Mexico?
This section shows you how to calculate the time frame for single payment compound interest transactions. You can apply this knowledge to any personal financial goal. Or in your career, if you work at a mid-size to large company, you might need to invest monies with the objective of using the funds upon maturity to pursue capital projects or even product development opportunities. So knowing the time frame for the investment to grow large enough will allow you to schedule the targeted projects.
Introductory scenarios examine situations where the number of compounding periods works out to be an integer. Then you will tackle more challenging scenarios involving time frame computations with non-integer compounding periods.
Integer Compounding Periods
Some applications of solving for the number of compounding periods include the following:
- Determining the time frame to meet a financial goal
- Calculating the time period elapsing between a present and future value
- Evaluating the performance of financial investments
The Formula
To solve for the number of compounds you need Formula 9.3 one more time. The only difference from your previous uses of this formula is that the unknown variable changes from \(FV\) to \(N\), which requires you to substitute and rearrange the formula algebraically.
How It Works
Follow these steps to compute the number of compounding periods (and ultimately the time frame):
Step 1: Draw a timeline to visualize the question. Most important at this step is to identify \(PV\), \(FV\), and the nominal interest rate (both \(IY\) and \(CY\)).
Step 2: Solve for the periodic interest rate (\(i\)) using Formula 9.1.
Step 3: Substitute into Formula 9.3, rearrange, and solve for \(N\). Note that the value of \(N\) represents the number of compounding periods. For example, if the compounding is quarterly, a value of \(N = 9\) is nine quarters.
Step 4: Take the value of \(N\) and convert it back to a more commonly expressed format such as years and months. When the number of compounding periods calculated in step 3 works out to an integer, handling step 4 involves applying the rearranged Formula 9.2 and solving for \(\text { Years }=\dfrac{N}{CY}\).
- If the Years is an integer, you are done.
- If the Years is a non-integer, the whole number portion (the part in front of the decimal) represents the number of years. As needed, take the decimal number portion (the part after the decimal point) and multiply it by 12 to convert it to months. For example, if you have \(\text{Years }= 8.25\) then you have 8 years plus \(0.25 \times 12 = 3\) months, or 8 years and 3 months.
Revisiting the opening scenario, your friend has saved $1,775 and needs it to become $1,998.94 at 8% compounded quarterly. How long will it take?
Step 1: The timeline illustrates this scenario. Note that \(IY=8 \%\) and \(CY=\text { quarterly }=4\).
Step 2: The periodic interest rate is \(i=8 \% / 4=2 \%\).
Step 3: Applying Formula 9.3, you have \(\$ 1,998.94=\$ 1,775(1+0.02)^{N}\) or \(N = 6\) (details of the algebra can be found in subsequent examples).
Step 4: Applying the rearranged Formula 9.2, \(\text { Years }=\dfrac{6}{4}=1.5\). Your friend will be headed to the Mayan Riviera in \(1\frac{1}{2}\) years. If you prefer to express this in months, it is 1 year plus \(0.5 \times 12=6\) months, or 1 year and 6 months.
Important Notes
You can use your financial calculator in the exact same manner as in Section 9.2 except that your unknown variable will be \(N\) instead of \(FV\). As always, you must load the other six variables and obey the cash flow sign convention. Once you compute \(N\), you will have to complete step 4 manually to express the solution in its more common format of years and months.
Things To Watch Out For
\(N\) rarely works out to an integer, and that causes a lot of grief. When the values of both the \(PV\) and \(FV\) are known, usually one of these numbers is rounded. Because of this rounding, the calculation of \(N\) almost always produces decimals, even if those decimals are very small. For example, in calculating \(N = 6\) it is possible to come up with a value of 5.999963 or 6.000018. How can you tell if those decimals result from a rounded value (meaning that \(N\) is really an integer) or if those decimals are significant and need to be dealt with as they stand (as you will see discussed below when we consider non-integer compounding periods)?
You determine which way to go by mentally rounding your calculated \(N\) to the third decimal place. If this results in an integer, then the decimals are due to the rounding of \(FV\) and from then on you can treat the \(N\) as an integer. If a non-integer is still present, then the decimals are meaningful and you will have to address a non-integer compounding period (as discussed later in this section). For example, 5.999963 when mentally rounded to three decimals is 6.000. Therefore, you consider the \(N\) to be 6 and not 5.999963. In contrast, 5.985996 when mentally rounded to three decimals is 5.986, which is a non-integer. This means the decimals are significant, so you will not be able to round off to the closest integer.
Paths To Success
In Section 9.6, the Rule of 72 allowed you to estimate the time it takes in years for money to double when you know the effective rate. When you work with any time frame that involves the doubling of money, this rule can allow you to quickly check whether your calculated value of \(N\) is reasonable.
- Listed are some values of \(N\). Determine if you should treat the \(N\) like an integer or a noninteger.
- 9.000345
- 5.993129
- 6.010102
- 11.999643
- 10.000987
- 3.999999
- Of $1,000 growing to $2,000 or $2,000 growing to $3,000, which will take longer at the same interest rate?
- Answer
-
-
- 9.000 integer
- 5.993 non-integer
- 6.010 non-integer
- 12.000 integer
- 10.001 non-integer
- 4.000 integer
- $1,000 growing to $2,000 will take longer since it involves a doubling of money, while $2,000 growing to $3,000 increases the amount by less than a doubling.
-
Jenning Holdings invested $43,000 at 6.65% compounded quarterly. A report from the finance department shows the investment is currently valued at $67,113.46. How long has the money been invested?
Solution
Determine the amount of time that the principal has been invested. This requires calculating the number of compounding periods (\(N\)).
What You Already Know
Step 1:
The principal, future value, and interest rate are known, as illustrated in the timeline.
\(IY\) = 6.65%; \(CY\) =4
How You Will Get There
Step 2:
Apply Formula 9.1.
Step 3:
Substitute into Formula 9.3, rearrange, and solve for \(N\). Recall from Section 2.6 that to solve an equation for an unknown exponent you take the logarithm of both sides. You will also need to apply the general rule for logarithm of a power \(\ln \left(x^{y}\right)=y(\ln x)\).
Step 4:
Apply Formula 9.2, rearrange, and solve for Years.
Perform
Step 2:
\[i=\dfrac{6.65 \%}{4}=1.6625 \% \nonumber \]
Step 3:
\[\begin{aligned} \$ 67,113.46 &=\$ 43,000(1+0.016625)^{N} \\ 1.560778 &=1.016625^{N} \\ \ln (1.560778) &=\ln \left(1.016625^{N}\right)=N \times \ln (1.016625) \\ 0.445184 &=N \times 0.016488 \\ N &=26.999996 \text { or } 27 \text { quarterly compounds } \end{aligned} \nonumber \]
Step 4:
\(\begin{align*} 27 &= 4\times \text {Years}\\ \text {Years} &= 6.75 \end{align*} \)
which is 6 years plus 0.75 × 12 = 9 months
Calculator Instructions
| N | I/Y | PV | PMT | FV | P/Y | C/Y |
|---|---|---|---|---|---|---|
| Answer: 26.999996 | 6.65 | -43000 | 0 | 67113.46 | 4 | 4 |
Jenning Holdings has had the money invested for six years and nine months.
Non-integer Compounding Periods
When the number of compounding periods does not work out to an integer, the method of calculating \(N\) does not change. However, what changes is the method by which you convert \(N\) and express it in more common terms (step 4 of the process). Typically, the non-integer involves a number of years, months, and days.
As summarized in the table below, to convert the compounding period into the correct number of days you can make the following assumptions:
| Compounding Period | # of Days in the Period |
|---|---|
| Annual | 365 |
| Semi-annual | 182* |
| Quarter | 91* |
| Month | 30* |
| Week | 7 |
| Daily | 1 |
How It Works
You still use the same four steps to solve for the number of compounding periods when \(N\) works out to a non-integer as you did for integers. However, in step 4 you have to alter how you convert the \(N\) to a common expression. Here is how to convert the value:
- Separate the integer from the decimal for your value of \(N\).
- With the integer portion, apply the same technique used with an integer \(N\) to calculate the number of years and months.
- With the decimal portion, multiply by the number of days in the period to determine the number of days and round off the answer to the nearest day (treating any decimals as a result of a rounded interest amount included in the future value).
The figure to the right illustrates this process for an \(N = 11.63\) with quarterly compounding, or \(CY = 4\). Thus, an \(N = 11.63\) quarterly compounds converts to a time frame of 2 years, 9 months, and 57 days. Note that without further information, it is impossible to simplify the 57 days into months and days since it is unclear whether the months involved have 28, 29, 30, or 31 days.
Tabitha estimates that she will need $20,000 for her daughter's postsecondary education when she turns 18. If Tabitha is able to save up $8,500, how far in advance of her daughter's 18th birthday would she need to invest the money at 7.75% compounded semi-annually?
Solution
Determine the amount of time in advance of the daughter's 18th birthday that the money needs to be invested. The number of compounding periods (\(N\)) must be calculated.
What You Already Know
Step 1:
The principal, future value, and interest rate are known, as illustrated in the timeline.
\(IY\) = 7.75%; \(CY\) =2
How You Will Get There
Step 2:
Apply Formula 9.1.
Step 3:
Substitute into Formula 9.3, rearrange, and use the rule for logarithm of a power to solve for \(N\).
Step 4:
Apply Formula 9.2, rearrange, and solve for Years.
Step 2:
\[i=\dfrac{7.75 \%}{2}=3.875 \% \nonumber \]
Step 3:
\[\begin{aligned} & \$ 20,000=\$8,500(1+0.03875)^{N} \\ 2.352941 &=1.03875^{N} \\ \ln (2.352941) &=N \times \ln (1.03875) \\ 0.855666 &=N \times 0.038018 \\ N &=22.506828 \; \text {semi-annual compounds } \end{aligned} \nonumber \]
Step 4:
Take the integer:
\(\begin{align*} 22 &= 2\times \text {Years}\\ \text {Years} &= 11 \end{align*} \nonumber \)
Take the decimal: semi-annual compounding according is 182 days, so 0.506828 × 182 days = 92 days
Calculator Instructions
| N | I/Y | PV | PMT | FV | P/Y | C/Y |
|---|---|---|---|---|---|---|
| Answer: 22.506828 | 7.75 | -8500 | 0 | 20000 | 2 | 2 |
If Tabitha invests the $8,500 11 years and 92 days before her daughter's 18th birthday, it will grow to $20,000.