6.3: Determining the Present Value

Determining the Present Value

[latex]PV[/latex] is the Present Value or Principal.  This is the new unknown variable.  If this is in fact the amount at the start of the financial transaction, it is also called the principal. Or it can simply be the amount at some earlier point in time than when the future value is known. In any case, the amount excludes the future interest.  To calculate this variable, substitute the values for the other three variables into the formula and then algebraically rearrange to isolate [latex]PV[/latex].

Solving for present value requires you to use the future value formula we introduced in Section 6.2 (Formula 6.2b[latex]FV=PV\times(1+i)^n[/latex]). We rearrange the future value formula to solve for [latex]PV[/latex].

[latex]\boxed{6.3}[/latex] Present Value (Principal)

[latex]\begin{align*}\Large\color{red}{PV}\color{black}{=}\frac{\color{blue}{FV}}{\left(1+\color{green}{i}\right)^{\color{purple}{n}}}\end{align*}[/latex]

[latex]\color{red}{PV}[/latex] is the Present Value or principal.

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

[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

You use the financial calculator in the exact same manner as described in Section 6.2. The only difference is that the unknown variable is [latex]PV[/latex] instead of [latex]FV[/latex]. You must still load the other six variables into the calculator and apply the cash flow sign convention carefully.

Example 6.3.1

Castillo’s Warehouse will need to purchase a new forklift for its warehouse operations three years from now, when its new warehouse facility becomes operational. If the price of the new forklift is [latex]\$38,000[/latex] and Castillo’s can invest its money at [latex]7.25\%[/latex] compounded monthly, how much money should it put aside today to achieve its goal?

Solution

Step 1: Given variables:

[latex]FV=38,000[/latex];        [latex]\text{I/Y}=7.25\%[/latex];        [latex]\text{C/Y}=12[/latex];        [latex]\text{Years}=3[/latex]

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

[latex]\begin{align*}i&=\frac{\text{I/Y}}{\text{C/Y}}\\[1ex]i&=\frac{7.25\%}{12}\\[1ex]i&=0.6041\overline{6}\%\\i&=0.006041\overline{6}\end{align*}[/latex]

Step 3: Calculate the number of compound periods, [latex]n[/latex].

[latex]\begin{align*}n&=\text{C/Y} \times (\text{Number of Years})\\n&=3 \times 12\\n&=36\end{align*}[/latex]

Step 4: Solve for the present value, [latex]PV[/latex].

[latex]\begin{align} PV&=\frac{FV}{(1+i)^n}\\[1ex] PV&=\frac{38,\!000}{(1+0.006041\overline{6})^{36}}\\[1ex] PV&=30,\!592.06 \end{align}[/latex]

Table 6.3.1 Calculator Instructions for Example 6.3.1

N	I/Y	PV	PMT	FV	P/Y	C/Y
36	7.25	?	0	38,000	12	12

Step 5: Write as a statement.

If Castillo’s Warehouse places [latex]\$30,592.06[/latex] into the investment, it will earn enough interest to grow to [latex]\$38,000[/latex] three years from now to purchase the forklift.

Present Value Calculations with Variable Changes

Addressing variable changes in present value calculations follows the same techniques as future value calculations. You must break the timeline into separate time segments, each of which involves its own calculations.

Solving for the unknown [latex]PV[/latex] at the left of the timeline means you must start at the right of the timeline. You must work from right to left, one time segment at a time using the formula for [latex]PV[/latex] each time. Note that the present value for one time segment becomes the future value for the next time segment to the left.

Your BAII Plus Calculator

To use your calculator efficiently in working through multiple time segments, follow a procedure similar to that for future value:

1. Load the calculator with all the known compound interest variables for the first time segment on the right.
2. Compute the present value at the beginning of the segment.
3. With the answer still on your display, adjust the principal if needed, change the cash flow sign by pressing the [latex]\pm[/latex] key, then store the unrounded number back into the future value button by pressing FV. Change the N, I/Y, and C/Y as required for the next segment.

Return to Step 2 for each time segment until you have completed all time segments.

Example 6.3.2

Sebastien needs to have [latex]\$9,200[/latex] saved up three years from now. The investment he is considering pays [latex]7\%[/latex] compounded semi-annually, [latex]8\%[/latex] compounded quarterly, and [latex]9\%[/latex] compounded monthly in successive years. To achieve his goal, how much money does he need to place into the investment today?

Solution

Starting from the right end of the timeline and working backwards:

Step 1: First time segment: 

[latex]FV_1=\$9,200[/latex];        [latex]\text{I/Y}=9\%[/latex];        [latex]\text{C/Y}=12[/latex];        [latex]\text{Years}=1[/latex]

 

[latex]\begin{align*}i&=\frac{\text{I/Y}}{\text{C/Y}}\\[1ex]i&=\frac{9\%}{12}\\[1ex]i&=0.75\%\\[1ex]\\[1ex]n&=\text{C/Y} \times (\text{Number of Years})\\n&=1 \times 12\\n&=12\end{align*}[/latex]

Find [latex]PV_1[/latex]

[latex]\begin{align} PV_1&=\frac{FV_1}{(1+i)^n}\\[1ex] PV_1&=\frac{9,\!200}{(1+0.0075)^{12}}\\[1ex] PV_1&=8,\!410.991026 \end{align}[/latex]

This becomes [latex]FV_2[/latex] in Step 2.

Step 2: Second line segment:

[latex]FV_2=PV_1=8,410.991026[/latex];        [latex]\text{I/Y}=8\%[/latex];        [latex]\text{C/Y}=4[/latex];        [latex]\text{Years}=1[/latex]

[latex]\begin{align*}i&=\frac{\text{I/Y}}{\text{C/Y}}\\[1ex]i&=\frac{8\%}{4}\\[1ex]i&=2\%\\[1ex]\\[1ex]n&=\text{C/Y} \times (\text{Number of Years})\\n&= 1 \times 4\\n&=4\end{align*}[/latex]

Find [latex]PV_2[/latex]

[latex]\begin{align} PV_2&=\frac{FV_2}{(1+i)^n}\\[1ex] PV_2&=\frac{8,\!410.991026}{(1+0.02)^{4}}\\[1ex] PV_2&=7,\!770.455587 \end{align}[/latex]

This becomes [latex]FV_3[/latex] in Step 3.

Step 3: Third line segment:

[latex]FV_3=PV_2=7,770.455587[/latex];        [latex]\text{I/Y}=7\%[/latex];        [latex]\text{C/Y}=2[/latex];        [latex]\text{Years}=1[/latex]

[latex]\begin{align*}i&=\frac{\text{I/Y}}{\text{C/Y}}\\[1ex]i&=\frac{7.5\%}{12}\\[1ex]i&=0.625\%\\[1ex]\\[1ex]n&=\text{C/Y}\times (\text{Number of Years})\\n&= 1 \times 2=2\end{align*}[/latex]

Find [latex]PV_3[/latex]

[latex]\begin{align} PV_3&=\frac{FV_3}{(1+i)^n}\\[1ex] PV_3&=\frac{7,\!770.455587}{(1+0.035)^{2}}\\[1ex] PV_3&=7,\!253.80 \end{align}[/latex]

The present value is [latex]\$7,253.80[/latex].

Table 6.3.2 Calculator Instructions for Example 6.3.2

Step	N	I/Y	PV	PMT	FV	P/Y	C/Y
1	12	9	?	0	9,200	12	12
2	4	8	?	0	±(PV from Step 1)	4	4
3	2	7	?	0	±(PV from Step 1)	2	2

Step 4: Write as a statement.

Sebastien needs to place [latex]\$7,253.80[/latex] into the investment today to have [latex]\$9,200[/latex] three years from now.

When you calculate the present value of a single payment for which only the interest rate fluctuates, it is possible to find the principal amount in a single division:

[latex]\begin{align*}PV=\frac{FV}{(1+i_1)^{n_1}\times (1+i_2)^{n_2}\times (1+i_3)^{n_3}\times \ldots(1+i_n)^{n_n}}\end{align*}[/latex]

where [latex]n[/latex] represents the time segment number.

In the previous example you can calculate the same principal as follows:

[latex]\begin{align*}PV&=\frac{\$9,200}{(1+0.0075)^{12}\times (1+0.02)^{4}\times (1+0.035)^{2}}\\[1ex]PV&=\$7,\!253.80\end{align*}[/latex]

Section 6.3 Exercises