Time Value of Money With Different Frequencies of Compounding

Time value of money calculations allow us to establish the future value of a given amount of money. The present value (PV) is the money you have today. On the other hand, the future value (FV) is the accumulated amount of money you get after investing the original sum at a certain interest rate and for a given time period, say, 2 years.

Fundamental Formulas in Time Value of Money Calculations

Let,

\(FV\) = Future value.

\(PV\) = Present value.

\(r\) = Interest rate.

\(N\) = Number of years.

Then the future value (FV) of an investment is given by:

$$FV=PV(1+r)^N$$

To find the present value of the investment, we rewrite the above formula so that:

$$PV=FV(1+r)^{-N}$$

Calculation Using a Financial Calculator

To calculate FV and PV using the BA II Plus™ Financial Calculator, use the following keys:

\(N\) = Number of years.

\(I/Y\) = Rate per period.

\(PV\) = Present value.

\(FV\) = Future value.

\(PMT\) = Payment.

\(CPT\) = Compute.

It is important to note that the sign of PV and FV will be opposite. For example, if PV is negative, then FV will be positive. Generally, an inflow is entered with a positive sign, while an outflow is entered as a negative sign in the calculator.

Example: Calculating Future Value

Assume that an individual invests $10,000 in a bank account that pays interest at 10% compounded annually. The future value after two years is closest to:

1. $12,000.
2. $12,100.
3. $22,000.

Solution

The correct answer is B.

Recall that:

$$FV=PV(1+r)^N$$

In this case, we have PV=10,000, r=10%, N=2 so that:

$$FV=10,000(1+10%)^2=12,100$$

$$\begin{aligned}&\textbf{Using the BA II Plus™ Financial Calculator}\\ &\begin{array}{l|l|l}
\textbf { Steps } & \textbf { Explanation } & \textbf { Display } \\
\hline \text { [2nd] [QUIT] } & \text { Return to standard calc mode } & 0 \\
\hline\left[2^{\text {nd }}\right][\mathrm{CLR} \text { TVM }] & \text { Clears TVM Worksheet } & 0 \\
\hline 2[\mathrm{~N}] & \text { Years/periods } & \mathrm{N}=2 \\
\hline 10[\mathrm{I} / \mathrm{Y}] & \text { Set interest rate } & \mathrm{I} / \mathrm{Y}=10 \\
\hline-10000[\mathrm{PV}] & \text { Set present value } & \mathrm{PV}=-10000 \\
\hline 0[\mathrm{PMT}] & \text { Set payment } & \mathrm{PMT}=0 \\
\hline[\mathrm{CPT}][\mathrm{FV}] & \text { Compute future value } & \mathrm{FV}=12,100 \\
\end{array}\end{aligned}
$$

To confirm our answer, we could work out the PV of a future value of $12,100 invested under similar terms, starting with the FV of $12,100, using the formula below:

$$PV=FV(1+r)^{-N}=12,100(1+10\%)^{-2}= $10,000$$

$$\begin{aligned} &\textbf{Using the BA II Plus™ Financial Calculator}\\ &\begin{array}{l|l|l}
\textbf { Steps } & \textbf { Explanation } & \textbf { Display } \\
\hline \text { [2nd] [QUIT] } & \text { Return to standard calc mode } & 0 \\
\hline\left[2^{\text {nd }}\right][\text { CLR TVM }] & \text { Clears TVM Worksheet } & 0 \\
\hline 2[\mathrm{~N}] & \text { Years/periods } & \mathrm{N}=2 \\
\hline 10[\mathrm{I} / \mathrm{Y}] & \text { Set interest rate } & \mathrm{I} / \mathrm{Y}=10 \\
\hline 12100[\mathrm{FV}] & \text { Set present value } & \mathrm{FV}=12100 \\
\hline 0[\mathrm{PMT}] & \text { Set payment } & \mathrm{PMT}=0 \\
\hline[\mathrm{CPT}][\mathrm{PV}] & \text { Compute present value } & \mathrm{PV}=10,000 \\
\end{array}\end{aligned}
$$

Time Value of Money With Different Frequencies of Compounding

Some types of investments accumulate interest more than once a year. This results from semi-annual, quarterly, monthly, or daily compounding. In turn, this leads to different present values (PV) or future values (FV) of an investment depending on the frequency of compounding employed. In calculating an investment’s present or future value with multiple compounding periods per year, the most important thing is ensuring that the interest rate used corresponds to the number of compounding periods present per year.

Future Value

The future value (FV) of an investment is given by:

$$
\mathrm{FV}_{N}=\mathrm{PV}\left(1+\frac{r_{s}}{m}\right)^{m N}
$$

Where;

\(r_s\) = Quoted annual rate.

\(N\) = Number of years.

\(m\) = Compounding periods (per year).

\(N\) = Number of years.

Present Value

To find the present value of an investment, make PV the subject of the above formula. You should find that:

$$
\mathrm{PV}=\mathrm{FV}\left\{\left(1+\frac{r_{s}}{m}\right)^{-m \times n}\right\}
$$

Example: Present Value With Monthly Compounding

Imagine that you wish to have $10,000 in your savings account at the end of the next 3 years. Further, assume that the account offers a return of 9 percent per year, subject to monthly compounding. How much would you need to invest now so as to have the specified amount of money in your account after three years?

Solution

First, we write down the formula to use,

$$ PV= FV \left\{ \left( 1+ \frac {r_s}{m} \right) \right\}^{ -m\times n}$$

Secondly, we establish the components that we already have:

\(r_q\) = 0.09,  \(m\) = 12 since compounding is monthly, \(n\) = 3 years.

Then, we factor everything into the equation to find our PV:

$$ \begin{align*} PV & = 10,000 \left\{ \left(1+\frac {0.09}{12} \right) \right\}^{-12\times 3} \\ & = 10,000\times 1.0075^{-36} \\ & = $7,641.50 \\ \end{align*} $$

Therefore, you will need to invest at least $7,642 in your account to ensure you have $10,000 after three years.

$$\begin{aligned}&\textbf{Using the BA II Plus™ Financial Calculator}\\ &\begin{array}{l|l|l}
\textbf { Steps } & \textbf { Explanation } & \textbf { Display } \\
\hline \text { [2nd] [QUIT] } & \text { Return to standard calc mode } & 0 \\
\hline\left[2^{\text {nd }}\right][\mathrm{CLR} \mathrm{TVM}] & \text { Clears TVM Worksheet } & 0 \\
\hline 36[\mathrm{~N}] & \text { Years/periods }(12 \times 3=36) & \mathrm{N}=36 \\
\hline 0.75[\mathrm{I} / \mathrm{Y}] & \text { Set the interest rate }(9 / 12=0.75) & \mathrm{I} / \mathrm{Y}=0.75 \\
\hline-10000[\mathrm{FV}] & \text { Set the future value } & \mathrm{FV}=-10000 \\
\hline 0[\mathrm{PMT}] & \text { Set the periodic payment } & \mathrm{PMT}=0 \\
\hline[\mathrm{CPT}][\mathrm{PV}] & \text { Compute the present value } & \mathrm{PV}=7,641.50 \\
\end{array}\end{aligned}
$$

Question 1

Elizabeth Mary invests $2,000 in a project that pays a rate of return of 8% compounded quarterly. The interest that Mary would have earned after investing in the project for two years is closest to:

1. $343.32.
2. $2,300.00.
3. $2,343.32.

Solution

The correct answer is A.

$$ \begin{align*} FV & = 2000 \left\{ \left(1+\frac {0.08}{4} \right) \right\}^{4\times 2} \\ & = 2,000\times 1.02^8 \\ & = $2,343.32 \\ \end{align*} $$

Therefore, interest gained = 2,343.32-2,000= $343.32

$$\begin{aligned} &\textbf{Using the BA II Plus™ Financial Calculator}\\ &\begin{array}{l|l|l}
\textbf { Steps } & \textbf { Explanation } & \textbf { Display } \\
\hline \text { [2nd] [QUIT] } & \text { Return to standard calculator mode } & 0 \\
\hline\left[2^{\text {nd }}\right][\text { CLR TVM }] & \text { Clears the TVM Worksheet } & 0 \\
\hline 8[\mathrm{~N}] & \text { Years/periods }(4 \times 2=8) & \mathrm{N}=8 \\
\hline 2[\mathrm{I} / \mathrm{Y}] & \text { Set the interest rate }(8 / 4=2) & \mathrm{I} / \mathrm{Y}=2 \\
\hline-2000[\mathrm{PV}] & \text { Set the present value } & \mathrm{PV}=-2000 \\
\hline 0[\mathrm{PMT}] & \text { Set the payment } & \mathrm{PMT}=0 \\
\hline[\mathrm{CPT}][\mathrm{FV}] & \text { Compute the future value } & \mathrm{FV}=2,343.32 \\
\end{array} \end{aligned}
$$

Question 2

Elizabeth Mary invests $2,000 dollars in a project that pays a rate of return of 8% compounded daily. The interest that Mary would have earned after investing in the project for two years is closest to:

1. $343.
2. $344.
3. $347.

Solution

The correct answer is C.

$$ \begin{align*} FV & = 2,000 \left\{ \left( 1+ \frac {0.08}{365} \right) \right\}^{365\times 2} \\ & = 2,000\times 1.00021918^{730} \\ & = $2,347 \\ \end{align*} $$

Similarly, the interest is \(2,347 – 2,000 = $347\).

You should notice that with a higher compounding frequency, the corresponding profit is also higher. This confirms that interest earned increases as the number of compounding periods per year increases.

$$\begin{aligned} &\textbf{Using the BA II Plus™ Financial Calculator}\\ &\begin{array}{l|l|l}
\textbf { Steps } & \textbf { Explanation } & \textbf { Display } \\
\hline \text { [2nd] [QUIT] } & \text { Return to standard calc mode } & 0 \\
\hline\left[2^{\text {nd }}\right] \text { [CLR TVM] } & \text { Clears TVM Worksheet } & 0 \\
\hline 730[\mathrm{~N}] & \text { Years/periods }(365 \times 2=730) & \mathrm{N}=730 \\
\hline 0.021918[\mathrm{I} / \mathrm{Y}] & \text { Set interest rate }(8 / 365=0.021918) & \mathrm{I} / \mathrm{Y}=0.021918 \\
\hline-2000[\mathrm{PV}] & \text { Set the present value } & \mathrm{PV}=-2000 \\
\hline 0[\mathrm{PMT}] & \text { Set the payment } & \mathrm{PMT}=0 \\
\hline[\mathrm{CPT}][\mathrm{FV}] & \text { Compute the future value } & \mathrm{FV}=2,347 \\
\end{array}\end{aligned}
$$

A is incorrect. It assumes daily compounding for one year.

$$FV=2,000\left[1+\frac{0.08}{365}\right]^{365}=$2,166.55$$

The interest gained will be \(\$2,167-\$2,000=\$167\).

B is incorrect. It assumes a monthly rate in the calculation of FV as opposed to a daily rate as follows.

$$ FV=2,000\left[1+\frac{0.08}{12}\right]^{12\times2}=\$2,345.78$$

The interest gained will be \(\$2,345-\$2,000=\$345\).

Note: We can convert our stated annual rates into the effective annual rate of interest, and arrive at similar answers. However, if you do that, you should ensure that you use years in the computation.