6.2: Determining the Future (Maturity) Value

Determining the Future (Maturity) Value 

The simplest future value scenario for compound interest is for all of the variables to remain unchanged throughout the entire transaction. To understand the derivation of the formula, continue with the following scenario. If [latex]\$4,000[/latex] was borrowed two years ago at [latex]12\%[/latex] compounded semi-annually, then a borrower will owe two years of compound interest in addition to the original principal of [latex]\$4,000[/latex]. That means [latex]PV=\$4,000[/latex]. The compounding frequency is semi-annually, or twice per year, which makes the periodic interest rate [latex]i=\frac{\text{I/Y}}{\text{C/Y}}=\frac{12\%}{2}=6\%[/latex]. Therefore, after the first six months, the borrower has [latex]6\%[/latex] interest converted to principal. This a future value, or [latex]FV[/latex], calculated as follows:

Principal after one compounding period (six months) = Principal plus interest:

[latex]\begin{eqnarray}FV &=&PV+{i}(PV)\\FV&=&\$ 4,000+0.06(\$4,000)\\FV&=&\$ 4,000+\$ 240\\FV&=&\$ 4,240\end{eqnarray}[/latex]

Now proceed to the next six months. The future value after two compounding periods (one year) is calculated in the same way.

Note that the equation [latex]FV = PV +i(PV)[/latex] can be factored and rewritten as [latex]FV = PV(1 +i)[/latex].

[latex]\begin{eqnarray*}FV_\textrm{after two compounding periods}&=&PV(1 +i)\\FV_\textrm{after two compounding periods}&=&\$4,240(1 + 0.06)\\FV_\textrm{after two compounding periods}&=&\$4,240(1.06)\\FV_\textrm{after two compounding periods}&=&\$4,494.40\end{eqnarray*}[/latex]

Since the [latex]PV = $4,240[/latex] is the result of the previous calculation where [latex]PV(1 + i) = $4,240[/latex], the following algebraic substitution is possible:

[latex]\begin{eqnarray*}FV_\textrm{after two compounding periods}&=&PV(1 +i )(1 + i)\\FV_\textrm{after two compounding periods}&=&\$4,000(1.06)(1.06)\\FV_\textrm{after two compounding periods}&=&\$4,240(1.06)\\FV_\textrm{after two compounding periods}&=&\$4,494.40\end{eqnarray*}[/latex]

Simplifying algebraically, you get:

[latex]\begin{eqnarray*}FV&=&PV(1 +i)(1 + i)\\FV&=&PV(1 +i)^2\end{eqnarray*}[/latex]

Do you notice a pattern? With one compounding period, the formula has only one [latex](1 + i)[/latex]. With two compounding periods involved, it has two factors of [latex](1 + i)[/latex]. Each successive compounding period multiplies a further [latex](1 + i)[/latex] onto the equation. This makes the exponent on the [latex](1 + i)[/latex]  exactly equal to the number of times that interest is converted to principal during the transaction.

The Formula

First, you need to know how many times interest is converted to principal throughout the transaction. You can then calculate the future value. Use Formula 6.2a below to determine the number of compound periods involved in the transaction.

[latex]\boxed{6.2\text{a}}[/latex] Number of Compound Periods

[latex]\color{red}{n}[/latex] is the total number of compounding periods.

[latex]\color{blue}{\text{C/Y}}[/latex] is the number of compounding periods per year.

[latex]\color{green}{\text{Number of Years}}[/latex] is the total number of years.

Once you know [latex]n[/latex], substitute it into Formula 6.2b, which finds the amount of principal and interest together at the end of the transaction, or the future (maturity) value, [latex]FV[/latex].

[latex]\boxed{6.2\text{b}}[/latex] Future (Maturity) Value

[latex]\color{red}{FV}[/latex] is Future or Maturity Value.

[latex]\color{blue}{PV}[/latex] is the Present Value or principal. This is the starting amount upon which compound interest is calculated.

[latex]\color{green}{i}[/latex] is the periodic interest rate from Formula 6.1[latex]\begin{align*}i=\frac{\text{I/Y}}{\text{C/Y}}\end{align*}[/latex].

[latex]\color{purple}{n}[/latex] is the number of compound periods from Formula 6.2a[latex]n=\text{C/Y}\times\text{Number of Years}[/latex].

Your BAII Plus Calculator

We will be using the function keys that are presented in the third row of your calculator, known as the [latex]TVM[/latex] row or (time value of money row). The five buttons located on the third row of the calculator are five of the seven variables required for time value of money calculations. This row’s buttons are different in color from the rest of the buttons on the keypad.

The table below relates each button (variable) to its meaning:

BAII Plus Memory

Your calculator has permanent memory. Once you enter data into any of the time value buttons it is permanently stored until

• • You override it by entering another piece of data and pressing the button;
  • You clear the memory of the time value buttons by pressing 2nd CLR TVM before proceeding with another question; or
  • The reset button on the back of the calculator is pressed.

Example 6.2.1

If you invested [latex]\$5,000[/latex] for [latex]10[/latex] years at [latex]9\%[/latex] compounded quarterly, how much money would you have? What is the interest earned during the term?

Solution

The timeline for the investment is below.

Step 1: Given information:

[latex]PV=5,000[/latex];         [latex]\text{I/Y}=9\%[/latex];        [latex]\text{C/Y}=4[/latex]

Step 2: Calculate the periodic interest rate, [latex]i[/latex].

[latex]\begin{eqnarray*}i&=&\frac{\text{I/Y}}{\text{C/Y}}\\[1ex]i&=&\frac{9\%}{4}\\[1ex]i&=&2.25\%\\i&=&0.0225\end{eqnarray*}[/latex]

Step 3: Calculate the total number of compoundings, [latex]n[/latex].

[latex]\begin{eqnarray*}n&=&\text{C/Y}\times (\text{Number of Years})\\n&=&4 \times 10\\n&=&40\end{eqnarray*}[/latex]

Step 4: Solve for the future value, [latex]FV[/latex].

[latex]\begin{eqnarray*}FV&=&\$5,000(1+0.0225)^{40}\\FV&=&\$12,175.94\end{eqnarray*}[/latex]

Step 5: Find the interest earned.

[latex]\begin{eqnarray*}I&=&FV-PV\\I&=&\$12,175.94-\$5,000\\I&=&\$7,175.94\end{eqnarray*}[/latex]

Calculator instructions:

Table 6.2.2

N	I/Y	PV	PMT	FV	P/Y	C/Y
40	9	-5,000	0	?	12	12

Step 6: Write as a statement.

After [latex]10[/latex] years, the principal grows to [latex]\$12,175.94[/latex], which includes your [latex]\$5,000[/latex] principal and [latex]\$7,175.94[/latex] of compound interest.

Future Value Calculations with Variable Changes

What happens if a variable such as the nominal interest rate, compounding frequency, or even the principal changes somewhere in the middle of the transaction? When any variable changes, you must break the timeline into separate time fragments at the point of the change. To arrive at the solution, you need to work from left to right one time segment at a time using Formula 6.2b[latex]FV=PV\times(1+i)^n[/latex]. 

Example 6.2.2

Five years ago Coast Appliances was supposed to upgrade one of its facilities at a quoted cost of [latex]\$48,000[/latex]. The upgrade was not completed, so Coast Appliances delayed the purchase until now. The construction company that provided the quote indicates that prices rose [latex]6\%[/latex] compounded quarterly for the first [latex]1\frac{1}{2}[/latex] years, [latex]7\%[/latex] compounded semi-annually for the following [latex]2\frac{1}{2}[/latex] years, and [latex]7.5\%[/latex] compounded monthly for the final year. If Coast Appliances wants to perform the upgrade today, what amount of money does it need?

Solution

The timeline below shows the original quote from five years ago until today.

 

Step 1: First time segment:

[latex]PV_1=\$48,000[/latex];        [latex]\text{I/Y}=6\%[/latex];        [latex]\text{C/Y}=4[/latex];        [latex]\text{Years}=2[/latex]

 

[latex]\begin{eqnarray*}i&=&\frac{\text{I/Y}}{\text{C/Y}}\\[1ex]i&=&\frac{6\%}{4}\\[1ex]i&=&1.5\%\\\\n&=&\text{C/Y}\times\left(\text{Number of Years}\right)\\n&=&4\times 1.5\\n&=&6\end{eqnarray*}[/latex]

Find [latex]FV_1[/latex]

[latex]\begin{eqnarray}FV_1&=&PV_1(1+i)^n\\FV_1&=&\$48,000(1+0.015)^6\\FV_1&=&\$24,500(1.015)^6\\FV_1&=&\$52,485.27667\end{eqnarray}[/latex]

This becomes [latex]PV_2[/latex] for the next calculation in Step 2.

Step 2: Second line segment:

[latex]PV_2=FV_1=\$52,485.27667[/latex];        [latex]\text{I/Y}=7\%[/latex];        [latex]\text{C/Y}=2[/latex];        [latex]\text{Years}=2.5[/latex]

 

[latex]\begin{eqnarray*}i&=&\frac{\text{I/Y}}{\text{C/Y}}\\[1ex]i&=&\frac{7\%}2\\[1ex]i&=&3.5\%\\\\n&=&\text{C/Y}\times\left(\text{Number of Years}\right)\\n&=&2\times2.5\\n&=&5\end{eqnarray*}[/latex]

Find [latex]FV_2[/latex]

[latex]\begin{eqnarray}FV_2&=&PV_2(1 + i)^n\\FV_2&=&\$52,485.27667(1+0.035)^5\\FV_2&=&\$62,336.04435\end{eqnarray}[/latex]

This becomes [latex]PV_3[/latex] for the next calculation in Step 3.

Step 3: Third line segment:

[latex]PV_3=FV_2=\$62,336.04435[/latex];        [latex]\text{I/Y}=7.5\%[/latex];        [latex]\text{C/Y}=12[/latex];        [latex]\text{Years}=1[/latex]

 

[latex]\begin{eqnarray*}i&=&\frac{\text{I/Y}}{\text{C/Y}}\\[1ex]i&=&\frac{7.5\%}{12}\\[1ex]i&=&0.625\%\\\\n&=&\text{C/Y}\times\left(\text{Number of Years}\right)\\n&=&12\times 1\\n&=&12\end{eqnarray*}[/latex]

Find [latex]FV_3[/latex]

[latex]\begin{eqnarray}FV_3&=&PV_3(1 + i)^n\\FV_3&=&\$62,336.04435 (1+0.00325)^{12}\\FV_3&=&\$67,175.35\end{eqnarray}[/latex]

The future value is [latex]\$67,175.35[/latex].

Calculator instruction:

Table 6.2.3

Step	N	I/Y	PV	PMT	FV	P/Y	C/Y
1	6	6	48,500	0	?	4	4
2	5	7	52,485.27667	0	?	2	2
3	12	7.5	62,336.04435	0	?	12	12

Step 4: Write as a statement.

Coast Appliances requires [latex]\$67,175.35[/latex] to perform the upgrade today. This consists of [latex]\$48,000[/latex] from the original quote plus [latex]\$19,175.35[/latex] in price increases.

Example 6.2.3

Two years ago Lorelei placed [latex]\$2,000[/latex] into an investment earning [latex]6\%[/latex] compounded monthly. Today she makes a deposit to the investment in the amount of [latex]\$1,500[/latex]. What is the maturity value of her investment three years from now?

Solution

The timeline for the investment is below.

 

Step 1: First time segment:

[latex]PV_1=\$2,000[/latex];        [latex]\text{I/Y}=6\%[/latex];        [latex]\text{C/Y}=12[/latex];        [latex]\text{Years=2}[/latex]

 

[latex]\begin{eqnarray*}i&=&\frac{\text{I/Y}}{\text{C/Y}}\\[1ex]i&=&\frac{6\%}{12}\\[1ex]i&=&0.5\%\\\\n&=&\text{C/Y}\times(\text{Number of Years})\\n&=&12\times2\\n&=&24\end{eqnarray*}[/latex]

Find [latex]FV_1[/latex]

[latex]\begin{eqnarray}FV_1&=&PV_1(1+i)^n\\FV_1&=&\$2,000(1+0.005)^24\\FV_1&=&\$2,000(1.005)^24\\FV_1&=&\$2,254.319552\end{eqnarray}[/latex]

 

[latex]\$2,254.319552+\$1,500=\$3,754.319552[/latex]

This becomes [latex]PV_2[/latex] for the second line segment in Step 2.

Step 2:  Second line segment:

[latex]PV_1=FV_1=3,754.319552[/latex];        [latex]\text{I/Y}=6\%[/latex];        [latex]\text{C/Y}=12[/latex];        [latex]\text{Years}= 3[/latex]

 

[latex]\begin{eqnarray*}i&=&\frac{\text{I/Y}}{\text{C/Y}}\\[1ex]i&=&\frac{6\%}{12}\\[1ex]i&=&0.5\%\\\\n&=&\text{C/Y}\times\text{(Number of Years)}\\n&=&12\times3\\n&=&36\end{eqnarray*}[/latex]

Find [latex]FV_2[/latex]

[latex]\begin{eqnarray}FV_2&=&PV_2(1+i)^n\\FV_2&=&\$3,754.319552(1+0.005)^{36}\\FV_2&=&\$4,492.72\end{eqnarray}[/latex]

The future value is [latex]\$4,492.72[/latex]

Calculator instructions:

Table 6.2.4 Calculator Instructions for Example 6.2.3

Step	N	I/Y	PV	PMT	FV	P/Y	C/Y
1	24	6	-2,000	0	?	12	12
2	36	6	-3,754.319552	0	?	12	12

Step 3: Write as a statement.

Three years from now Lorelei will have [latex]\$4,492.72[/latex]. This represents [latex]\$3,500[/latex] of principal and [latex]\$992.72[/latex] of compound interest.

Section 6.2 Exercises