###### Defining Properties of Probability

Defining properties of a probability refers to the rules that constitute any given... **Read More**

Some types of investments are known to accumulate interest more than once a year. This results from semi-annual, quarterly, monthly or daily compounding. This, in turn, leads to different present values (PV) or future values (FV) of an investment depending on the frequency of compounding employed.

We have previously seen that the effective annual rate of interest increases as the number of compounding periods per year increases. In calculating the present value or future value of an investment with multiple compounding periods per year, the most important thing is to ensure that the interest rate used corresponds to the number of compounding periods present per year.

**Future Value **

$$ FV= PV \left\{ \left( 1+ \frac {r_q}{m} \right) \right\}^{ m*n}$$

Where:

r_{q} is the quoted annual rate;

m represents the number of compounding periods (per year); and

lastly, n is the number of years

**Present Value**

Suppose you make PV the subject of the above formula, you should find that:

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

You wish to have $10,000 in your savings account at the end of the next 3 years. 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 after the three years?

**Solution**

First, we write down the formula to use,

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

Secondly, we establish the components that we already have:

r_{q =} 0.09, m = 12 since compounding is monthly, n = 3 years; and

then, we factor everything into the equation to find our PV.

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

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

Question 1Elizabeth Mary invests $2,000 in a project that pays a rate of return of 8% compounded quarterly. How much interest will Mary have earned after investing in the project for two years?

A. $2,300

B. $2,343.32

C. $343.32

SolutionThe correct answer is C.

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

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

Question 2What if the project paid a rate of return of 8% compounded daily? How much interest would Elizabeth Mary earn?

A. $2,347

B. $347

C. $2,340

SolutionThe correct answer is B.

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

Similarly, the interest = 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.

**Note**

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

*Reading 6 LOS 6d*

*Solve time value of money problems for different frequencies of compounding.*