# 9.7 Determining the Number of Compounds

# Determining the Number of Compounds

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.

The number of compounding periods could work out to be an integer. More challenging scenarios involve time frame computations with non-integer compounding periods.

## How It Works

Follow these steps to compute the number of compounding periods, n.

**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 I/Y and C/Y).

**Step 2**: Solve for the periodic interest rate (i) using the formula

[latex]i=\frac{I/Y}{C/Y}[/latex]

**Step 3**: Use the formula for the future value, 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, you an solve for the number of years using the formula

[latex]\text{Years} =\frac{n}{C/Y}[/latex].

- 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 Years = 8.25 then you have 8 years plus 0.25 × 12 = 30.25 × 12 = 3 months, or 8 years and 3 months.

### Example 9.7.1: Integer Compounding Period Investment

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).

**Step 1:** Given variables:

PV = $43,000; I/Y = 6.65%; C/Y = quarterly = 4; FV = $67,113.46

**Step 2:** Solve for the periodic interest rate, i.

[latex]i=\frac{I/Y}{C/Y}=\frac{6.65\%}{4}=1.6625\%[/latex]

**Step 3:** Use the formula for the future value, rearrange, and solve for n.

[latex]\begin{align} FV&=PV(1+i)^n\\ \$67,\!113.46&=\$43,\!000(1+0.016625)^n\\ 1.560778&=1.016625^n\\ \ln(1.560778)&=\ln(1.016625)^n\;\text{(by taking ln of both sides)}\\ \ln(1.560778)&=n \ln(1.016625)\; \text{(by using the property} \ln(x)^n=n\ln(x))\\ n&=\frac{\ln(1.560778)}{\ln(1.016625)}\\ n&=\frac{0.445184}{0.016488}\\ n&=26.99996\;\text{or}\; 27\;\text{quarterly compounds} \end{align}[/latex]

**Step 4:** Convert n to years and months.

[latex]\begin{align} \text{Years}&=\frac{n}{C/Y}\\ &=\frac{27}{4}\\ &=6.75\; \text{years}\\ &= 6\;\text{years}\; \text{and}\; .75\times 12=9\;\text{months} \end{align}[/latex]

**Calculator instructions:**

N | I/Y | PV | PMT | FV | P/Y | C/Y |
---|---|---|---|---|---|---|

Answer: 26.999996 | 6.65 | -43,000 | 0 | 67,113.46 | 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.

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.

- 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 as we discussed before.
- 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).

#### Example 9.7.2: Saving for Postsecondary Education

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? Answer in years and days. Round to the nearest day.

**Solution:**

**Step 1:** Given information:

The principal, future value, and interest rate are known, as illustrated in the timeline.

PV = $8,500; I/Y = 7.75%; C/Y = semi-annually = 2; FV = $20,000

**Step 2:** Solve for the periodic interest rate, i.

[latex]i=\frac{I/Y}{C/Y}=\frac{7.75\%}{2}=3.875\%[/latex]

**Step 3:** Use the formula for the future value, rearrange, and solve for n.

[latex]\begin{align} FV&=PV(1+i)^n\\ \$20,\!000&=\$8,\!500(1+0.03875)^n\\ 2.352941&=1.03875^n\\ \ln(2.352941)&=\ln(1.03875)^n\;\text{(by taking ln of both sides)}\\ \ln(2.352941)&=n \ln(1.03875)\; \text{(by using the property} \ln(x)^n=n\ln(x))\\ n&=\frac{\ln(2.352941)}{\ln(1.03875)}\\ n&=\frac{0.855666}{0.038018}\\ n&=22.506828\;\text{semi-annual compounds} \end{align}[/latex]

**Step 4:** Convert n to years and days.

Take the integer:

[latex]\text{Years}=\frac{n}{C/Y}=\frac{22}{2}=11[/latex]

Take the decimal:

[latex]\text{Days}=0.506828 \times 182= 92[/latex]

**Calculator instructions:**

N | I/Y | PV | PMT | FV | P/Y | C/Y |
---|---|---|---|---|---|---|

Answer: 22.506828 | 7.75 | -8,500 | 0 | 20,000 | 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.

## Exercises

In each of the exercises that follow, try them on your own. Full solutions are available should you get stuck.

- You just took over another financial adviser’s account. The client invested $15,500 at 6.92% compounded monthly and now has $24,980.58. How long (in years and months) has this client had the money invested? (
**Answer:**6 years and 11 months)

- Your organization has a debt of $30,000 due in 13 months and $40,000 due in 27 months. If a single payment of $67,993.20 was made instead using an interest rate of 5.95% compounded monthly, when was the payment made? Use today as the focal date. (
**Answer:**15 months from today)

- A $9,500 loan requires a payment of $5,000 after 1½ years and a final payment of $6,000. If the interest rate on the loan is 6.25% compounded monthly, when should the final payment be made? Use today as the focal date. Express your answer in years and months. (
**Answer:**3 years and 2 months)

## Image Description

**Figure 9.7.2:** Timeline showing PV = $8,500 on the left at Time = ?. FV = $20,000 on the right at 18th Birthday. 7.75% semi-annually throughout. [Back to Figure 9.7.2]