Calculating Present Value, Interest Rate, or Number of Compounding Periods

Example 7

Phoebe is planning to borrow money at an annual interest rate of $7\%$, compounded quarterly. She will not be repaying any of the loan for two years, and does not want it to accumulate to more than $$10~000$. How much money can Phoebe borrow?

Solution

Quarterly interest rate: $\dfrac{7\%}{4}=1.75\%$
$i=0.0175$

A Formula for Present Value

Re-arrange the compound interest formula so that we have a general formula for $P$, instead of $A$.

$A=P(1+i)^n$, or $P(1+i)^n=A$

Example 8

Ten years ago, Raheem invested some of his money in an account with an interest rate of $4.2\%$, compounded annually. The balance in his account is now $$1961.65$. How much interest has he made over the past ten years?

Solution

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Check Your Understanding 5

So far, we have explored how to calculate future value or present value using the compound interest formula. In the next example, we will use the formula to calculate the interest rate.

Example 9

What annual interest rate is required:

To earn $$1000$ in interest over one and a half years on an initial investment of $$10~000$, if interest is compounded monthly.
For an investment to double in value over five years, if interest is compounded annually.

Solution — Part A

$P=10~000$ (the amount of the initial investment)
$A=11~000$ (the initial investment, plus $$1000$ of interest)
$n=18$ (the number of months in one and a half years)

From the compound interest formula,

$\begin{align*} A&=P(1+i)^n \\ 11~000 &= 10~000 (1+i)^{18} \end{align*} $

Solve this equation for $i$.

Divide both sides by $10~000$:
$1.1 =(1+i)^{18}$ or $(1+i)^{18} = 1.1$

Take the $18$th root of both sides of the equation. (Use the $n$th root button on your calculator.) Since $18$ is even, there is technically a positive and a negative root.

Calculator - elsen029/iStock/Getty Images

$\begin {align*} 1+i &= \pm \sqrt[18]{1.1} \\ 1+i &= \pm 1.005309 \dots \end{align*}$

However, $1+i$ must be positive in this context, so $1+i = 1.005309 \dots$ and $i \approx 0.005309$.

Remember that $i$ is the interest rate per compounding period. Since interest is compounded monthly, the monthly interest rate is approximately $0.5309\%$.

For the annual interest rate: $0.5309\% \times 12 \approx 6.37\%$.

Therefore, to earn $$1000$ on an initial investment of $$10~000$ over one and half years, the annual interest rate needs to be $6.37\%$, if interest is compounded monthly.

Solution — Part B

We have not been given the value of the investment. However, if the investment doubles, then the accumulated value is twice the value of the principal, or $A=2P$.
$n=5$ (interest is compounded once a year, for five years)

From the compound interest formula,

$\begin{align*}A&= P(1+i)^n \\ 2P &= P(1+i)^5 \\\end{align*}$

Divide both sides of the equation by $P$:

$2=(1+i)^5$, or $(1+i)^5 =2$

Take the $5$th root of both sides of the equation. (Note that $5$ is odd so there is only one root). Then, solve for $i$.

$\begin{align*}1+i &= \sqrt[5]{2} \\ 1 + i &= 1.148698 \dots \\ i &\approx 0.1487\end{align*}$

This means the interest rate per compound period is $14.87 \%$. Since the interest is compounded annually, this is the yearly rate.

Therefore, an annual interest rate of $14.87\%$ is required to double the value of an investment over five years, if interest is compounded annually.

Check Your Understanding 6

Example 10

A loan of $$11~000$ is borrowed at an annual interest rate of $8.5\%$, compounded semi-annually. If none of the loan is paid back, how long does it take for the balance of the loan to grow to $$15~000$?

Solution

$\dfrac{8.5\%}{2}=4.25\%$, so $i=0.0425$

Example 10 Continued

A loan of $$11~000$ is borrowed at an annual interest rate of $8.5\%$, compounded semi-annually. If none of the loan is paid back, how long does it take for the balance of the loan to grow to $$15~000$?

Solution — By Graphing

To solve $15~000= 11~000(1.0425)^n$:

So $(1.0425)^{7.452} \approx 1.364$, then $n\approx 7.452$.

Example 10 Continued

A loan of $$11~000$ is borrowed at an annual interest rate of $8.5\%$, compounded semi-annually. If none of the loan is paid back, how long does it take for the balance of the loan to grow to $$15~000$?

Solution — Alternate Graph

To solve $15~000= 11~000(1.0425)^n$:

Example 10 Continued

A loan of $$11~000$ is borrowed at an annual interest rate of $8.5\%$, compounded semi-annually. If none of the loan is paid back, how long does it take for the balance of the loan to grow to $$15~000$?

Solution — Conclusion

$n\approx 7.452$

If partial compounding periods were not allowed, note that $n=7$ produces a balance less than $$15~000$, while $n=8$ produces a balance greater than $$15~000$. The solution would be $n=8$.

Example 10 — Guess and Check

To solve $15~000= 11~000(1.0425)^n$:

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Check Your Understanding 7

In addition to all of the techniques discussed in this lesson, problems involving compound interest can also be solved using a TVM Solver.

Using a TVM Solver

A TVM Solver (or "Time Value of Money" Solver) is a feature found on some graphing calculators, on financial calculators, or online. It can be used to solve problems involving compound interest.

The TVM solver we will use in this lesson looks similar to those found on a graphing calculator. There are typically seven values to consider - if you enter six of them, the TVM solver will be able to calculate the seventh. A TVM solver is typically used for calculations relating to annuities (which you will see in a later lesson); however, we can also use it for simpler compound interest questions.

For compound interest, use the TVM solver as follows:

$N$ represents the number of years (note the difference from $n$).
$I\%$ represents the yearly interest rate. It is entered as a percent, not as a decimal.
$PV$ represents the present value.
$PMT$ is left equal to $0$ for this lesson.
$FV$ represents the future value.
$P/Y$ is left equal to $1$.
$C/Y$ represents the number of compounding periods in a year.

On a TVM solver:

For an investment, $PV$ is entered or displayed as a negative value, while $FV$ is entered or displayed as a positive value. This is because at the start of an investment, you are paying money into an investment, while at the end of the investment you are receiving money back.

For a loan, $PV$ is entered or displayed as a positive value, while $FV$ is entered or displayed as a negative value. This is because at the start of a loan, the lender pays you, while at the end of the loan, you pay the lender.

In this last example, we will solve for the number of compounding periods, both by graphing, and by using a TVM solver.

Example 11

How long does it take for an investment to triple, at an annual interest rate of $7.2\%$, compounded monthly?

Solution

The investment triples if $A=3P$.
$\dfrac{7.2\%}{12}=0.6\%$, so $i=0.006$.
$n$ is unknown.

Method 1: Set up an equation and solve by graphing

Substitute into the compound interest formula:

$\begin{align*} A &= P(1+i)^n \\ 3P &= P (1.006)^n\end{align*}$

Dividing both sides by $P$ gives $(1.006)^n=3$. This is an equation that we can solve by graphing.

Consider the graph of $f(x)=(1.006)^x$ and find the point that has a $y$-coordinate of $3$. Remember that any points found using graphing technology will likely have been rounded.

The point $(183.651, 3)$ approximately lies on $f(x)=(1.006)^x$, which means that $(1.006)^{183.651}\approx3$.

Thus, the solution to the equation $(1.006)^n=3$ is $n \approx 183.651$. This is the number of months it takes for the investment to triple.

Divide by $12$ to determine the number of years: $\dfrac{183.651}{12}\approx 15.3$

Therefore, it takes approximately $15.3$ years (or $15$ years, $4$ months) for an investment to triple at an annual interest rate of $7.2\%$, compounded monthly.

Method 2: Use a TVM solver

We need to decide which values to enter into the TVM solver.

We are not given a present value, but the solution will be the same for any size of investment.

So let's pick an amount of principal, say $$100$, and calculate how long it takes for it to triple to $$300$.

Then, here are the values that need to be entered into the solver:

Nothing for $N$, since that is what we are trying to find.
$I\%=7.2$ (Recall that this input is always the annual interest rate).
$PV=-100$ (Negative, since we would pay $$100$ into the investment).
$FV=300$ (Positive, since at the end of the investment we would receive $$300$ back).
$C/Y=12$ (Interest is compounded monthly, so $12$ times per year).

$PMT$ and $P/Y$ are not used, so the values in the solver should not be changed.

$\texttt{N=0}$
$\texttt{I%=7.2}$
$\texttt{PV=-100}$
$\texttt{PMT=0}$
$\texttt{FV=300}$
$\texttt{P/Y=1}$
$\texttt{C/Y=12}$
$\texttt{PMT: }\class{PMT}{\texttt{END}} \texttt{ BEGIN}$

Now we use the TVM solver to determine the value for $N$. The calculator will now display that value.

$\texttt{N=15.3}$
$\texttt{I%=7.2}$
$\texttt{PV=-100}$
$\texttt{PMT=0}$
$\texttt{FV=300}$
$\texttt{P/Y=1}$
$\texttt{C/Y=12}$
$\texttt{PMT: }\class{PMT}{\texttt{END}} \texttt{ BEGIN}$

The display indicates that $N=15.3$. Recall that this is the number of years, and it is consistent with the answer we obtained by graphing.

It turns out that a TVM solver can help us find any of $PV$, $FV$, $N$, or $I\%$ if we know the values of the other three (and the number of compounding periods per year). Although it is important to know how to use the algebraic or graphical techniques discussed in this lesson, a TVM solver is a quick way to check your answers!

Check Your Understanding 8

Created with GeoGebra. Author: University of Waterloo. CC BY-NC-SA 4.0. https://ggbm.at/gvatgwj9#

Calculating Present Value, Interest Rate, or Number of Compounding Periods

Slide Notes

Glossary

All Slides

Navigation

Example 7

Solution

A Formula for Present Value

Example 8

Solution

Check Your Understanding 5

Example 9

Solution — Part A

Solution — Part B

Check Your Understanding 6

Slide Notes

Glossary

All Slides

Navigation

Example 10

Solution

Example 10 Continued

Solution — By Graphing

Example 10 Continued

Solution — Alternate Graph

Example 10 Continued

Solution — Conclusion

Example 10 — Guess and Check

Check Your Understanding 7

Using a TVM Solver

Example 11

Solution

Check Your Understanding 8